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First year Wilkinson Microwave Anisotropy Proberesults: Cosmological parameters and implications for inflation

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FIRST Y EA R W IL K IX S O X M IC R O W AVE A X IS O T R O P Y
PR O B E RESULTS: COSMOLOGICAL PARAM ETERS
A N D IMPLICATIONS FOR INFLATION
Hiranya V. Peiris
A D IS S E R T A T IO N
PRESENTED TO TH E FACULTY
O F P R IN C E T O N U N IV E R S IT Y
IN C A N D ID A C Y FO R T H E D E G R E E
O F D O C T O R O F P H IL O S O P H Y
R E C O M M E N D E D FOR A C C E P T A N C E
BY T H E D E P A R T M E N T OF
A S TR O P H Y S IC A L SC IE N C E S
N o ve m be r. 2003
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UMI Number: 3101056
Copyright 2003 by
Peiris, Hiranya Vajramani
All rights reserved.
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0
C o p y rig h t 2003 b y H ira n y a V . P eiris
A ll rig h ts reserved.
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I c e r tify th a t I have read th is thesis and th a t in m y
o p in io n it is fu lly a d e q ua te, in scope a n d in q u a lity , as a
d is s e rta tio n fo r tin* degree o f D o c to r o f P h ilo so p h y.
^
._______
D a v id N . Spergel
( P rin c ip a l A d v is o r)
I c e rtify th a t 1 have read th is thesis and th a t in m y
o p in io n it is fu lly a d e q ua te, in scope and in q u a lity , as a
d is s e rta tio n fo r th e degree o f D o c to r o f P h ilo so p h y.
M ich a e l A . S trauss
I c e r tify th a t I have read th is thesis and th a t in m y
o p in io n it is fu lly ade q ua te, iti scope a n d in q u a lity , as a
d is s e rta tio n fo r th e degree o f D o c to r o f P h ilo so p h y.
/ r a u l .1. S te in h a rd t
A p p ro v e d fo r th e P rin c e to n U n iv e rs ity G ra d u a te
S chool:
Dean o f th e G ra d u a te S chool
iii
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For M y Parent*
I (jo. but I return: I would I were
The pilot of the darkness and the dream.
Audley Court.. A lfre d . L o rd T e n n yso n (1842)
IV
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A ck n o w led gm en ts
Life i.s' an unfnldmcnt. and the f u r t h e r irr tnircl. the more truth irr run comprehend.
To understand the thint/s that are at our door is the best preparation f o r understandiiu/
those that he bei/ond.
a ttr ib u te d to H y p a tia o f A le x a n d ria
T h is thesis deals w ith tin* b e g in n in g o f th e u niverse, and it is a p p ro p ria te to
b eg in it by a c k n o w le d g in g those whose s u p p o rt, e n c o u ra g e m e n t. h elp, and a d vice
has been in v a lu a b le to me. I d e d ica te th is thesis w ith love to m y m o th e r. X im a la .
and m y fa th e r. T h ila k . F ro m a ve ry yo u n g age. m y p a re n ts trie d to im p a rt to me a
sense o f c u rio s ity a b o u t th e w o rld , perseverence in tin ' face o f p ro b le m s, and a love
o f life lo n g le a rn in g th a t has served me w e ll in m y research career. T h e y believed in
th e va lu e o f a goo d e d u c a tio n , and w hen m y h o m e la n d was to rn w ith s trife , th e y
m oved th e fa m ily h a lf w ay across th e w o rld so m y s is te r and I c o u ld have one. I am
g ra te fu l b e yo n d m easure fo r th e ir co n s ta n t s u p p o rt d u r in g m y lo n g acad e m ic career.
I also th a n k m y s is te r K a la n i fo r b e in g th e re fo r me a t a ll th e im p o rta n t tim e s o f m y
life , and fo r b e in g such a g o o d frie n d to me.
I c re d it th e b e g in n in g o f m y in te re s t in a s tro n o m y to C a rl Sagan, whose
"C o sm o s" e n th ra lle d me as a ve ry s m a ll c h ild : h is w o rd s a n d th e s tu n n in g v isu a ls
o f th is series s t ill h a u n t m e decades la te r.
A s a b u d d in g a m a te u r a s tro n o m e r,
m y in te re s t was foste re d b y th e Y o u n g A s tro n o m e rs " A s s o c ia tio n o f S ri L a n ka , a
re m a rk a b le g ro u p , m a n y o f whose n u m b e r have gone on to be a s tro n o m e rs and
v
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s c ie n tis ts.
M a n y teachers enco u ra ge d and in flu en ce d m e in school: I m ust sin g le o u t tw o fo r
sp ecial g ra titu d e . In m id d le school. M rs M e n d is came witEi a fo rm id a b le re p u ta tio n ,
h a v in g ta u g h t m y m o th e r before me. She ta u g h t a ve ry general s u b je c t ca lle d
"S cience” , e x p la in in g physics, c h e m is try and b io lo g y w ith great s k ill and e nth usiasm .
She alw a ys trie d to a nsw er m y co n sta n t stre a m o f "w h y " and "h o w " q u e stio ns w ith
great p atie n ce, even w hen it m ust have been ra th e r a n n o y in g .
A t h ig h school,
the in c o m p a ra b le D r E gan s ta rte d m y love a ffa ir w ith m a th e m a tic a l physics.
He
conveyed his s u b je c t o f A p p lie d M a th e m a tic s w ith great s k ill, w it a n d good h u m o u r.
F ro m m y u n d e rg ra d u a te years at C a m b rid g e . I th a n k m y d ire c to r o f stu d ie s at
N ew H a ll. O w en S a x to n , fo r his s u p p o rt, d ilig e n ce , a n ti s k ill as a tea ch er o f physics.
He was a lw a ys th e re fo r his s tu d e n ts , h o ld in g o ffice h o u rs and discussions a b o u t
physios p ro b le m s o fte n la te in to th e n ig h t. I am g ra te fu l to M a lc o lm L o n g a ir. fo r his
in s p irin g e n th u sia sm a b o u t physics (he once w a lke d o ff a h ig h stage in his e xcite m e n t
in a firs t ye a r le c tu re a b o u t g ra v ity ) - he b ro u g h t th e same e n th u sia sm in m y fin a l
y e a r to make* gen e ra l r e la t iv it y su p e rvisio n s fu n . and in s tille d in me a sense th a t one
can play w ith physics. A n th o n y Lasenby. A n d y F a bian and M a r tin Rees’ fa s c in a tin g
te a m -ta u g h t co sm o lo g y a n d a stro p h ysics le ctu re s s ta rte d me t h in k in g se rio u sly a b o u t
a d o in g a g ra d u a te degree in th is to p ic . I also th a n k a ll m y frie n d s at C a m b rid g e ,
b u t e s p e c ia lly D a w n A rd a a m i T z e -L e i Poo. m y best frie n d s a t N e w H a ll. T h e y
h e lp e d me get th ro u g h th e im p e n d in g d o o m o f T rip o s exam s and em erge re la tiv e ly
u n sca th e d ! I a m fo re ve r in d e b te d to J u d ith P e rry, w h o s ta rte d o ff as m y firs t te rm
M a th s s u p e rv is o r a n d ended u p as a great frie n d : she is one o f th e m ost b r illia n t
a nd fa s c in a tin g p eople I have ever m e t. and a ro le m o d e l b o th as a w om an a n d a
s c ie n tis t. I w ill a lw a ys re m e m b e r co n ve rsa tio n s over s h e rry lo n g in to th e e vening,
and h o u s e s ittin g in h e r b e a u tifu l house w ith h e r gorgeous cats. I f I do even h a lf the
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tilin g s she has a cco m p lish e d d u r in g her life . I w ill d ir happy.
D u rin g th is rim e . I
had m y firs t research e xperience at .Y .-lS .A ’s Jet P ro p u ls io n L a b o ra to ry , w o rk in g fo r
r h r Galileo P P R tr a in . T h r s r m a g ical su m m e rs were lit e r a lly life-c h a n g in g fo r me:
1 expe rie n ced firs t h a n d the* a b so lu te t h r i l l o f seeing s o m e th in g no h u m a n eye had
ever seen. I owe g re at th a n k s to T e rry M a r tin . G le n n O rto n and Leslie T a m p a rri fo r
m em ories I w ill alw a ys ch erish.
C o n s id e rin g m y time* a t P rin c e to n . I am e s p e c ia lly g ra te fu l to m y a d v is o r D a v id
Spergel fo r his co n s ta n t encouragem ent and th e b e lie f he showed in me* when he
pushed fo r a m ere g ra d u a te s tu d e n t to be g ra n te d access to th e p recious U '.\/.A P
d a ta . He has re m a rk a b le sc ie n tific in t u itio n and p e o p le -in tu irio n . He knew w hen to
let me struggle* w ith p ro b le m s, and w hen to give me a h in t. It goes w ith o u t sayin g
th a t D a v id is a b r illia n t ro le -m o d e l as a s c ie n tis t, b u r he* is also cares g e n u in e ly
and d e e p ly a b o u t th e w elfare o f stu d e n ts. In s p ite o f his in c re d ib ly busy schedule,
he a lw ays fo u n d time* to ta lk to me. T h e greatest c re d it fo r a n y success I m ig h t
have had a t P rin c e to n can be d ire c tly a ttr ib u te d to his s u p e rb m e n to rs h ip . I also
owe a huge d e b t to m y sem ester p ro je c t research a d viso rs. It was in tim id a tin g to
be assigned to .John B a h c a ll u p o n a rriv a l here, g ive n his to w e rin g sta tu re 1 in the1
c o m m u n ity . B u t I s h a ll never forge t his guidance1, patience*, a n d kindness in m a k in g
me feel a t hom e. S co tt T re m a in e im pressed me* w ith th e care*, a tte n tio n te> d e ta il,
and c la r ity o f th o u g h t he p u ts in to re>search. He* also w rite's b e a u tifu lly . These are*
a ll q u a litie s th a t I hope to e m u la te in m y ow n w o rk .
T h e courses I have taken a t P rin c e to n have been a m o n g th e finest I have ever
e xpe rie n ced .
B ohclan P a czyn ski and .Jeremy G o o d m a n m ade th e s ta id to p ic o f
S te lla r S tru c tu re corne a liv e w it h th e ir fa s c in a tin g le ctu re s a n d in te re s tin g p ro b le m
sets. S c o tt T re m a in e 's S te lla r S ystem s le ctu re s were as g re a t a pleasure as w o rk in g
th ro u g h th e w o n d e rfu l te x tb o o k by B in n e v and T re m a in e . T o me. th e IS M is lik e
v ii
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b it te r m e d icin e , b u t B ru ce D ra in e ’s a b ilit y to e x p la in th e th e u n ify in g p hysics b e h in d
th o u sa n d s o f d is p a ra te o b s e rv a tio n a l facts m ade it go d o w n sw eeter. T h e in im ita b le
•Jim G u n n , h ig h ly o ccup ie d w ith SDSS. lo ca liz e d h is w a v e fu n e tio n in P e yto n 10G fo r
a w h o le sem ester to im p a rt to us his e n o rm o u s w is d o m on m a tte rs o f E x tra g a la c tic
A s tro n o m y , a nd set us a h ig h ly in te re s tin g class p ro je c t to m ake a re a lis tic m odel
o f the G a la x y . M ic h a e l S trauss has fille d m a n y roles fo r me. in c lu d in g d ire c to r o f
stu d ie s, te a c h in g A s tro 203 to w h ich I owe* m y re a ch in g e xp e rie n ce , thesis c o m m itte e
m e m b e r. R eader o f th is thesis, a n d se rv in g as a m e n to r fo r an SDSS o b s e rv a tio n a l
p ro je c t I w o rke d on w ith R ita K im . I owe a g re at d e a l to h im . D it t o .Jill K n a p p ,
w ho served as d ire c to r o f stu d ie s, teacher o f a m e m o ra b le o b s e rv a tio n a l s e m in a r
(w h e re we had to a c tu a lly do a real p ro je c t w ith SDSS d a ta a nd th e n speak a b o u t
it ) and p ro v id e r o f g o o d food and m a n y s c u rv y -p re v e n tin g c itr u s fr u its d u r in g SDSS
lunches.
M a n y o th e r fa c u lty have tou ch ed m y life here, th ro u g h th e ir re a chin g
s k ill, th e ir s c ie n tific kn ow led g e a nd th e ir in s p irin g e x a m p le : I e s p e c ia lly th a n k X e ta
B a h c a ll. Ed T u rn e r. J e rry O s trik e r. R ich G o t t . and in tin* p hysics d e p a rtm e n t. P aul
S te in h a rd t. L’ ros S e lja k and H e rm a n n V e rlin d e . I o ffe r sincere th a n k s to th e fa c u lty
c o lle c tiv e ly fo r g ra n tin g me an e x tra ye ar d u r in g w h ic h to ta ke classes a t th e physics
d e p a rtm e n t. I le a rne d so m uch d u r in g th is tim e . E ven th e stre s s fu l p hysics generals
exam served me to g ro w s c ie n tific a lly .
A m o n g th e research s ta ff. I am g ra te fu l to R o b e rt L u p to n . Z e ljk o Ivezic.
D a v id B ow en a nd T o d d T rip fo r h e lp in g m e o u t o f s c ie n tific a n d c o m p u te r-re la te d
q u a n d a rie s. D e p a rtm e n t M a n a g e r Susan D aw son ke p t me fin a n c ia lly so lve n t th ro u g h
m a n y trip s , a nd w it h great kin d ne ss, to o k in Jo e y th e hom eless d o g t i l l he fo u n d
a good hom e.
G ra d u a te se cre ta ry M in d y L ip m a n h elpe d me get th ro u g h the
p re p a ra tio n fo r th e F P O . S up e r lib ra ria n s Jane H o lm q u is t a n d C a rrie Swanson
p ro v id e d in v a lu a b le assistance. T h e sysa d m in s, th e fearsom e a n d u ltra c o o l M ich a e l
W a y a nd th e im p o s s ib ly d e d ica te d Steve H u s to n , k e p t th e C P C p o w e r tic k in g over.
v iii
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and to o k tim e o u t o f th e ir very busy schedules to h e lp m o o u t w ith a few personal
c o m p u tin g p ro b le m s. M ik e C a rr fixed m y car. lent me to o ls , and h e lp e d me create
th e P in g Pong T o u rn a m e n t tro p h y .
T h e m a jo r ity o f th e w o rk presented in th is thesis was c a rrie d o u t in a re la tiv e ly
s h o rt p e rio d o f tim e w ith a h ard d e a d lin e . It w o u ld have been im p o s s ib le to do it
w ith o u t th e a id o f th e tw o best c o lla b o ra to rs one c o u ld hope to have. L ic ia Verde
and E iic h iro K o m a ts u . S p e a kin g as someone w h o does n o t t h in k o f h e rs e lf as a team
p la ye r, th e e xp e rie n ce o f w o rk in g so closely w ith th e m was e x tre m e ly pleasant b o th
on a in te lle c tu a l and p ersonal level. It was a u n iq u e e xperience fo r m e to be able to
" c lic k ” so w e ll w ith th e m in m y tho u gh t-p ro ecsse s. and th e y e n rich e d m y last tw o
years at P e yto n in a w ay I cannot begin to d escribe. L ic ia and E iic h iro . I a d m ire
yo u tre m e n d o u s ly , and I s h a ll a lw a ys cherish o u r frie n d s h ip . In a d d itio n . I o ffe r m y
deepest g r a titu d e to th e e n tire W M A P Science T eam : e s p e c ia lly C’b uck B e n n e tt,
fo r ta k in g me on b o a rd : M ik e N o lta . fo r p ro v id in g essential s u p p o rt fo r th e sm o o th
fu n c tio n in g o f th e M a rk o v C h a in m e th o d : L y m a n Page, fo r his a d v ic e as p a rt o f m y
thesis c o m m itte e a n d th e in s p irin g care and d e d ic a tio n w ith w h ich he approaches his
w o rk , a ll th e P M A P p e rs fo r c o m p a n io n s h ip a nd m o ra l s u p p o rt w h e n we were not
allo w e d to discuss o u r w o rk w ith th e rest o f th e w o rld : at th e G o d d a rd Space F lig h t
C e n te r. G a ry H in s h a w . fo r q u ic k responses to d e sp e ra te q u e stio n s e m a ile d a t 2 a .m .:
M ik e G reason, w h o ke p t th e O rig in 300s b la z in g o u t re su lts: .Janet W e ila n d and
B r it t G ris w o ld , whose e x p e rt g ra p h ic s s k ills enhanced some o f th e fig u re s seen in the
fo llo w in g ch a p te rs. A m i once m ore, th a n k s so m uch to yo u . D a v id . I have le a rne d so
m uch fro m yo u a n ti I co n sid e r m y s e lf ve ry lu c k y to have had th is e xpe rie n ce. F in a lly .
I th a n k D a v id W ilk in s o n , w h o p u ts th e 11” in W M A P . He passed a w ay soon a fte r
I jo in e d th e W M A P T e a m , b u t in m y e a rlie r years he ta u g h t m e h o w to use th e
telescope u p on th e d o m e , and in s p ire d me w ith h is O S E T I p ro g ra m . I t was c le a r
th a t science was a w a y o f life fo r h im . and I d e e p ly re g re t n o t h a v in g k n o w n h im
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be'tte'r.
I th a n k aga in m y rhe*sis c o m m itte e '. D a v id Sperged. L y m a n Page1, a nd M ich a e l
S trauss, and o ffe r m y g ra titu d e ' to P aul S te in h a rd t. J im G u n n , and N e ta B a h c a ll fo r
agree-ing to re'ad it on such s h o rt n o tice .
In a d d itio n to the* in te lle c tu a l fire p o w e r c o n c e n tra te d in the fa c u lty a nd p ostd ocs,
life* a t P e yto n H a ll com es w ith th e adeleel bonus o f a c o m m u n ity o f g ra d u a te s tu d e n ts
w h o are ch 've r. kn o w le d g e a b le , e asy-going and m u lti-fa c e te d .
M y (Jehus, whose*
shoes are d iffic u lt to f ill, m ade m y e a rly life* here v e ry e n jo ya b le . In p a r tic u la r. I
th a n k Peng O h . M ik e B la n to n . E ric k Lee*. Dave* G o ld b e rg . R ita K im . X ia o h u i Fan.
A rie lle P h illip s . K en N a g a m in e . Przeunyslaw W o z n ia k and L e o n id M a ly s h k in . w ho
to o k me u n d e r th e ir c o lle c tiv e w in g , and were' w illin g to discuss a lm o s t a n y s c ie n tific
q u e s tio n (a n d likedv p rovide1 an in s ig h tfu l answ er).
S tro n g p e rs o n a litie s a ll. th e y
m ade life in th e baseunent a fu n a nd e 'v e r-in te re stin g experieuice. b o th s o c ia lly and
s c ie n tific a lly .
I th a n k m y class. B a rt P inelor. Q in g ju n n Y u . R o m a n R a fik o v . Lei
Hao a nd Y eong L o ll, a nd m y aclopte'd class. Iskra S tra te v a and E ric F o rd , fo r th e ir
frie n d s h ip a n d c o m p a n io n s h ip in th e adventure* o f g ra d u a te school. The* yo u n g e r
fo lk s , in p a r tic u la r N iayesh A fs h o rd i. Joe H e n n a w i. N a d ia Zakam ska. S im o n D eD eo.
N ic k B o n d , a nd Feng D o n g , have also p ro v id e d a g re a t a tm o sp h e re in m o re re'ceuit
years. T h e basem ent is a h o tb e d o f c u ltu r a l in te rch a n g e : it is lik e h a v in g th e w h o le
w o rld com e to you.
I m u st o ffe r huge h e lp in g s o f g ra titu d e to D r C a ro ly n T o rre a t M cC o sh . w h o
d iagnosed m y h y p o th y ro id is m in m y t h ir d ye a r here w ith n ot m a n y s y m p to m s to
go on. a n d to D r A la n F e ld m a n a t P M C fo r g u id in g me back to h e a lth . W ith o u t
these p ro fe ssio n als, m y illne ss w o u ld have progressed unchecked a n d there* is a ve ry
re a l p o s s ib ility th a t I w o u ld n o t have been a ble to c a rry o u t th is w o rk . I owe m a n y
th a n k s to rny Yoga te a ch e r. F a ith C o n ge r, fo r ke e p in g me p h y s ic a lly a n d e m o tio n a lly
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fit d u rin g th e last co u p le o f years. I also th a n k th e ta le n te d fo lk s at M u ta n t E nem y
fo r keeping me e n te rta in e d a n d p ro v id in g me w ith escapist fun d u r in g the stressful
few m o n th s before th e W M A P firs t ye a r d a ta release*.
Last, b u r by no m eans least. I th a n k M a rk Jackson. He has been m y cheering
se ction and life -s u p p o rt system , th ro u g h goo d tim e's and bad. fo r over se*ve»n years.
H is c re a tiv ity . in te*g rity and passion fo r life*, b o th as a s c ie n tis t a n d a h um a n be'ing.
c o n tin u e to inspire' tne\ T h is w o rk co u ld n ot have* bee'n a c c o m p lish e d w ith o u t his
unw ave'ring s u p p o rt, a n d I am ve*ry lu c k y to have such a kindre'd s p ir it w ith w hom
to share m y existence.
xi
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A b stra ct
T h is thesis describes th e a n a ly s is o f th e firs t ye a r Wilkinson Micnnvnvv Anisotropy
P ro b e (W ’A P IP ) d a ta , and in ve stig a te s its im p lic a tio n s fo r co sm o log y and in fla tio n .
W e begin by p re s e n tin g o u r m e th o d o lo g y fo r c o m p a rin g tin* W M A P m easurem ents
o f th e cosm ic m icro w a ve b a c k g ro u n d (C’ M B ) and o th e r c o m p le m e n ta ry d a ta sets to
th e o re tic a l m odels. W e d e scrib e o u r use o f th e lik e lih o o d fu n c tio n , sh o w in g how th e
s ta tis tic a l and s y s te m a tic u n c e rta in tie s in th e m o d e l and th e d a ta are p ro p a g a te d
th ro u g h th e fu ll a n a lysis. I 's in g the M o n te C a rlo M a rk o v C h a in (M C ’ M C ) te ch n iq u e ,
we e x p lo re th e lik e lih o o d o f th e d a ta given a m o d e l to d e te rm in e th e best fit
c o s m o lo g ica l p a ra m e te rs a n d th e ir u n c e rta in tie s . We fin d th a t th e e m e rg in g s ta n d a rd
m o d e l o f cosm ology, a Hat A —d o m in a te d u nive rse seeded by a n e a rly sca le -in v a ria n t
a d ia b a tic G au ssia n flu c tu a tio n s , fits th e W M A P d a ta alone. T h is s im p le m o d el
is also co nsiste n t w ith a h o st o f o th e r a s tro n o m ic a l m easurem ents o b ta in e d w ith a
v a rie ty o f d iffe re n t te ch n iq u e s, a t d iffe re n t scales a n d re d sh ifts. A n u n e xp e cte d re su lt
fro m th is m o d e l, d riv e n by W M A P ' s m e a surem e n t o f the te m p e ra tu re -p o la riz a tio n
( T E ) cross-pow er s p e c tru m , is th a t th e u niverse was re io n ize d e a rlie r th a n p re v io u s ly
th o u g h t, r u lin g o u t w a rm d a r k m a tte r.
B y c o m b in in g W M A P d a ta w ith o th e r
a s tro n o m ic a l d a ta , we c o n s tra in th e g e o m e try o f th e universe, th e e q u a tio n o f s ta te
o f th e d a rk energy, a n d th e ene rg y d e n s ity in s ta b le n e u trin o species. T h e n , we
c o n fro n t th e p re d ic tio n s o f in fla tio n , th e c u rre n tly d o m in a n t th e o ry o f s tru c tu re
fo rm a tio n , w ith th e W M A P d a ta , b o th a lo ne a n d in c o m b in a tio n w ith la rg e scale
x ii
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s tru c tu re d a ta . In fla tio n is s u p p o rte d by th e flatness o f the u n ive rse , th e G a u s s ia n ity
and n e a r-sca lc-in va ria n ce o f p r im o r d ia l p ow er sp e ctra a nd th e a d ia b a tic s u p e rh o riz o n
flu c tu a tio n s at th e rim e o f d e c o u p lin g im p lie d by the* T E d a ta .
I's in g c o n s tra in ts
on th e shape o f the sca la r pow er s p e c tru m and th e a m p litu d e o f g r a v ity waves, we
in v e s tig a te the p a ra m e te r space o f in fla tio n a ry m o d els th a t is co n siste n t w ith the
d a ta , p u ttin g u p p e r lim its on th e te n so r m odes and c o rre la te d is o c u rv a tu re m odes.
F in a lly , we in v e s tig a te fu tu re p ro sp e cts fo r co sm o log y using W M A P re s u lts a fte r 4.
(i and 8 years o f o p e ra tio n .
x iii
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P reface
T h is thesis consists o f five ch a p te rs.
T h e firs t c h a p te r is m a in ly a e s th e tic and
p ro vid e s an in tro d u c tio n to th e firs t ye a r Wilkinson Mi crowave
\nisotrof)y P ro b e
( W M A T ) d a ta , and a b ib lio g ra p h y o f tin* th irte e n c o m p a n io n [ta p e rs to th e firs t year
U A M P d a ta release. T h e m id d le th re e C h a p te rs provide* the* b a ckb o n e o f th is thesis.
These* ch a p te rs re*pre>eluee pape*rs w h ich are* alreaely aeee*pte*el fo r 66
,
!’_ a tio n in the*
Astrophysicnl Journal Supplement. w ith tw o m in o r ehange*s: one* e*xtra e'epiation has
be*e*n adeie*el to C’hapte*r 2 fo r c la rifie a tie m . anel A[)pe*nelix B. w h ic h was w r itte n a fte r
th e o rig in a l [)fipe*r was aeee*pteel. has be*e*n aelele*el to C'hapte*r A. Cha[)te*r 2 pre>viele*s a
de*taile*d e le scrip tion o f the* parame*te*r e*stim ation m e'thoehdogy use*d in the* a n a ly s is
de*seribe*el in the* fo llo w in g twet e h a p te rs. C h a p te r 3 eliseusse*s th e im p lie a tie m s o f
th e a n a lysis fo r cetsmetlogy. anel C h a p te r A feu-use's on its irn p lic a tie m s fo r in fla tio n .
C h a p te rs 2 A are m u lti- a u th o r p ap e rs, in w hieh th e e e w m tlm rs a p p e a rin g before
the* alphabetize*el lis t have earrie*d o u t th e w o rk fo r th a t speedfic p a j)e r. M y specific
c o n triltu tio n s have* be*en to th e M arkejv C h a in M o n te Carle) ( M C M C ) e-e>ele*s.
e jp tim iza tie m anel ee)nverge*nce technic|iies fo r M C M C . likedihemel a n a ly s is (C h a p te r
2 ): p a ra m e te r e*stimatiem using M C M C fo r a v a rie ty o f c o s m o lo g ic a l moelels (C h a p te r
3 ): and th e a n a lysis on in fla tie m (C h a p te r A). C o -a u th o rs D aviel Spe*rgel. L ic ia W rele.
E iic h iro K o m a ts u a n d M ic h a e l X o lta have provieied e*ssential in p u t on th e s tru c tu re ,
c o n te n t, and s c ie n tific in s ig h t in to these ch a p te rs: w ith o u t th e m , g ive n th e a m o u n t
o f we>rk in vo lve d anel th e s h o rt tim e p e rio d in w h ic h it was c a rrie d o u t. c o m p le tio n o f
x iv
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these ch ap ters w o u ld not have heeti possible. T h e e n tire W M A P te a m , c o m p ris in g in
a d d itio n to those above. C h a rle s B e n n e tt [P I]. M a rk H a lp e rn . G a r y H in sh a w . N o rm
J a ro s ik . A1 K o g u r. M ich e le L im o ti. S te p h a n M eyer. L y m a n Page. G re g T u c k e r. Ed
W o lla c k . and Ned W r ig h t, has p ro v id e d s c ie n tific g u id a n c e a nd d e ta ile d c o m m e n ts on
th e fin a l m a n u s c rip ts . T h e e p ilo g u e . C h a p te r •">. in ve s tig a te s tin* fu tu re p ro spe cts fo r
cosm ology, g iv in g p re d ic tio n s were U W /.A P to operate* fo r 1. G a n d 8 years. B elow . I
give* the* correspondences be*twe'e*n Chapte*r numbe*rs anel the* accep ted pape*rs. The*
chapte*rs c o n ta in inte*rnal references to V erde e*t al. (2 0 0 3 ). Spe*rge*l e*t al. (2 0 03 ).
anel Pe*iris e*t al. (2 0 03 ). w h ich shendel be* lindersteioei to re*fe*r re» the* c o rre s p o n d in g
C'hapte*rs.
C h . 2 "First Year Wilk inson Mi cr owave Anisotropy Probr ( W M A P ) Obsecrations:
Par ame te r Estimation Methodoloyy". Verde. L .. P e i r is , H . V . . Spe*rge*l. D. N..
N o lta . M . R.. Be*nne*tt. C’ . L .. H a lp e rn . M .. H in s h a w . G .. J a ro s ik . N .. K o g u r.
A .. L im o n . M .. M e ye r. S. S.. Page*. L .. T u cke r. G . S.. W o lla c k . E.. A: W rig h t.
E. L. 2003. a stre )-p h /0 3 0 2 2 1 8 . Ap.JS. to a p p e a r in v l 18 n l
Ch. 3
"First Yr ar Wilkinson Mi cr owave Anisotropy Probe ( W M A P ) Obsecrations:
De te rmi nati on of Cosmoloyical P ar ame te rs ". S pe rg e l. D. N \. Verele. L .. P e i r is ,
H . V . . K o m a ts u . E.. N o lta . M . R .. B e n n e tt. C . L .. H a lp e rn . M .. H in s h a w .
G .. J a ro s ik . N .. K o g u r. A .. L in to n . M .. M eyer. S. S.. Page. L .. T u cke r. G . S..
W eilanei. J. L .. W o lla c k . E ..
W r ig h t. E. L. 2003. a s tro -p h /0 3 0 2 2 0 9 . A p J S . to
a p p e a r in v l4 8 n l
C h . 4 "Fi rst Year Wilkinson M ic row av e Anisotropy Probe ( W M A P ) Observations:
Implications f o r I n f l a t i o n " . P e i r is , H . V . . K o m a ts u . E .. V erde . L .. S pergel.
D . N .. B e n n e tt. C. L .. H a lp e rn . M .. H in sh a w . G .. J a ro s ik . N .. K o g u t. A ..
L in to n . M .. M e ye r. S. S.. Page. L .. T u cke r. G . S.. W o lla c k . E ..
2003. a s tr o p h /0 3 0 2 2 2 5 . A p J S . to a p p e a r in v l4 8 n l
xv
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W r ig h t. E. L.
C o n ten ts
A ck n o w led g m en ts
v
A b stra ct
xii
P reface
xiv
1
In tro d u ctio n
1
2
P aram eter E stim a tio n M eth o d o lo g y
5
1
I n t r o d u c t i o n ....................................................................................................................
0
2
L ik e lih o o d A n a ly s is o f W M A P A n g u la r P ow er S p e c tra
............................
7
2.1
L ik e lih o o d F u n c t io n ......................................................................................
8
2.2
C u rv a tu re M a t r i x ..........................................................................................
12
2.3
L ik e lih o o d fo r th e T E a n g u la r pow er s p e c t r u m ................................
16
M a rk o v C h a in s M o n te C a rlo L ik e lih o o d A n a ly s is ..........................................
17
3.1
M a rk o v C h a in M o n te C a r l o ...................................................................
18
3.2
C o nvergence a n d M i x i n g ...........................................................................
20
3
xvi
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3
3.3
M a rk o v C h a in s in P r a c tic e '........................................................................
22
3.4
Im p ro v in g M C M C E f fic ie n c y ....................................................................
23
3.0
The’ C'hoiee o f P r i o r s ...................................................................................
27
3.6
M C M C O u tp u t A n a ly s is ............................................................................
28
4
E x te rn a l C M B D a ta S e*rs..........................................................................................
31
0
A n a ly s is o f Large’ Seale’ S tructure* D a t a .............................................................
33
o .l
The’ 2.1FCRS P ow er S p e c t r u m ................................................................
34
6
L y m a n o Forest D a t a .................................................................................................
46
7
C o n c lu s io n s ....................................................................................................................
49
C o sm o lo g ica l P a ra m eters
66
1
I n t r o d u c t i o n ....................................................................................................................
67
2
Baye'sian A n a ly s is o f C o s m o lo g ic a l D a ta
.........................................................
69
3
P ow er L a w A C D M M o d e l anel th e W M A P D a t a ..........................................
71
4
C o m p a ris o n w ith A s tro n o m ic a l P r e d ic t io n s .....................................................
73
4.1
H u b b le C o n s t a n t ...........................................................................................
74
4.2
A m p litu d e o f F lu c t u a t io n s ........................................................................
7o
4.3
B a rv o n A b u n d a n c e .......................................................................................
77
4.4
C o s m ic A g e s ......................................................................................................
79
4.5
L arge Scale S t r u c t u r e ....................................................................................
81
4.6
S up e rn o va D a t a ...............................................................................................
82
x v ii
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4.7
3
G
4
R e io n iz a tio n N' S m a ll Seale P o w e r ........................................................
S3
C o m b in in g D a ta S e t s ................................................................................................
So
3.1
Pow er Law AC’ D M M o d e l
.......................................................................
93
3.2
R u n n in g S p e c tra l In d e x AC’ D M M o d e l ................................................
93
B eyond th e AC’ D M M o d e l ......................................................................................
97
G. I
D a rk E n e r g y .....................................................................................................
97
G.2
N o n -F la t M o d e l s ..........................................................................................
99
G.3
M assive N e u trin o s
G.4
Tensors
......................................................................................
100
............................................................................................................
102
7
In tr ig u in g D is c r e p a n c ie s ..........................................................................................
103
8
C o n c lu s io n s ....................................................................................................................
10G
Im p lica tio n s for In flation
137
1
I n t r o d u c t i o n .....................................................................................................................
2
Im p lic a tio n s o f U ’.M A P " T E " D e te c tio n fo r th e In fla tio n a r y P a ra d ig m
3
S in g le F ie ld In fla tio n M o d e ls ...................................................................................
4
138
139
143
3.1
I n t r o d u c t i o n .....................................................................................................
144
3.2
F ra m e w o rk fo r d a ta a n a ly s is ....................................................................
143
3.3
D e te rm in in g th e p o w e r s p e c tru m p a r a m e te r s ...................................
149
3.4
S in gle fie ld m o d e ls c o n fro n t th e d a t a ..................................................
134
M u lt ip le F ie ld In fla tio n M o d e l s ............................................................................
167
x v iii
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4.1
F ra m e w o rk
.....................................................................................................
4.2
C o rre la te d A d ia h a tie /Is o c u rv a tu re F lu c tu a tio n s fro m D o u b le F ie ld I n f l a t i o n ................................................................................................
5
1G8
0
S5479 1
...............................................................
173
G
C o n c lu s io n s ....................................................................................................................
173
A
In fla tio n a ry F lo w E q u a t io n s ...................................................................................
178
B
'ss o f t in 1 In H a to n P o te n tia l
10
A n e s tim a te o f D o n
................................................................................................
F u tu re P r o sp e c ts w ith UA/AP
1
181
199
M o t iv a t i o n ........................................................................................................................
19!)
1.1
Bayesian E vid en ce
......................................................................................
200
1.2
T h e R a z o r .........................................................................................................
201
2
C re a tin g Test D a t a ......................................................................................................
204
3
P a ra m e te r E s tim a tio n R e s u lt s ..................................................................................
207
4
3.1
A C D M M o d e l .................................................................................................
208
3.2
R u n n in g In d e x A C D M M o d e l ................................................................
208
3.3
"S in g le F ie ld In fla tio n " M o d e l .................................................................
210
3.4
R e su lts in th e ( u \ f i r„ ) a n d ( ir .
h) P la n e s ..........................................
211
3.3
R e su lts in th e ( f i A. f i m) P l a n e .................................................................
211
C o n c lu s io n s .....................................................................................................................
211
x ix
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C h ap ter 1
In tro d u ction
The knoirn i.s finite. the nnknotrn infinite: intellectuallfi ire stand on an islet in the
midst of an illimitable ocean of inexplicability. O u r business in erery ye iteration is
to reclaim a little more land.
T .H . H u x le y ( 18S7)
T h e Wilkinson Microwave Anisotropy Prohe ( W M A P ) e x p e rim e n t [ B lj has
c o m p le te d one ye a r o f o b s e rv in g th e C o sm ic M icro w a ve B a c k g ro u n d (C’ M B ). h e lp in g
us to re c la im a lit t le m o re la n d in tIn* c o u n try o f th e U n k n o w n . T h e C 'M B is a
p r im a r y to o l fo r d e te rm in in g th e g lo b a l c h a ra c te ris tic s , c o n s titu e n ts , h is to ry and
e v e n tu a l fa te o f th e U niverse.
W M A P was launched on 30 Ju ne 2001 fro m C ape
C a n a v e ra l and observes th e C 'M B sky fro m an o r b it a b o u t th e second L ag ra n ge
p o in t o f th e E a rth -S u n system . L>. In th is c h a p te r, we w ill in tro d u c e th e W M A P
e x p e rim e n t and p ro v id e references to th e co m p a n io n p ap e rs th a t a cco m p a n ie d the
release o f its firs t ye a r o f d a ta .
T h e c e n tra l design p h ilo s o p h y o f th e W M A P m ission was to m in im iz e sources o f
s y s te m a tic m e a surem e n t e rro rs. T o achieve th is goal. W M A P u tiliz e s a d iffe re n tia l
design : it observes th e te m p e ra tu re differences betw een tw o d ire c tio n s in th e sky.
u sin g a b a c k -to -b a c k set o f n e a rly id e n tic a l o p tic s [B 3. 4].
These o p tic s focus
1
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C h a p t e r 1:
2
In tro d u c tio n
rh e ra d ia tio n in to h o rn s [B4] th a t feed d iffe re n tia l m ic ro w a v e ra d io m e te rs [B 2. 3j.
F u ll-s k y m aps in five fre q u e n cy hands fro m 23 94 G H z art* p ro d u c e d fro m th e
ra d io m e te r d a ta o f te m p e ra tu re differenees m easured over th e fu ll sky. A C’ M B m a p
is th e m ost co m p a ct re p re s e n ta tio n o f C’ M B a n is o tro p y w ith o u t loss o f in fo r m a tio n .
T ] presents th e m aps, th e ir p ro p e rtie s , a n d a synopsis o f th e basic re s u lts o f U ’.M A P 's
firs t y e a r o f o p e ra tio n .
C a lib r a tio n e rro rs are < 0 . 3 '/ . and tin* low level o f s y s te m a tic e rro r is f u lly
c h a ra c te riz e d in [2.3.4.-")]. T h e m u lti-fre q u e n c y d a ta enables th e s e p a ra tio n o f the
C’ M B s ig n a l fro m tin* fo re g ro u n d d u s t, s y n c h ro tro n a n d free-free e m issio n [G]. since
the* C’ M B sig n a l does n ot v a ry w ith fre q u e n cy b u t th e fo re g ro u n d e m issio n does.
The* C’ M B obeys G aussian s ta tis tic s [7]. and th e re fo re rhe in fo r m a tio n in the
C 'M B m aps can be* represented by a p ow er s p e c tru m .
W M A P has m easured the*
te m p e ra tu re ( T T ) p ow er s p e c tru m to 2 < t < 900. c o s m ic -v a ria n c e -lim ite d fo r
f < 334 [8j. The* te m p o ra tu re -p o la riz a tio n ( T E ) cro ss-p o w e r s p e c tru m has also been
m easured to ( ^
450 [91.
T h e m e th o d o lo g y used to a n a lyze th is w e a lth o f new
in fo r m a tio n is presented in [10]. and [11.12.13] discuss som e o f rhe co s m o lo g ic a l
im p lic a tio n s o f th e W M A P re su lts. F in a lly . [14] p ro v id e s d e ta ile d in fo r m a tio n a b o u t
the* W M A P in -flig h t o p e ra tio n s and d a ta p ro d u c ts .
L in k s to th e fo llo w in g papers, as w e ll as th e d a ta p ro d u c ts and so ftw a re fo r th e
firs t y e a r d a ta release, can be fo u n d o n -lin e a t th e Legacy Archive for Microwave
Background D a t a Analysis (L A M B D A : h ttp ://la m b d a .g s fc .n a s a .g o v ).
F ir s t ye ar d a ta s c ie n tific papers:
1. “ Fi rst Year Wilkinson Mi crowave Anisotropy Probe ( W M A P ) Observations:
Ma ps and Basic Results". B e n n e tt. C . L . et a l. 2003. A p .IS . to a p p e a r in v l4 8
n1
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
C h a p te r I :
In tro d u c tio n
3
2. "Fi rst Year Wilkinson M i r r o i r a r c Anisotropy Probe ( W M A P ) Obsecrations:
D a ta Processing and Systematic E n 'o r Li mi ts ". H in sh a w . G . et al. 2003. A p.IS .
to a p p e a r in vl-48 n l
3. "Fi rst Year Wilkinson M ic roi ra re Anisotropy Probe ( W M A P ) Observations:
O n - O r b it Radi omet er Char acter izati on". J a ro s ik . X . <’t a l. 2003. A p .IS . to
a p p e a r in vl-18 n I
1.
"Fi rst Year Wilkinson M ic roi ra re Anisotropy Probe ( W M A P ) Obsecrations:
Beam Profiles and Wi ndow Functions". Page. L. et al. 2003. A p.IS . to a p p e a r
in vl-18 n 1
o. "First Year Wilkinson M i c ro i ra r e Anisotropy Probe ( W M A P ) Obsecrations:
Galactic Signal Cont ami na ti on f r om Sidelobe Pickup". B arnes. C‘ . et a l. 2003.
A p.IS . to a p p e a r in vl-18 n l
G. "Fi rst Year Wilkinson M i c ro i ra r e Anisotropy Probe ( W M A P ) Observations:
Foreground Emi ss ion". B e n n e tt. C’ . L. et al. 2003. A p .IS . to a p p e a r in v l4 8 n l
7. "First. Year Wilkinson M i c ro ir ar e Anisotropy Probe ( W M A P ) Observations:
Tests of Gaussianity" . K o m a ts u . E. et al. 2003. A p .IS . to a p p e a r in v l4 8 n l
8. " Fi rst Year Wilkinson M i c r oi r a r e Anisotropy Probe ( W M A P ) Obsecrations:
The Angul ar P o we r Spectrum". H in s h a w . G . et a l. 2003. A p .IS . to a p p e a r in
v l4 S n l
9. "Fi rst Year Wilkinson Microwave. Anisotropy Probe ( W M A P ) Observations:
T E Po l ar i za t io n" . K o g u t. A . et al. 2003. A p.IS . to a p p e a r in v !4 8 n l
10. " Fi rs t Year Wilkinson Mi crowave Anisotropy Probe ( W M A P ) Observations:
P a r a m e t e r Est imati on Methodology". V erde. L. et a l. 2003. A p .IS . to a p p e a r in
v l4 8 n l
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
C h a p te r I:
I n t r o d u c t io n
4
11. "Fi rst Year Wilkinson M i r r o i r n r e Anisotropg Probe ( W M A P ) Observations:
De te rmi nati on of Cosmological Par ame te rs ". S pergel. D . X . et al. 2003. A p.IS .
to a pp e a r in v l4 8 t i l
12. "Fi rst Year Wilkinson M i r r o t r a r e Anisotropg Probe ( W M A P ) Observations:
Implications f o r I n f l a t i o n " . P e iris. H. V . et al. 2003. Ap.JS. to a p p e a r in v l4 8
nl
13. "First Year Wilkinson M i r r o t r a r e Anisotropg Probe ( W M A P ) Observations:
Interpretation of the T T and T E Angul ar Potrer Spectrum Peaks". Page. L. <*t
al. 2003. A p .IS . to a p p e a r in v l4 8 n l
14. "First. Year Wilkinson Mi crowave Anisotropg Probe ( W M A P ) Observations:
Explanatory Supplement". L itn o n . M . et a l. 2003 (a v a ila b le at L A M B D A o n ly )
S c ie n tific papers s u b m itte d before th e firs t ye a r d a ta release:
B1
"The Microwave Anisotropg Probe ( M A P ) Mission". B e n n e tt. C. L. et a l..
2003. A p .I. 583. 1
B 2 "Design. Impl ementati on and Testing of the M A P Radi omet er s". .Jarosik. N.
et a l.. 2003. A p.IS . 145. 413
B 3 " T he Optical Design and Characterization o f the Mi crowave Anisotropy P r o b e ' .
Page. L. et a l. 2003. A p .I. 585. 566
B 4 " Th e M A P Satellite Feed H o r n s " . B arnes. C . et a l.. 2002. Ap.JS. 143. 567
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C h ap ter 2
P aram eter E stim a tio n
M et ho d o lo g y
Abstract
W e d escribe o u r m e th o d o lo g y fo r c o m p a rin g th e W M A P m e a surem e n ts o f
th e co sm ic m icrow ave b a ckg ro u n d (C 'M B ) and o th e r c o m p le m e n ta ry d a ta sets to
th e o re tic a l m odels. T h e unp re ce de n te d q u a lity o f th e W M A P d a ta , a n d th e tig h t
c o n s tra in ts on c o sm o lo g ica l p a ra m e te rs th a t are d e riv e d , re q u ire a rig o ro u s a n a ly s is
so th a t th e a p p ro x im a tio n s m ade in th e m o d e lin g d o n ot lead to s ig n ific a n t biases.
W e d escribe o u r use o f th e lik e lih o o d fu n c tio n to c h a ra c te riz e th e s ta tis tic a l
p ro p e rtie s o f th e m icro w a ve b a ckg ro u n d sky. W e o u tlin e th e use o f th e M o n te C a rlo
M a rk o v C h a in s to e x p lo re th e lik e lih o o d o f th e d a ta g ive n a m o d e l to d e te rm in e th e
best f it c o sm o lo g ica l p a ra m e te rs and th e ir u n c e rta in tie s .
W e add to th e W M A P d a ta th e f ^ 700 C B I a n d A C 'B A R m e a surem e n ts o f th e
C M B . th e g a la x y p o w e r s p e c tru m a t : ~ 0 o b ta in e d fro m th e 2 d F g a la x y re d s h ift
su rv e y (2 d F G R S ). a n d th e m a tte r p ow er s p e c tru m a t r ~ 3 as m easured w ith th e
L y m a n o fo re st. These la st tw o d a ta sets c o m p le m e n t th e C M B m e a surem e n ts by
5
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C'hnptvr 2:
6
P n r n m r t r r Est imati on Mf 'thodol oyy
p ro b in g th e m a tte r p ow er s p e c tru m o f the n ea rb y u niverse . C o m b in in g C M B and
2
dF~GRS re q uires th a t we in c lu d e in o u r a n a ly s is a m o d e l fo r
g a la x y
bias, re d s h ift
d is to rtio n s , and tin* n o n -lin e a r g ro w th o f s tru c tu re . W e show how th e s ta tis tic a l and
s y s te m a tic u n c e rta in tie s in the m o d e l and th e d a ta are p ro p a g a te d th ro u g h th e fu ll
a na lysis.
1.
In tro d u ctio n
C M B e x p e rim e n ts are p o w e rfu l co sm o lo g ic a l pro be s because tin* e a rly u niverse
is p a r tic u la r ly s im p le and because the* flu c tu a tio n s o ver a n g u la r scales H >
0
C2
art* d escribe d by lin e a r th e o ry (Peebles X: Y u . 1970: B o n d iC E fs ta th io u . 1981:
Z a ld a rria g a
S eljak. 2000). E x p lo itin g th is s im p lic ity to o b ta in precise c o n s tra in ts
on co s m o lo g ica l p a ra m e te rs re q u ire s th a t we a c c u ra te ly c h a ra c te riz e th e p e rfo rm a n c e
o f th e in s tru m e n t (J a ro s ik et a l.. 2003b: Page et a l.. 2003a: B arne s et a l.. 2003:
H in s h a w et a l.. 2003a). th e p ro p e rtie s o f th e fo re g ro u n d s (B e n n e tt et a l.. 2 0 0 3 b ). and
th e s ta tis tic a l p ro p e rtie s o f the* m icro w a ve sky.
T h e p rim a ry goal o f th is [ta p e r is to present o u r a p p ro a ch to e x tr a c tin g the
c o s m o lo g ica l p a ra m e te rs fro m th e te m p e ra tu re -te m p e ra tu re a n g u la r p ow er s p e c tru m
( T T ) and th e te rn p e ra tu re -p o la riz a tio n a n g u la r cro ss-p o w e r s p e c tru m ( T E ) . In
c o m p a n io n papers, we present th e T T (H in s h a w et a l.. 2003a) a n d T E (K o g u t et a l..
2003) a n g td a r p ow er sp e c tra a n d show th a t th e C M B flu c tu a tio n s m a y be tre a te d as
G au ssia n (K o m a ts u et a l.. 2003).
O u r basic a p p ro a ch is to c o n s tra in c o s m o lo g ic a l p a ra m e te rs w ith a lik e lih o o d
a n a ly s is firs t o f th e W M A P T T a n d T E s p e c tra a lo n e , th e n jo in t ly w ith o th e r
C M B a n g u la r p ow er s p e c tru m d e te rm in a tio n s a t h ig h e r a n g u la r re s o lu tio n , a n ti
fin a lly o f a ll C M B p o w e r s p e c tra d a ta jo in t ly w ith th e p ow er s p e c tru m o f th e
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C h a p te r 2:
P a ra m e te r E s t im a tio n M e t h o d o lo g y
la rge-scale s tru c tu re (L S S ). In ^2 we d escribe th e use o f th e lik e lih o o d fu n c tio n
fo r th e a n a lysis o f m icro w a ve b a ckg ro u n d d a ta . T h is b u ild s on the* H in sh a w et al.
(20031)) m e th o d o lo g y fo r d e te rm in in g tin* T T s p e c tru m and its c u rv a tu re m a tr ix ,
a n d K o g u t et al. (2003) w h o d escribe o u r m e th o d o lo g y fo r d e te rm in in g th e T E
s p e c tru m .
In fj3 we d escribe o u r use o f M a rk o v C h a in s M o n te C a rlo (M C M C )
tech n iq u e s to e va lu a te th e lik e lih o o d fu n c tio n o f m o d el p a ra m e te rs. W h ile \ \ . \ / . - \ P ‘s
m e a surem e n ts are a p o w e rfu l p ro be o f cosm ology, we can s ig n ific a n tly enhance th e ir
s c ie n tific value by c o m b in in g the W M A P d a ta w ith o th e r a s tro n o m ic a l d a ta sets.
T h is p a p e r also presents o u r app ro a ch fo r in c lu d in g e x te rn a l C M B d a ta sets (1)4).
LSS d a ta ($3) and L y m a n o forest d a ta (*jG). W h e n in c lu d in g e x te rn a l d a ta sets the
re a de r sh o u ld keep in m in d th a t th e physics and the in s tru m e n ta l effects in v o lv e d
in th e in te r p r e ta tio n o f these e x te rn a l d a ta sets (e sp e cia lly
2
d F G R S and L y m a n
n ) are m uch m ore c o m p lic a te d a n d less w e ll u n d e rs to o d th a n fo r \ \ M A P d a ta .
N evertheless we a im to m a tc h the rig o ro u s tre a tm e n t o f u n c e rta in tie s in th e W A /A P
a n g u la r p ow er s p e c tru m w ith the in c lu s io n o f kn o w n s ta tis tic a l and s y s te m a tic
e ffects ( o f th e d a ta and o f th e th e o ry ), in th e c o m p le m e n ta ry d a ta sets.
2.
L ik elih ood A n a ly sis o f W M A P A ngular P ow er S p ectra
T h e firs t g oal o f o u r a n a lysis p ro g ra m is to d e te rm in e th e values and confid e nce
levels o f th e co sm o lo g ica l p a ra m e te rs th a t best d escribe th e W M A P d a ta fo r a
g ive n c o s m o lo g ica l m o d e l. We also w ish to d is c rim in a te betw een d iffe re n t classes o f
c o s m o lo g ic a l m odels, in o th e r w o rd s to assess w h e th e r a co sm o lo g ica l m o d e l is an
a c c e p ta b le f it to W M A P d a ta .
T h e u ltim a te goal o f th e lik e lih o o d a n a ly s is is to fin d a set o f p a ra m e te rs th a t
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C h a p t e r 2:
S
P a r a m e te r E s t im a tio n M e th o d o lo g y
g iv e a il e s tim a te o f (C,-). tin* ensem ble average o f w h ic h th e re a liz a tio n on o u r s k y 1 is
C^k> . T h e lik e lih o o d fu n c tio n . £(C /|C Jll( n ) ). y ie ld s th e p r o b a b ility o f th e d a ta given
a m o d e l and its p a ra m e te rs (n) . In o u r n o ta tio n Ct denotes o u r bust e s tim a to r o f
C7k> (H in s h a w et a l.. 2003a) and C}h is th e th e o re tic a l p re d ic tio n fo r a n g u la r pow er
s p e c tru m . F ro m Daves’ T h e o re m , we can s p lit th e expression fo r th e p r o b a b ility o f a
m o d e l g ive n th e d a ta as:
•P(o!Cf) = £ ( C i C ; ' ' ( o ) ) P ( o ) .
(2-1)
w h e re 'P (o ) describes o u r p r i or .s on co sm o lo g ica l p a ra m e te rs and we have neglected
a n o rm a liz a tio n fa c to r th a t does not depend on th e p a ra m e te rs . O nce th e choice
o f th e p rio rs are sp ecified , o u r e s tim a to r o f (Ct) is given by C,'h e v a lu a te d at th e
m a x im u m o f V ( n \ C f ) .
2.1.
L ik elih ood F u nction
O ne o f th e gen e ric p re d ic tio n s o f in fla tio n a r y m o d els is th a t flu c tu a tio n s in the
g r a v ita tio n a l p o te n tia l have G aussian ra n d o m phases. Since th e p hysics th a t governs
th e e v o lu tio n o f th e te m p e ra tu re a nd m e tric flu c tu a tio n s is lin e a r, th e te m p e ra tu re
flu c tu a tio n s are also G aussian.
c <
10
I f we ig n o re th e effects o f n o n -lin e a r physics at
and th e effect o f fo re g ro u n d s, th e n a ll o f th e c o s m o lo g ica l in fo r m a tio n in th e
m ic ro w a v e sky is encoded in the th e rn p e ra tu re a n d p o la riz a tio n p ow er s p e ctra . T h e
le a d in g -o rd e r lo w -re d s h ift a s tro p h y s ic a l effect is e xp e cte d to be g r a v ita tio n a l le a s in g
o f th e C M B by fo re g ro u n d s tru c tu re s . W e ig n o re th is effect here as it generates a
< 1{X co va ria n ce in th e T T a n g u la r p ow er s p e c tru m on W M A P a n g u la r scales (H u .
2001) (see S pergel et al. 2003. §3).
‘ T h ro u g h o u t th is p a p e r we use th e c o n v e n tio n th a t Ct = ( { ( + l ) 0 / ( 2 r r ) .
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C h a p t e r 2:
9
P a r a m e te r E s t im a t i o n M e t h o d o lo g y
T h e re are several e xpe cted sources o f n o n -co sm o lo g ica l sig n al and o f nonG a u s s ia n ity in th e m icro w a ve sky. T h e m ost s ig n ific a n t sources on t in 1 fu ll sky are
G a la c tic fo re g ro u n d em issio n , ra d io source's, a n d g a la x y clu sters.
B e n n e tt et al.
(2 0 03 a ) show th a t these c o n trib u tio n s are1 g re a tly reduced i f we re s tric t o u r a n a lysis
to a c u t sky th a t m asks b rig h t sources and re g ion s o f b rig h t G a la c tic em issio n . T h e
re s id u a l c o n tr ib u tio n o f these fo re g ro u n d s is fu r th e r reduced by th e use* e x te rn a l
te m p la te s to s u b tra c t fo re g ro u n d e m ission fro m th e Q . V and W band m aps.
K o m a ts u et al. (2003) fin d no e vid en ce fo r d e v ia tio n s fro m G a u s s ia n ity on th is
te m p la te -c le a n e d cu t sky. W h ile th e sky c u t g re a tly re'duces fo re g ro u n d em ission,
if has th e u n fo rtu n a te ' e'ffect o f c o u p lin g m u lfip o le moeh's on the sky so th a t the'
p ow er sp e'ctrum co varian ce m a tr ix is no longe*r d ia g o n a l. The 1 goal o f th is se'ction is
to in c lu d e th is co varian ce in the* lik e lih o o d fu n c tio n .
T h e lik e lih o o d fu n c tio n fo r th e te m p e ra tu re flu c tu a tio n s obse>rve'd by a noisele'ss
e'.\pe*rimenr w ith fu ll sky coverage' has the 1 fo rm :
C (f\C lh)
X
w h e re T den o te s o u r te m p e ra tu re m a p : a nd S tJ =
+ l ) O h^ V (". • h j ) / { 4” ).
w h e re th e P( are th e Lege*nelre p o ly n o m ia ls and it, is th e p ix e l p o s itio n on th e m ap.
I f we e x p a n d th e te m p e ra tu re m a p in s p h e ric a l h arm onie s: T ( i i ) = Y.tnP'fm'i r,„- th e n
th e lik e lih o o d fu n c tio n fo r each <i(m has a s im p le fo rm :
£ ( f i c ; h) x n
S ince we assum e th a t th e u n ive rse is is o tro p ic , th e lik e lih o o d fu n c tio n is in d e p e n d e n t
o f m. T h u s , we can sum o ver m a n d re w rite th e lik e lih o o d fu n c tio n as
- 2 1 n £ = y i ‘2 l + I )
In I
4-
+ C( / C lf h -
1
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(2-4)
C h a p t e r 2:
P a r a m e te r E s t im a tio n M e t h o d o lo g y
10
u p to an irre le v a n t additive* c o n s ta n t. Here, fo r a fu ll sky. noiseless e x p e rim e n t we
have id e n tifie d
\nf,„\2/ { ~ l +- 1) w ith O - N o te th a t th e lik e lih o o d fu n c tio n depends
o n ly t>n th e a n g u la r pow er s p e c tru m .
In th is lim it , th e umjidnr power spectrum
encodes a ll o f th e co sm o lo g ica l in fo rm a tio n in the* C M B .
C h a ra c te ris tic s o f the in s tru m e n t are also in c lu d e d in th e lik e lih o o d a na lysis,
.la ro s ik et al. (‘2 003a) show th a t th e d e te c to r noise is G aussian (see th e ir F ig u re G
and 1(3.4): co n se q u e n tly th e p ix e l noise in the sky m a p is also G aussian (H in s h a w
et a l.. 2 0 0 3 b ). T h e re s o lu tio n o f W M A P is q u a n tifie d w ith a w in d o w fu n c tio n . trf
(P age et a l.. 20031 >). T h us, th e lik e lih o o d fu n c tio n fo r o u r C M B m a p has th e same
fo rm as e q u a tio n (2 -2 ). h u t w ith S replaced by C = S r N w here N is the n e a rly
d ia g o n a l noise c o rre la tio n m a t r ix -’ a n d S tJ = £ a ('If -t-
1
)C)'"irf Pt {n, • h j) / ( 4 ~ ) .
I f fo re g ro u n d rem oval d id not re q u ire a sky cu t and the noise wen* u n ifo rm and
p u re ly d ia g o n a l, the n the lik e lih o o d fu n c tio n fo r th e W M A P e x p e rim e n t w o u ld have
th e fo rm (B o n d et a l.. 2000):
-2
In £ =
5
j
2
f + l;
.
(C} " + . V f \
"1
c,
c,
,
when* th e effec tiv e bias .V) is re la te d to th e noise* b ia s ,\V as , \ ' f =
a n d Cf = f ( ( ■+■ l ) / ( 2 ~ ) 5 I „ .
\> _ ;
J c +x ,
+ l) / ( 2 ” )
|“ /(*2/' + l) / « v . N o te th a t .V ) a n d C fh a p p e a r to g e th e r
in e q u a tio n (2 -5 ) because th e noise a n d co sm o lo g ic a l flu c tu a tio n s have th e same
s ta tis tic a l p ro p e rtie s , th e y b o th are G aussian ra n d o m fields.
Because o f th e fo re g ro u n d s k y -c u t. d iffe re n t m u ltip o le s are c o rre la te d and
o n ly a fra c tio n o f th e sky. / skv. is used in th e a n a lysis.
In th is case, it becomes
c o m p u ta tio n a lly p ro h ib itiv e to c o m p u te th e e xa ct fo rm o f th e lik e lih o o d fu n c tio n .
/ / noise m akes a n o n -ra n d o m phase c o n tr ib u tio n to th e d e te c to r noise a n d leads
to o ff-d ia g o n a l te rm s in th e noise m a tr ix . B y m a k in g th e noise .V0 a fu n c tio n o f f
-1
(d e n o te d by \ f ) we in c lu d e th is effect to le a d in g o rd e r (H in s h a w et a l. 2003)
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C h a p t e r 2:
11
P a ra m e te r E s t im a tio n M e t h o d o lo g y
T h e re an* several d iffe re n t a p p ro x im a tio n s used in th e C M B lite r a tu r e fo r the
lik e lih o o d fu n c tio n . A t la rg e (. e q u a tio n (2 -o ) is o fte n a p p ro x im a te d as G aussian:
ln £<;,„s, X - ^ ( C ; h
-C n Q u d C -Cr) -
( —-G )
~ (c
w h e n ' Q i e . th e c tirv a tu n * m a trix , is th e inverse o f t in 1 p ow er s p e c tru m co varian ce
m a tr ix .
T h e p ow er s p e c tru m covariance encodes th e u n c e rta in tie s in the p ow er s p e c tru m
due to cosm ic va ria nce , d e te c to r noise, p o in t sources, the sky c u t. and s y s te m a tic
e rro rs . H in sh a w et al. (2003a) and *j(2.2) d escrib e th e va rio us te rm s th a t e n te r in to
th e p ow er s p e c tru m co variance m a trix .
S ince th e lik e lih o o d fu n c tio n fo r th e p ow er s p e c tru m is s lig h tly n o n -G a u ssia n .
e q u a tio n (2-G) is a s y s te m a tic a lly biased e s tim a to r. B o n d et al. (2000) suggest u sin g
a lo g n o rm a l d is tr ib u tio n .
(B o n d et a l.. 2000: Sievers et a l.. 2002):
- 2 In £ , . \ =
- zf ) Q ( r ( $ ' - c>).
(2 -7 )
re
w h e re
= ln(CJh
- t-
A y ) . ^ = ln (C r + A r ) a n d Q m is th e lo ca l tra n s fo rm a tio n o f th e
c u rv a tu re m a tr ix Q to th e lo g n o rm a l v a ria b le s : r .
Qie = (C( + A f)Qff(Cc -+-A r )•
(2-8)
W e fin d th a t, fo r th e W M A P d a ta , b o th e q u a tio n s ( 2 - 6 ) a nd (2-7) are biased
e s tim a to rs . We use an a lte rn a tiv e a p p ro x im a tio n o f th e lik e lih o o d fu n c tio n fo r th e
C( s (e q u a tio n
2
- 1 1 ) m o tiv a te d by th e fo llo w in g a rg u m e n t.
W e can e xp a n d th e exact expression fo r th e lik e lih o o d (e q u a tio n 2--1) a ro u n d its
m a x im u m by w r itin g C( = CJh( l + ()■ T h e n , fo r a sin g le m u ltip o le f.
- 2 \ n C t = ( 2 f + l)[e - l n ( l + 0 ] = ( 2 f + 1) ( j
- j
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
.
(2-9)
C h a p t e r 2:
12
P a ra m e te r E s t im a tio n M e t h o d o lo g y
W e not«* th a t th e G aussian lik e lih o o d a p p ro x im a tio n is e q u iv a le n t to th e above
expressio n tru n c a te d at r 2:
- 2
In
x ( 2 1 -+- l ) / 2 [(CV — C)h) / C ) h\2 — (2 f +
T lie B o n d et al. (1998) expression fo r th e lo g n o rm a l lik e lih o o d fo r the equal
v a ria n c e a p p ro x im a tio n is
C(
■2 In £ | \ = ----- —
f-’
e1
" " c
T h u s o u r a p p ro x im a tio n o f lik e lih o o d fu n c tio n is g ive n by th e fo rm .
In C = ^ In ^ In C\ N .
w hen*
C\ Nhas th e 62051339
c
r t
(2 - 1 1 )
; (2-7) a p a rt fro m Q tf th a t n ot g ive n by e q u a tio n
( 2 - 8 ) b u t by
Q f f = (C'f h + X f )QndC'f!' + A H .
W e tested th is fo rm o f th e
lik e lih o o d by m a k in g 100.000 fu ll sky re a liz a tio n s
(2-12)
o f the
T T a n g u la r p ow er s p e c tru m C'/‘ . F o r each re a liz a tio n , th e m a x im u m lik e lih o o d
a m p litu d e o f flu c tu a tio n s in th e u n d e rly in g m o d e l was fo u n d a nd th e m ean value
was c o m p u te d . Since we kept a ll o th e r m o d e l p a ra m e te rs fixe d , th is one d im e n s io n a l
m a x im iz a tio n was c o m p u ta tio n a lly t r iv ia l. T h e G au ssia n a p p ro x im a tio n (e q u a tio n
2
- 6 ) was fo u n d to s y s te m a tic a lly o v e re s tim a te th e a m p litu d e o f th e flu c tu a tio n s by
~
0
.8 / f . w h ile th e lo g n o rm a l a p p ro x im a tio n u n d e re s tim a te s it b y ~ 0.2'X. E q u a tio n
( 2 - 1 1 ) was fo u n d to be a ccu ra te to <
2.2.
0.1
(X .
C u rvatu re M a trix
W e o b ta in th e c u rv a tu re m a t r ix in a fo rm th a t can be used in th e lik e lih o o d
a n a ly s is fro m th e p ow er s p e c tru m c o va ria n ce m a t r ix fo r C( c o m p u te d in H in s h a w
et a l. (2 0 03 a ). T h e m a tr ix is com posed o f several te rm s o f th e fo llo w in g fo rm :
—t r = \ J D f Df> ( d" , — r / c ) -I- (((>.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2 -1 3 )
C h a p t e r 2:
13
P a r a m e te r E s t im a tio n M e t h o d o lo g y
w h e re f((' i-s th«» c o u p lin g in tro d u c e d by tin * beam u n c e rta in tie s and p o in t source's
s u b tra c tio n (fft> = 0 i f ( = (') . i)K den o te s th e K ro n e c k e r d e lta fu n c tio n , and D f
d e n o te s the' d ia g o n a l te rm s.
=
.
(2 -1 4 )
The* q u a n tity r ff encodes th e m ode c o u p lin g due1 to th e s k y -c u t and is th e d o m in a n t
o ff-d ia g o n a l te rm ( it is set to be
0
if ( = (') . T h e m o d e -c o u p lin g c o e ffic ie n t. r t f . is
m ost e a sily d efined in te rm s o f the* c u rv a tu re m a tr ix . Q (r = D t Sj), -*- r i r / s / D f D f
(se»' H in s h a w et al. 2 0 0 3 1).
T h e sky cu t has tw o s ig n ific a n t effects on th e p ow er s p e c tru m covariance 1
m a tr ix . Because* le>ss d a ta is useel. the* co va ria n ce m a tr ix is increased by a fa c to r o f
/„k y
A n a d d itio n a l fae to r o f / „ ky arises fro m the* c o u p lin g to n e a rb y t tnoeles. The*
a d d itio n a l te>rm does n o t le>ael to a loss e>f in fo r m a tio n as n ea rb y I mode's are* s lig h tly
a n ti-c o rre la te d .
H in s h a w <*t a l. (2 0 03 b ) elescribe> th e beam u n c e rta in ty anel p o in t semrce te rm s
in c lu d e d in A 7 anel f ( r . T h e beam a n d c a lib ra tio n u n c e rta in tie s d e p e n d on th e
re a liz a tio n o f the 1 a n g u la r p ow er s p e c tru m on th e s k y C)k>. n o t on th e th e o re tic a l
a n g u la r p ow er spent ru m CJh. th u s th e y sh o u ld n ot change as. in e x p lo rin g th e
lik e lih o o d surface. we change C}h in th e expressiem fo r D f . T h is eliffers fro m o th e r
a p p ro a che s (e.g.. B rie lle et a l. (2 0 0 2 )).
R e sca lin g a ll th e c o n trib u tio n s to th e o ff
e lia go n ai te rm s in th e covarian ce m a t r ix w ith C l( h is n o t c o rre c t and le*ads to a
2
{/f
bias in o u r e s tim a to r o f {Ct) w h ich p ro p a g a te s, fo r e x a m p le , in to a ~ 2(/< e rro r on
th e m a tte r d e n s ity p a ra m e te r S2rn or ~ 2(X e rro r on th e s p e c tra l slo p e />,.
:,In th is e q u a tio n we have set to zero th e b ea m a n d p o in t sources u n c e rta in tie s .
T h is is because th e c o u p lin g co e ffic ie n t is c o m p u te d fo r an id e a l c u t sky.
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C h a p t e r 2:
14
P a ra m e te r E s t im a tio n M e th o d o lo g y
We fin d the c u rv a tu n ' m a tr ix b y in v e rtin g e q u a tio n (2-13)
*'■ = V ' * "
'
l & r
+ A
■
C2Ar,}
w here we have assum ed th a t the o ff-d ia g o n a l te rm s are s m a ll.
For co sm o lo g ic a l
m o d e ls th a t have UJh very d iffe re n t fro m the best fit C,. e q u a tio n 2-13 does not
y ie ld th e inverst' o f (2-13): in these cases th e in ve rsio n o f I / c
needs to be co m p u te d
e x p lic ite ly .
We do not p ro p a g a te tin ' W A /.4 P O.V/i c a lib ra tio n u n c e rta in ty in th e co variance
m a tr ix as th is u n c e rta in ty dot's not affect co sm o lo g ica l p a ra m e te rs d e te rm in a tio n s .
T h is s y s te m a tic o n ly affects th e p o w e r s p e c tru m a m p litu d e c o n s tra in t at th e 0 .3 '/
level, w h ile th e s ta tis tic a l e rro r on th is q u a n tity is ~
10
'/ .
Calibration with Mont e Carlo Simulations
T h e a n g u la r p ow er s p e c tru m is c o m p u te d u sin g th re e d iffe re n t w e ig h tin g s :
u n ifo rm w e ig h tin g in the s ig n a l-d o m in a te d re g im e {( <
2 0 0
). an in te rm e d ia te
w e ig h tin g scheme fo r 200 < ( < 430. a n d .Y„fc.s w e ig h tin g (fo r th e n o is e -d o m in a te d
re g im e 430 < f < 900 (H in s h a w et a l.. 2 0 0 3 b )). U n ifo rm w e ig h tin g is a m in im u m
va ria n c e w e ig h tin g in th e s ig n a l-d o m in a te d re g im e and .Vnf)s w e ig h tin g is a m in im u m
va ria n c e in th e noise d o m in a te d re g im e . H ow ever, in th e in te rm e d ia te re g im e the
w e ig h tin g schemes are not necessarily o p tim a l a n d th e a n a ly tic e xp re ssio n fo r the
co va ria n ce m a tr ix m ig h t th u s underestimate th e e rro rs. T o ensure th a t we have the
a p p ro p ria te e rro rs , we c a lib ra te th e co varian ce m a tr ix fro m 100.000 M o n te C a rlo
re a liz a tio n s o f th e sky w ith th e W M A P noise level, s y m m e triz e d beam s and th e K p 2
s k y c u t. A g o o d a p p ro x im a tio n o f th e c u rv a tu re m a tr ix can be o b ta in e d by using
e q u a tio n s (2-13) (2 -1 5 ). b u t s u b s titu tin g A ) and / sky w ith .V )‘,r a n d
fro m th e M o n te C a rlo s im u la tio n s , as show n in F ig u re s 2.1 a n d 2.2.
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c a lib ra te d
C h a p t e r 2:
15
P a ra m e te r E s t im a tio n M e t h o d o lo g y
W e fin d th a t fo r f < 200 th e w e ig h tin g scheme is n e a rly o p tim a l. T h e pow er
s p e c tru m co varian ce m a tr ix (2 -1 3 ) gives a co rre ct e s tim a te o f th e e rro r bars, th u s
we do not need to c a lib ra te A r o r f >ky. W e have c o m p u te d an e ffe c tiv e reduced
ch i-s q u a re d *
=
— 2
In C / n when* u is th e n u m b e r o f degrees o f fre e d o m . T h e
e ffe c tiv e reduced ch i-sq u a re d fro m th e M o n te -C a rlo s im u la tio n s in th is I range is
co nsiste n t w ith u n ity .
In the* in te rm e d ia te re g im e o u r nnsntz pow er s p e c tru m co va ria n ce m a tr ix
(e q u a tio n (2 -1 3 )) s lig h tly u n d e re s tim a te s th e e rro rs.
T h is can be c o rre c te d by
c o m p u tin g th e covarian ce m a tr ix fo r an e ffe ctive fra c tio n o f th e sky
as show n in
F ig u re 2.1. T h e ja g g e d lin e is th e r a tio o b ta in e d fro m th e M o n te C a rlo s im u la tio n s ,
tin * s m o o th curve shows th e fit to
we i.89! ;
f v ft
= ().8 1 3 + ().0 0 1 9 1 4 /-7 .4 ()5 x l ( r V - ' + 8 .0 5 x 1 (C "C
(fo r 200 < ( < 450). (2-16)
/s k y
F o r I > 450. in th e n o is e -d o m in a te d re g im e , th e w e ig h tin g is a s y m p to tic a lly
o p tim a l fo r I — ►?c. H ow ever, since we are using a s m a lle r fra c tio n o f th e sky. we
need a ga in the co rre ct th e f sky fa c to r. T h is n u m e ric a l fa c to r d escribes th e re d u c tio n
in e ffe ctive sky coverage (hie to w e ig h tin g th e w e ll observed e c lip tic poles m o re
h e a v ily th a n th e e c lip tic p la ne (see F ig u re 3 o f B e n n e tt et al. (2 0 0 3 b )). W e f it th is
fa c to r to th e n u m e ric a l s im u la tio n s o f th e T T s p e c tru m co varian ce m a tr ix . K o g u t
et al. (2003) notes th a t th is same fa c to r is also a goo d fit to th e M o n te C a rlo
s im u la tio n s o f th e T E s p e c tru m co va ria n ce m a tr ix . F o r th e n o is e -d o m in a te d regim e,
we d efine an e ffe ctive sky fra c tio n f * kv = /sky/
A ? ff = y S J ? " ( / * £ .)
(2
( + l
) /2
1
-14 and an e ffe ctive noise g ive n by
— C f m. w h ic h can be o b ta in e d fro m th e noise bias o f
th e m aps A # bv a noise c o rre c tio n fa c to r . V / fr/A v . T h is is show n in F ig u re
2 .2
w here
'T h is is n o t e x a c tly th e re d uce d ch i-sq u a re d because th e lik e lih o o d is non -g a ussia n
e s p e c ia lly a t low i.
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C h a p t e r 2:
1C
P a ra m e te r E s t im a t io n M e t h o d o lo g y
th e s m o o th cu rve is th e fit wo a d o p t to th is o o rro o tio n fa c to r.
VI
= 1.04G - 0.0()()234G(t - 450) + 3.204 x
10
7( / - 4 50 )-
fo r I > 450 .
(2-17)
T h is c a lib ra tio n o f tin* co varian ce m a tr ix fro m th e M o n te C a rlo s im u la tio n s
a llo w s us to use the e ffe ctive reduced o ili-s q u a re d as a to o l to assess goodness o f fit.
It can also be used to d e te rm in e the re la tiv e lik e lih o o d o f d iffe re n t m o d els (e.g
2.3.
L ikelihood for th e T E angular p ow er sp e ctru m
Since th e T E sig n a l is n o is e -d o m in a te d , we ;891 ; * a G au ssia n lik e lih o o d , w here
th e c u rv a tu re m a tr ix is given by
(2-18)
T h e expression fo r D f h is given by e q u a tio n (10) o f K o g u t et a l. (2 0 0 3 ). g ive n here
a g a in fo r c la rity :
w h e re .V *rr
N
ee
arp
T T and T E noise bias te rm s . f ^ y = 0.85 is th e
fra c tio n a l sky coverage fo r th e K p 2 m ask, a n d / r'jjfv = / Sky /E 1 4 fo r n o is e -w e ig h tin g .
T h e c o u p lin g co e ffic ie n t due to th e s k y c u t. r f f - . is o b ta in e d fro m 100.000 M o n te
C a rlo re a liz a tio n s o f th e sky w it h W M A P m a sk and noise level. T h e T E s p e c tru m
is c o m p u te d w ith noise inverse w e ig h tin g : in th is re g im e rtc dep e nd s o n ly on th e
d iffe re n ce _V = f — (' a n d is set to be 0 a t se p a ra tio n s A / > 15. W e use a ll n iu ltip o le s
2 < f < 450. as c o m p a ris o n w ith the M o n te C a rlo re a liz a tio n s show s th a t in th is
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C h a p t e r 2:
P a r a m e te r E s t im a tio n M e th o d o lo g y
17
re g im e e q u a tio n (2 -1 8 ) c o rre c tly e s tim a te s rh<* T E u n c e rta in tie s .
We have also
v e rifie d on th e s im u la tio n s th a t th e G aussian lik e lih o o d is an unbiased e s tim a to r,
and th a t th e e ffe ctive reduced \ J is centered a ro u n d
1.
T h e a m p litu d e o f th e co varian ce between T T and T E p ow er s p e c tra is
~ r / ( I -i- n {l h j C \ h ) w here r is th e c o rre la tio n te rm (C'j h
1 ) _l ~ 0.2. Since
C j ’ l' / i i f h < < 0.25 fo r l- y r d a ta , we neglect th is te rm , b u t we w ill in c lu d e it in tin*
2
* y r a n a lysis as it becomes in c re a s in g ly im p o rta n t.
W e p ro v id e a s u b ro u tin e th a t reads in a set o f C}1' ( T T . o r T E o r b o th ) and
re tu rn s th e lik e lih o o d fo r tin* U ’A /.AP d ata set in c lu d in g a ll the effects d escribe d in
th is se ctio n . T h e ro u tin e is a v a ila b le at h t t p : / / l a m b d a . g s f c . n a s a . g o v .
3.
M arkov C h ains M o n te C arlo L ikelihood A n alysis
T h e a n a lysis d escribed in S pergel et al. (2003) and P eiris et a l. (2003) is
n u m e ric a lly d e m a n d in g . A t each p o in t in th e s ix o r m ore d im e n s io n a l p a ra m e te r
space a new m o d el fro m C M B F A S T ’ (S e lja k X: Z a ld a rria g a . 199G) is c o m p u te d . O u r
ve rsio n o f th e code in c o rp o ra te s a n u m b e r o f c o rre c tio n s and uses th e RECFAST
(Seager et a l.. 1999) re c o m b in a tio n ro u tin e . M o st o f th e lik e lih o o d c a lc u la tio n s were
done w ith fo u r shared m e m o ry 32 C P U S G I O r ig in 300 w ith 600 M H z processors.
W it h
8
processors p e r c a lc u la tio n , each e v a lu a tio n o f C M BFAST fo r ( < 1500 fo r a
Hat re io n ize d A d o m in a te d u niverse re q u ire s 3.6 seconds. (T h e sca lin g is n o t lin e a r:
w ith 32 processors each e v a lu a tio n re q u ire s 1.62 seconds.)
A g rid -b a se d lik e lih o o d a n a lysis w o u ld have re q u ire d p ro h ib itiv e a m o u n ts o f
T,W e used th e p a ra lle liz e d version 4.1 o f C M BFAST developed in c o lla b o ra tio n w ith
U ros S e lja k a nd M a tia s Z a ld a rria g a .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C ' l ui p tr r 2:
18
Pnrnmctix Estimatiim Mrtlioriology
C P U rim e . For e xa m p le . a coarse g rid
p a ra m e te rs re q uires
20 g rid p o in ts p e r d im e n s io n ) w ith six
6.4 x 10‘ e v a lu a tio n s o f th e p ow er sp e ctra . A t l.G seconds per
e v a lu a tio n , th e c a lc u la tio n w o u ld take ~ 1200 days. C h riste n se n A M e ye r
(2000)
p ro po se d u sin g M a rk o v C h a in M o n te C a rlo ( M C M C ) to in v e s tig a te th e lik e lih o o d
space.
T h is a p p ro a ch has becom e th e s ta n d a rd to o l fo r C M B analyses (e.g..
C h ris te n s e n et a l.. 2001: K n o x et a l.. 2001: Lew is A' B rid le . 2002: K o so w sky et a l..
2002) and is the backb o ne o f o u r a n a ly s is e ffo rt. For a Hat re io n ize d A d o m in a te d
universe , we can e va lu a te th e lik e lih o o d —
1 2 0 .0 0 0
tim e s in <
2
days u sin g fo u r sets
o f e ig ht processors. A s we e x p la in below , th is is ade q ua te fo r fin d in g th e best fit
m o d e l and fo r re c o n s tru c tin g th e
1-
and
2
-rr confid e nce levels fo r th e c o sm o lo g ic a l
p a ra m e te rs.
We re fer th e re a de r to C lilks et a l. ( 199G) fo r m o re in fo rm a tio n a b o u t M C M C .
H ere, we w ill o n ly p ro v id e a b r ie f in tr o d u c tio n to tin* su b je ct and c o n c e n tra te on the
issue o f convergence.
3 .1 .
M arkov C h ain M o n te C arlo
M C M C is a m e th o d to s im u la te p o s te rio r d is trib u tio n s .
In p a r tic u la r , we
s im u la te ' o b se rva tio n s fro m th e p o s te rio r d is tr ib u tio n 'P (o |.r). o f a set o f p a ra m e te rs
n g ive n event x. o b ta in e d v ia Bayes' T h e o re m .
= J P u p P M _
(2_19)
J V { . r jo ) V ( n )(/o
w here V ( x \ n ) is th e lik e lih o o d o f e vent x given th e m o d el p a ra m e te rs o and V { n )
is th e p r io r p r o b a b ility d e n sity. F o r o u r a p p lic a tio n th e W M A P n d e n o te s a set
o f c o s m o lo g ic a l p a ra m e te rs (e.g.. fo r th e s ta n d a rd , fla t A C D M m o d e l these c o u ld
be. th e c o ld -d a rk m a tte r d e n s ity p a ra m e te r f>r . th e b a ry o n d e n s ity p a ra m e te r Q&.
th e s p e c tra l slope n ,. th e H u b b le c o n s ta n t
in u n its o f 100 k m s _l M p c )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
h. th e
C h a p t e r 2:
P a r a m e te r E s t im a t io n M e t h o d o lo g y
19
o p tic a l d e p th r and the' p ow er s p e c tru m a m p litu d e ' .4). and e*vent ./• w ill he> the> set
o f observe'd Cf.
The' M C’ M C ge*ne*rate>s ra n d o m d ra w s (i.e*. s im u la tio n s ) fro m the* p o ste 'rio r
d is t r ib u t io n th a t are* a "fa ir" sample* o f the* lik e lih o o d surface'. F ro m th is sample', we*
can estim ate* a ll e>f the* c ju a n titie s o f inte*re*st a b o u t the* p o ste 'rio r d is tr ib u tio n (m ean,
variance*, conheh'tice* leve*ls). T h e M C M C tnerhoel scale's a p p ro x im a te ly lin e a rly w ith
the* n u m b e r o f param ete*rs. th u s a llo w in g us to pe*rform like d ih o o d a n a ly s is in a
reasonable* a m o u n t o f time*.
A prope*rly ele*rived anel impleme*nte*d MC’ M C d ra w s fro m th e jo in t p o s te rio r
d e n s ity P ( n j. r ) once* it has e-onverged to the* s ta tio n a ry d is tr ib u tio n . The* p rim a ry
e-onsieleration in imple*me*nting M C M C ’ is de*te*rmining whe*n the* c h a in has converged.
Afte*r an in it ia l ""h n m - i n " p e rio d , a ll fu r th e r sam ples can be* th o u g h t o f as c o m in g
fro m the* s ta tio n a r y d is tr ib u tio n . In othe*r w o rd s the ch a in has no eh*pe*nele'nce' on the
s ta r tin g lo c a tio n .
A nothe*r fundam e*ntal proble*m o f inferenc e* fro m M a rk o v ch a in s is th a t the*re
are a lw a ys are*as o f th e ta rg e t d is tr ib u tio n th a t have n ot be*e'n covered by a fin ite'
c h a in . I f th e M C M C is ru n fo r a v e ry lo n g tim e , th e e *rg od icity o f the* M a rk o v ch ain
guarantc'es th a t e v e n tu a lly the* ch a in w ill cover a ll th e ta rg e t d is tr ib u tio n , b u t in th e
s h o rt te rm th e s im u la tio n s c a n n o t te ll us a b o u t areas where* th e y have* n o t been.
I t is th u s c ru c ia l th a t th e ch a in achieves g o o d "mixing". I f th e M a rk o v c h a in does
n o t m ove r a p id ly th ro u g h o u t th e s u p p o rt o f th e targe't d is t r ib u t io n because o f p o o r
mixing, it m ig h t ta ke a p r o h ib itiv e a m o u n t o f tim e fo r th e c h a in to f u lly e x p lo re th e
lik e lih o o d su rface . T h u s it is im p o r ta n t to have a convergence c r ite r io n a n d a m ix in g
d ia g n o s tic .
P lo ts o f th e sa m p le d M C M C p a ra m e te rs o r lik e lih o o d value's versus
ite r a tio n n u m b e r are c o m m o n ly used to p ro v id e such c r ite r ia (le ft p anel o f F ig u re
2 .3 ).
H ow ever, sam ples fro m a c h a in are ty p ic a lly s e ria lly c o rre la te d : v e ry h ig h
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C h a p t e r 2:
P a r a m e te r E s t im a tio n M e th o d o lo g y
20
a u to -c o rre la tio n loads to litrIt* m ovem ent o f th e ch a in and th u s makes th e ch a in to
"a p p e a r" to have converged. For a m ore d e ta ile d d iscussion see G iik s et al. (199G).
I's it ig a M C M C th a t has nor fu lly e x p lo re d th e lik e lih o o d su rface fo r d e te rm in in g
c o s m o lo g ica l p a ra m e te rs w ill y ie ld urony re su lts. W e d escrib e below th e m e th o d we
use to ensure convergence and good m ix in g .
3.2.
C on vergen ce and M ixin g
We use tin* m e th o d p roposed by G e lm a n
R u b in ( 1992) to test fo r convergence
a nd m ix in g . T h e y advocate* c o m p a rin g several sequences d ra w n fro m d iffe re n t
s ta r tin g p o in ts and ch e ckin g to see* th a t th e y are in d is tin g u is h a b le . T h is m e th o d not
o n ly rests convergence b u t can also dia gn o se p o o r m ix in g . F o r any analysis o f the
W M A P data, ire stronyly cncouruye the use of a convenience criterion.
Let us co n sid e r M ch a in s (th e analyses in S pergel et al. (2003) a nd FV iris
et al. (2003) use 4 ch ains unless o th e rw is e s ta te d ) s ta r rin g a t w e ll-se p a ra te d p o in ts
in p a ra m e te r space (fo r each c h a in , we use a ra n d o m 3-rr ste p in a ll p a ra m e te rs ,
aw ay fro m a " fid u c ia l" m o d e l th a t is a reasonable f it to th e d a ta by eye): each has
2
.Y e le m en ts, o f w h ic h we co n s id e r o n ly th e last N: {if1
, } w here / =
j = I
I
Y and
\ l . i.e. y den o te s a ch a in ('le n ie n t (a p o in t in p a ra m e te r space) th e in d e x
i ru n s over th e e le m en ts in a ch a in th e in d e x j ru n s over th e d iffe re n t ch ains. We
d efin e th e m ean o f th e ch a in
>r =
* ' i= i
( 2 ~ 2 0 )
a n d th e m ean o f th e d is tr ib u tio n
i
•'
-v.w
«J=t
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( 2 - 2 1 )
C h a p te r 2:
P a ra m e te r E s t im a t io n M t 't h o r io lo g y
21
W e th e n d efin e th e va ria n ce betw een ch a in s as
a n d th e va ria n ce w ith in a ch ain as
W =
M ( -V -
1)
o
T h e q u a n tity
R = —
is th e r a tio o f tw o e stim a te s o f th e va ria n ce in tin ' ta rg e t d is tr ib u tio n : th e n u m e ra to r
is an e s tim a te o f th e va ria nce th a t is u nbiased i f th e d is tr ib u tio n is s ta tio n a ry , b u t is
o th e rw is e an overestim ate'. T h e d e n o m in a to r is an u n d e re s tim a te o f the* variance' o f
the* ta rg e t d is tr ib u tio n i f the1 in d iv id u a l sequences eliei m>t have* time* to converge.
The* convergence o f the' M a rk o v ch a in is th e n me>nite>reel by re c o rd in g the'
q u a n tity R fo r a ll the* p a ra m e te rs anel ru n n in g th e s im u la tio n s u n til the* value's fo r
R are* a lw a ys < 1.1. G e lm a ti (2000) sugge’st to use* value's fo r R < 1.2. He're*. we*
ee)nservativelv a d o p t th e e rite rie m /? <
1.1
as o u r elefinitiem o f cemve'rge'nee'. We
have fo u n d th a t th e femr chains w ill som e'tim es ge» in and o u t e>f ee)iivergence' as theyv
e'xpleire th e likeliheioel surface, e'specially i f th e n u m b e r o f p o in ts alreaelv in th e ch ain
is s m a ll. T o a vo id th is , one e-oulel ru n m a n y ehains sim ultane'eaisly o r ru n e>ne' ch a in
fo r a v e ry lo n g tim e (e.g.. P a n te r et al. (2 0 0 2 )). D ue te) C’ P U -tim e c o n s tra in ts , we
ru n fo u r chains u n til th e y f u lf ill be>th o f th e fo llo w in g c r ite r ia a) then- have* reached
convergence, anel b) each ch a in c o n ta in s a t le*ast 30.000 pennts.
In a d d itio n to
m in im iz in g chance d e v ia tio n s fro m convergence, we fin d th a t th is m a n y p o in ts are
needed to be able te> ro b u s tly re c o n s tru c t th e
lik e lih o o d fo r a ll th e p a ra m e te rs.
1 -a n d
2
-rr levels o f th e m a rg in a liz e d
Fe)r m o st ch ains, th e b u rn -in tim e is re la tiv e ly
ra p id , so th a t we o n ly d is c a rd th e firs t
2 0 0
p o in ts in each c h a in , h ow ever th e re s u lts
a re n o t se n sitive to th is p ro ce d u re .
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C h a p t e r 2:
P a r a m e te r E s t im a t io n M e t h o d o lo g y
3.3.
M arkov C h ain s in P r a ctice
In th is seer ion we e x p la in th e necessary steps to ru n a M C ’M C ’ fo r C M B
tem pe ra ture* p ow er s p e c tru m . It is s tra ig h tfo rw a rd to gen e ra lize these* in s tru c tio n s
to in c lu d e the* re m p e *ratu re *-p oIa riza tio ri powe*r s p e c tru m and othe*r d ata sets. The*
M C M C is e s s e n tia lly a ra n d o m w a lk in p a ra m e te r space*. whe*re* the* p r o b a b ility o f
be*itig at a ny p o s itio n in the* space is p ro p o rtio n a l to the* p o s te rio r p r o b a b ility .
He*re is o u r basic app ro a ch:
1
) S ta rt w ith a set o f c o sm o lo g ica l p a ra m e te rs {cm }- c o m p u te the* Cj and the*
lik e lih o o d £ t = C(C}"'\Cf ).
2) Take a ra n d o m step in paratne*te*r space* to o b ta in a ne*w se*r o f c o s m o lo g ic a l
parame*te*rs {o_>}.
The* p r o b a b ility d is tr ib u tio n o f the* step is ta ke n to be*
G au ssia n in each d ire c tio n / w ith r.m .s g ive n by rr,. We w ill re*fe*r b elow to a,
as th e
"step size*". The* choice o f th e ste*p size* is im p o rta n t to optim ize* the*
c h a in e*fficie*ncy (se*e* ?j3.4)
3) C o m p u te the* Cf'*' fo r th e new set o f co sm o lo g ic a l p a ra m e te rs and th e ir
lik e lih o o d £_>.
4
.a)
I f C > / C i > 1. "take the* ste p ” i.e. save th e new set o f co sm o lo g ic a l p a ra m e te rs
{ e h } as p a rt o f th e ch a in , th e n go to ste p
4.b )
2
a fte r the* s u b s titu tio n { o i } — >
I f C , / C \ < 1. d ra w a ra n d o m n u m b e r x fro m a u n ifo rm d is t r ib u t io n fro m
0 to
. I f x > C - i ! £ \ ” (h j n o t ta ke th e s te p ” , i.e. save th e p a ra m e te r set { a j
as
1
p a rt o f th e ch a in and re tu rn to ste p 2. I f x < C, ^!L\ . " ta ke th e s te p ” , i.e. do
as in 4 .a ).
5) F o r each co sm o lo g ica l m o d e l ru n fo u r c h a in s s ta r tin g at ra n d o m ly chosen,
w e ll-se p a ra te d p o in ts in p a ra m e te r space. W h e n th e convergence c r ite r io n is
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C h a p t e r 2:
2.5
P a r a m e te r E s t im a t io n M e t h o d o lo g y
sa tis fie d and the chains have enough p o in ts to p ro v id e reasonable sam ples fro tn
th e a p o s te rio ri d is tr ib u tio n s (i.e. enough p o in ts to lie able to re c o n s tru c t the
1-
and 2 -rr levels o f tin ' m a rg in a liz e d lik e lih o o d fo r a ll th e p a ra m e te rs) s to p the
chains.
It is c le a r th a t th e M C M C a p p ro a ch is e a sily g en e ra lize d to c o m p u te the jo in t
lik e lih o o d o f W’.U.AP d a ta w ith o th e r d ata sets.
3 .4 .
Im p rovin g M C M C E fficiency
T h e M a rk o v ch ain e fficie n cy can be im p ro v e d in d iffe re n t ways. W e have tu rn 'd
o u r a lg o r ith m by re p a ra m e te riz a tio n and o p tim iz a tio n o f th e stop size.
Rrpararnetenzdt 1 o n
D egeneracies and p o o r p a ra m e te r choices slo w th e rate* o f convergence and
m ix in g o f th e M a rk o v C h a in . T h e n * is one n e a r-e xa ct degeneracy (th e g e o m e tric
deg e ne ra cy) and several a p p ro x im a te degeneracies in th e p a ra m e te rs d e s c rib in g the
C M B p o w e r s p e c tru m (B o n d et a l.. 1994: E fs ta th io u
B o n d . 1999). T h e n u m e ric a l
effects o f these degeneracies are reduced by fin d in g a c o m b in a tio n o f c o s m o lo g ic a l
p a ra m e te rs (e.g.. f 2 r .
1 2 *.
h. e tc .)
th a t have e s s e n tia lly o rth o g o n a l effects on th e
a n g u la r p o w e r s p e c tru m . T h e use o f such p a ra m e te r c o m b in a tio n s rem oves o r reduces
degeneracies in th e M C M C and hence speeds u p convergence and im p ro v e s m ix in g ,
because th e c h a in does n o t have to spend tim e e x p lo rin g degeneracy d ire c tio n s .
K o s o w sky e t a l. (2002) in tro d u c e d a set o f re p a ra m e te riz a tio tis to d o ju s t th is . In
a d d itio n , these new p a ra m e te rs re fle ct th e u n d e rly in g p h y s ic a l effects d e te rm in in g
th e fo rm o f th e C M B p ow er s p e c tru m (w e w ill re fe r to these as p h y s ic a l p a ra m e te rs ).
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C h a p t e r 2:
24
P a r a m e te r E s tim a tio n M e t h o d o lo g y
T h is leads to p a r tic u la r ly in tu itiv e and tra n s p a re n t p a ra m e te r dependencies o f th e
C M B p ow er s p e c tru m .
F o llo w in g K oso w sky et al. (2002). we use a core set o f s ix p h ysica l p a ra m e te rs .
T h e re are tw o p aram ete rs fo r the* p h ysica l energy d e n sitie s o f co ld d a rk m a tte r.
.j, = < },/)'. and baryons.
= M ft/r. T h e n * is a p a ra m e te r fo r th e c h a ra c te ris tic
a n g u la r scale o f the a cou stic peaks.
w here a,u, is th e scale fa c to r a t d e c o u p lin g .
-
('"!)('
hJ s L
)
1
■_»
ds
(2-26)
is th e sound h o riz o n at d e co u p lin g , and
) =
77-
/A)
/ l
[ ( 1 - V.)r2
4-
U ms + <>r
J "1J d.r
(2 -2 7 )
1
is th e a n g u la r d ia m e te r d ista n ce at d e c o u p lin g , w here H {) denotes th e H u b b le
c o n s ta n t a n d c is th e speed o f lig h t. H ere {}m = i l r 4- 52*,. { } \ denotes tin* va cu u m
e n e rg y d e n s ity param ete rs, i l = <?„, 4-2r;«i =
4-
and th e ra d ia tio n d e n s ity p a ra m e te r
U-.. V.v are th e the p h o to n and n e u trin o d e n s ity p a ra m e te rs
re s p e c tive ly. For re io n iz a tio n we use th e p h ysica l p a ra m e te r Z = e x p ( —2 r ) w here r
den o te s th e o p tic a l d e p th to th e la st s c a tte rin g su rface (n o t th e d e c o u p lin g su rfa ce ).
T h e re m a in in g tw o core p a ra m e te rs are th e s p e c tra l slope o f th e sca la r p rim o rd ia l
d e n s ity p e r tu rb a tio n pow er s p e c tru m . n s. and th e o v e ra ll a m p litu d e o f th e p r im o r d ia l
p o w e r s p e c tru m .4. B o th are n o rm a liz e d a t k = 0.05 M p c ~ l ( f
700).
F o r m o re c o m p le x m odels we add o th e r p a ra m e te rs as d e scrib e d in S pergel et al.
(2003) and P e iris et al. (2003) a n d in §5. T o in v e s tig a te n o n -fla t m o d e ls we use th e
v a c u u m energy.
= f l \ h 2. O th e r e xa m p le s in c lu d e th e te n so r in d e x . n t . th e te n s o r
to s c a la r ra tio , r . and the ru n n in g o f th e sca la r s p e c tra l in d e x . d n s/ d l n k .
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C h a p t e r 2:
P a r a m e te r E s t im a tio n M e t h o d o lo g y
Here. we re la te th e in p u t p a ra m e te r fo r th e o v e ra ll n o rm a liz a tio n . .4. as in th e
C M B F A S T code (ve rsio n 4.1 w ith I ’ N N O R M o p tio n ), to th e a m p litu d e o f p r im o rd ia l
e o m o v iiiff c u rv a tu re p e rtu rb a tio n s 1Z. A ^ fA 'o ) = (A‘'* /2 :rJ )(j7S jJ). W e also re la te o u r
c o n v e n tio n fo r th e ten so r p e rtu rb a tio n s to th e one in th e code. C M B F A S T c a lc u la te s
(2-28)
where' 'I* is th e N e w to n ia n p o te n tia l. uriik') is the ra d ia tio n tra n s fe r fu n c tio n , and
TI) = 2.725 x U ) ' 1 is th e C’ M B te m p e ra tu re in u n its o f //K .
The* tild e 1 ele»ne»te*s
th a t S j , ( k ) is use'el in C M B F A S T . b u t diffe>rs fro m o u r cemveuitieui.
A j, = A /1 6 .
The 1 c o m o v in g c u rv a tu re p e r tu rb a tio n . Tv. is re'late'd to
= —(3 /5 )7 S : th u s. A ’^ (A ’ ) = (2 5 /9 )A '4 (A -).
where 1
by
N o te th a t th is re la tio n heilds fro m
ra d ia tio n d o m in a tio n to m a tte 'r d o m in a tio n w ith a ccu ra cy b e tte r th a n 0.5 rA .
C M B F A S T uses .4 te> p a ra m e te riz e A ^ lA o ) .
T h e te'nse>r p e rtu rb a tie u is are1
c a lc u la te d a c c o rd in g ly . T h e re la tio n s are
(2-30)
(2-31)
T h e re fo re , one o b ta in s
A
k
(A-o ) = 2.95 x 10-<J.4.
T h e a m p litu d e .4 is n o rm a liz e d a t
(2-32)
= 0.05 M p c - 1 a nd th e te n so r to sca la r
r a tio r is e v a lu a te d a t A-0 = 0.002 M p c - 1 . unless o th e rw is e sp ecified . T o c o n ve rt
.4 (Aq) to .4(A-[). we use
n, (Aco)—1* kotn,/d In k) ln(fci fko)
(2-33)
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C h a p t e r 2:
26
P a r a m e te r E s t im a t i o n M e t h o d o lo g y
Step Size Optimization
T h e choir** o f tii* ' step size in the M a rk o v C h a in is c ru c ia l to im p ro v e th e ch a in
e ffic ie n c y a n d speed up convergence. I f th e step size is to o b ig . th e accep tan ce ra te
w ill
1)*'
v e ry s m a ll: i f th e step size is to o s m a ll th e acceptance ra te w ill be h ig h b u t
th e ch a in w ill e x h ib it p o o r m ix in g . B o th s itu a tio n s w ill lea*I to slow convergence.
F o r o u r in it ia l step sizes fo r each p a ra m e te r we us*' tin* s ta n d a rd d e v ia tio n fo r
each p a ra m e te r w hen a ll th e o th e r p a ra m e te rs are held fixe d a t th e m a x im u m
lik e lih o o d value. These are easy to fin d one*' a p re lim in a ry c h a in has been ru n and
th *' lik e lih o o d su rface has been fitte d , as e xp la in e d in $3.-1. I f a given p a ra m e te r is
ro u g h ly o rth o g o n a l to a ll th e o th e r p a ra m e te rs, it is not necessary to a d ju s t th*' step
size fu r th e r: in th** presence o f severe degeneracies the step size e s tim a te needs to bo
increased by a "banana c o rre c tio n ” fa c to r w h ic h is a p p ro x im a te ly th e r a tio o f the
p ro je c tio n o f th*'
1 -rr
e rro r a lo n g the degeneracy to th e p ro je c tio n p e rp e n d ic u la r to
t he degeneracy.
W ith tin 's*' o p tim iz a tio n s th e convergence c rite rio n is sa tis fie d fo r th e 4 ch ains
a fte r ro u g h ly 30.000 stops (2 .Y = 3 0 .0 0 0 ) fo r a m o d e l w ith G p a ra m e te rs. O n a
O r ig in 300 m a ch in e th is takes ro u g h ly 32 h rs ru n n in g each ch a in on
8
processors.
T hese n u m b e rs serve o n ly as a ro u gh in d ic a tio n : convergence speed d ep e nd s on th e
m o d e l a n d on th e d a ta -s e t: fo r a fixe d n u m b e r o f p a ra m e te rs, convergence can be
s ig n ific a n tly slow er i f th e re a re severe degeneracies am o ng th e p a ra m e te rs : a d d in g
m o re d a ta s e ts m ig h t slow d o w n the e v a lu a tio n o f a single ste p in th e c h a in , b u t can
a lso speed u p convergence b y b re a k in g degeneracies.
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C h a p t e r 2:
P a ra m e te r E s t im a t io n M e t h o d o lo g y
Likelihood Surface Fitting
T h e lik e lih o o d surface e xp lo re d by the MC’ M C was fou n d to be fu n c tio n a lly w e ll
a p p ro x im a te d by a q u a rtie e xpa n sio n o f th e co sm o lo g ica l p a ra m e te rs (fo r e xa m p le .
{<■>,} = {^v-V- ti -fl A. Z . .4}):
Here (/ are fit co e fficie n ts and fi, are re la te d to th e co sm o lo g ica l p a ra m e te rs v ia
= (o , — <■>'*) / n , . w h e n 1 o )1 is th e m a x im u m -lik e lih o o d value o f tin* p a ra m e te r.
L o w e r-o rd e r expansions were u n a b le to re p ro d u ce th e lik e lih o o d surface.
p a ra m e te rs th e re are A // = 210 fit co efficie n ts.
W it h G
W r itin g (2-11) as ;/ = if ■x. the
m in im u m Ieast-squares e s tim a to r fo r if is
w here .V is th e .V x M f m a tr ix . \ tJ = x {j ' . .V th e n u m b e r o f u n iq u e p o in ts in th e
ch a in .
W e ru n p r e lim in a ry M C M C chains w ith "g u e s s tim a te d ” step sizes u n t il th e re
are
1000
u n iq u e p o in ts in to ta l. T h e n we use e q u a tio n 2-34 to c u t th ro u g h th e
lik e lih o o d su rface a t th e m a x im u m lik e lih o o d value to fin d th e lr r level in each
p a ra m e te r d ire c tio n (see § 3.4). T h is defines o u r “ ste p size" fo r subsequent chains.
3.5.
T h e C h oice o f P riors
F ro m B ayes’ T h e o re m (e q u a tio n 2-19) we can in fe r V { n t \x). th e p r o b a b ility
o f th e m o d e l p a ra m e te rs o , give n th e event x (i.e . o u r o b s e rv a tio n o f th e p ow er
s p e c tra ), fro m th e lik e lih o o d fu n c tio n once th e p r io r is sp ecified. I t is re a son a ble to
ta k e p r io r p ro b a b ilitie s to be e qual w hen n o th in g is kn ow n to th e c o n tra ry (B a ye s’
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C h a p t e r 2:
P a ra m e te r E s t im a tio n M e t h o d o lo g y
28
p o s tu la te ).
I ’ nlcss o th e rw ise state»d we assume* u n ifo rm p rio rs on th e p a ra m e te rs
g ive n iti T a b le I. N o te th a t we assume u n ifo rm p rio rs on _j,..
u n ifo rm p rio rs on
and f)A ra th e r th a n
a tu l / / „ .
Exc ept fo r the* p rio rs on r . and ir (th e e q u a tio n o f state* o f the* d a rk ene rg y
c o m p o n e n t), th e MC’ M C s never h it th e im p o sed b o u n d a rie s, th u s m ost o f o u r choices
fo r p rio rs have no effect on the* o u tco m e . A d e ta ile d discussion a b o u t th e p r io r on r
is presented in Spergel et al. (2003). W e set lo w e r b o u n d on ir at —3.2 ( — 1.2) b u t
we d is c a rd the* region o f p a ra m e te r space* w here ir < —3 (i f < — I ) . T h is is necessary
because o u r b e s t-fit value* fo r th is p a ra m e te r is close* to the* b o u n d a ry .
I f we* had
in s te a d set the* p rio r to be* //■ > —3 (i r > —1). them the* chains w o u ld fa il to be* a fa ir
repre*se*ntation o f the* poste*rior e iis trib u tio u in the* re'gion o f p a ra m e te r space* whe*re>
the* d is ta n c e fro m th e b o u n d a ry is com parable* to th e ste'psize*.
3.6.
M C M C O u tp u t A n alysis
W e rne'rge* th e 4 converged M C M C chains ( ^
120.000 p o in ts ) in to one*. F ro m
th is we g ive the* co sm o lo g ica l p a ra m e te rs th a t y ie ld o u r be*st e*stim ate o f Cf a n d we
g iv e the* m a rg in a liz e d d is tr ib u tio n o f th e p a ra m e te rs. We* c o m p u te th e m a rg in a liz e d
d is t r ib u t io n fo r one p a ra m e te r, and th e jo in t d is tr ib u tio n fo r tw o p a ra m e te rs ,
o b ta in e d m a rg in a liz in g over a ll th e o th e r p a ra m e te rs.
Since th e M C M C passes
o b je c tiv e te*sts fo r convergence and m ix in g , th e d e n s ity o f p o in ts in p a ra m e te r space
is p r o p o rtio n a l to th e p o s te rio r p r o b a b ility o f th e p aram ete rs.
T h e m a rg in a liz e d d is tr ib u tio n is o b ta in e d b y p ro je c tin g th e MC’ .MC p o in ts . For
th e m a rg in a liz e d p a ra m e te rs values o ,. S pergel et a l. (2003) q u o te th e e x p e c ta tio n
va lu e o f th e m a rg in a liz e d lik e lih o o d , f Cct,dn, = l / . V J ^ r q . , . Here. .V is th e n u m b e r
o f p o in ts in th e m erged ch a in anel o t , denotes th e value o f p a ra m e te r o , a t th e t~th
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h n p t r r 2:
P a ra m e te r Est imati on Mt'thodology
T able 2.1.
P rio rs fo r B ayesian A na lyse s
Ite m
0 <
< I
< a.-* < 1
0.005 < f)A < 0.1
0
0 < r < 0.3
0.5 < A < 2.5
D < "s\k„ < ->
0
0
<
<
H is ., <
/is .,
<
2
3 ( M )«
—0.5 < t h i / d X u k < 0.5
—3 < riy < 3
0 < r < 2.5
0 < - . • „ <
1
—3.2( - 1 .2 )a< ir < 0
0 <
< 1
a\Ve w ill present
tw o
sets o f re su lts, one w ith
th e p rio r i f > —1 . 2 th e
o th e r w ith it: > —3.0
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 2:
P a r a m e te r E s t im a tio n M e th o d o lo g y
30
ste p o f th e c h a in . The* last e q u a lity becom es cle a r i f we co n sid e r th a t the* M C ’ M C
gives to each p o in t in p a ra m e te r space* a " w e ig h t” p ro p o rtio n a l to the* n u m b e r o f
steps th e c h a in has spent a t th a t p a r tic u la r lo c a tio n . T h e !()()( l-2 p )'/t co nfid e nce
in te rv a l [o;).C |_ pj fo r a p a ra m e te r is e s tim a te d b y s e ttin g ep to the* p ' h q u a n rile o f
o ,,.t = 1
V and <p__x to th e
(1
— p ) tfl c |iia n tile . The* proce*dure is s im ila r fo r
m n ltic lim e 'n s io n a l c o n s tra in ts : the* d e n s ity o f p o in ts in th e n-elime'nsiemal space* is
p ro |> o rtio n a l to the* lik e lih o o d and m u lti-d im e n s io n a l cemfidence* le*ve*ls c an be* fo u n d
as illu s tra te 'e l in [jlo .G o f Pre*ss e*r a l.(1 9 9 2 ).
We* note* th a t the* g lo b a l m a x im u m lik e lih o o d value* fe>r the* param ete*rs eloe*s not
necessarily coincide* w ith the* e*xpe*ctation value* o f th e 'ir m arginali/.e*d d is tr ib u tio n if
the* lik e lih o o d surface* is not a m u lti-v a ria te G au ssia n . We* fin d th a t, fo r m ost o f th e
parame*te*rs. the* m a x im u m lik e lih o o d value's o f th e g lo b a l je»irit fit are consiste*nr w ith
the* expe'ctatiem value's e>f the* m a rg in a lize 'd e lis trih u tio n .
A v ir tu e o f the* MC’ M C m e th o d is th a t th e aelelition o f e x tra elata sets in the* jo in t
a n a ly s is ra n e ffic ie n tly be done w ith m in im a l c o m p u ta tio n a l e ffo rt freun the* M C M C
o u tp u t i f th e in c lu s io n o f e*xtra d a ta se*t cloe*s n o t re q u ire the* in tr o d u c tio n e>f e x tra
p a ra m e te rs e»r does n ot d riv e the p a ra m e te rs s ig n ific a n tly aw ay freun th e c u rre n t be*st
fit. Feu e x a m p le , we acid L y m a n n p ow er s p e c tru m c o n s tra in t to M C M C ’ s o u tp u ts ,
b u t we c a n n o t d o th is feu th e 2 d F G R S . since th is re q u ire s th e in tr o d u c tio n o f tw o
e x tra p a ra m e te rs (d. rrp. se*e ^5.1 b e le w fo r meue d e ta ils ).
I f th e lik e lih o o d su rface fo r a subset o f p a ra m e te rs fro m an e x te rn a l (in d e p e n d e n t)
d a ta set is k n o w n , o r i f a p rio r needs to be added a posteriori, th e jo in t lik e lih o o d
su rface can be o b ta in e d b y m u ltip ly in g th e lik e lih o o d w ith th e p o s te rio r d is tr ib u tio n
o f th e M C M C o u tp u t. In Spergel et a l. (2003) we fo llo w th is m e th o d to o b ta in th e
jo in t c o n s tra in t o f C M B w ith S upernovae la (R iess e t a l.. 1998. 2001) d a ta a n d C M B
w it h H u b b le K e y p ro je c t H u b b le c o n s ta n t (F re e d m a n et a l.. 2001) d e te rm in a tio n .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 2:
P a r a m e te r E s t im a tio n M e t h o d o lo g y
31
T h c ro is yet a n o th e r a dva n ta g e o f th e M C M C te ch n iq ue . T h e c u rre n t version
o f C M B F A S T w ith th e n o m in a l in te r p o la tio n s e ttin g s is a ccu ra te to V Z . h u t
random n u m e ric a l e rro rs can so m e tim e s exceed th is . As th e p re cisio n o f the C M B
m e a surem e n ts im prove', these effects can becom e p ro b le m a tic fo r a ny a p p ro a ch th a t
c a lc u la te s d e riv a tiv e s as a fu n c tio n o f p a ra m e te rs.
Because* M C M C c a lc u la tio n s
average* ove'r ~ 100.000 C M B c a lc u la tio n s , the* M C M C tee-hnique is m uch le>ss sensitive*
th a n e'ithe-r griel-base'd lik e lih o o d c a lc u la tio n s o r m e th o d s th a t n u m e ric a lly ca lcu late '
the* Fishe-r m a trix .
4.
E x tern a l C M B D a ta S ets
The* C B I (M aso n e*f a l.. 2002: Sie'vers e*r a l.. 2002: Pe'arson e't a l.. 2002) and
th e AC’ B A R (K u o e*t a l.. 2002) expe*rime'nts com ple'tnent W M A P by p ro b in g th e
a m p litu d e * o f C M B te'mpe'rature* p ow er s p e c tru m a t ( > 900. The'se* obse'rvatiems
probe 1 th e S ilk e la tu p in g ta il atiel im p ro v e o u r a n a lysis in 2 ways: a) improve* o u r
a b ilit y to c o n s tra in the> b a ryo n d e n sity, the* a m p litu e h ' o f H u e tn a tio n s and th e slope
e>f the* m a tte r pe>we*r spe'etrum . anel b) im p ro v e converge'tice* by p re v e n tin g th e ch a in s
fro m s p e n d in g lo n g pe>rie)ds o f tim e' in large, m o d e ra te 'ly lo w -lik e lih o o d regions o f
p a ra m e te r space.
T h e C B I d a ta set is elescribed in M ason e t a l. (2002). Pearson e*t al. (2002) anel
em th e ir web s ite F). We use d a ta fro m th e C B I mosaic d a ta set (Pe'arsem e*t a l.. 2002)
a n ti elo n o t in c lu d e th e deep d a ta set as th e tw o elata sets are n ot in d e p e n d e n t. W e
use th e th re e b an d p o w e rs fro m th e even b in n in g a t c e n tra l f values o f 876. 1126.
1301. th u s e n s u rin g th a t th e chosen b ands p o w e r can be co nside re d in d e p e n d e n t fro m
Kh t t p : / / w w w . a s t r o . e a ite c h .e d u / ~ t jp / C B I/ d a ta / in d e x . h t m l
(la st
u p d a te
2002 )
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A ugust
C h a p t e r 2:
P a r a m e te r E s t im a tio n M e t h o d o lo g y
th e W M A P d a ta . A t ( ^
32
Io(K). th e C B I e x p e rim e n t d e te cte d excess p ow er. I f th e
rm s a m p litu d e o f mass flu c tu a tio n s on scales o f 8 h _l M p c . is rrs
1. th e n th is excess
p ow er can be in te rp re te d as due to S u n a ye v-Z e ld o vich d is to r tio n fro m u n d e te cte d
g a la x y c lu s te rs (M a so n et a l.. 2002: B o tu i et a l.. 2002: K o m a ts u A" S e lja k. 2002). We
s im p lify o u r a na lyst's by not using th e C B I d a ta on scales w here th is effect can be
im p o r ta n t. T h e c o rre la tio n s between d iffe re n t band pow ers are take n in to a ccount
w ith th e fu ll co varian ce m a trix : we use th e lo g n o rm a l fo rm o f th e lik e lih o o d (as in
Pearson et al 2002). In a d d itio n , we m a rg in a liz e over a
10
1/ c a lib ra tio n u n c e rta in ty
(C B I beam u n c e rta in tie s are n e g lig ib le ).
T h e AC’ B A R d a ta set is d escribed in K u o et a l. (2 0 02 ).
We use th e 7
b a n d -p o w e rs a t m u ltip o le s 842. 98G. 1128. 1279. 142G. loSO. 171G. A s show n in
F ig u re 2.4. these p o in ts do not o ve rla p w ith th e W M A P p ow er s p e c tru m except at
i
800 w here U W /.4 P is n o is e -d o m in a te d . As show n in F ig u re 2.4 th e AC’ B A R
e x p e rim e n t is less se n sitive to S u n v a e v -Z e ld o v ic h c o n ta m in a tio n th a n C B I. We
c o m p u te th e lik e lih o o d a na lysis fo r co sm o lo g ica l p a ra m e te rs fo r th e A C B A R d a ta set
fo llo w in g G o ld s te in et al. (2002) a n d u sin g th e e rro r bars g ive n in A C B A R web s ite '.
In a d d itio n we m a rg in a liz e over co n s e rv a tiv e beam a n d c a lib ra tio n u n c e rta in tie s
(B . H o lz a p fe l 2002. p riv a te c o m m u n ic a tio n ). In p a r tic u la r we assum e a c a lib ra tio n
u n c e rta in ty o f
2 0
‘zf (th e d o u b le o f th e n o m in a l va lu e) and T/c beam u n c e rta in ty
(GO'/f la rg e r th a n th e n o m in a l va lu e).
T h e A C B A R a n d C B I d a ta are c o m p le te ly in d e p e n d e n t fro m each o th e r (th e y
m a p d iffe re n t re gions o f th e sky) a n d fro m th e W M A P d a ta (th e b a n d -p o w e rs we
c o n s id e r span d iffe re n t f ranges). T o p e rfo rm th e jo in t lik e lih o o d a n a lysis, we s im p ly
m u lt ip ly th e in d iv id u a l lik e lih o o d s .
7ht t p : / / co sm o log y.b erkele v.e du / g r o u p / s w lh / a c b a r / d a t a /
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C h a p t e r 2:
33
P a r a m e te r E s tim a tio n M e t h o d o lo g y
5.
A n a ly sis o f Large S cale S tr u c tu r e D a ta
Wo can enhance th e s c ie n tific value o f t in ' C M B d a ta
fro m c
c o m b in in g it w ith m easurem ents o f tin ' lo w re d s h ift u niverse.
su rve ys a llo w us to m easure the g a la x y p ow er s p e c tru m a t c ~~
1089 by
G a la x y re d s h ift
0
and o b s e rv a tio n s o f
L y m a n o a b s o rp tio n o f a b o u t 30 q ua sa r sp e c tra (L y m a n o fo re st) a llo w us to probe
th e d a rk m a tte r p ow er s p e c tru m at re d s h ift c ~ 3.
W e use th e A n g lo -A u s tra lia n Telescope T w o D egree F ie ld G a la x y R e d s h ift
S u rve y (2 d F G R S ) (C'olless et a l.. 2001) as c o m p ile d in F e b ru a ry 2001. T h is su rve y
probes th e u niverse at re d sh ift c,.fr ~
0.1
and probes th e p n w rr s p e c tru m
c o rre s p o n d in g to 0.022 < A’ < 0.2 (w h e n 1 A- is in u n its o f h M p c
S loan D ig ita l S ky S urvey (G u n n
011
scales
*. T h e a n tic 40
:, /
1
K n a p p . 1993) p o w e r s p e c tru m w ill be an
im p o r ta n t co m p le m e n t to 2 d F G R S . We also use* th e lin e a r m a tte r p ow er s p e c tru m
as recovered by C ro ft et al. (2002) fro m L y m a n n forest o b se rva tio n s. T h is p ow er
s p e c tru m is re c o n s tru c te d at an e ffe ctive re d s h ift ; — 2.72 and probes scales A- > 0.2
h M p c 1. T o g e th e r these d a ta sets a llo w us n ot o n ly to p ro b e a w id e range o f
p h y s ic a l scales
fro m
A- ~ I x
F ig u re 2 .3 ). b u t
also to probe th e e v o lu tio n o f a
1 0
“ * (30000 M p c h - 1 ) to A-1 (3 M p c h _ l )
(see
g ive n scale w ith re d s h ift asw e ll.
W h e n in c lu d in g LSS d a ta sets one s h o u ld keep in m in d th a t th e u n d e rly in g
physics fo r these d a ta sets is m uch m o re c o m p lic a te d a n d less w e ll u n d e rs to o d th a n
fo r W M A P d a ta , and s y s te m a tic and in s tru m e n ta l effects are m uch m o re im p o rta n t.
We a tte m p t here to in c lu d e a ll th e kn o w n (u p to d a te ) u n c e rta in tie s and s y s te m a tic ^
in o u r a n a lysis. In w h a t follo w s, we illu s tr a te o u r m o d e lin g o f th e " r e a l- w o r ld " effects
o f LSS su rve ys and how we p ro p a g a te s y s te m a tic a n d s ta tis tic a l u n c e rta in tie s in to
th e p a ra m e te rs e s tim a tio n . T h e g oal o f o u r m o d e lin g is to re la te n o t ju s t th e shape
b u t also th e amplitude o f the observed p ow er s p e c tru m to th a t o f th e lin e a r m a tte r
p o w e r s p e c tru m as c o n s tra in e d by C M B d a ta . T h e reason fo r th is w ill be c le a r in
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 2:
P a r a m e te r E s t im a tio n M e t h o d o lo g y
34
§ o .l: by u sin g th e in fo rm a tio n in the pow er s p e c tru m a m p litu d e we can b re a k sonic 1
o f the* degenerac ies a m o n g co sm o lo g ica l p aram c'ters.
5.1.
T h e 2d F G R S P ow er S p ectru m
Thc> 2 d F G R S pow er spec-trum . as released in .June 2002. has been c a lc u la te d
fro m the* F e b ru a ry 2001 ca ta lo g th a t in clu d e s 140.000 g a laxie s (P e rc iv a l et a h. 2001).
The* fu ll su rv e y is com posed o f 220.000 g alaxie s b u t is not yet a v a ila b le . The 1 sample 1
is m a g n itu d e - lim ite d at h, = 19.43 anel th u s p ro be s the* universe at c,.tr
0.1 and
th e p o w e r s p e c tru m on scale's c o rre s p o n d in g to k > 0.013 h Mpc ' 1. The* in p u t
c a ta lo g is an e xte n d e d version o f the 1 A u to m a tic P late 1 M achine 1 ( A P M ) g a la x y
c a ta lo g (M a d d o x et ah. 1990b.a. 1990) w h ic h in clu d e s a b o u t 3 m illio n g alaxie s to
bj = 20.3. T h e A P M c a ta lo g was used p re v io u s ly to recover the 1 3-d pow er s p e c tru m
o f g alaxie s by in v e rtin g the c lu s te rin g p ro p e rtie s o f the 1
2
-ci g a la x y d is tr ib u tio n
(B a u g h A’ E fs ta th io u . 1993: E fs ta th io u A M o o d y . 2001). These tech n iq ue s, however.
arc 1 affec te d by sa m p le va ria nce and u n c e rta in tie s in th e p h o to m e try : a fu ll 3-d
a n a ly s is is th u s more 1 re lia b le .
T h e p o w e r s p e c tru m o f th e g a la x y d is t r ib u t io n as m easured b y LSS surveys,
such as th e 2 d F G R S . ca nn o t be1 d ir e c tly co m p a re d to th a t o f th e in it ia l d e n s ity
flu c tu a tio n s as p re d ic te d by th e o ry , o r recovered fro m W M A P or. th e c o m b in a tio n
o f U 'A /.A P -f-C B I-t-A C B A R d a ta -se ts. T h is is d u e to a n u m b e r o f in te rv e n in g effects
th a t can be b ro a d ly d iv id e d in tw o classes: effe cts due1 to th e s u rv e y g e o m e try
(i.e .. w in d o w fu n c tio n , se le ction fu n c tio n effects) a nd effects in trin s ic to th e g a la x y
d is t r ib u t io n (e.g.. re d sh ift-sp a ce d is to rtio n s , b ia s, n o n -lin e a ritie s ).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 2:
P a r a m e te r E s t im a tio n M e th o d o lo g y
Surrey Geometry
G a la x y surveys such as the* 2 d F G R S a rc m a g n itu d e -lim ite d ra th e r th a n
v o lu m e -lim ite d , th u s m ost nearby g alaxies are in c lu d e d in tin* c a ta lo g w h ile o n ly th e
b rig h te r o f th e m o re d is ta n t galaxies an* selected. T h e se lection fu n c tio n accou n ts
fo r th e fact th a t few er galaxies are in c lu d e d in th e su rve y as th e d is ta n c e (o r th e
re d s h ift) increases.
A n a d d itio n a l effect arises fro m the fact th a t th e c lu s te rin g
p ro p e rtie s o f b rig h t g alaxies m ig h t la* d iffe re n t fro m th e average c lu s te rin g p ro p e rtie s
o f th e g a la x y 05
, ,
'a tio n as a w hole. T h e se le ctio n fu n c tio n does n ot take th is in to
accou n t (we w ill re tu rn to th is p o in t in ^ o .l) .
M o re o ve r, th e c^ :
1
‘ *ness across th e sky is not co n sta n t a nd th e survey
can o n ly cover a fra c tio n o f th e w h o le sky. so m e tim e s w ith a very c o m p lic a te d
g e o m e try d escribe d by th e w in d o w fu n c tio n .
In p a r tic u la r, fo r the d a ta we use.
uno b served fie ld s m ake th e su rve y com pleteness a s tro n g ly v a ry in g fu n c tio n o f
p o s itio n . T h e m easured F o u rie r co e fficie n ts are th e re fo re th e tru e co e fficie n ts o f th e
g a la x y d is tr ib u tio n co nvo lve d by th e F o u rie r tra n s fo rm o f th e se le ction fu n c tio n (in
th e d ire c tio n o f th e lin e o f s ig h t) and o f th e w in d o w fu n c tio n (on th e p la ne o f th e
s k y ). In th is se ctio n , we fo llo w th e s ta n d a rd n o ta tio n used in LSS analyses a nd re fer
to a ll o f these effects as w in d o w effects.
T h e w in d o w n o t o n ly m o d ifie s the m easured p ow er s p e c tru m b u t also in tro d u c e s
s p u rio u s c o rre la tio n s between F o u rie r m odes. (See P e rciva l et al. (2001) fo r m o re
d e ta ils ).
F o r th e 2 d F G R S these effects have been q u a n tifie d by M o n te C a rlo
s im u la tio n s o f m o ck ca ta lo g s o f th e s u rv e y 8. W e in c lu d e th e m in o u r a n a lysis by
HF o r W M A P d a ta , we deconvolve th e ra w m easured C \ by th e effect o f th e w in d o w
(th e m a s k ), th u s le a v in g th e effect o f th e w in d o w fu n c tio n a n d th e m ask o n ly in th e
fis h e r m a tr ix . F or LSS we w ill co nvo lve th e th e o ry w ith th e w in d o w , p ro je c t th e p ow er
s p e c tru m in to re d s h ift space and co m p a re th is to th e observed p ow er s p e c tru m .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 2:
P a r a m e te r E s t im a tio n M e t h o d o lo g y
36
c o n v o lv in g th e th e o ry p ow er s p e c tru m w ith th e w in d o w "k e rn e l'', and by in c lu d in g
o ff d ia g o n a l te rm s in th e covarian ce m a trix .
Effect.s- I ntrinsic to the Galaxy Distribution
L in e a r g r a v ita tio n a l ('v o lu tio n m o d ifie s th e a m p litu d e h u t not the shape' o f the
u n d e rly in g p o w e r s p e c tru m . However, in th e n o n -lin e a r re g im e (w h e re th e a m p litu d e '
o f flu c tu a tio n s is d p /p
1) th is is no lo n g e r th e case'.
N o n -lin e a r g ra v ita tio n a l
e v o lu tio n changes th e shape' o f th e pe>wer s p e c tru m a n d in tro d u c e s c o rre la tio n s
betw een F o u rie r mode's. T h is effe'ct be'comes im p o rta n t on scales k
0.1
h M p c -1 .
b u t th e e xa ct scale* at w h ic h it app e ars a m i its eletaile'd c h a ra c te ris tic d ep e nd on
c o s m o lo g ic a l p a ra m e te rs. M ost o f th e c lu s te rin g s ig n a l fro m g a la x y surveys such as
2dFCJRS com es fro m th e regim e w here n o n -lin e a ritie s are n o il-n e g lig ib le because' shot
noise* is the* d o m in a n t source' o f e rro r at k ^ 0.3 h M p c - 1 and th e n u m b e r d e n s ity
o f mode's scale’s as A 1. These* nem lirif'a ritie 's encode* a d d itio n a l in fo rm a tio n a b o u t
c o s m o lo g y a n d m o tiv a te ’s th e ir in c lu s io n in th e pre'sent a n a lysis. T h is a p p ro a ch
is c o m p lic a te d by th e fa ct th a t an a c c u ra te d e s c rip tio n o f th e fu lly non-line*ar
e v o lu tio n o f th e galaxy p ow er s p e c tru m is c o m p lic a te d . In th e lite ra tu r e , th e re are
several d iffe re n t approaches to m o d e lin g th e n o n -lin e a r e v o lu tio n o f th e u n d e rly in g
dark ma tt er powe*r sp e'etrum in re a l space: ( 1 ) line’a r (a n d extende’d ) p e r tu rb a tio n
th e o ry : (2) s e m i-a n a ly tic a l m o d e lin g a nd (3 ) n u m e ric a l s im u la tio n .
A ll o f these
a pp ro a che s y ie ld co n s is te n t re su lts on th e scales used in o u r a n a lysis. We w ill use the
s e m i-a n a ly tic a l a p p ro a ch developed b y H a m ilto n et a l. (1991) a n d Peacock 4: D o d ds
(1 9 9 6 ).
In p a r tic u la r , we use th e M a et a l. (1999) fo r m u la tio n o f th e n o n -lin e a r
p o w e r s p e c tru m . F ig u re 2.6 shows th e effect o f n o n -lin e a ritie s on th e m a tte r p ow er
s p e c tru m on th e scales o f in te re st (co m p a re s o lid a n d dashed lin e s ).
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P a r a m e te r E s t im a tio n M e t h o d o lo g y
T h e o ry p re d ic ts th e s ta tis tic al p ro p e rtie s o f th e c o n tin u o u s m a tte r d is tr ib u tio n .
w hile 1 o b s e rv a tio n s arc 1 concerned w ith the 1 g a la x y d is t r ib u t io n , w h ich is discrete1.
M o re o v e r, g a la xie s m ig h t n ot be fa ith fu l tra c e rs o f the* mass d is tr ib u tio n (i.e. the 1
g a la x y d is t r ib u t io n m ig h t be biased). In the 1 a n a lysis o f g a la x y surveys it is assum ed
th a t g a la xie s fo rm a Poisson sainpliny o f an u n d e rly in g c o n tin u o u s Held w h ic h is
re la te d to the 1 m a tte r H u ctua ticm fie ld v ia the 1 bias. It is possible 1 to fo r m a lly re la te
the 1 d is c re te g a la x y fie ld anel its c o n tin u o u s c o u n te rp a rt.
F o r the* p ow er s p e c tru m ,
th is co nsists o f the 1 s u b tra c tio n fro m th e m easured g a la x y pow er s p e c tru m o f the 1
shot noise1 c o n tr ib u tio n . T h e p u b lis h e d p ow er s p e c tra fro m g a la x y surveys a lre a d y
have th is c o n tr ib u tio n s u b tra c te d , b u t are s t ill biased w ith respec t to the 1 u n d e rly in g
mass pow er sp e ctra .
The 1 idea th a t g a la xie s are biased tra c e rs o f the 1 mass d is tr ib u tio n even on large
scale's was in tro d u c e d by K a is e r (1984) to e x p la in the* p ro p e rtie s o f A b e ll clu ste rs.
N e vertheless, the 1 fact th a t galaxie s o f d iffe re n t m o rp h o lo g ie s have* d iffe re n t c lu s te rin g
p ro p e rtie s (hence 1 d iffe re n t pow er s p e c tra ) was kn o w n m u ch before (e.g.. H u b b le
(1 9 36 ): D re ssie r (1 9 8 0 ): P o stm a n A: G e lle r (1 9 8 4 )). S ince th e c lu s te rin g p ro p e rtie s
o f d iffe re n t ty p e s o f g alaxie s are d iffe re n t, th e y c a n n o t a ll be good tra ce rs o f the
u n d e rly in g mass d is t r ib u t io n 9 .
In th e s im p le s t b ia s in g m o d e l, th e lin e a r bias m o d e l, th e mass and g a la x y
frac tio n a l o v e rd e n s ity fie ld s 6 and dg are re la te d by <)g( x ) = bd{x). T h is im p lie s th a t
on a ll scales
r g( k) = b2P ( k ) .
(2-36)
“ G a la x ie s are lik e ly to be fo rm e d in th e ve ry h ig h -d e n s ity regions o f th e m a tte r
flu c tu a tio n fie ld , th u s th e y are fo rm e d v e ry biased a t ; > > 0 (e.g.. L y m a n b reak
g a la x ie s ). B u t th e n g r a v ita tio n a l e v o lu tio n s h o u ld m ake th e g a la x y d is tr ib u tio n less
a n d less biased as tim e goes on (e.g.. F ry (1 9 9 6 ))
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P a r a m e te r E s t im a tio n M e t h o d o lo g y
T h is s im p le m o d e l (a lth o u g h ju s tifie d by th e K a is e r (1984) a s s u m p tio n th a t galaxie s
fo rm on th e h ig he st peaks o f th e mass d is tr ib u tio n ) c a n n o t be tru e in d e ta il fo r
tw o reasons. The* firs t is th a t, on a fu n d a m e n ta l level, the g a la x y H u e tu a tio n fie ld
on s m a ll s m o o th in g scales co u ld becom e <),, <
— 1
w h ich corre sp on d s to a n e g a tive
g a la x y d e n sity. T h e second is th a t, fro m an observ;345^89
*'
1
t
* o f vie w , th is scheme
leaves the* shape' o f th e pow er s p e c tru m unchanged w h ile not a ll g a la x y 45
, , C a tio n s
have the1 same' observed pow er s p e c tru m shape, a lth o u g h th e difference's are not
la rg e (e.g.. Peacock V D odds (1 9 94 ): N o rb e rg et al. (2 0 0 1 )).
M a n y d iffe re n t and
m ore c o m p lic a te d b ia sin g schemes have been in tro d u c e d in the lite ra tu r e . F o r o u r
purpose's it is im p o rta n t to note th a t the* bias o f a sample* o f galaxie's depends on the
sample' se le ctio n c r ite r ia and on the* w e ig h tin g scheme' used in the* a na lysis. T h u s
d iffe re n t surve ys w ill have differe'tit biase's. and care m ust be* taken whe'ti c o m p a rin g
the* d iffe re 'tit g a la x y powe*r sp ectra .
T lie 're are> se*veral in d ic a tio n s th a t large-scales g a la x y bias is scale* inele'pendent on
large* scales (e.g.. H o c k s tra et al. (2 0 02 ): Verde <*t al. (2 0 0 2 )). T h is ju s tifie s a elop tin g
e*t|uation 2-3G. Fe>r th e 2 d F G R S . the* bias o f g alaxie s has bc'e'ti m easured by V erde et
al. (2 0 0 2 ). by u sin g h ig h e r-o rd e r c o rre la tio n s o f th e g a la x y Hue tu a tio n Held. T h e y
assum e a g e n e ra liz a tio n o f the s im p le lin e a r b ia s in g sche'tne. dg = bid 4- b>/26~. T h e y
fin d no evid en ce fo r scale-dependent bias a t least on line*ar anel m ild ly n em -lin e ar
scale's (i.e . k < 0.4 h M p c - 1 ) and b> co n siste n t w ith 0. T h is fin d in g fu r th e r s u p p o rts
th e use o f e q u a tio n 2-36. In p a r tic u la r, thew fin d b\ = 1.06 ±
0
. 1 1 . In o u r a n a ly s is
we w ill assum e lin e a r b ia sing .
T h e V erde e t al. (2002) bias m easurem ent has to be in te rp re te d w ith care. It
a p p lie s to 2 d F G R S g alaxie s w e ig h te d w ith a m o d ific a tio n o f th e F e ld tn a n et al.
(1 9 94 ) w e ig h tin g scheme as d escribe d in P e rciva l et a l. ( 2
0 0 1
). I t is im p o r ta n t to
n o te th a t, close to th e observer, d im g a la xie s are in c lu d e d in th e su rve y: th e g a la x y
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C h a p t e r 2:
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P a r a m e te r E s t im a tio n M e t h o d o lo g y
d e n s ity is h ig h . b u t a s m a ll v o lu m e o f th e sky is covered. O n th e o th e r h a n d , fa r
a w ay fro m tin* o bserver, o n ly very b rig h t g ala xie s are in c lu d e d in th e su rve y: a large
v o lu m e is p ro b e d b u t th e g a la x y d e n s ity is lo w . As a consequence, c lu s te rin g o f d im
g a la xie s in a s m a ll v o lu m e close to th e o bse rve r c o n ta in m ost o f th e s ig n a l fo r the
p ow er s p e c tru m at s m a ll scales. W h ile ra re , b rig h t g a la xie s in a la rg e v o lu m e enclose
m ost o f th e in fo r m a tio n a b o u t the p ow er s p e c tru m on la rg e scales. A n " o p tim a l"
w e ig h tin g schem e w o u ld th u s w e ig h t d im g a la xie s on s m a ll scales and b rig h t g alaxies
on la rg e scales. T h is w e ig h tin g scheme is. u n fo rtu n a te ly , biased.
B rig h t g a la xie s
are m ore s tro n g ly c lu s te re d (i.e. m o re biased) th a n d im ones. T h is effect is kn ow n
as " lu m in o s ity b ia s '. T h e p ow er s p e c tru m recovered fro m such a w e ig h tin g scheme
w ill have o p tim a l e rro r bars, b u t w ill e x h ib it sca le -de p en d e nt bias. T h e w e ig h tin g
schem e used in P e rciva l et a l.
(2
0 0 1
by lu m in o s ity bias (\V . .1 . P erciva l
) is n ot o p tim a l, b u t is v ir t u a lly u n a ffe cte d
2 0 0 2
. in p re p a ra tio n ). T h e p ow er s p e c tru m so
o b ta in e d is th a t o f 2 L . g alaxie s on v ir t u a lly a ll scales, and th e e ffe ctive re d s h ift
fo r th e p ow er s p e c tru m is c,.(f = 0.17. s lig h tly la rg e r th a n th e effective* re d s h ift o f
th e su rve y as d efin e d by th e se le ctio n fu n c tio n (P e rc iv a l et a l..
2 0 0 1
2 0 0 1
: Peacock et a l..
).
T h e fin a l c o m p lic a tio n is th a t g a la x y c a ta lo g s use th e re d s h ift as th e t h ir d
s p a tia l c o o rd in a te . In a p e rfe c tly hom ogeneous F rie d m a n u niverse , re d s h ift w o u ld
be an a c c u ra te d is ta n c e in d ic a to r.
In h o m o g e n e itie s , th o u g h , p e r tu r b th e H u b b le
flo w a n d in tro d u c e p e c u lia r v e lo c itie s . A s K a is e r (1 9 87 ) e m p ha size d, th e p e c u lia r
v e lo c itie s d is to r t th e c lu s te rin g p a tte rn n o t o n ly on s m a ll scales w here v iria liz e d
o b je c ts p ro d u c e "F in g e rs -o f-G o d " . b u t also on la rg e scales w here co he re n t flow s
p ro d u c e la rg e scale d is to r tio n co m p o n e n ts.
O n la rg e (lin e a r) scales th e re d s h ift-s p a c e e ffect on an in d iv id u a l F o u rie r
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P a r a m e te r E s t im a t io n M e t h o d o lo g y
40
component of the d e n s ity H u e tu a tio n fie ld d* can he m o d ele d by.
4
— *
= d k(l + V )
•
(2-37)
w h o re th e s u p e rs c rip t *■ refers to th e q u a n tity in re d s h ift space and // is th e cosine
o f th e a ng le betw een th e A’-v e c to r and th e lin e o f s ig h t.
is th e lin e a r re d s h ift space d is to r tio n p a ra m e te r.
/
T h e K a is e r fa c to r, d.
O ne defines d = f j b . w here
= (I In i ) / d In n . w ith d = d p /p and a = (L + c ) _ l : l> is th e lin e a r bias p a ra m e te r.
T h e expressio n fo r / ( ; ) is a kn ow n fu n c tio n o f <}m. A a nd c ( L a h a v et a l.. 1991).
’ r
\ . = ) = .Y
1[
(1
*A ;) - ’
2
( 1- >1 -
1
~
(1
+ -)
1 A'
1
-
[ '
J<>
.V - \ i «
(2-38)
w here A ' =
1
—
* . \ ( n 2 — 1 ). a m i can be a p p ro x im a te d b y 10 d
a n a ly s is o f th e 2dFC iR S (P eacock et a l.. 2001: P e rciva l et a l..
2 0 0 1
V. ^' /b. T h e
) c o n s tra in s /
a t th e e ffe c tiv e re d s h ift o f th e survey. T h e effective* re d s h ift o f the* su rve y depends
on the* g a la x y w e ig h tin g schem e a d o p te d to com pute* th e p ow er spe'e tr u m ft>r the
above w o rk ( r,.(r
0 .1 7 ).
T h is p e c u lia r v e lo c ity in fa ll cause's the* o v e rd o n s ity to
appe*ar squashed a lo n g th e lin e o f s ig h t. T h e net effect on th e angle*-ave*rage'd powe*r
sp e'etru m in th e s m a ll a ng le a p p ro x im a tio n is
P*{k) = P( k) ( l + ~ d + i. f - ’ ) .
(2-39)
T h u s on la rg e scale's th e re d s h ift space d is to r tio n s b oost th e p ow er sp e'etrum i f d > 0.
O n s m a lle r scales, v iria liz e d m o tio n s p ro d u ce a ra d ia l s m e a rin g a n d th e
associate d "F in g e rs -o f-G o d " effect c o n ta m in a te s th e w a ve le n g th s we are in te re s te d
in . T h is is d iffic u lt to tre a t e x a c tly , b u t as it is a s m e a rin g e ffect, it produce’s a m ild
d a m p in g o f th e pow er, a c tin g in th e o p p o s ite d ire c tio n to th e la rge-scale b o o s tin g
“ ’ In o u r a n a ly s is we use th e e xa ct e xpression fo r d as in e q u a tio n (2 -3 8 ).
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C h a p t e r 2:
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P a ra m e te r E s t im a tio n M e t h o d o lo g y
by th e K a is e r effect {see fo r e x a m p le M a ts u b a ra (1 9 0 4 )).
O n these scales, th e
re d s h ift space c o rre la tio n fu n c tio n is w e ll m o d e le d as a c o n v o lu tio n o f tin * real space
is o tro p ic c o rre la tio n fu n c tio n w ith some d is tr ib u tio n fu n c tio n fo r th e lin e o f sig h t
v e lo c itie s (e.g. D a vis A' Peebles (198.?): C’ole et a l. (1994): F ish e r (1 9 9 -j)). Since t in 1
c o n v o lu tio n in real space is e q u iv a le n t to m u ltip lic a tio n in F o u rie r space, th e re d s h ift
space p ow er s p e c tru m on sm a ll scales is m u ltip lie d by tin* square o f th e F o u rie r
tra n s fo rm o f tin* v e lo c ity d is tr ib u tio n fu n c tio n (e.g.. Peacock A' Dot his (1 9 9 4 )).
P'(A-./i) = m > ( l + d p - ) D ( k < t,,//) .
w here rrp denotes th e lin e -o f-s ig h t p a irw is e v e lo c ity d is p e rs io n .
(2-40)
I f tin* p a irw is e
v e lo c ity d is tr ib u tio n is taken to be an e x p o n e n tia l (e.g. B a llin g e r ef a l. (1995. 1996):
H a tto n A- C o le (1 9 9 8 )) w h ich seems to be s u p p o rte d by s im u la tio n s (e.g.. Z u re k
et a l. (1 9 9 4 )) a nd o b se rva tio n s (e.g.. M a rzke et a l. (1 9 9 5 )). th e n th e d a m p in g fa c to r
is a L o re n tz ia ti (see also K a n g et al. (2 0 0 2 )).
We a d o p t th is fu n c tio n a l fo rm as it is used by Peacock et al. (2001) in d e te rm in in g
th e re d s h ift space d is to r tio n p a ra m e te rs A and <rp fro m th e 2 d F G R S . T h e o ve ra ll
effect fo r th e p ow er s p e c tru m in a th in sh ell in A-space is g ive n by
(n'pk- - .f) .f
P*(k) =
4
^
2
J-
\/2{k~(T~ - 2 .f) * a r c ta n ( k npJ \ / 2 )
+ 3U p +
P(k)
(2-42)
o b ta in e d b y a v e ra g in g over p in e q u a tio n (2 -4 0 ) w ith the d a m p in g fa c to r g ive n In ­
e q u a tio n (2 -4 1 ). F ig u re 2.G show th e effect o f re d s h ift space d is to r tio n s (e q u a tio n
2-42) on th e scales o f in te re s t.
T h is m o d e l is s im p lis tic fo r several reasons. T h e m ost im p o r ta n t is th a t, because
o f th e c o m p lic a te d g e o m e try o f th e survey, th e s im p le a ngle average o p e ra tio n
p e rfo rm e d to o b ta in e q u a tio n (2 -4 2 ) m ig h t n o t be s t r ic t ly c o rre c t. A ls o , e q u a tio n
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P a ra m e te r E s t im a tio n M v t h o r io lo g y
(2 -4 2 ) is o b ta in e d in th e p la n e -p a ra lle l (also k n o w n as s m a ll-a n g le ) a p p ro x im a tio n
(i.e. as i f th e lin t's o f sig h t to d iffe re n t g alaxie s on tin* sky wen* p a ra lle l).
We have p e rfo rm e d extensive te s tin g o f ('({n a tio n (2-42) u sin g m o ck 2dFC iR S
ca ta lo g s o b ta in e d fro m the H u b b le v o lu m e s im u la tio n . We fin d th a t th e fo rm o f
('((n a tio n (2 -4 2 ) We fin d th a t th e s im u la tio n s re d s h ift-s p a c e p ow er s p e c tru m is
c o n s is te n t, given th e errors, w ith (‘({n a tio n (2-42) w here P ( k ) is the s im u la tio n s
real-spaee p ow er s p e c tru m up to k < 0.4 h M p c 1. even fo r th e c o m p lic a te d
g e o m e try o f th e 2 d F G R S . T h is means th a t u p to k
by c(|.
0.4 th e s ysfe m a tie s in tro d u c e d
(2 -4 2 ) are s m a lh 'r th a n th e s ta tis tic a l e rro rs : in th e a n a lysis we use o n ly
k ^ 0 .1 5 ."
H ow ever, th e value fo r .i in (‘({n a tio n (2 -4 2 ) needs to b(> c a lib ra te d on
M o n te C a rlo re a liz a tio n s o f th e survey. We fin d th a t T ',,r = 0 .8 5 .1 We have ve rifie d
th a t o u r re s u lts fo r th e co sm o log ica l p a ra m e te rs are in s e n s itiv e to tin* exact choice
o f th e c o rre c tio n fa c to r. Peacock et al. ( 2
0 0 1
) m easured th e p a ra m e te rs .i and rrp
a n d th e ir jo in t p ro b a b ility d is tr ib u tio n fro m th e su rve y o b ta in in g .1 = 0.43 and
a p = 385 k m s _ l . T h is m easurem ent has been o b ta in e d by u sin g th e fu ll a n g u la r
dependence o f th e p ow er s p e c tru m and th e re fo re recovers d ir e c tly .i and n ot . l ,r.
H a w k in s et a l. (2002) m easured these p a ra m e te rs fro m a la rg e r sa m p le th a n th e one
fro m Peacock et a l. ( 2
0 0 1
). o b ta in in g a s lig h tly d iffe re n t re s u lt. T h is is m o s tly due
to a s h ift in th e recovered value fo r crp. Since m o st o f th e g a la xie s in th e H a w kin g s et
al. ( 2
0 0 2
) sa m p le are in the Peacock et al. ( 2
0 0 1
) sa m p le , we c o n s e rv a tiv e ly e xte n d
o u r e rro r-b a rs on .i and a by 10(/< and 30(/{ re sp e c tiv e ly , to in c lu d e th e new value
w ith in th e
1-a
m a rg in a liz e d confid e nce c o n to u r, a n d to in c lu d e a p ossible e rro r in
th e d e te rm in a tio n o f Tofr. F ig u re 2.6 illu s tra te s th e im p o rta n c e o f in c lu d in g a ll the
above effects in o u r analysis.
In o u r a n a lysis we co nside r d a ta in th e k ra n ge 0.02 < k < 0.2 h M p c - 1 . O n
la rg e scales th e lim it is set by th e a ccu ra cy o f th e w in d o w fu n c tio n m o d e l: on s m a ll
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C h a p t e r 2:
P a ra m e te r E s t im a tio n M e t h o d o lo g y
43
scales th e lim it is sot by where th e covariance 1 m a tr ix has been e x te n s iv e ly rested. In
th is re g im e we also have1 a weak dependence on the 1 v e lo c ity d is p e rs io n p a ra m e te r rrp.
th e p a ra m e te r w ith the largest s y s te m a tic u n c e rta in ty .
Motivation f o r this Motleliny
T h e m o tiv a tio n b e h in d the c o m p lic a te d m o d e lin g o f jp . 1-”). 1 is to be able
to in fe r th e a m p litu d e o f the m a tte r pow er s p e c tru m fro m th e observed g a la x y
c lu s te rin g p ro p e rtie s .
F ig u re s 2.7 and 2.8 illu s tra te how th e m o d e lin g o f f jo .l- o .l helps in b re a k in g
degeneracies a m o n g co sm o log ica l p a ra m e te rs. For illu s tr a tio n , we co n sid e r tw o cases
b elow : th e degeneracy o f the d a rk energy e q u a tio n o f s ta te , tv. (H u e y et a l.. 1999)
w ith V.t, a m i n , and th e
- h degeneracy, w here
= M Jr.
F ig u re 2.7 shows tw o m odels th a t are v ir t u a lly in d is tin g u is h a b le w ith C M B
d a ta , b u t w h ic h p re d ic t d iffe re n t a m p litu d e s fo r th e m a tte r p ow er sp e c tra a t c ~
0
.
T h is is because th e lin e a r g ro w th fa c to r a n d th e shape p a ra m e te r F are d iffe re n t fo r
th e tw o cases. T h e tw o m odels d iffe r in th e values o f ^.7,. n s and tr. T h e s o lid lin e is
a m o d e l w it h tv = —0.4 w h ile th e d o tte d lin e is a m o d e l w ith tv — — I.
In F ig u re 2.8 we show tw o sets o f co s m o lo g ic a l p a ra m e te rs th a t d iffe r o n ly in
th e values o f th e n e u trin o mass a n d th e H u b b le c o n s ta n t. These tw o m o d e ls are
v ir t u a lly in d is tin g u is h a b le w ith C M B o b se rva tio n s. B u t th e m a tte r p ow er s p e c tru m
in th e tw o cases is d iffe re n t in shape and a m p litu d e . S ince re d s h ift-s p a c e d is to r tio n s
and w in d o w fu n c tio n a ffect th e p ow er s p e c tru m shape, e x tra in fo r m a tio n a b o u t
c o s m o lo g ic a l p a ra m e te rs is encoded in its a m p litu d e .
B y u sin g th is in fo r m a tio n .
S pergel et a l. (2003) o b ta in a c o s m o lo g ica l u p p e r b o u n d on th e n e u trin o mass th a t
is ~ 4 tim e s b e tte r th a n c u rre n t c o s m o lo g ic a l c o n s tra in ts (E lg a ro v et a l.. 2002).
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C h a p t e r 2:
44
P a r a m e te r E s tim a tio n M e t h o d o lo g y
F o r co m p lete n ess. we have show n th e p ow er s p e c tru m also fo r scales p ro b e d by
th e L y m a n n forest (see *jG). T h e e rro r bars in F ig u re s 2.7 and 2.8 are e xam ple s o f
th e size o f tin* 2 d F G R S a nd L y m a n n pow er sp e ctra s ta tis tic a l u n c e rta in tie s in one
d a ta - p o in t. sh o w in g th a t th e tw o m o d els can be d is tin g u is h e d i f tin* observed p ow er
s p e c tru m can be re la te d to th e lin e a r m a tte r p ow er s p e c tru m w ith o u t in tro d u c in g
la rg e a d d itio n a l u n c e rta in tie s .
Practical Approach
T in 1 p ro c e d u re we ;89! ; ‘ in o rd e r to co m p a re th e observed g a la x y pow er
s p e c tru m w ith th e th e o ry p re d ic tio n s is o u tlin e d b elow (th e p u b lis h e d 2dFC !R S
g a la x y p ow er s p e c tru m has been a lre a d y co rre cte d fo r shot noise). F or a given set o f
c o s m o lo g ic a l p a ra m e te rs and a p a ir-w is t' v e lo c ity d is p e rs io n p a ra m e te r we proceed
as fo llo w s:
1) T h e M C M C selects a set o f co sm o lo g ica l p a ra m e te rs and values fo r .1 and rrp.
C M BFAST co m p u te s the th e o re tic a l lin e a r m a tte r p ow er s p e c tru m a t c =
0
.
2) We evolve th e th e o re tic a l lin e a r m a tte r p ow er s p e c tru m to o b ta in th e n o n -lin e a r
m a tte r p ow er s p e c tru m at th e e ffe ctive re d s h ift o f th e survey, fo llo w in g th e
p re s c rip tio n o f M a et al. (1999).
3) W e th e n o b ta in th e re d sh ift-sp a ce p ow er s p e c tru m fo r the mass by usin g
e q u a tio n 2-42 w ith d ',fr c a lib ra te d fro m M o n te C a rlo re a liz a tio n s o f th e ca ta lo g .
4) T h e bias is c o m p u te d fro m d a n d Q, n u sin g e q u a tio n (2 -3 8 ). T h e g a la x y p ow er
s p e c tru m is o b ta in e d by c o rre c tin g fo r bias, e q u a tio n 2-36.
5) T h e re s u ltin g p ow er s p e c tru m is co nvo lve d w ith th e g a la x y w in d o w fu n c tio n .
W e use th e ro u tin e p ro v id e d on th e 2 d F G R S w eb s ite to p e rfo rm th is
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C h a p t e r 2:
P a r a m e te r E s t im a tio n M e th o d o lo g y
n u m e ric a lly .
T h is is the p ow er s p e c tru m th a t can he co m p a re d w ith th e
q u a n tity m easured fro m a g a la x y survey.
G) W e can now e va lu a te the lik e lih o o d using th e f u ll co variance m a tr ix as p ro vid e d
b y th e 2 d F G R S tea m .
We a p p ro x im a te 1 th e lik e lih o o d to he G aussian as
it was done by th e team .
In p rin cip le ' th is is n ot s t r ic t ly co rre ct since* in
th e line'ar re‘gime> th e pow er s p e c tru m is an e x p o n e n tia l d is tr ib u tio n and in
the* n o n -lin e a r re g im e the* d is tr ib u tio n has c o n trib u tio n s fro m h ig h e r-o rd e rs
c o rre la tio n s .
How ever, due* to the* size* o f the- survey we are conside'ring. the
c e n tra l lim it the*e»re*m e*nsure*s th a t th e lik e lih o o d is we'll <le*se-ril>e*e 1 by the*
G au ssia n a p p ro x im a tio n (e*.g.. S co e cim a rro e*t al. (1 9 0 9 )).
M o re o ve r. the*
co va ria n ce m a tr ix fe>r the* 2elFG R S p ow er spe'etrum has be'e'ti c o m p u te d by the*
2elFG R S fe'am uneh'r the1 a s s u m p tio n th a t the* lik e lih o o d is G aussian.
We* assume' th a t the* like lih o o e l fo r the* bias param c*tt‘r is G aussian, ce nte re d on
b = 1.04 w ith d is p e rs io n nh =
0
. 1 1 . T h is is a c o n se rva tive ewere'stim ate e»f the* e rro r
on th e b ia s paratne'te'r. as ne)te*el in Verele et a l. (2 0 0 2 ). T h e d e 'te rm in a tie m o f b is
co rre la te 'd w ith A and rrp and the* e>rror epiote'd has a lre a d y be>en m a rg in a lize 'd over the
u n c e rta in tie 's in the*se tw o p aram ete rs. In p ra c tic e . fo r e*ach ste'p in th e M a rk o v chain
we c o m p u te th e lik e lih o o d a c c o rd in g to ite m s 1 th ro u g h
6
above. T h e bias p a ra m e te r
is d e te rm in t'd once A. erp a nd th e o th e r co s m o lo g ic a l parame'te'rs are* chosen. We
th e n m u lt ip ly th e lik e lih o o d by th e jo in t lik e lih o o d fo r A and rrp. as in F ig u re 4 o f
P eacock et a l. ( 2
0 0 1
). and by th e lik e lih o o d fo r th e bias p a ra m e te r. In e ffe ct, we use
th e d e te rm in a tio n o f A. op. and b as p rio rs . B y m u ltip ly in g th e lik e lih o o d we assume
t h a t th e measurements o f th e re d s h ift space d is to r tio n p a ra m e te rs, bias, and the
2 d F G R S p o w e r s p e c tru m are in d e p e n d e n t. W e ju s t if y th is a s s u m p tio n below .
T h e p a ra m e te rs needed to m a p th e real-space n o n -lin e a r m a tte r p o w e r s p e c tru m
o n to th e re d s h ift-s p a c e g a la x y s p e c tru m are: A. a p a n d b. These three p a ra m e te rs are
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C h it[)({'r 2:
P a rn tn t'ttT E s t im a tio n M e t h o d o lo g y
4G
n o t in d e p e n d e n t, not o n ly is .1 x l / b h u t. m o re im p o r ta n tly , th e th re e p a ra m e te rs are
m easured fro m th e same c a ta lo g w h ic h we are u sin g to c o n s tra in o th e r co sm o lo g ica l
p a ra m e te rs . H ow ever, th e in fo rm a tio n we use to c o n s tra in c o sm o lo g ica l p a ra m e te rs
is a ll encoded in th e shape and a m p litu d e o f th e angle-averaged pow er s p e c tru m .
T h e in fo r m a tio n used to m easure .i and rtp is a ll encoded in t in 1 dependence o f the
F o u rie r c o e ffic ie n ts (i.e.. o f th e p ow er s p e c tru m ) on th e a ngle fro m th e lin e o f sig h t.
T h u s we can tre a t the d e te rm in a tio n s o f .i a nd erp as in d e p e n d e n t fro m the lik e lih o o d
fo r c o s m o lo g ic a l p aram ete rs. T in * a n a ly s is o f V erde et al. ('2002) to m easure the
bias p a ra m e te r fro m the 2 d F G R S uses b o th in fo r m a tio n a b o u t th e a m p litu d e o f the
F o u rie r c o e ffic ie n ts and th e ir a n g u la r dependence. T h is dependence, how ever, is not
th a t in tro d u c e d by re d s h ift-s p a c e -d is to rtio n s . b u t is th e c o n fig u ra tio n dependence
o f th e b is p e c tru m .
T h u s, in p rin c ip le we s h o u ld n ot tre a t th is m easurem ent as
c o m p le te ly in d e p e n d e n t.
H ow ever, m ost o f th e s ig n a l fo r tin ' bias m easurem ent
com es fro m th e Arrange o f 0.2 < A- < 0.4 h M p c - 1 w h ile th e sig n a l fo r th e present
a n a ly s is com es fro m A- < 0.2 h M p c - 1 . N o te th a t th e c o n fig u ra tio n dependence o f
th e b is p e c tru m is la rg e ly in d e p e n d e n t o f cosm o log y. T h is a llo w s us to e a sily in c lu d e
a p r io r fo r th e bias p a ra m e te r in th e a n a lysis.
6.
L ym an n F orest D a ta
T h e L y m a n o forest tra ces th e flu c tu a tio n s in th e n e u tra l gas d e n s ity a lo n g the
lin e o f s ig h t to d is ta n t quasars. Since m o st o f th is a b s o rp tio n is p ro d u ce d by low
d e n s ity unsh o cked gas in th e voids o r in m ild ly overdense re g ion s th a t are th o u g h t
to be in io n iz a tio n e q u ilib r iu m , th is gets is assum ed to be an a c c u ra te tra c e r o f
th e la rg e-scale d is tr ib u tio n o f d a rk m a tte r. In th is epoch and on these scales the
c lu s te rin g o f d a r k m a tte r is s t ill in th e lin e a r re g im e .
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C h a p t e r 2:
P a r a m e te r E s t im a tio n M e t h o d o lo g y
47
S ince th e L y m a n o forest o b s e rv a tio n s are p ro b in g tin* d is tr ib u tio n o f m a tte r
at c
3. th e y are an im p o rta n t c o m p le m e n t to tin* C M B d a ta and th e g a la x y
su rve ys d a ta . Because o f th e ir im p o rta n c e , th e re has been extensive* n u m e ric a l and
o b s e rv a tio n a l w o rk te s tin g th e n o tio n th a t th e y trace* the* large*-seale> structure*. In
o u r analyses. we* fin d th a t the* a d d itio n o f L y m a n o forest elata a p p e a r te» c o n firm
tre'ticls se*en in othe*r d a ta sets anel tig h te n s c o s m o lo g ic a l c o n s tra in ts . He)we*ve*r. more*
o b s e rv a tio n a l anel the*ore*tical w o rk is s t ill ne*e*de*d to c o n firm the* v a lid ity e»f the*
e*me*rging consensus th a t the* L y m a n o forest elata trace's the* LSS.
Rece*nt papers use tw o diff«*re*nt approaches fo r a n a ly s is o f th e L y m a n o fore*st
powe*r spe'etrum elata. M c D o n a ld e*t al. ( 2
0 0 0
) anel Z a ld a rria g a e*t al. ( 2
0 0 1
) d ire c tly
compare* the* obse*rve*d tra n s m is s io n s p e c tra to the* p re d ic tio n s fro m co sm o lo g ica l
m o d e ls.
(2
0 0 2
W e fo llo w the* app ro a ch o f C ro ft e*t al. (2 0 02 ) anel G n e d in
H a m ilto n
) ( G H ) w h o use* an a n a ly tic a l f it t in g fu n c tio n to re*cove*r the* tnatte*r powe*r
sp e'etru m fro m the* tra n s m is s io n s p e c tru m 11.
G H factorize* the* lin e a r p ow er s p e c tru m in to fo u r te*rms.
r , . ( k ) = P VM' ( k ) Q u Q r Q r •
(2-43)
w h e re P {Ml ( k) is a q u a n tity th a t is in d e p e n d e n t o f m o d e lin g anel is is a lm o st d ire c tly
m e asured. T h e o th e r p a ra m e te rs c o n v e rt th is q u a n tity in to th e lin e a r m a tte r p ow er
s p e c tru m and encode th e dependence on co s m o lo g y a n d th e m o d e lin g o f th e in te r
g a la c tic m e d iu m ( IG M ) . In o u r tre a tm e n t, we use th e value's o f P )bs(A ) (th e e s tim a to r
fro m L y m a n n fo rt's t o b se rva tio n s o f p fa ,t) fro m G H a n d th e ir p a ra m e te riz a tio n in
11 A f te r th e present p a p e r was s u b m itte d , a p re p r in t a pp e ared (S e lja k et a l. 2003)
c la im in g th a t th e tre a te m e n t o f G H a nd C r o ft et a l. (2 0 02 ) s ig n ific a n tly u n d e re s tim a te
th e e rro rs . G iv e n th e im p o rta n c e o f th is d a ta set to tig h te n co s m o lo g ic a l c o n s tra in ts ,
th e L y m a n o fo re st c o m m u n ity sh o u ld reach a consensus on th e in te r p r e ta tio n o f
these o b s e rv a tio n s and on th e level o f s y s te m a tic c o n ta m in a tio n
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 2:
48
P a r a m e te r E s tim a tio n M e t h o d o lo g y
te rm s o f e q u a tio n (2-43) because it a llo w s us to e x p lic it ly in c lu d e th e dependence
o f th e recovered lin e a r m a tte r pow er s p e c tru m on th e co sm o lo g ica l param ete rs.
Qn encodes th e dependence o f the recovered lin e a r p o w e r s p e c tru m on th e m a tte r
d e n s ity p a ra m e te r a t c = 2.72. For Qn we use th e G H a nsa tz of.
•>
(2-44)
Qi
— 2 0 .0 0 0 K /7 o (71)
2 0 .0 0 0 K ) p a ra m e te rize s th e dependence on th e m ean
te m p e ra tu re o f th e IG M . Q - ~ 1.11 p a ra m e te rize s th e dependence on th e assumed
m ean o p tic a l d e p th .
In a d d itio n to th e s ta tis tic a l e rro rs . G H q u o te a s y s te m a tic
u n c e rta in ty th a t we add to th e s ta tis tic a l one. F in a lly , th e u n c e rta in tie s in Q n . Q r
and Q r c o n tr ib u te to the o v e ra ll n o rm a liz a tio n u n c e rta in ty . W e use th e C ro ft et al.
(2002) p re s c rip tio n to p a ra m e te rize th is u n c e rta in ty as ln 'P (.4 ) = — 1 /2 (.4 -
1
)J/ ' Tr.v„
w here i f .4 < 1 th e n n r 7> - 0.23 w h ile i f .4 > 1. n / ,/o = 0.29.
N -b o d y s im u la tio n s are used to co n ve rt th e Hux pow er s p e c tru m to the d a rk
m a tte r p o w e r s p e c tru m and c a lib ra te th e fo rm o f e q u a tio n (2 -4 3 ). T in * tw o d iffe re n t
g ro u p s. G H and C ro ft et al. (2002). have done th is in d e p e n d e n tly . T h e re s u ltin g
pow er s p e c tra agree w ell w ith in th e
1 -rr
e rro rs fo r a ll d a ta p o in ts except tin* last
three . W e th u s increase the \ a u n c e rta in tie s on th e last th re e d a ta p o in ts to m ake
th e tw o d e te rm in a tio n s o f Pi ,(k) co n siste n t a nd use th is as a ro u g h m easure o f the
in tr in s ic s y s te m a tic u n c e rta in tie s in th e L y m a n o d a ta .
G H p o in t o u t th a t the c o rre la tio n in flu x m easured fro m th e L y m a n a forest
sam ples p o w e r o ve r a fin ite band o f w a ve nu m b ers.
T h e e ffe c tiv e b a n d -p o w e r
w in d o w s are ra th e r bro ad due to th e p e c u lia r v e lo c itie s th a t sm ear pow er on scales
c o m p a ra b le to th e
1 -d
v e lo c ity d is p e rs io n . T h u s th e recovered lin e a r pow er s p e c tru m
is e ffe c tiv e ly s m o o th e d w ith an w in d o w th a t becom es b ro a d e r a t s m a lle r scales. In
p rin c ip le , th e re s u ltin g covariance betw een e s tim a te s o f p o w e r a t d iffe re n t k needs
to be ta k e n in to a ccount to d o a fu ll lik e lih o o d a n a ly s is to e x tra c t co sm o lo g ica l
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p te r 2:
P a ra m e te r E s tim a tio n M e t h o d o lo g y
49
p a ra m e te rs. H ow ever, the fu ll ro va ria n e e m a tr ix is n ot a v a ila b le . Since th e L y m a n o
d a ta are such a p o w e rfu l to o l we ju s t p e rfo rm a s im p le
fit and c a u tio n th e reader
th a t in te r p r e tin g th e reduced \ * as a m easure o f goodness o f fit fo r th is d a ta set is
n o t m e a n in g fu l since the d a ta are s tro n g ly c o rre la te d .
T o m a rg in a liz e over th e o ve ra ll n o r m a liz a tio n u n c e rta in ty , we take a dva n ta g e o f
th e M C M C ' a p p ro a ch . In p rin c ip le we c o u ld m a rg in a liz e o ver it a n a ly tic a lly , its we
do fo r the c a lib ra tio n u n c e rta in ty . In ste a d , a t each ste p o f th e ch a in we c o m p u te th e
best fir a m p litu d e .4 as done fo r [jo in t sources (H in s h a w et a l.. 20031)).
(2-4"))
A =
Lk
Lk
The
lik e lih o o d
fo r the
Lym an
n d a ta
l n £ / . (/<i = In C( P " ,1SL4. P /_) - r ln P ( . 4 ) .
o b ta in e d fro m th e M C M C o u tp u t.
fo r th e
m o d e l is g ive n
by
T h e m a rg in a liz a tio n is th e n a u to m a tic a lly
In o th e r w o rd s, th e a n a ly tic m a rg in a liz a tio n
c o m p u te s J P ( d a ta jm o d e l) 'P ( A ) d A w h ile we c o m p u te an e s tim a to r o f th is g ive n by
f 'P (d a ta |m o d e l)'P (.4 )(/.4 .
7.
C o n clu sio n s
In th is p a p e r, we have presented th e basic fo rm a lis m th a t we use fo r o u r
lik e lih o o d a n a lysis. T h is p a p e r shows th e fin a l ste p on th e p a th fro m tim e -o rd e re d
d a ta to co sm o lo g ic a l p aram ete rs.
It p ro v id e s th e fra m e w o rk fo r th e a n a lysis o f
c o s m o lo g ica l p a ra m e te rs and th e ir im p lic a tio n s fo r cosm o log y.
T h e u np re ce de n te d q u a lity o f th e W M A P d a ta a n d th e tig h t c o n s tra in ts on
c o s m o lo g ica l p a ra m e te rs th a t are d e riv e d re q u ire a rig o ro u s a n a ly s is so th a t th e
a p p ro x im a tio n s m ade in th e m o d e lin g d o n o t p ro p a g a te in to s ig n ific a n t biases and
s y s te m a tic e rro rs . W e have d e rive d an a p p ro x im a tio n to th e e xa ct lik e lih o o d fu n c tio n
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C h a p t e r 2:
P a r a m e te r E s t im a t io n M e t h o d o lo g y
fo r th e Cf w h ic h is a c c u ra te to b e tte r th a n
0
50
. 1 ' / . and we have c a re fu lly c a lib ra te d
th e te m p e ra tu re p ow er s p e c tru m co va ria n ce m a t r ix w ith M o n te C a rlo s im u la tio n s .
T h is enables us to use th e e ffe ctive ch i-sq u a re d p e r degree o f fre ed o m as a to o l to
test w h e th e r o r n ot a m o d e l is an acce,02‘
1 1
fit to th e d a ta .
W e im p le m e n t o u r lik e lih o o d a n a lysis w it h th e M C M C . We have c o n c e n tra te d
on th e issue o f convergence and m ix in g , e m p h a s iz in g how im p o rta n t these* issues are
in re co vering co sm o lo g ica l p a ra m e te rs value's anel th e ir confideuice 1 lewels freun the*
M C M C o u tp u t.
To the* W M A P data-sets ( T T anel T E a n g u la r powe»r spe'ctra) we1 have* adeled
the- C B I a m i A C B A R me'asuremeuit o f the* C M B on s m a lh 'r a n g u la r scale's. the'
2 d F G R S g a la x y penver s p e c tru m at c ~
0
. a n d th e L y m a n o forest m atte’r p ow er
spe'etrum at c ~ .}. The'se e xte 'rna l d a ta sets s ig n ific a n tly enhance the* s c ie n tific value 1
e>f the1 W M A P m easure'tne'tit. by a llo w in g us te> bre'ak p a ra m e 'te r dege'tierae-ies. W h ile '
the> u n d e 'riy in g p liy s ic s fo r the'se* d a ta sets is m u ch more* c o m p lic a te d and le*ss we'll
u n d e rs to e d th a n fo r W .U .A P d a ta . a n d system atic- and in s tru m e n ta l e'ffe’cts are' m uch
m ore im p o r ta n t, we* feel we have m ade a s ig n ific a n t ste p to w a rd s im p ro v in g th e rig o r
o f the* a n a lysis o f these elata sets. W e have in c lu d e d a d e ta ile d m o d e lin g o f g a la x y
bias, re d s h ift d is to r tio n s a n d th e n o n -lin e a r g ro w th o f struc tu re . We also in c lu d e
kn o w n (as to th e present d a y ) s y s te m a tic a n d s ta tis tic a l u n c e rta in tie s in trin s ic to
these o th e r d a ta sets.
A ck n o w led g m en ts
W e th a n k B ill H o lz a p fe l fo r in v a lu a b le discussions a b o u t th e A C B A R d a ta . We
th a n k th e 2 d F G R S te a m fo r g iv in g us access to th e M o n te C a rlo re a liz a tio n s o f th e
2 d F G R S . T h e m o ck c a ta lo g s o f th e 2 d F G R S w ere c o n s tru c te d a t th e In s titu te fo r
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 2:
P a ra m e te r E s t im a tio n M e t h o d o lo g y
C o m p u ta tio n a l C o sm o lo g y at D u rh a m . W e th a n k W ill P erciva l fo r discussions and
fo r p ro v id in g us w ith the covariance m a tr ix o f th e 2 d F G R S pow er s p e c tru m . LV is
s u p p o rte d by N A S A th ro u g h C h a n d ra F e llo w s h ip P F 2-30 0 22 issued by th e C h a n d ra
X -ra y O b s e rv a to ry ce nte r, w h ich is o p e ra te d by th e S m ith s o n ia n A s tro p h y s ic a l
O b s e rv a to ry fo r an on b e h a lf o f N A S A u n d e r c o n tra c t N A S 8-39073.
LY also
acknow ledges R u tge rs U n iv e rs ity fo r s u p p o rt d u r in g th e in it ia l stages o f th is w o rk.
H Y P is s u p p o rte d by a D o d ds fe llo w s h ip g ra n te d by P rin c e to n U n iv e rs ity .
The
W M A P m issio n is made* possible by the s u p p o rt o f th e O ffice o f Space Sciences
at N A S A H e a d q u a rte rs and by th e h a rd a nd c a p a b le w o rk o f scores o f scie n tis ts ,
engineers, te ch n icia n s, m a ch in ists, d a ta a n a ly s ts , b ud g e t a n a lysts, m anagers,
a d m in is tr a tiv e staff, and review ers.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 2:
P a r a m e te r E s t im a tio n M e t h o d o lo g y
•52
1.00
0 .9 8
>s
Jt
l/l
0 .9 6
Jt>N
(S)
0 .9 4
0 .9 2
0 .9 0
200
250
400
300
450
I
F ig . 2.1.
R a tio o f th e effective .sky coverage to th e a c tu a l sky coverage. T h is
c o rre c tio n fa c to r c a lib ra te s th e expression fo r th e F is h e r m a tr ix to th e va lu e o b ta in e d
fro m th e M o n te C a rlo app ro a ch. H ere we sh ow th e r a tio o b ta in e d fro m 100.000
s im u la tio n s (ja g g e d lin e ), the sm o o th cu rve show s th e f it we use. e q u a tio n (2 -1 6 ).
N o te th a t, since we are s w itc h in g betw een w e ig h tin g schemes, th e c o rre c tio n fa c to rs
are n o t e xp e cte d to s m o o th ly in te rp o la te betw een regim es.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 2:
o
P a r a m e te r E s t im a tio n M e t h o d o lo g y
53
0 .9 5
0 .9 0
500
700
600
800
900
L
F ig . 2.2.
C o rre c tio n fa c to r fo r th e noise. T h e lin e s are as in F ig u re 2 . 1 .N o te th a t,
since we are s w itc h in g betw een w e ig h tin g schem es, th e c o rre c tio n fa c to rs are n ot
e x p e c te d to s m o o th ly in te rp o la te betw een re g im e
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p te r 2 :
P a r a m e te r E s t im a tio n M e t h o d o lo g y
-700
1.20
-7 2 0
1.101
54
:sj
.*/•
I
I
-740
1.00 I
-760
0.90 r
c
•
•/
•
i<* •* ••
-780
•
_
0.80
-8 0 0
0
2000
1000
step
3000
0.70 L
0.96
0.98
1.00
1.02
1.04
1.06
n
F ig . 2.3.
U n c o n v f'rg f’d M a rk o v chains. T h e le ft p anel show n a truer plot o f the
lik e lih o o d values versus ite ra tio n n u m b e r fo r one M C M C (these arc* th e firs t 3000
steps fro m one o f o u r A C D M m o d el ru n s ). N o te th e burii-in fo r th e firs t ~ 100 steps.
In th e rig h t p a n e l, red d o ts are p o in ts o f th e c h a in in th e (n. .4) p lane a fte r d is c a rd in g
th e b u r n - in . G reen d o ts are fro m a n o th e r M C M C fo r th e same d a ta -se t and th e same
m o d e l. I t is c le a r th a t, a lth o u g h th e tra c e p lo t m a y a p p e a r to in d ic a te th a t the
c h a in has co nve rg e d, it has n ot fu lly e x p lo re d th e lik e lih o o d surface. L’sin g e ith e r o f
these tw o c h a in s a t th is stage w ill give incorrerA re su lts fo r th e best f it co sm o lo g ica l
p a ra m e te rs a n d th e ir e rro rs.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p te r 2:
P a r a m e te r E s tim a tio n M e t h o d o lo g y
55
10000 F
AC BAR
CBI
+
100
" "S 2 c o n trib u tio n a t CBI fre q u e n c y
SZ c o n trib u tio n a t ACBAR f r e q u e n c y
700
2000
1000
I
F ig . 2.4.
Ti»e C 'M B a n g u la r pow er s p e c tru m (in p K 2) fo r o u r best fit A C 'D M
m o d e l fo r f > 800 and th e S u n a ye v-Z e l'd o vich c o n tr ib u tio n fo r a;.* = 0.98 fo r C B I
w a ve le n g th s (d o tte d ) and fo r A C B A R (d ashed). T h e v e rtic a l lin e shows the a d o p te d
c u to ff fo r C B I a nd A C B A R .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 2:
P a r a m e te r E s t im a tio n M e t h o d o lo g y
06
10000
V *
t=
CN
+
1000
1 0 0
10000
1000
Ql
1 0 0
0.0001
0.0010
0.0100
0.1000
k / ( M p c / h ) -1
1.0000
F ig . 2.5.
T h e c o m b in e d C M B and LSS d a ta set. T h e C 'M B a n g u la r p ow er s p e c tru m
in //K * (to p p a n e l) as a fu n c tio n o f k w here k is re la te d to ( by / = r/ 0 k (w h e re
//o ~ 14400 M p e is th e d is ta n c e to th e la st s c a tte rin g su rfa ce ). B la ck p o in ts are th e
W A /.4 P d a ta , red p o in ts C B I. b lu e p o in ts A C B A R . T h e LSS d a ta (b o tto m p a n e l).
B la c k p o in ts are th e 2 d F G R S m ea surem e n ts a n d green p o in ts are th e L y m a n o
m ea surem e n ts. B o th LSS p ow er s p e c tra are in u n its o f (M p c / t ~ 1 ) 3 and have been
rescaled to : = 0. T h is p lo t o n ly illu s tra te s th e scale coverage o f a ll th e d a ta sets we
co nside r. T h e v a rio u s LSS p ow er s p e c tra as p lo tte d here c a n n o t be d ir e c tly co m p a re d
w ith th e th e o ry because o f th e effects o u tlin e d in § 0 (p.g.. re d sh ift-sp a ce d is to rtio n s ,
n o n -lin e a ritie s , bias a n d w in d o w fu n c tio n effect e tc .).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 2:
57
P a r a m e te r E s t im a t io n M e t h o d o lo g y
10000
1000
0.1
k /(M p c
h 1) ' 1
F ig . 2.6.
T h e m a tte r p o w e r s p e c tru m in (M p c / j~ ') '* ). lin e a r in real space (s o lid
lin e ), n o n -lin e a r in re a l space (dashed lin e ) a n d n o n -lin e a r in re d s h ift space (d o tte d
lin e ). T h e e rro r bars on th e d o tte d lin e show th e size o f th e s ta tis tic a l e rro r-b a rs
p e r A>bin o f th e 2 d F G R S g a la x y p ow er s p e c tru m . T h e p o w e r s p e c tru m is in u n its o f
(M p c / i ) 1.
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 2:
58
P a r a m e te r E s t im a tio n M e t h o d o lo g y
6000
5000
4
0
X
CM
\
Ly clpna
4-000
3000
0
2000
0
3
2
1000
o1
0
500
’ 000
I
1500
0.01
0.10
k/(M p c/h )
1
F ig . 2.7.
T w o co sm o lo g ica l m odels:
= 0.0235. - rri = 0.143. n , = 0.978 r = 0 . 1 1 .
ir = —0.426. h = 0.53 (so lid lin e )
= 0.0254.
= 0.137. n s = 1.024. r = 0.08
if = — 1. /» = 0.77 (d o tte d lin e ). T h e tw o m o dels are in d is tin g u is h a b le w ith in c u rre n t
e rro r-b a rs fro m th e C’ M B a n g u la r p ow er s p e c tru m (le ft p an e l, u n its fo r the pow er
s p e c tru m are pK~) . How ever th e y can e asily be d is tin g u is h e d i f we can re la te th e
observed p ow er s p e c tru m to the u n d e rly in g m a tte r p ow er s p e c tru m (r ig h t panel,
u n its fo r th e p ow er s p e c tru m are (M p c / t _ 1)'*). T h e e rro r bars on th e s o lid lin e are
e xam ple s o f th e size o f th e 2 dF G R S and L y m a n o p ow er s p e c tra s ta tis tic a l e rro r-b a rs
fo r one d a ta p o in t a t d iffe re n t scales. T h e re are 4 e rro r b ars fo r 2 d F G R S and 4 fo r
Lym an o .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 2:
59
P a ra m e te r E s t im a tio n M e t h o d o lo g y
6000
4
5000
0
o
0
3000
0
3
2
2000
o1
1000
0
0
1000
500
1500
0.01
I
F ig .
2.8.
T w o co sm o lo g ica l m odels:
0
0. 1 0
k /(M p c /n )
Qm =
0.26. -Cb =
0.02319.
1.00
'
r
=
0.12.
n s = 0.953.
= 0. h = 0.714 (so lid lin e ) a n d
= 0.26. .Jb = 0.02319. r = 0.12.
n x = 0.953.
— 0.02. h = 0.6 (dashed lin e ). A s b e fo re tin* tw o m odels are v ir t u a lly
in d is tin g u is h a b le fro m th e C M B a n g u la r p o w e r s p e c tru m (le ft panel, u n its fo r th e
p ow er s p e c tru m are p K 2). b u t th e y can e a sily be d is tin g u is h e d i f th e m a tte r p ow er
s p e c tru m a m p litu d e is kn ow n (rig h t panel . u n its fo r th e p o w e r s p e c tru m are (M p c
f r l ):i). T h e e rro r bars on th e s o lid lin e are e xa m p le s o f th e size o f th e 2 d F G R S and
L y m a n n p ow er s p e c tra s ta tis tic a l e rro r-b a rs fo r one d a ta p o in t. T h en * are 4 e rro r
bars fo r 2 d F G R S and 4 fo r L y m a n a .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
B ib lio g ra p h y
B a llin g o r. W . E.. Heavens. A . F.. A- T a y lo r. A . N. 1995. M N R A S . 276. L59
B a llin g e r. \V . E.. Peacock. .1. A .. A Heavens. A . F. 199G. M N R A S . 282. 877
B arnes. C\ et a l. 2008. A p.IS . to a p p e a r in \T 4 8 n l
B a u g h . C’ . M . A E fs ta th io u . Cl. 1993. M N R A S . 2G5. 143
B e n n e tt. C . L .. et al. 2003a. Ap.JS. to a p p e a r in v !4 8 n l
. 20031). A p .I. 883. 1
B o n d . I. R .. C ritte n d e n . R .. D avis. R. L .. E fs ta th io u . C».. A S te in h a rd t. P. .1. 1994.
P hvs. R ev. L e tt.. 72. 13
B o n d . .1. R. A E fs ta th io u . G . 1984. A p .I. 283. L45
B o n d . .1. R .. Jaffa. A . H .. A K n o x . L. 1998. P hvs. R ev. D . 57. 2117
. 2000. A p .I. 533. 19
B o n d . .1. R. et al. 2002. a s tro -p h /0 2 0 5 3 8 6
B rid le . S. L .. C ritte n d e n . R .. M e lc h io rri. A .. H o b son . M . P.. K n e is s l. R.. A Lasenbv.
A . N . 2002. M N R A S . 335. 1193
60
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
B IB L IO G R A P H Y
C h ris te n s o n . N .. M e ye r. R .. K n o x . L .. A L uo y. B. 2001. C la ssica l and Q u a n tu m
G ra v ity . 18. 2677
C h ris te n s o n . X .. M e ye r. R. p re p rin t (asrr-ph/0O 0G -101)
C ole. S.. F ish e r. K . B.. A; W e in b e rg . D. H. 1994. M N R A S . 267. 783
C’olless. M .. et al. 2001. M N R A S . 328. 1039
C 'ro ft. R. A . C \. W e in b e rg . D. H.. B o lte . M .. B u rie s. S.. H e rn q u is t. I... K a tz . X ..
K irk m a n . D .. A T y tle r . D . 2002. A p .I. 381. 20
D a vis. M . A" Peebles. P. •). E. 1983. A p .I. 267. 463
D ressier. A . 1980. A p .I. 236. 331
E fs ta th io u . G . At B on d . .1. R. 1999. M N R A S . 304. 73
E fs ta th io u . G . A M o o d y . S. .1. 2001. M N R A S . 323. 1603
E lg a ro y . O .. et a l. 2002. Phvs. Rev. L e tt.. 89. 61301
F e ld m a n . H. A .. K a ise r. N .. A Peacock. .J. A . 1994. A p .I. 426. 23
F ish e r. K . B. 1993. A p .I. 448. 494
F reedm an. W . L .. et al. 2001. A p .I. 333. 47
F ry . .1. N. 1996. A p .I. 461. L 6 5 +
G e lm a n . A . A R u b in . D . 1992. S ta tis tic a l Science. 7. 437
G ilk s . W . R .. R ic h a rd s o n . S.. A S p ie g e lh a lte r. D . .J. 1996. M a rk o v C h a in M o n te
C a rlo in P ra c tic e (L o n d o n : C h a p m a n a nd H a ll)
G n e d in . N. V . A H a m ilto n . A . .J. S. 2002. M N R A S . 334. 107
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61
62
B IB L IO G R A P H Y
G o ld s te in . .1. H. <*t al. 2002. A p .I. s u b m itte d (a s tio -p h /0 2 1 2 5 1 7 )
G u n n . .1. E. A’ K n a p p . G . R. 1993. in A S P C o n f. Ser. 43: S ky Surveys. P ro to s ta rs to
P ro to g a la x ie s . 267
H a m ilto n . A . .1. S.. M a tth e w s . A .. K u m a r. P.. A L u . E. 1991. A p .I. 374. L I
H a tto n . S. A- C o le . S. 1998. M N R A S . 296. 10
H a w k itis . E. et a l. 2002. M N R A S . s u b m itte d ( a s tr o - p h /0 2 12375)
H in sh a w . G . F. et al. 2003a. A p.IS . to a p p e a r in v l4 8 n l
. 2003b. A p.IS . to a p p e a r in v l4 8 n l
H o e k s tra . H .. van W aerbeke. L .. G la d d e rs. M . D .. M e llie r. Y .. A Yee. H. K . C. 2002.
A p .I. 377. 604
H n. W . 2001. P hvs. Rev. D. 64. 83003
H u b b le . E. 1936. A p .I. 84. 317
H u e y. G .. W a n g . L .. Dave. R .. C a ld w e ll. R. R .. A S te in h a rd t. P. J. 1999.
P hvs. R ev. D . 39. 63003
J a ro s ik . N . et a l. 2003a. A p .IS . 145
. 2003b. A p .IS . to a p p e a r in v l4 8 n l
K a is e r. N. 1984. A p .I. 284. L9
. 1987. M N R A S . 227.
1
K a n g . X .. .Jing. V . P.. M o . H . J.. A B o rn e r. G .
K n o x . L .. C h ris te n s e n . N .. A S kordis. C .
2 0 0 1
2 0 0 2
. M N R A S . 336. 892
. A p .I. 563. L95
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
B IB L IO G R A P H Y
K o g u t. A . et al. 2003. A p.IS . to a p p e a r in v l4 8 n l
K o m a ts u . E. A S e lja k . L". 2002. M N R A S . 33G. 1250
K o m a ts u . E. et al. 2003. A p.IS . to a p p e a r in v l4 8 n l
K o so w sky. A .. M ilo s a v lje v ic . M .. A' Jim e n ez. R. 2002. P hvs. Rev. D . GG. G3007
K u o . C'. L. c*t al. 2002. A p .I. a s tr o - p h /0 2 12289
L a h a v . ( ) .. Rees. M . .1.. L ilje . P. B .. A' P rim a c k . .1. R. 1991. M N R A S . 251. 128
Lew is. A . A' B rid le . S. 2002. P hvs. Rev. D. GG. 103511
M a . C \. C a ld w e ll. R. R .. Bode. P.. A W ang. L. 1999. A p .I. 521. L I
M a d d o x . S. .J.. E fs ta th io u . CL. A S u th e rla n d . W . .1. 1990a. M N R A S . 246. 433
. 199G. M N R A S . 283. 1227
M39
11
x. S. .1.. E fs ta th io u . CL. S u th e rla n d . W . .1.. A L ove d ay. .1. 1990b. M N R A S .
243. G92
M a rzke . R. O .. G e lle r. M . .J.. d a C o sta . L. N .. A H u e h ra . J. P. 1995. A .I. 110. 477
M a so n. B. S. et al. 2002. A p .I. s u b m itte d (a s tro -p h /0 2 0 5 3 8 4 )
M a ts u b a ra . T . 1994. A p .I. 424. 30
M c D o n a ld . P.. M ira ld a -E s c u d e . .1.. R auch. NL. S a rg e n t. \V . L. \V .. B a rlo w . T . A ..
C en. R .. A O s trik e r. J. P. 2000. A p .I. 543. I
N o rb e rg . P.. et a l. 2001. M N R A S . 328. 64
Page. L . et a l. 2003a. A p.IS . to a p p e a r in v l4 8 n l
— . 2003b. A p .I. 585. in press
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
B IB L IO G R A P H Y
P a n te r. B .. Heavens. A . F .. A' .Jimenez. R. 2002. M N R A S . s u b m itte d (a s tro p h / 0 2 l 1546)
Peacock. .J. A . A D o d ds. S. .J. 1994. M N R A S . 2G7. 1020
. 199G. M N R A S . 280. L19
Peacock. .1. A ., et al. 2001. N a tu re . 410. 1G9
P earson. T . .J.. et al. 2002. Ap.J. s u b m itte d (a s tro -p li/0 2 0 5 3 8 8 )
Peebles. P. .J. E. A Yu. J. T . 1970. Ap.J. 162. 815
P e iris. H. et al. 200.1. Ap.JS. to a p p e a r in v l4 8 n l
P e rc iv a l. \V . .J.. et al. 2001. M N R A S . 327. 1297
P o s tm a n . M . A G e lle r. M . .J. 1984. Ap.J. 281. 95
Riess. A . Ci.. et a l. 1998. A.J. 11G. 1009
. 2001. Ap.J. 560. 49
S e o ccim a rro . R .. Z a ld a rria g a . M .. A H tii. L. 1999. Ap.J. 527. 1
Seager. S.. Sasselov. D . D .. A S c o tt. D. 1999. Ap.J. 523. L I
S e lja k . I '. A Z a ld a rria g a . M . 199G. Ap.J. 469. 437
S e lja k . L \ et a l. 2003. p re p rin t (a s tro -p h /0 3 0 2 5 7 1 )
Sievers. .J. L . et a l. 2002. p re p rin t (a s tro -p h /0 2 0 5 3 8 7 )
S pe rg e l. D . N . et al. 2003. Ap.JS. to a p p e a r in v l4 8 n l
V erde. L .. et a l. 2002. M N R A S . 335. 432
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64
B IB L IO G R A P H Y
Z a ld a rria g a . M .. H u i. L .. A: T e g m a rk . M . 2001. Ap.J. 557. 519
Z a ld a rria g a . M . A: S eljak. L . 2000. Ap.JS. 129. 431
Z u re k . \V . H .. Q u in n . P. .J.. S a lm o n . .J. K .. A: W a rre n . M . S. 1994. Ap.J. 431. 559
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05
C h ap ter 3
C osm ological P a ra m eters
Abstract
W M A P p re cisio n d a ta enables a c c u ra te te s tin g o f c o s m o lo g ic a l m o dels. W e fin d th a t
th e e m e rg in g s ta n d a rd m o d e l o f cosm ology, a fia t A —d o m in a te d universe seeded by a
n e a rly s c a le -in v a ria n t a d ia b a tic G au ssia n flu c tu a tio n s , fits th e U ’A M P d a t a . For th e
W M A P d a ta o n ly, th e best fit p a ra m e te rs are /; = 0.72 ± O.Oo. i h j r = 0.02-1 ±
U , „ f r = 0.14 ±
0 .0 2
. r = 0 . 1 6 6 : f f i . /i, = 0.99 ± 0.04. a n d rr* = 0.9 ±
0
0
.0
0 1
.
. 1 . W ith
p a ra m e te rs fixer I o n ly b y W M A P d a ta , we can fir fin e r scale C 'M B m easurem ents
and m e asurem ents o f la rg e scale s tru c tu re (g a la x y surve ys a n d th e L y m a n n
fo re s t).
T h is s im p le m o d e l is also consistent w ith a host o f o th e r a s tro n o m ic a l
m easurem ents: its in fe rre d age o f th e universe is co n s is te n t w ith s te lla r ages, the
b a r y o n /p h o to n r a tio is co n siste n t w it h m e a surem e n ts o f th e [ D ] / [H ] ra tio , and th e
in fe rre d H u b b le c o n s ta n t is c o n siste n t w ith lo c a l o b s e rv a tio n s o f th e e xp a n sio n ra te .
W e th e n fit th e m o d e l p a ra m e te rs to a c o m b in a tio n o f W M A P d a ta w ith o th e r fin e r
scale C’ M B e x p e rim e n ts (A C B A R a n d C B I) .
2
d F G R S m e a su re m e n ts a n d L y m a n
a forest d a ta to fin d th e m o d e l’s best f it co sm o lo g ic a l p a ra m e te rs : h = 0.71
n 6/ r = 0.0224 ± 0 .0 0 0 9 . Q mh2 = 0.135lg;gg§. r = 0.17 ± 0.06. n ,(0 .0 5 M p c ' 1) =
0.93 ± 0.03. a n d <r8 = 0.84 ± 0.04. W M A P ' s best d e te rm in a tio n o f r = 0.17 ± 0.04
66
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C h a p te r 3:
67
C o s m o lo g ic a l P a ra m e te rs
arises d ir e c tly fro m th e T E d a ta and n ot fro m th is m o d e l fit. h ilt th e y are co n siste n t.
These p a ra m e te rs im p ly th a t th e age o f th e u niverse is 13.7 ± 0.2 G y r.
W ith the
L y m a n o forest d a ta , th e m o d e l favors h u t does n ot re q u ire a s lo w ly v a ry in g s p e c tra l
in d e x . T h e s ig n ific a n c e o f th is ru n n in g in d e x is s e n sitive to th e u n c e rta in tie s in the
L y m a n o forest.
B y c o m b in in g W M A P d a ta w ith o th e r a s tro n o m ic a l d a ta , we c o n s tra in the
g e o m e try o f th e u niverse: U ,„/ =
1 .0 2
±
0
.0 2 . and th e e q u a tio n o f s ta te o f th e d a rk
energy. //' < - 0 . 7 8 (9-V/( confidence lim it a ssu m in g ir > —I.) .
T h e c o m b in a tio n
o f W M A P and 2 d F G R S d a ta c o n s tra in s th e energy d e n s ity in s ta b le n e u trin o s :
i l „ h - < 0.007G (OCX co nfid e nce lim it ) . For 3 d eg e ne ra te n e u trin o species, th is lim it
im p lie s th a t th e ir mass is less th a n 0.23 eV (O-i'X co n fid e n ce lim it ) . T h e 11 AI A P
d e te c tio n o f e a rly re io n iz a tio n rules o u t w a rm d a rk m a tte r.
1.
In tro d u ctio n
O v e r th e past c e n tu ry , a s ta n d a rd c o sm o lo g ica l m o d e l has em erged:
W ith
re la tiv e ly few p a ra m e te rs , the m o d e l describes th e e v o lu tio n o f th e U n ive rse and
a s tro n o m ic a l o b s e rv a tio n s on scales ra n g in g fro m a few to th o u s a n d s o f M egaparsecs.
In th is m o d e l th e U n ive rse is s p a tia lly Hat. hom ogeneous and is o tro p ic on large
scales, com posed o f ra d ia tio n , o rd in a ry m a tte r (e le c tro n s , p ro to n s , n e u tro n s and
n e u trin o s ), n o n -b a ry o n ie co ld d a rk m a tte r, and d a rk energy. G a la x ie s and large-scale
s tru c tu re g re w g r a v ita tio n a lly fro m tin y , n e a rly s c a le -in v a ria n t a d ia b a tic G aussian
flu c tu a tio n s . T h e W ilk in s o n M ic ro w a v e A n is o tro p y P ro b e (1V A /.A P ) d a ta o ffe r a
d e m a n d in g q u a n tita tiv e test o f th is m o d el.
T h e W M A P d a ta are p o w e rfu l because th e y re s u lt fro m a m issio n th a t was
c a re fu lly designed to lim it s y s te m a tic m e a surem ent e rro rs (B e n n e tt et a l.. 2003a.b:
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C h a p te r 3:
G8
C o s m o lo g ic a l P a ra m e te rs
H in s h a w t't a l.. 200 3 b).
A c r it ic a l elem ent o f th is design in c lu d e s « liffe re n tia l
m e a stire m en ts o f th e fu ll sky w ith a c o m p le x sky scan p a tte rn .
T h e n e a rly
u n c o rre la te d noise betw een p a irs o f p ixe ls, th e a ccu ra te in -flig h t d e te rm in a tio n o f th e
beam p a tte rn s (Page et a l.. 2003c.a: B arnes et a l.. 2003). and th e w e ll-u n d e rs to o d
p ro p e rtie s o f th e ra d io m e te rs (.la ro s ik et a l.. 2003a.b) are in v a lu a b le fo r th is ana lysis.
O u r basic a p p ro a ch in th is a n a lysis is to begin by id e n tify in g th e s im p le s t m o d el
th a t fits th e W M A P d a ta and d e te rm in e th e best fit p a ra m e te rs fo r th is m o d e l using
W M A P d a ta o n ly w ith o u t th e use o f a n y s ig n ific a n t p rio rs on p a ra m e te r values.
We th e n co m p are th e p re d ic tio n s o f th is m o d el to o th e r d a ta sets and fin d th a t tlie*
m o d e l is b a s ic a lly co nsiste n t w ith these d a ta sets. We then fit to c o m b in a tio n s o f the
W M A P d a ta and o th e r a s tro n o m ic a l d a ta sets and fin d th e best fit g lo b a l m o d el.
F in a lly , we place c o n s tra in ts on a lte rn a tiv e s to th is m o d el.
W e b egin by o u tlin in g o u r m e th o d o lo g y (§2). V erde et a l. (2003) d escribes the
d e ta ils o f th e a p p ro a ch used here to co m p a re th e o re tic a l p re d ic tio tis o f co sm o lo g ic a l
m odels to d a ta . In §3. we fit a sim p le , s ix p a ra m e te r A C 'D M m o d e l to th e W M A P
d a ta -se t (te m p e ra tu re -te m p e ra tu re and te m p e ra tu rc -p o ia riz a tio n a n g u la r p ow er
s p e c tra ). In §4 we show th a t th is s im p le m o d el p ro vid e s an a cce p ta b le fit n o t o n ly
to th e W M A P d a ta , b u t also to a host o f a s tro n o m ic a l d a ta . W e use th e co m p a ris o n
w ith these o th e r d a ta se ts to te st the v a lid ity o f th e m o d el ra th e r th a n fu r th e r
c o n s tra in th e m o d e l p a ra m e te rs. In §5. we in c lu d e la rg e scale s tru c tu re d a ta fro m
th e 2 d F G a la x y R e d s h ift S u rve y (2 d F G R S . C olless et al. (2 0 0 1 )) and L y m a n o
forest d a ta to p e rfo rm a jo in t lik e lih o o d a na lysis fo r th e c o s m o lo g ic a l p a ra m e te rs.
W e fin d th a t th e d a ta favors a s lo w ly v a ry in g s p e c tra l in d e x . T h is seven p a ra m e te r
m o d el is o u r best fit to th e f u ll d a ta set.
In §6 . we re la x som e o f th e m in im a l
a s s u m p tio n s o f th e m o d e l by a d d in g e x tra p a ra m e te rs to th e m o d e l. W e e xa m in e
n o n -fla t m odels, d a rk energy m o d e ls in w h ic h th e p ro p e rtie s o f th e d a rk ene rg y are
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C h a p te r 3:
C o sm o lo g ica l P a ra m e te rs
G9
p a ra m e te riz e d by an effective* e q u a tio n o f sta te , and m o d els w ith g r a v ity waves. B y
a d d in g e x tra p aram ete rs we in tro d u c e degenerate sets o f m o d e ls c o n siste n t w ith the
W M A P d a ta alone. We lif t these degeneracies by in c lu d in g a d d itio n a l microwave*
b a c k g ro u n d d ata -se ts (C B I. A C B A R ) and o b se rva tio n s o f la rg e-scale s tru c tu re . We
use* these co m b in e d d a ta sets to p la ce s tro n g lim its on th e g e o m e try o f th e universe*,
the* neutrine) mass, the* energy d e n s ity in g ra v ity wave's, anel the* prope*rtie*s e>f the*
d a rk energy. In *j7. we* note* an in tr ig u in g discre*pancy be*twe*e*n the* stan d arel moeh'l
anel the* W M A P d a ta on the* largest a n g u la r scales anel speculate* on its o rig in . In $8 .
we* conclude* anel pre*se*nt parame*te*rs fo r o u r be*st fit mealed.
2.
B ayesian A n a ly sis o f C osm ological D a ta
The* basic- approac h o f th is p a p e r is to fin d the* s im p le st mode*! co n siste n t w ith
c o s m o lo g ica l d a ta . We* b egin by f it t in g a simple* six parame*fe*r mealed firs t to the*
W M A P elata anel the*n to o th e r co sm o lo g ica l d a ta se*ts. We th e n co n s id e r more*
compIe*x co sm o log ica l m odels and e va lu a te whe*ther th e y are a be*rte*r <le*scription
o f th e co sm o lo g ica l elata.
Since K o m a ts u et a l. (2003) fo u n d no e'vidence fo r
n o n -G a u s s ia n ity in th e W M A P d a ta , we assume th e p r im o r d ia l flu c tu a tio n s are
G au ssia n ra n d o m phase th ro u g h o u t th is p ap e r. For e*ach m o d e l s tu d ie d in the* p aper,
we* use a Monte* C a rlo M a rk o v C h a in to e x p lo re th e lik e lih o o d su rface . We assume
Hat p rio rs in o u r basic p a ra m e te rs, im p o se p o s itiv ity c o n s tra in ts on th e m a tte r and
b a ry o n d e n s ity (the*se lim it s lie a t such lo w lik e lih o o d th a t the*y a re u n im p o rta n t
fo r th e m odels. We* assum e a Hat p r io r in r . th e o p tic a l d e p th , b u t b o u n d
t
< 0.3.
T h is p r io r has lit t le effect on th e Hts b u t keeps th e M a rk o v C h a in o u t o f u n p h y s ic a l
re g ion s o f p a ra m e te r space. For each m o d e l, we d e te rm in e th e be*st fit p a ra m e te rs
fro m th e peak o f th e N -d im e n s io n a l lik e lih o o d surface. F o r each p a ra m e te r in the
m o d e l we also c o m p u te its one d im e n s io n a l lik e lih o o d fu n c tio n b y m a rg in a liz in g over
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C h a p te r 3:
70
C o s m o lo g ic a l P a ra m e te rs
a ll o th e r p a ra m e te rs: we th e n q u o te th e ( [-d im e n s io n a l) e x p e c ta tio n v a lu e 1 as o u r
best estim ate' fo r th e p a ra m e te r:
(3 -1 )
w here o denotes a p o in t in t in ' X -d im e n s io n a l p a ra m e te r space' (in o u r a p p lic a tio n
these are' [jo in ts
se'ts o f co sm o lo g ic a l param e'te'rs
in the' o u tp u t o f the* M a rk o v
C h a in ). C denotes the 1 lik e lih o o d (in o u r a p p lic a tio n the1 "w e ig h t" g ive n by th e ch a in
to e-ach p o in t). The- W M A P te 'tn p e ra tu re ( T T ) a n g u la r [tow e r sp e'ctrum anel the
U ’.M.AP te 'in p e 'ra tu re '-p o la riz a tio n ( T E ) a n g u la r powe>r s p e c tru m are* o u r core* d a ta
se'ts fo r the* lik e lih o o d a n a lysis.
H in s h a w «'t al. (20031)) and K o g u t e>t al. (2003)
de'scribe how to o b ta in the 1 te'inpe*rature' and te 'u i[)e 'ra tu re '-[)o la riza tio n a n g u la r
|)owe’r sp e ctra re s p e c tiv e ly fro m the* m aps. V erde e't al. (2003) describes o u r basic
m e th o d o lo g y fo r e w a lu a tin g th e lik e lih o o d fu n c tio n s using a Monte* C a rlo M a rk o v
C h a in a lg o rith m a n d fo r in c lu d in g d a ta -s e ts o th e r th a n W M A P in o u r a n a lysis . In
a d d itio n to U W /.A P d a ta we use rece*nt re su lts fro m the* C B I (Pe'arson e*t a l.. 2002)
a nd A C B A R (K u o et a l.. 2002) e x p e rim e n ts . We also use th e 2dFCJRS m e a surem e n ts
o f th e pow er s p e c tru m (P e rc iv a l et a l..
2 0 0 1
) anel th e bias p a ra m e te r (V e rd e et a l..
2002). m e asurem ents o f th e L y m a n n p ow er s p e c tru m (C ro ft et a l.. 2002: G n e d in
■V H a m ilto n . 2002). su p e rn o va la m e a surem e n ts o f the a n g u la r d ia m e te r d is ta n c e
re la tio n (G a rn a v ic h et a l.. 1998: Riess et a l.. 2001). and tin* H u b b le Space Telescope
K e y P ro je c t m e a surem e n ts o f th e lo c a l e xp a n sio n ra te o f th e u niverse (F re e d m a n
et a l..
2 0 0 1
).
‘ In a M o n te C a rlo M a rk o v C h a in , it is a m o re ro b u st q u a n tity th a n th e m o d e o f
th e a p o s te rio ri m a rg in a liz e d d is tr ib u tio n .
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C h a p te r 3:
C o sm o lo g ica l P a ra m e te rs
3.
Pow er Law A C D M
71
M o d e l a n d th e W M A P D a ta
W e b egin by c o n s id e rin g a basic c o s m o lo g ica l m o d el: a fla t I'n iv e rs e w ith
ra d ia tio n , baryons. co ld d a rk m a tte r and co sm o lo g ica l c o n s ta n t, and a p ow er-la w
p ow er s p e c tru m o f a d ia b a tic p r im o r d ia l flu c tu a tio n s . A s we w ill see. th is m odel does
a re m a rk a b ly goo d jo b o f d e s c rib in g W M A P T T and T E p ow er sp e c tra w ith o n ly six
p a ra m e te rs: th e H u b b le co n s ta n t h (in u n its o f 100 k m /s /M p e ) . th e p hysica l m a tte r
and b a ryo n den sitie s //•„, =
surface.
7.
1
2 a nd i/y, = S h ,/r. the optic al d e p th to the* d e co u p lin g
th e sca la r s p e c tra l in d e x n s a n d .4. the' n o rm a liz a tio n p a ra m e te r in the*
C M B F A S T code version 4.1 w ith o p tio n I ’ N N O R M . Verde et al. (2003) discusses
thc> re la tio n s h ip betw een .4 and th e a m p litu d e ' o f c u rv a tu re flu c tu a tio n s at h orizon
cro ssin g. ] A /? |' = 2.95 x 10 '*.4. In fj l. we show th a t th is m o d e l is also in a cceptable
agreem ent w ith a w id e range' o f a s tro n o m ic a l d a ta .
T h is s im p le m o d el p ro vid e s an acceptable' fit te» b o th the* W M A P T T anel
T E d a ta (see F ig u re 3.1 anel 3 .2 ).
T h e reduced"’ \ j j j fo r th e fu ll fit is 1.0G0 fo r
1342 d e'gm 's o f free'dotn. w h ich has a p r o b a b ility o f ~ 7f/ . F o r th e T T d a ta alone'.
\tjf/v =
1-09. w h ic h fo r 893 degree's o f fm 'd o rn has a p r o b a b ility o f '.VZ. M ost
o f the* excess \~yy is due tee th e in a b ilit y o f th e m o d el to fit sh a rp fea tu re s in the
powe>r s p e c tru m ne>ar / ~ 120. th e firs t T T peak anel a t I ~ 350. In Figure' 3.4 we
show th e c o n trib u tio n to \* y y p e r m u ltip o le . T h e o v e ra ll e'xcc'ss v a ria n ce is lik e ly
due te> e>ur n o t in c lu d in g several e ffects, each c o n trib u tin g ro u g h ly 0.5 — VZ to o u r
p o w e r s p e c tru m co varian ce near th e firs t pe'ak and tro u g h : g r a v ita tio n a l tensing e)f
th e C M B (H u . 2001). th e s p a tia l v a ria tio n s in th e e ffe ctive beam o f th e W M A P
-H e re . \ j j j = —2 In £ a n d u is n u m b e r o f d a ta m in u s th e n u m b e r o f p aram ete rs.
W e have used
1 0 0 .0 0 0
M o n te C a rlo re a liz a tio n o f th e W M A P d a ta w ith o u r m ask,
noise a n d angle-averaged beam s a n d fo u n d th a t th e ( —2 In C/is) =
te m p e ra tu re d a ta .
1
fo r th e s im u la te d
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p te r 3:
C o sm o lo g ica l P a ra m e te rs
72
e x p e rim e n t due to v a ria tio n s in o u r scan o rie n ta tio n between t in ' e c lip tic p ole and
p la n e regions (Page et a l.. 2003a: H in sh a w et a l.. 2003a). and n o n -G a u s s ia n ity in
tin* noise m aps due to tin*
1/
/ s tr ip in g . In c lu d in g these effects w o u ld increase o u r
estim ate* o f th e pow er s p e c tru m u n c e rta in tie s and im p ro v e o u r estim ate*
C ff
O u r n ext d a ta release* w ill inc lude’ th e corre*ctions and e*rrors associate*d w ith the
he*am a sym m e trie s. The* fea tu re s in the* m easured p ow er s p e c tru m c o u ld he* due* to
u n d e rly in g feature's in the* p rim o rd ia l p ow er spe*ctrum (se*e* §5 o f Pe*iris e*t al. (2 0 0 3 )).
b u t we* do not yet a tta c h co sm o lo g ic a l s ig n ifica n ce to theun.
Table*
1
lis ts the* be*st fit p a ra m e te rs u sin g the* W M A P d a ta alone* fo r th is mode*l
and F ig u re (3 .3) shows the* m a rg in a liz e d p ro b a b ilitie s fo r each of t he* basic- p aram ete rs
in the* m o d e l. The* value's in the* second c o lu m n o f Table* 3.1 (a n d the* subsequent
param ete*r table's) are* e x p e c ta tio n value's fo r the* m a rg in a liz e d d is tr ib u tio n o f each
pararnete*r a nd the* e rro rs are th e G8 (T confidence* infe*rval. The* value's in the* rhirel
c o lu m n are* the* value's at the* pe*ak o f th e likcd ih o o d fu n c tio n . Since* we* are* p ro je c tin g
a h ig h d im e n s io n a l lik e lih o o d fu n c tio n , the* pe*ak o f the* lik e lih o o d is n ot th e same as
the* e x p e c ta tio n value o f a p a ra m e te r. M o st o f th e basic p a ra m e te rs are re m a rk a b ly
w e ll d e te rm in c 'd w ith in th e c o n te x t e>f th is m o d e l. O u r m ost s ig n ific a n t p a ra m e te r
degeneracy (see F ig u re 3.5) is a d egeneracy betwee*n n s and r . The* T E elata favors
t
~~ 0.17 (K o g u t et a l.. 2003): on th e o th e r h a n d , th e lo w value o f the* q u a d ru p o le
(se*e F ig u re 3.1 and §7) a n d th e re la tiv e ly lo w a m p litu d e o f flu c tu a tio n s fo r / < 10
d is fa v o rs h ig h r as re io n iz a tio n produce’s a d d itio n a l la rg e sc ale a n is o tro p ie s . Because
o f th e c o m b in a tio n o f the*se tw o effects, th e lik e lih o o d su rface is q u ite fla t a t its peak:
th e lik e lih o o d changes b y o n ly 0.05 as r changes fro m
0 .1 1
- 0.19. T h is p a r tic u la r
shape depends upon th e assum ed fo rm o f th e p ow er s p e c tru m : in §5.2. we show th a t
m o d e ls w ith a scale-dependent s p e c tra l in d e x have a n a rro w e r lik e lih o o d fu n c tio n
th a t is m o re centered a ro u n d
t
= 0 .1 7 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p te r 3:
C o sm o lo g ica l P a ra m e te rs
73
S ince th e W M A P d a ta a llo w s us to a c c u ra te ly d e te rm in e m a n y o f th e basic
c o s m o lo g ic a l p aram ete rs, we can now in fe r a n u m b e r o f im p o rta n t d e riv e d q u a n titie s
to v e ry h ig h accuracy: we d o th is by c o m p u tin g these q u a n titie s fo r each m o d e l in the*
M C M C a nd use the c h a in to d e te rm in e th e ir e x p e c ta tio n value's a n d u n c e rta in tie s .
T a b le 3.2 lists c o s m o lo g ica l p a ra m e te rs based on f it t in g a p ow er la w ( P L ) C’ D M
m o d e l to the* W’.\/.-VP d a ta o n ly. The' p a ra m e te rs t,i,, and
are d e te rm in e d by
u sin g th e C’ M B F A S T code (S e lja k A: Z a ld a rria g a . 199G) to c o m p u te the 1 re d s h ift o f
th e C’ M B "p h o to s p h e re " (th e peak in th e p h o to n v is ib ilit y fu n c tio n ). We d e te rm in e
the' th ickn e ss o f the* d e c o u p lin g su rface by m e a s u rin g A c ,/,r and A /,/,, , th e f u ll- w id t h
at h a lf m a x im u m o f the 1 v is ib ilit y fu n c tio n . T h e age o f th e L’ niverse is d e riv e d by
in te g ra tin g the F rie d m a n n ('({n a tio n , a nd rrs (th e lin e a r th e o ry p re d ic tio n s fo r th e
a m p litu d e o f flu c tu a tio n s w it h in
s p e c tru m at r =
4.
0
8
M p c / h spheres) fro m t in 1 lin e a r m a tte r p o w e r
is c o m p u te d by C 'M B F A S T .
C o m p a rison w ith A stro n o m ica l P r e d ic tio n s
In th is se ction , we co m p a re th e p re d ic tio n s o f th e best f it p o w e r la w A C D M
m o d e l to o th e r co sm o lo g ica l o b s e rv a tio n s . W e also lis t in T a b le 3.10 t in ' best fit
m o d e l to th e fu ll d a ta set: a A C D M m o d e l w ith a ru n n in g s p e c tra l in d e x (see §5.2).
In p a r tic u la r we co n sid e r d e te rm in a tio n s o f th e lo c a l e xp a n sio n ra te (i.e . th e H u b b le
c o n s ta n t), th e a m p litu d e o f flu c tu a tio n s on g a la x y scales, th e b a ry o n a b u n d a n c e ,
ages o f th e oldest sta rs, la rg e scale s tru c tu re d a ta a nd su p e rn o va la d a ta . W e also
c o n s id e r i f o u r d e te rm in a tio n o f th e re io n iz a tio n re d s h ift is co n s is te n t w it h th e
p re d ic tio n fo r s tru c tu re fo r m a tio n in o u r best fit U n ive rse a nd w it h recent m o d e ls
o f re io n iz a tio n . In §5 a nd
6
. we a d d som e o f these d a ta sets to th e W M A P d a ta to
b e tte r c o n s tra in p a ra m e te rs a n d c o s m o lo g ic a l m o d els.
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C h a p te r 3:
C o s m o lo g ic a l P a ra m e te rs
4 .1 .
74
H u b b le C on stan t
C’ M B o b se rva tio n s do n o t d ire c tly m easure th e lo ca l e xp a n sio n ra te o f the
U n ive rse ra th e r th e y m easure th e c o n fo rm a l d ista n ce to th e d e c o u p lin g surface and
th e m a tte r - r a d ia tio n r a tio th ro u g h th e a m p litu d e o f tin ' e a rly In te g ra te d Sachs W o lfe
(IS W ) c o n tr ib u tio n re la tiv e to th e h e ig h t o f the firs t peak. F o r o u r p o w e r la w A C D M
m o d e l, th is is enough in fo r m a tio n to
"p re d ic t" th e lo c a l e xp a n sio n ra te . T h u s , lo ca l
H u b b le co n s ta n t m e a surem e n ts art* an im p o rta n t test o f o u r basic m o d e l.
T h e H u b b le K e y P ro je c t (F re e d m a n et al.. 2001) has c a rrie d o u t an e xte n sive
p ro g ra m o f using C’epheids to c a lib ra te several d iffe re n t se con d ary d is ta n c e in d ic a to rs
(T y p e la supernovae. T u lly -F is h e r. T y p e I I supernovae, and surface* b rig h tn e s s
flu c tu a tio n s ). W ith a d is ta n c e m o d u lu s o f 18. j fo r th e L M C . th e ir c o m b in e d estimate*
fo r th e H u b b le c o n s ta n t is Ho = 72 ± 3 (s ta t.) ± 7( s y s te m a tic ) k m / s / M p c .
The*
agree*me*nt be*twe*e*n th e H S T K e y P ro je c t value* and e>ur value, h = 0.72 ± 0.05.
is s tr ik in g . give*n th a t th e tw o m e th o d s re ly on eliffe*re*nt observables, d iffe re n t
u n d e *rlyin g physics, a n d d iffe re n t m o d e l a ssu m p tio n s.
A s we w ill show in §6 . m o d e ls w ith e q u a tio n o f s ta te fo r th e d a rk e n e rg y ve*ry
d iffe re n t fro m a co sm o lo g ic a l c o n s ta n t (i.e .. ir = —1) o n ly f it th e W M A P d a ta i f the
H u b b le c o n s ta n t is m uch s m a lle r th a n th e H u b b le K e y P ro je c t value. A n in d e p e n d e n t
d e te rm in a tio n o f th e H u b b le c o n s ta n t th a t makes d iffe re n t a s s u m p tio n s th a n th e
t r a d itio n a l d is ta n c e la d d e r can be o b ta in e d by c o m b in in g S u n y a e v -Z e l’d o v ic h and
X -ra y H ux m e a surem e n ts o f c lu ste rs o f g ala xie s, u n d e r th e a s s u m p tio n o f s p h e ric ity
fo r th e d e n s ity and te m p e ra tu re p ro file o f clu ste rs.
T h is m e th o d is se n s itiv e to
th e H u b b le c o n s ta n t a t in te rm e d ia te re d s h ifts ( r ~ 0 .5 ). ra th e r th a n in th e n ea rb y
u niverse .
Reese et a l. ( 2
0 0 2
). .Jones et a l. (2 0 01 ). a n d M ason et a l. ( 2
0 0 1
) have
o b ta in e d values fo r th e H u b b le c o n s ta n t s y s te m a tic a lly s m a lle r th a n , th e H u b b le
K e y P ro je c t and W M A P A C D M m o d e l d e te rm in a tio n s , b u t a ll c o n s is te n t a t th e lcr
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C h a p te r ■I :
C o s m o lo g ic a l P a ra m e te rs
level. T a b le (3 .3) su m m a rize s recent H u b b le c o n s ta n t d e te rm in a tio n s and co m p ares
th e m w ith tin* W M A P A C D M m o d e l value.
4 .2 .
A m p litu d e o f F lu ctu a tio n s
T h e o v e ra ll a m p litu d e o f flu c tu a tio n s on large-scale s tru c tu re scales has
been re c e n tly d e te rm in e d fro m weak le a sing surveys, clu ste rs n u m b e r co u n ts and
p e c u lia r v e lo c itie s fro m g a la x y surveys. W eak tensing surveys and p e c u lia r v e lo c ity
m e a surem ents are m ost sensitive* to th e c o m b in a tio n cr>
cl ust er abundance* at
low re d s h ift is sensitive* to a ve*rv s im ila r parame*te*r c o m b in a tio n rrs{Vjri'. b u t co u n ts
o f h ig h re*dshift clu ste rs ra n bre*ak the* ele*ge*ne*racy.
Weak Lcnsin.fi
We*ak le asing d ir e c tly probes th e a m p litu d e o f mass flu c tu a tio n s a lo n g the* line*
e>f s ig h t te> th e b a c k g ro u n d g alaxies. O nce the* re*elshift d is tr ib u tio n e>f the b a c k g ro u n d
galaxie*s is kn o w n , th is te ch n iq u e elire*ctly p robes g ra v ita tio n a l pote>ntial flu c tu a tio n s ,
a nd th e re fo re can be e a sily co m p a re d w ith o u r C M B m o d e l p re d ic tio n s fo r th e
a m p litu d e o f d a rk m a tte r flu c tu a tio n s . S everal g ro u p s have re p o rte d weak shear
m e a surem e n ts w ith in th e past ye ar (see T a b le 3.4 a nd V an W aerbeke et al.
(2002a)
fo r recent re v ie w ): w h ile th e re is s ig n ific a n t s c a tte r in th e re p o rte d a m p litu d e , the
best f it m o d e l to th e W M A P d a ta lies in th e m id d le o f th e re p o rte d range. A s these
shear m e asurem ents c o n tin u e to im p ro v e , th e c o m b in a tio n o f W M A P o b s e rv a tio n s
a n d le a sin g m e a surem e n ts w ill be a p o w e rfu l p ro b e o f c o s m o lo g ica l m odels.
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C h a p te r 3:
C o s m o lo g ic a l P a ra m e te rs
7G
Galasij celerity fields
T h e g a la x y v e lo c ity fields an* a n o th e r im p o rta n t p ro b e o f th e large scale
d is tr ib u tio n o f m a tte r.
T h e W illic k A: S trau ss (1998) a n a lysis o f th e M a rk I I I
v e lo c ity fields and the IR A S re d s h ift su rve y y ie ld s
1ll<As = 0.50 ± 0.04.
IR A S
g a la x ie s are less clu ste re d th a n o p tic a lly selected galaxies: F ish e r et a l. (1994) fin d
'TsW‘ ' > — d-69 ± 0.04 im p ly in g rr"“1'’sU^f' = 0.84a ± 0.05. co nsistent w ith o u r A C D M
m o d e l value o f 0.44 ± 0.10.
Cluster Number Counts
O u r best fit to the- W M A P d a ta is
= 0.48 ±
0
. 1 2 . B a h c a ll et a l. ( 2
recent s tu d y o f th e mass fu n c tio n o f 300 c lu ste rs a t re d s h ifts
0 .1
< c <
0 .2
0 0 2
b)
in the
e a rly SDSS d a ta release y ie ld s <7nil";* = 0.33 ± 0.03. T h is d iffe re n ce m a y re fle ct the
s e n s itiv ity o f th e c lu s te r m ea surem e n ts to th e co nve rsio n o f c lu s te r richness to mass.
O b s e rv a tio n s o f the mass fu n c tio n o f h ig h re d s h ift clu ste rs break th e d egeneracy
betw een rrH a n ti V.m. T h e recent B a h c a ll A: B ode (2002) a n a lysis o f th e a b u n d a n c e
o f m assive c lu ste rs at c = 0.3 — 0.8 y ie ld s rrs = 0.95 ± 0.1 fo r
c lu s te r a n a lysis y ie ld d iffe re n t values:
B o rg a n i et a l. ( 2
0 0 1
= 0.25. O th e r
) best f it values fo r a
la rg e sa m p le o f X -ra y clu s te rs are rrH = 0.667°;o5 an<l --»* = 0 .3 5 7 °;{q. O n th e o th e r
h a n d . R e ip ric h A: B o h rin g e r ( 2
0 0 2
) fin d v e ry d iffe re n t values: rrH = 0.9GToIi2 a n (l
Q rn — 0 .1 2 1°;“ . P ie rp a o li et a l. (2002) discuss th e w id e range o f values th a t d iffe re n t
X -ra y analyses fin d fo r rrH. W it h th e la rg e r R E F L E X sa m ple. S chuecker et al.
(2003a) fin d rrH = 0.71 llo lo iu -o !i 6 ° a n t^
= O
-o!orl- w here th e second set
o f e rro rs in c lu d e th e s y s te m a tic u n c e rta in tie s . T h e best fit W M A P values lie in th e
m id d le o f th e re le van t range.
M e a su re m e n ts o f th e c o n tr ib u tio n to th e C M B p ow er s p e c tru m on s m a ll scales
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C h a p te r 3:
C o s m o lo g ic a l P a ra m e te rs
fro m th e S u n ya ev-Z el d o vich effect also p ro b e the1 n u m b e r d e n s ity o f h ig h re d s h ift
c lu s te rs. T h e recent C’ B l d e te c tio n o f excess fin e rn a tio n s (M a so n et a l.. 2001: B o n d
<>t a l.. 2002) at ( > 1■">()() im p lie s rrH = 1.04 ± 0.12 (K o m a ts u
S e lja k . 2002). if the1
s ig n a l is due to th e S u n ya e v-Z e l’d o vich e ffe ct.
4 .3.
B aryon A b u n d an ce
B o th th e a m p litu d e o f th e a co u stic peaks in th e C 'M B s p e c tru m (B o n d A'
K fs ta th io u . 1984) and th e p rim o rd ia l a b u n d a n ce o f D e u te riu m ( B oesgaard A
S te ig m a n . 1985) are se nsitive fu n c tio n s o f th e c o s m o lo g ica l b a ry o n d e n s ity . Since th e
h eigh t and p o s itio n o f the a co u stic peaks d ep e nd u po n th e p ro p e rtie s o f th e cosmic'
p la sm a 372.000 years a fte r th e B ig B a n g a nd t in 1 D e u te riu m abu n da n ce 1 depends
on physics o n ly th re e m in u te s a fte r t in 1 B ig B a n g , c o m p a rin g th e b a ry o n d e n s ity
c o n s tra in ts in fe rre d fro m these tw o d iffe re n t p robes p ro vid e s an im p o r ta n t test o f
the1 B ig B ang m o d e l. T h e best fit b a ry o n a b u n d a n ce based on W M A P d a ta o n ly
fo r the1 P L L C D M m o d e l. H fc/r = 0.024 ± 0.001. im p lie s a b a r y o n /p h o fo n ra tio
o f if = (G.ulJJ ',) x l( ) “ lu . F o r th is a b u n d a n ce , s ta n d a rd b ig b a n g n ucleo syn th e sis
(B u rie s e t a l.. 2001) im p lie s a p r im o r d ia l D e u te riu m abundance* re la tiv e to H yd ro g e n :
[D ]/[H ] = 2 .3 7 7 ”
21
x 10“ ’ ■ A s it w ill be c le a r fro m §5 a nd G. th e best f it
va lu e
fo r o u r fits is re la tiv e ly in s e n s itiv e to c o s m o lo g ic a l m o d e l a nd d a ta set c o m b in a tio n
as it depends p r im a r ily on th e r a tio o f th e firs t to second peak h e ig h ts (P age et a l..
2 0 0 3 b ). F o r th e ru n n in g s p e c tra l in d e x m o d e l discussed in §5.2. th e best fit b a ry o n
a b u n d a n ce . U bh'2 = 0.0224 ± 0.0009. im p lie s a p r im o rd ia l [ D ]/[H ] = 2 .6 2 Io ‘>o x 10“ ’ .
H o w does th e p rim o rd ia l D e u te riu m a b u n d a n ce in fe rre d fro m C M B co m p a re
w ith th a t observed fro m th e IS M ? G a la c tic c h e m ic a l e v o lu tio n d e s tro y s D e u te riu m
because th e D e u te riu m nucleus is re la tiv e ly fra g ile a nd is e a s ily d e s tro y e d in sta rs.
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C h a p te r 3:
C o s m o lo g ic a l P a ra m e te rs
T im s . measurem<*nts o f the D e u te riu m a b u n d a n ce w ith in th e G a la x y an* u s u a lly
tre*ated as lowe*r lim its on the p r im o rd ia l abundance' (E p s te in et al.. 197C). Local
m e a surem e n ts o f D and H a b s o rp tio n fin d [D /H ] a bu n da n ce near I.a x 10
while* m o re d is ta n t m easurem ents by IM A P and F l'S E fin d s ig n ific a n t v a ria tio n in
D e'U terinm abunelances sugge*sting a comple*x G alactic- che'tnical h is to ry (.Jenkins
et a l.. 1999: S on n eb o rn et al.. 2000: M oos et a l.. 2002).
O b s e rv a tio n s o f L y m a n o clo u d s reduce* the need to corre'ct the* D eute*rium
a b u n d a n ce fo r s te lla r processing as these* system s have* lo w (b u t non-zere>) m e ta l
abu n da n ces. These* obse*rvations re’tjuire* iele*ntifying gas system s th a t do not have*
se*rious inte*rfere*nce* fro m the* L y m a n o fore*st. The* K irk m a n e*t al. (2003) a na lysis
o f Q S O HS 2 4 3 + 3 0 5 7 y ie ld s a D / H r a tio e>f 2.12
J-J x 1 0 ~ ’ . T h e y co m b in e th is
rne>asure'ine*nt w ith fo u r othe*r D / H me*a.sure*me*nts ( Q 0 130-4021: D / H < G.8 x 10 ’ .
Q 1009+ 295G :
3.98 ± 0.70 x 1 0 - \ P K S 1937-1009:
3.25 ± 0.28 x 10
anel
Q S O H S 0 1 0 5 4 -1G19: 2.5 ± 0.25 x 1 0 ~ ’ ). to o b ta in th e 'ir curre*tit be*st D / H ra tio :
2 .7 8 ro :;y
X
10-*’ im p ly in g Uh
lr = 0.0214 ± 0.0020.
D O d o rie -o e*t al. (2001) fin d
2.24 ± 0.G7 x 1 0 " ’ fro m th e ir obse'rvatiems e>f Q 0347-3819 (a lth o u g h a re a n a lysis
o f th e syste m by Levshakov et al.
(2003) fim ls a highe*r D / H value: 3.75 ± 0.25.
P e ttin i A Bowe*n (2001) re p o rt a D / H a b u n d a n ce o f 1.65 ± 0.35 x 10“ ’ fro m S T IS
m e a surem e n ts o f Q S O 2206-199. a lo w r n e ta llic ity ( Z — 1 /2 0 0 ) D a m p e d L y m a n
n syste'tn. T h e W M A P value lie's between th e P e ttin i A’ Benven (2001) e s tim a te
fro m D L A s . 12fc/r = 0.025 ± 0.001. and th e K irk m a n et al. (2003) e*stim ate o f
f i f t / r = 0.0214 ± 0.0020 T h e re m a rk a b le agreem ent betw een the* baryem d e n s ity
in fe rre d fro m D / H value’s and o u r m easurem ents is an im p o rta n t tr iu m p h fo r the
basic B ig B a n g m o d el.
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C h a p te r 3:
C o s m o lo g ic a l P a ra m e te rs
4.4.
79
C osm ic A ges
T h e ago o f th e I ’ niverse based on th e best fit to W M A P d a ta o n ly.
/o =
13.4 ± 0.3 G y r.
However. th e a d d itio n o f o th e r d a ta sets (see ^5) im p lie s a
lo w e r m a tte r d e n s ity and a s lig h tly la rg e r age. T h e best fit age fo r th e p o w e r law
m o d e l based on a c o m b in a tio n o f W M A P . 2 d F G R S a n d L y m a n o forest d a ta is
G =
13.G ± ().2 G yr. T in * best fit age fo r th e sam e d a ta set fo r th e ru n n in g in d e x
m o d e l o f jj5.2 is G = 13.7 ± 0 .2 G y r. (See Hu et al. 2001 H u (F u k u g ita ) and K n o x .
C h ris te n s e n A; S ko rd is
(2001) fo r discussions o f using C’ M D d a ta to d e te rm in e
c o s m o lo g ic a l ages.)
A lo w e r lim it to tin* age o f th e u niverse can in d e p e n d e n tly be o b ta in e d fro m
d a tin g th e oldest s te lla r 84
, t C a tio n s . T h is has been done tr a d itio n a lly by d a tin g the
old est s ta rs in th e M ilk y W ay (see e.g.. C’ h a b u ye r (1998): Jim enez (1 9 9 9 )). F o r th is
p ro g ra m , g lo b u la r clu s te rs are an excellen t la b o ra to ry fo r c o n s tra in in g th e age o f the
u niverse : each c lu s te r has a c h e m ic a lly hom ogeneous 6^5384
, , '
.
r sta rs a ll b o rn
n e a rly s im u lta n e o u s ly . T in 1 m a in u n c e rta in ty in th e age d e te rm in a tio n com es fro m
th e p o o rly kn o w n d is ta n c e (C h a b o y e r. 1993). W e ll-u n d e rs to o d s te lla r p o p u la tio n s
are usefu l to o ls fo r c o n s tra in in g c lu s te r d ista n ce s: R e n z in i et al. (1996) used th e
w h ite d w a r f sequence to o b ta in an age o f 14.5 ± 1.5 G y r fo r N G C 6752. Jim e n ez
et a l. (1 9 96 ). u sin g a d is ta n c e -in d e p e n d e n t m e th o d d e te rm in e d th e age o f th e oldest
g lo b u la r c lu s te rs to be 13.5 ± 2 G y r. U sin g th e lu m in o s ity fu n c tio n m e th o d . Jim enez
A: P a d o a n (1998) fo u n d an age o f 12.5 ± 1.0 G y r fo r M 5 5 . T h is m e th o d gives a jo in t
c o n s tra in t on th e d is ta n c e and th e age o f th e g lo b u la r c lu s te r. O th e r g ro u p s fin d
c o n s is te n t ages: G r a tto n et al. (1997) e s tim a te an age o f l l . S l j i G y r fo r th e oldest
G a la c tic g lo b u la rs : V a n d e n B e rg et a l. (2002) e s tim a te s an age o f ~
M 9 2 . C h a b o y e r A: K ra u ss
13.5 G y r fo r
(2003) re vie w th e g lo b u la r c lu s te r a n a ly s is a n d q u o te a
best fit age o f 13.4 G y r.
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C h a p te r 3:
C o s m o lo g ic a l P a ra m e te rs
80
O b s e rv a tio n s o f e c lip s in g d o u b le lin e sp e ctro sco p ic b in a rie s e na b le g lo b u la r
c lu s te r age d e te rm in a tio n s th a t a vo id th e co n sid e ra b le u n c e rta in ty associated
w ith th e g lo b u la r c lu s te r d is ta n c e scale (P a c z y n s k i. 1997).
T h o m p s o n et al.
(2001) wen* able to o b ta in a h ig h p re cisio n mass e s tim a te fo r the d etached d o u b le
lin e s p e ctro sco p ic b in a ry . O G L E G C -1 7 in x - C e n .
C sing th e a g e /tu r n o ff mass
re la tio n s h ip , th e K a lu z n y et al. (2002) a n a lysis o f th is system yie ld e d an age fo r th is
b in a ry o f 11.8 ± 0 . 0 G y r. C h a b o y e r
K ra uss (2002) re -a n a lysis o f th e a g e /tu r n o ff
mass re la tio n s h ip fo r th is system y ie ld s a s im ila r age e s tim a te :
1 1.1 i t 0.07 G y r.
T h e W M A P d e te rm in a tio n o f th e age o f th e u nive rse im p lie s th a t g lo b u la r c lu s te rs
fo rm w ith in 2 G y r a fte r th e B ig B an g , a reasonable e stim a te 1 th a t is co nsiste n t
w ith s tru c tu re 1 fo rm a tio n in the1 A C D M cosm ology. W hite- d w a r f e la tin g p ro vid e s an
a lte rn a tiv e 1 a p p ro a ch to the1 tra c litio n a l stuelie's o f the1 m a in sequence tu rn -o ff. R ic h e r
et a l. (2002) anel Hansem et al. (2002) fin d an age fo r the1 g lo b u la r c lu s te r M 4 o f
12.7 ± 0 .7 G y rs (2 n e rro rs. ± 0 .3 5 at the- 1 a level a ssu m in g G aussian e’rn>rs) using
th e w h ite elwarfs c o o lin g sequence m et heal. The-se re su lts, w h ich y ie ld an age close1
te> th e co sm o lo g ic a l age*, are p o te n tia lly ve ry usefu l: fu r th e r te>sts o f th e assum ptiem s
o f th e w h ite d w a rf age e la tin g m e th o d w ill c la r ify its s y s te m a tic u n c e rta in tie s .
O b s e rv a tio n s o f ne>arby h a lo s ta rs enable1 a s tro n o m e rs to o b ta in spe*ctra o f
v a rio u s ra d io -is o to p e s .
B y m e a s u rin g is o to p ic ra tio s , th e y in fe r s te lla r age*s th a t
are in d e p e n d e n t o f m uch o f th e physics th a t d e te rm in e s m a in sequence t u r n - o ff
(see T h ie le m a n n et a l. (2002) fo r a recent re v ie w ). The>se stu d ie s y ie ld s te lla r ages
c o n s is te n t w ith b o th th e g lo b u la r c lu s te r ages a n d th e ages in o u r best fit m odels.
C la y to n
(1988) u sin g a range o f ch e m ic a l e v o lu tio n m o d els fo r th e G a la x y fin d s
ages betw een 12 - 20 G y r. S chatz e t a l. (2002) s tu d y T h o r iu m a n d C ra n iu m iti CS
31082-001 a n d e s tim a te an age o f 15.5 ± 3.2 G y r fo r th e r-process e le m en ts in th e
s ta r. O th e r g ro u p s fin d s im ila r e stim a te s: th e C a y re l et a l. (2001) a n a ly s is o f U -238
in th e o ld h a lo s ta r CS 31082-001 y ie ld s an age o f 12.5 ± 3 G y r. w h ile H ill e t a l.
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C iiH p tr r 3:
81
C o s m o lo g ic a l P n n im c to rs
(2002) fin d an age o f 14.0 ± 2.4 G y r. S tu d io s o f o th e r o ld h a lo stars y ie ld s im ila r
e s tim a te s: C o w a n et al. (1 9 9 9 )tw o stars C'S 22892-052 and H D 11."3444 fin d 1-3.6 ± 4.G
G y r.
T a b le 3.6 su m m a rize s th e lo w e r lim it s on th e ago o f th e u niverse fro m various
a s tro n o m ic a l m easurem ents. W h ile the e rro rs on these* m e asurem ents re m a in to o
large* to e*ffeetively c o n s tra in p a ra m e te rs, th e y provide* an im p o rta n t co n siste n cy
ehe'ok on o u r basic co sm o log ica l m o d e l.
4.5.
Large S ca le S tru ctu re
The* large* sc ale* struc tu re o b s e rv a tio n s a n d the* L y m a n o
forest d a ta e30 ; V m e tit
the* C M B m e a surem e n ts by m e a su rin g s im ila r p h y s ic a l sc ales at ve ry d iffe re n t e*pochs.
The* W M A P a n g u la r p ow er s p e c tru m has th e sm a lle st u n c e rta in tie s near I ~ 300.
w h ic h corre'sponel to w avenum bers k
0.02 M p c " 1. W ith the* A C B A R re*sults. o u r
C M B d a ta set extenels to f ~ 1800. c o rre s p o n d in g to k ~ 0.1 M p c ' 1. I f we assume*
th a t g r a v ity is the* p rim a ry force d e te rm in in g th e large-sc ale d is tr ib u tio n o f m a tte r
and th a t g a la xie s tra ce mass at least on large* scales, th e n we* can d ir e c tly com pare
o u r best fit A C’ D M m o d el ( w ith p a ra m e te rs f it to th e \ \ AI A P d a ta ) to o b se rva tio n s
o f la rg e sc ale d is tr ib u tio n o f galaxie s. T h e re are c u r r e n tly tw o m a jo r o n g o in g large
scale s tru c tu re surveys: th e A n g lo - A u s tr a lia n Telescope tw o degree fie ld G a la x y
R e d s h ift S u rv e y (2 d F G R S ) (C o lless et a l.. 2 00 1 ). a n d th e S loan D ig ita l S ky S u rv e y '
(S D S S ). L a rg e sc ale s tru c tu re d a ta sets a re a p o w e rfu l to o l fo r b re a k in g m a n y o f the
p a ra m e te r degeneracies associated w ith C M B d a ta . In §3. we m ake e xte n sive use o f
th e 2 d F G R S d a ta set.
‘ w w w .sd ss.o rg
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p te r 3.
82
C o sm o lo g ica l P ara m e te rs
F ig u re 3.6 shows th a t th e A C D M m o d e l o b ta in e d fro m th e W M A P d a ta alo ne
is an a cce p ta b le fit to the 2 d F G R S pow er s p e c tru m .
T h e best fit has .i = 0.45
co n s iste n t w ith Peacock et al. (2001) m easured va lu e o f .1 - 0.43 ± 0.07.
T h e L y m a n a forest o b se rva tio n s are an im p o rta n t c o m p le m e n t to C M B
o b s e rv a tio n s since th e y p ro b e the lin e a r m a tte r p o w e r s p e c tru m at r = 2 — 3
(C r o ft et a l.. 1998. 2002). These o b se rva tio n s are se n sitive to s m a ll le n g th scales,
inaccessible to C M B e xp e rim e n ts.
U n fo rtu n a te ly , tin* re la tio n s h ip between the
m easured flu x p ow er s p e c tru m and th e lin e a r p o w e r s p e c tru m is c o m p le x (G n e d in
A: H a m ilto n . 2002: C ro ft et a l.. 2002) and needs to be c a lib ra te d by n u m e ric a l
s im u la tio n s . In V erde et al. (2003). we describe o u r m e th o d o lo g y fo r in c o rp o ra tin g
th e L y m a n o forest d a ta in to o u r lik e lih o o d a p p ro a ch .
F ig u re 3.0 com pares tin '
p re d ic te d pow er sp e ctra fo r th e best fit A C D M m o d e l to th e lin e a r p ow er s p e c tra
in fe rre d by G n e d in A- H a m ilto n (2002) a n ti by C ro ft et al. (2002).
4.6.
S u p ern ova D a ta
O v e r th e past decade. T y p e la supernovae have em erged as im p o rta n t
c o s m o lo g ic a l probes. O nce su pe rn o va lig h t curve s have been co rre cte d u sin g th e
c o rre la tio n betw een decline ra te and lu m in o s ity ( P h illip s . 1993: Riess et a l.. 1995)
th e y a p p e a r to be re m a rk a b ly g oo d s ta n d a rd candles. S y s te m a tic stu d ie s b y th e
s u p e rn o va co sm o lo g y p ro je c t (P e rlrn u tte r et a l.. 1999) a nd by th e h ig h ; su p e rn o va
search tea m (R iess et a l.. 1998) p ro v id e e vidence fo r an a c c e le ra tin g universe . T h e
c o m b in a tio n o f th e large scale s tru c tu re . C M B a n ti su pe rn o va d a ta p ro v id e s tro n g
e vid en ce fo r a fla t universe d o m in a te d by a c o s m o lo g ic a l c o n s ta n t (B a h c a ll e t a l..
1999). Since th e su pernova d a ta p ro be s th e lu m in o s ity d is ta n c e versus re d s h ift
re la tio n s h ip a t m o d e ra te re d s h ift c < 2 a nd th e C M B d a ta probes th e a n g u la r
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C h a p te r 3:
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C o s m o lo g ic a l P aram ete rs
d ia m e te r d is ta n c e re la tio n s h ip to h ig h re d s h ift (z
1089) . th e tw o d a ta sets are
c o m p le m e n ta ry . T h e supernova c o n s tra in t on c o s m o lo g ic a l p a ra m e te rs are co n siste n t
w ith th e A C D M W M A P m o d el. As we w ill see in th e d iscussion o f n on -H a t m odels
and quintessence m o dels, the S N Ia lik e lih o o d su rface in th e <>„, - f> \ and in the
1>r„ — ic planes p ro vid e s useful a d d itio n a l c o n s tra in ts on c o s m o lo g ica l p a ra m e te rs .
4.7 .
R eio n iza tio n & S m all S cale Pow er
T h e \ \ AI A P d e te c tio n o f re io n iz a tio n (K o g u t et a l.. 2003) im p lie s tin* existence
o f an e a rly g e n e ra tio n o f stars a ble to re io n ize th e I'n iv e rs e at c ~ 20. Is th is e a rly
s ta r fo rm a tio n <951
w ith o u r best fit A C D M co sm o lo g ica l m o d e l? W e can
e v a lu a te th is effect by first c o m p u tin g th e fra c tio n o f co llapsed o b je c ts .
at a
give n re d s h ift:
(3-2)
w here «t>(.\/. : ) is th e S heth <C T o rm e n (1999) mass fu n c tio n .
T h e firs t sta rs
c o rre sp o n d to e x tre m e ly rare flu c tu a tio n s o f th e o v e rd e n s ity fie ld :
Eq.
(3 -2 ) is
ve ry s e n sitive to th e ta il o f th e mass fu n c tio n . T h u s th e ve ry s m a ll change* in th e
m in im u m mass needed fo r s ta r fo rm a tio n re s u lts in a s ig n ific a n t change in the
fra c tio n o f co lla pse d o b je cts. T h e m in im u m h a lo mass fo r s ta r fo rm a tio n .
is c o n tro v e rs ia l and depends on w h e th e r m o le c u la r h yd ro g e n ( H j) is a v a ila b le as a
c o o la n t. I f th e gas te m p e ra tu re is fixe d to th e C M B te m p e ra tu re , th e n th e .lean
M ass. M J = 10**M . . I f m o le c u la r h yd ro g e n is a v a ila b le , th e n th e Jeans mass before
re io n iz a tio n is M J ~ 2.2 x 10-,[u.,6 ///( x ,rn) ] 1 ° ( 1
- ) / 1 0 fo r c < 150 (Y e n ka te sa n
et a l.. 2001). A t c > 150. the e le ctro n s are th e r m a lly co u p le d to th e C M B p h o to n s .
H ow ever, as H a iin a n et al. (1997) p o in t o u t. a s m a ll U Y b a c k g ro u n d g e n e ra te d by
th e firs t sources w ill d isso ciate H>. th u s m a k in g th e m in im u m mass m u ch la rg e r
th a n th e Jeans mass. T h e y suggest u sin g a m in im u m mass th a t is m uch h ig h e r:
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C h a p te r 3:
C o s m o lo g ic a l P a ra m e te rs
M " ^ 1-(~) =
l()'s( l •+■ c ) / 1 0 ) - '
84
O n th e o th e r han d i f th e firs t stars g en erated
a signific a nt flu x o f X -ra y s (O h . 2001) th e n th is w o u ld have p ro m o te d m o le c u la r
h y d ro g e n fo rm a tio n (H a im a n et a l.. 2000: V enkatesan et a l.. 2001: C’en. 2002). T h u s
lo w e rin g the* m in im u m mass back to M J.
F o llo w in g T e g m a rk A" S ilk (1995) we e stim a te 1 the* rate* o f r e io n i/a tio n by
m u lt ip ly in g the* co llapse fa c to r by an e ffic ie n c y fac to r.
the1 universe1.
fa lls in to the1 n o n -lin e a r struc ture's.
f h.
A frac tio n o f b a ryo n s in
W e assume1 f t , =
f p u
(i
.
c o n s ta n t b a ry o n /e la rk m a tte r r a tio ). A c e rta in fra c tio n o f tlmse1 b aryon s fo rm stars
o r epiasars.
f b um
■ w h ic h emiit I V
ra d ia tio n w ith som e c'ffiedency.
f i
\
■ Some1 o f th is
ra d ia tio n escapes in to the- in te rg a la c tic m c'diu m p h o to io n i/in g it : h o w e w r. the1 ne-t
n u m b e r o f io n i/a tio tis peT I V
p h o to n s .
f „ m .
is expec-te'd to be1 le>ss th a n u n ity (due1
to c o o lin g and ree-otnbinations). F in a lly the1 in terga la c t ic m e d iu m m ig h t be c lu m p y ,
m a k in g the1 p h o to io n iz a tio n proce'ss less e ffic ie n t. T h is e'ffe'ct is co u n te d fo r by the*
c lu m p in g fa c to r C ri,uni>. T h u s in th is a p p ro x im a tio n the* io n iz a tio n fra c tio n is giveui
1)\ :
S f
= -3.8
X
10
frirtfh
W
llC'TC1 J
nrt
—
f h u m f t ‘ V‘
/ C rtump'
TllC1 fa c to r >3.8
X
10
arises because 7.3 x 1 0 " 1 o f th e rest mass is released in th e b u rn in g o f h yd ro g e n to
h e liu m a n d we assum e th e p rim o rd ia l h e liu m mass fra c tio n to be 24% . W e assume1
/burn ^
fnrt
/ , , r ^ 30% .
f(
V
<£ 50% .
$ 90% . a ild 1 ^ C /un.p $
190. t flllS
^ 5.6 x 1 0 - '.
F ig u re 3.7 shows the1 fra c tio n o f co lla p se d o b je c ts a nd th e m a x im u m io n iz a tio n
fra c tio n as a fu n c tio n e>f re d s h ift fo r o u r best f it W M A P A C D M m o d e l.
s o lid lines co rre s p o n d to M , nin =
The
w h ile th e dashed lines co rre s p o n d to
-Mrnin = \ I J. T h e W A Z A P d e te c tio n o f re io n iz a tio n a t h ig h re d s h ift suggests th a t H>
c o o lin g lik e ly p layed an im p o r ta n t ro le in e a rly s ta r fo rm a tio n .
Because e a rly re io n iz a tio n recjuires th e e xisten ce o f s m a ll scale flu c tu a tio n s , th e
W’A /A P T E d e te c tio n has im p o r ta n t im p lic a tio n s fo r o u r u n d e rs ta n d in g o f th e n a tu re
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p te r 3:
85
C o s m o lo g ic a l P aram ete rs
o f th e d a rk m a tte r. B a rka n a et al. (2001) n o te th a t tlit* d e te c tio n o f re io n iz a tio n at
' > 10 rules o u t w a rm d a rk m a tte r as a v ia b le c a n d id a te fo r t in ’ m issin g mass as
s tru c tu re fo rm s ve ry la te in these m odels. W a rm d a rk m a tte r can not c lu s te r on
scales s m a lle r th a n th e d a rk m a tte r Jeans' mass. T h u s , th is lim it a pp lie s regardless
o f w h e th e r th e m in im u m mass is M , n a - o r M J.
5.
C om b in in g D a ta S ets
In th is se ctio n , we co m b ine tin* W M A P d a ta w ith o th e r C M B e x p e rim e n ts
th a t p ro b e s m a lle r a n g u la r scab’s (A C 'B A R and C 'B I) 1 a nd w ith a s tro n o m ic a l
m e a surem e n ts o f the p ow er s p e c tru m (th e 2dFCJRS and L y m a n n
fo re s t).
We
begin by e x p lo rin g how in c lu d in g these d a ta sets affects o u r best fit pow er law
A C D M m o d e l p a ra m e te rs ($5.1).
T h e a d d itio n o f d a ta sets th a t p ro be s m a lle r
scab’s s y s te m a tic a lly p u lls dow n th e a m p litu d e o f th e flu c tu a tio n s in th e best fit
m o d e l. T h is m o tiv a te s o u r e x p lo ra tio n o f an e xte n sio n o f th e p ow er la w m o d e l, a
m o d e l w here t in ’ p rim o rd ia l pow er s p e c tru m o f sca la r d e n s ity flu c tu a tio n s is fit by a
ru n n in g s p e c tra l in d e x (K o so w sky A- T u rn e r. 1995):
(3-3)
w here we fix th e sca la r s p e c tra l in d e x and slope a t A0 = 0 .0 5 M p c _ I. N o te th a t
th is d e fin itio n o f the ru n n in g in d e x m a tches th e d e fin itio n used in H a n n csta d et al.
(2002) a n a lysis o f ru n n in g s p e ctra l in d e x m o d els and d iffe rs by a fa c to r o f 2 fro m the
K o s o w sky A T u rn e r (1995) d e fin itio n . A s in th e scale in d e p e n d e n t case, we d efine
'I n th e fo llo w in g sections, we re fe r to th e co m b in e d U W /.4 P . A C B A R a n d C B I
d a ta sets as W M A P e x t.
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C h a p te r 3:
86
C o s m o lo g ic a l P aram eters
W e e x p lic it ly assum e th a t ip n s/ d l u k 1 = 0. so th a t
n ,( h ) = m ( h „ ) +
y In (
(I III A’
] .
y A’o )
(3- ' j )
In |jo.2. wo show th a t the ru n n in g s p o rtra l in d e x m o d e l is a b e tte r tit th a n the
pure p ow er la w m o d e l to the c o m b in a tio n o f W M A P and o th e r d a ta sets. P eiris
et al. (2003) e xp lo re s th e im p lic a tio n s o f th is ru n n in g s p e c tra l in d e x fo r in fla tio n .
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C lu ip tc r 3:
C o s m o lo g ic a l P a ra m e te rs
T a b le . } . ! .
Power Law A C D M M o d e l P a ra m e te rs-
P a ra m e te r
B a rv o n D e n s itv
87
\ P D a ta O n ly
M ean (G8 'Z eoiifide n ee ra tin e )
M a x im u m L ik e lih o o d
0.024 ± 0.001
0.1 1 ± 0 .0 2
0.72 ± 0.05
0.9 ± 0 . 1
0.02.5
M a tte r D e n s itv
H u l)b le C o n s ta n t
V-'Jiti
A m p litu d e
O p tic a l D e p th
S p e c tra l In d e x
7
o.icc:!J;3f?
0 .1 0
n>
0.99 ± 0.04
0.97
A
0.13
0 .G8
0.78
1 431/1342
aF it to H A /.4 P d a ta o n ly
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p te r 3:
C o s m o lo g ic a l P a ra m e te rs
T a b le 3.2.
88
D e rive d C o s m o lo g ic a l P aram ete rs
P a ra m e te r
M ean (GSVt confidence range)
A m p litu d e o f G a la x y F lu c tu a tio n s
(7* =
C h a ra c te ris tic A m p litu d e o f V e lo c ity F lu c tu a tio n s
B a ry o n D e n s ity /C r itic a l D e n s ity
M a tte r D e n s ity /C r itic a l D e n s ity
A ge o f th e U n ive rse
rrAA",!' = ( ) - 4 4 ±
n h = 0.047 ± 0.00G
<>„, = 0.29 ± 0.07
R e d s h ift o f R e io n iz a t io n 1’
R e d s h ift at D e c o u p lin g
A ge o f tin* U n ive rse at D e c o u p lin g
T h ic k n e ss o f S urfa ce o f Last S c a tte r
T h ic k n e s s o f S urfa ce o f Last S c a tte r
0 .0
±
0.1
t» = 13.4 ± 0.3 G y r
cr = 17 ± 3
:,lr, = 1 0 8 8 :!
t,u,- = 372 ± 14 k y r
A ;,/,, = 194 ± 2
R e d s h ift a t M a tte r /R a d ia tio n E q u a lity
S o u n d H o riz o n at D e c o u p lin g
A n g u la r D ia m e te r D ista n ce to th e D e c o u p lin g S urface
A c o u s tic A n g u la r Scale1-
At,I,-, = 113 ± 3 k y r
4 4 3 4 ~ ,v'
-r(/ — oaoa..|c|_»
r \ = 144 ± 4 M p c
(1 \ = 13.7 ± 0.3 G p c
C u rre n t D e n s ity o f B a ryo n s
f A = 299 ± 2
tti, = (2.7 ± 0 . 1 ) x
B a r y o n /P h o to n R a tio
>1 = (6 .3 :|j;;{) x
F it to th e W M A P d a ta o n ly
bA ssum es io n iz a tio n fra c tio n . x r =
1
r I a = ~dr [ r s
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1 0
10
- '"
~‘ r t iC
1
C h a p te r 3:
80
C o s m o lo g ic a l P a ra m e te rs
T a b le 3..'5.
M e th o d
H u b b le K e y P ro je c t
S Z E + X -ra v
R e m it H u b b le C o n s ta n t D e te rm in a tio n s
M ean (GSM co n fid e n ce range)
72 ± 3 ± 7
g o ± 4 :1 ,!
gg : |
W M A P P L A C D M m o d el
[' ± 13
72 ± 3
Reference
Freedm an et al. ( 2 0
Reese et al. (2002)
M ason et al. (2001)
§3
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0 1
]
C h a p te r 3:
C o s m o lo g ic a l P a ra m e te rs
T a b le 3.4.
M e th o d
90
A m p litu d e o f F lu c tu a tio n s . rrs
M ean and G8 (Z
Reference
confidence range 1
PL ACDM + W M A P
W eak L e a sin g '1'1’
0.9 ± 0.1
0.72 ± 0.18
o.sG
n.G 9:;i;i;i
0.90 ± 0.12
0.92 ± 0.2
B ro w n et a l. (2002)
H o e k s tra et al. (2002)
J a rv is et a l. (2002)
Bacon et a l. (2002)
R e fre g ie r et al. (2002)
V an W’aerbeke et al. (2002b)
G a la x v V e lo c itv F ie ld s h
0.98 ± 0.12
0.73 ± 0.1
C B I SZ d e te c tio n
H ig h re d s h ift d u s te rs 1’
1.04 ± 0 .1 2‘
W 'illic k A S trauss (1998)
K o m a ts u A S eljak (2002)
0.93 ±
B a h c a ll A’ B ode (2002)
0.1
“ Since m ost weak le a sin g pap e rs re p o rt Oo'X co nfid e n ce lim it s in th e ir papers, the
ta b le lis ts th e 95 (/ co n fid e n ce lim it fo r these e x p e rim e n ts .
hA ll o f th e (th m e a surem e n ts have been n o rm a liz e d to U m = 0.287. th e best fit value
fo r a fit to th e W M A P d a ta o n ly.
r 95cX co nfid e nce lim it
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p te r 3:
C o s m o lo g ic a l P a ra m e te rs
T a b le 3.5.
91
M easured ra t it) o f D e u te riu m to H yd ro g e n
Reference
Q uasar
[D ]/[H ]
QO 1.30-403
P K S 1937-1009
Q 1009+299
H S 0105+1G 19
Q220G -199
Q0.34 7-383
< 0 . 8 x 1 0 -"’
3.25 ± 0.3 x II)--"'
Q 1 2 3 4 + 30 4 7
2 .4 2 :1 {;^ * H >"‘‘
4.0 ± 0.05 x 1 0 "'
2.5 + 0.25 x K ) - ' 1
1.05 ± 0.35 x 10 ’’
3.75 ± 0.25 x 10 '*
T a b le 3.0.
K irk m a n et al. ( 2
0 0 0
)
B u rie s A: T y t le r (1998a)
B urie s A: T y t le r (1998b)
O ’ M eara et al. ( 2 0 0 1 )
P e ttitii X: Bowen (2001)
L evsh a kov et al. (2003)
K irk m a n et al. (2003)
C’o sm ie Age
A ge
M e th o d
W M A P d a ta (A C D M )
\ V M A P e x t + LSS
13.4 ± 0.3 G y r
13.7 ± 0 .2 G v r
G lo b u la r C lu s te r Ages
W h ite D w a rf
> 1 1
O G L E G C -1 7
R a d io a c tiv e d a tin g
— 10 G vr
> 12.7 ± 0.7 G v r
> 10.4 - 12.8 G v r
> 9.5 — 2 0 G y r
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p te r 3:
C o s m o lo g ic a l P a ra m e te rs
92
t. rr
re
Xf - —
‘
—;
7“
1 'T
'
-H
*M
r-
r-
-H
-H —
34
zn
Tl
4-
c: = ac
-ft -H
1-
Zl
is i)
3C s - i -
M ?£
^ t ’is
I.“7
-H —
*
2
-H V
-H f ^
-H -H
~
X
r.
—
—*
2J
-H
4
3
-H -H
M
r-
0
—
-rl
+
—
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p te r 3:
93
C o s m o lo g ic a l P a ra m e te rs
5 .1 .
P ow er Law A C D M M o d e l
T h e p ow er la w AC’ D M m o d el is an a cce p ta b le fit to rhr* W M A P d a ta . W h ile
it o v e rp re d ic ts th e a m p litu d e o f flu c tu a tio n s on large a n g u la r scales (see jjG). th is
d e v ia tio n m a y be due to cosm ic va riance at these la rg e scales. In tr ig u in g ly . it also
o v e rp re d ic ts th e a m p litu d e o f H u ctu a tio n s on s m a ll a n g u la r scales.
T a b le (3 .7 ) shows rh«* best fit p aram ete rs fo r th e p o w e r la w A C D M m o d el fo r
d iffe re n t c o m b in a tio n o f d a ta sots. As we add m ore a n d m o re d a ta on s m a lle r scales,
tin* best fit value fo r th e a m p litu d e o f flu c tu a tio n s at k = 0.05 M p c
d ro ps: W hen we fit to th e W M A P d a ta alone, the best fit is 0.9 ±
th e C B l. A C B A R a nd 2 d F G R S d a ta , the best fit va lu e d ro p s to
0
1
g ra d u a lly
. 1 . W hen we add
0 .8
± 0.1. A d d in g
th e L y m a n o d a ta fu r th e r reduces .1 to O.TotoJl*. T in * best fit s p e c tra l in d e x shows
a s im ila r tre n d : th e a d d itio n o f m ore and m ore s m a ll scale d a ta d riv e s th e best fit
s p e c tra l in d e x to also change by n e a rly 1 o fro m its best fit va lu e fo r W M A P d a ta
o n ly : 0.99 ± 0.04 ( W M A P o n ly ) to 0.96 ±
0 .0 2
( U 'A M Pox t + 2d FG R S + L y o ) • W h e n
th e a d d itio n o f new d a ta c o n tin u o u s ly p u lls a m o d e l aw ay fro m its best fit value, th is
is o fte n th e s ig n a tu re o f th e m o d e l re q u irin g a new p a ra m e te r.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p te r 3.
C o s m o lo g ic a l P a ra m e te rs
94
zC
"f" |
IT. I
T'l
*
i
8
X.
X.
zC
-h
-H := 2
-h -H
X cn d
~ 7Z
C C X
-H -H —
*M
'M
X
X
•M
CM
^
r - urt 'M
5 feS= - H ________
-H I
5
; r -H -H -H i i ^
X. "t: i-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p te r 3:
C o s m o lo g ic a l P a ra m e te rs
5.2.
95
R u n n in g S p ectral In d ex A C D M M od el
In fla tio n a ry m o d e ls p re d ic t th a t th e s p e c tra l in d e x o f flu c tu a tio n s sh o u ld be
a s lo w ly v a ry in g fu n c tio n o f scale.
P eiris et al. (2003) discusses the in fla tio n a ry
p re d ic tio n s a n d shows th a t a p la u s ib le set o f m o d els p re d ic ts a d e te c ta b le v a ry in g
s p e c tra l in d e x. T h e re are classes o f in fla tio n a ry m o d e ls th a t p re d ic t m in im a l ten so r
m odes. T h is se ctio n e xplore s th is class o f m odels. In *jG.4. we e xp lo re a m ore general
m o d e l th a t has b o th a ru n n in g s p e c tra l in d e x and te n so r m odes.
T a b le 5.3 shows th e best fit p a ra m e te rs fo r th e ru n n in g s p e c tra l in d e x m odel
as a fu n c tio n o f d a ta set. N o te th a t th e best fit p a ra m e te rs fo r these m odels b a re ly
change as we add new d a ta sets: how ever, tin* e rro r bars s h rin k . W h e n we in clu d e
a ll d a ta sets, th e best fit value o f th e ru n n in g o f th e s p e c tra l in d e x is —0.031 iJJJJ|V:
few er th a n 5 ''/ o f tin * m o dels have (his/ ( l \ u k >
0
.
F ig u re 3.9 show s th e th e p ow er s p e c tru m as a fu n c tio n o f scale. T in * fig ure
show s th e re s u lts o f o u r M a rk o v ch ain a n a lysis o f th e c o m b in a tio n o f W M A P . C B I.
A C 'B A R . 2 d F G R S and L y m a n o d a ta . A t each w a ve n u m b e r. we c o m p u te the range
o f values fo r th e p o w e r law in d e x fo r a ll o f th e p o in ts in tin * M a rk o v ch a in . T h e
G8 rA and 95 rA c o n to u rs a t each k va lu e are show n in F ig u re 3.9 fo r th e fit to th e
\V M A P e x t-t-2 d F G R S + L y m a n n d a ta sets.
O v e r th e c o rn in g ye ar, new d a ta w ill s ig n ific a n tly im p ro v e o u r a b ilit y to m easure
(o r c o n s tra in ) th is ru n n in g s p e c tra l in d e x. W h e n we c o m p le te o u r a n a ly s is o f the
E E p ow er s p e c tru m , th e W M A P d a ta w ill place s tro n g e r c o n s tra in ts on r . Because
o f th e ns — r degeneracy, th is im p lie s a s tro n g c o n s tra in t on n„ on la rg e scales. T h e
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C h a p te r 3:
96
C o s m o lo g ic a l P a ra m e te rs
SDSS c o lla b o ra tio n w ill soon release its
g a la x y
s p e c tru m and its m easurem ents o f
th e L y m a n o forest. These o b se rva tio n s w ill s ig n ific a n tly im prove' o u r m easurem ents
o f n y on s m a ll scale's. Pe'iris et al. (2003) shows th a t the* ele'te'ctiem o f a ru n n in g
s p e v tra l in d e x a n d p a r tic u la r ly the' d e te c tio n e»f a spe'ctral inelex th a t varie's frejni
n, >
1
on large* scales to
<
1
on sm a ll scale's w o u ld severe*ly c o n s tra in in fla tio n a ry
mode'ls.
The' ru n n in g spe'ctral in d e x m o d el p re d ic ts a s ig n ific a n tly h>wer a m p litu d e o f
flu c tu a tio n s em s m a ll scale's th a n the* s ta n d a rd A C D M mode'l (se*e figure* 3.9). T h is
suppressiem o f s m a ll sc ale* p o w t'r has several im p o rta n t a s tro n o m ic a l im p lic a tio n s :
(a) the* re d u c tio n in s m a ll scale* powe*r make's it mote* d iffic u lt to reionize 1 th e unive'rse
unless H> c o o lin g enable's mass d a rk halos to collapse* and fo rm galaxie*s (se*e> ((3.7
and Figure' 3 .1 0 ): (b ) a re d u c tio n in the s m a ll scale' p ow er redue-e's the* a m o u n t o f
s u b s tru c tu re w it h in g a la c tic halos (Zem tner Aj B u llo c k .
2 0 0 2
) (c) since s m a ll obje'cts
fo rm la te r, th e ir elark m a tte r halos w ill be less c o n c e n tra te d as rhe're is a m o n o to n ic
re la tio n s h ip be'twe'em collapse tim e' and h a lo c e n tra l c o n c e n tra tio n (N a v a rro e*t al..
1997: Eke et a l.. 2001: Z e n fn e r <k: B u llo c k . 2002: We'chsle'r e>t a l., 2002: H u ffe n b e rg e r
S e lja k. 2003).
T h e re d u c tio n in th e a m o u n t o f s u b s tru c tu re w ill also reduce
a n g u la r m o m e n tu m tra n s p o rt betw een d a rk m a tte r a n d b a ryo n s and w ill also reduce
th e ra te o f d is k d e s tru c tio n th ro u g h in fa ll (T o th .k: O s trik e r. 1992). W e suspect th a t
o u r proposer! m o d ific a tio n o f th e p rim o rd ia l pow er s p e c tru m w ill resolve m a n y o f
th e lo n g -s ta n d in g p ro b le m s o f th e C D M m o d e l on s m a ll scale's (see M o o re (1994)
and Spergel
S te in h a rd t (2000) fo r discussions o f th e fa ilin g s o f th e p ow er law A
C D M m o d e l on g a la x y scales).
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C h a p te r 3:
C o s m o lo g ic a l P a ra m e te rs
6.
97
B eyon d th e A C D M M o d el
In th is se ctio n . \vc co n sid e r va rio u s e xte n sio n s to th e A C 'D M m o d e l. In (jG.l. we
co n s id e r d a r k ene rg y m odels w ith a co n sta n t e q u a tio n o f s ta te . In
n o n -fla t m o dels. In
?}G .3.
(jG .2 .
we consider
we co n sid e r m odels w ith a m assive lig h t n e u trin o . In ^ G .4.
we in c lu d e te n so r m odes.
In th is se ction o f th e p a p e r, we co m b in e th e U '.W A P d a ta w ith e x te rn a l d a ta
sets so th a t we can b re a k degeneracies and o b ta in s ig n ific a n t c o n s tra in ts on the
va rio u s e xte n sio n s o f o u r s ta n d a rd co sm o lo g ica l m o d e l.
6.1.
D ark E n ergy
T h e p ro p e rtie s o f th e d a rk energy, th e d o m in a n t c o m p o n e n t in o u r universe
to d a y , is a m y s te ry .
is quintessence.
T h e m ost p o p u la r a lte rn a tiv e to tin ' co sm o lo g ic a l co n sta n t
W e tte ric h
(1 9 88 ). R a tra A: Peebles (1 9 8 8 ) a n d Peebles A* R a tra
(1988) suggest th a t a ro llin g sca la r fie ld c o u ld p ro d u c e a tim e -v a ria b le d a rk energy
te rm , w h ic h leave a c h a ra c te ris tic im p r in t on th e C M B and on large scale s tru c tu re
(C a ld w e ll et a l.. 1998). In these quintessence m o d e ls, th e d a r k energy p ro p e rtie s are
q u a n tifie d b y th e e q u a tio n o f s ta te o f th e d a rk e ne rg y: ir = p / p . w here p and p are
th e pressure a n d th e d e n s ity o f the d a rk energy. A c o s m o lo g ic a l c o n s ta n t has an
e q u a tio n o f s ta te , tr = — 1 .
S ince th e space o f p ossible m odels is q u ite la rg e, we o n ly co n sid e r m o dels w ith
a c o n s ta n t e q u a tio n o f sta te . W e now increase o u r m o d e l space so th a t we have 7
p a ra m e te rs in th e c o s m o lo g ic a l m o d el (.4. nx. h. D rn. f>ft. r a n d tr). W e a n a lyze the
d a ta u sin g tw o approaches: (a ) we b eg in by r e s tr ic tin g o u r a n a ly s is to tr > —1
m o tiv a te d b y th e d iffic u ltie s in c o n s tru c tin g s ta b le m o d e ls w it h tr < —I (C a rro ll
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C h a p te r 3:
98
C o s m o lo g ic a l P a ra m e te rs
et a l.. '2003) a n d (b ) re la x th is c o n s tra in t and co n sid e r m o d e ls th a t v io la te the weak
ene rg y c o n d itio n (S clm ecker et al. . 2003b). F u rth e r a n a ly s is is needed fo r m odels
when* ic a n d th e quinte ssen ce sound speed are a fu n c tio n o f tim e (D edeo et al..
2003). T h e a d d itio n o f a new p a ra m e te r in tro d u c e s a new degeneracy between U r„ .
h. and tr th a t can not bo b ro ke n b y C M B d a ta alone (H u e y et a l.. 1999: Verde et a l..
2003): m o d els w ith th e same values o f
QbhJ and firs t peak p o s itio n have
n e a rly id e n tic a l a n g u la r p o w e r sp ectra .
For example*, a m o d e l w ith V.„, = 0.-17. tr = —1 /2 a n d it = 0.57 has a n e a rly
id e n tic a l a n g u la r p ow er s p e c tru m to o u r A C D M m o d e l. N o te , how ever, th a t th is
H u b b le C o n s ta n t value' d iffe rs by
2
rr fro m the 1 H S T K e y Projec t value 1 and the1
p re d ic te d shape* o f the* powe>r s p e c tru m is a p o o r fit to the* 2 d F G R S o bse rva tio ns.
T h is mealed is also a worse 1 fit to th e supe rn o va a n g u la r e lia m e te r distance* re la tio n .
We* c o n s id e r femr eliffe*re*nt c o m b in a tio n s o f astrem em iical elata sets:
(a)
W M A P e *xt elata co m b in e d w ith the* supe rn o va o b se rva tio n s: (b ) W M A P e *xt d a ta
e-emibine'd w ith H S T elata: (c) W M A P e *xt elata ce)mbinc*el w ith the* 2 d F G R S large
scale s tru c tu re elata: (el) a ll elata sets co m b in e d .
T h e C M B peak p o s itio n s c o n s tra in th e c o n fo rm a l d is ta n c e to th e ele'cempling
surface.
T h e a m p litu d e o f th e e a rly IS W s ig n a l d e te rm in e s th e m a tte r elensitv.
T h e c o m b in a tio n e)f the*se tw o me*asurements s tro n g ly c o n s tra in s Q ( t r ) and
h( t r ) (see Figure's 3.11 a n d 3 .1 2 ). T h e H S T K e y P ro je c t me*asurement o f Ho agree*s
w ith th e in fe rre d C M B value* i f tr = —1 . A s tr increases, th e best fit H 0 value
fo r th e C M B d ro p s b e lo w th e K t*y P ro je c t value. O u r jo in t a n a ly s is o f C M B -F
H S T K e y P ro je c t d a ta im p lie s th a t tr < —0.5 (959c co n fid e n ce in te rv a l). I f fu tu re
o b s e rv a tio n s can reduce th e u n c e rta in tie s associated w ith th e d is ta n c e to th e L M C .
th e Ho m e a su re m e n ts c o u ld place s ig n ific a n tly s tro n g e r lim it s on tr. F ig u re s 3.11 and
3.12 show th a t th e c o m b in a tio n o f e ith e r C M B + s u p e rn o v a d a ta o r C .M B + la rg e scale
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C h a p te r -3:
99
C o s m o lo g ic a l P a ra m e te rs
s tru c tu re d a ta place s im ila r lim its o il d a rk energy p ro p e rtie s . For o u r c o m b in e d d a ta
set. we m a rg in a liz e over a ll o th e r p a ra m e te rs and fin d t h a t tr < - 0 . 7 8 (9•"//< C’ L )
w ln 'ii we impose' t in 1 p r io r th a t tr > — I.
I f we d ro p th is p rio r, th e n a ll o f th e
co m b in e d d a ta sets a p p e a r to favo r a m o d el w here the p ro p e rtie s o f th e d a rk energy
are close to th e p re d ic te d p ro p e rtie s o f a co sm o lo g ica l c o n s ta n t (tr = —0.98 ± 0 .12).
6.2.
N o n -F la t M od els
T h e p o s itio n o f tin ' firs t peak c o n s tra in s th e u nive rse to be n e a rly fla t
(K a m io n k o w s k i et a l.. 1994): lo w d e n s ity m odels w ith V.\ — 0 have th e ir first peak
p o s itio n at /20011 “ , 1 ' . How ever, i f we a llo w fo r th e p o s s ib ility th a t th e u niverse is
non-H at and th e re is a co sm o lo g ica l c o n s ta n t, th e n the n * is a g e o m e tric degeneracy
(E fs ta th io u .C B o n d . 1999): a lo n g a lin e in <!,„ w ith n e a rly id e n tic a l a n g u la r pow er sp ectra .
space, there' is a set o f m odels
W h ile th e a llo w e d range o f
is
re la tiv e ly s m a ll, th e re is a w id e range in 1>„, values c o m p a tib le w ith th e C M B d a ta
in a non-H at universe.
I f we place no p rio rs on co sm o lo g ica l p a ra m e te rs, th e n th e re is a m o d el w ith
nA =
0
co n siste n t w ith th e W .M .4P d a ta ( A \ * =
6 .6
re la tiv e to th e Hat m o d e l).
H ow ever, th e c o s m o lo g ic a l p a ra m e te rs fo r th is m o d e l ( H n = 32.5 k m / s / M p c . and
Qi„t = 1.28) are v io le n tly in co n siste n t w ith a host o f a s tro n o m ic a l m easurem ents.
T h e fla t Q m = l . A = 0 s ta n d a rd C D M m o d el is in c o n s is te n t w ith th e W M A P d a ta
a t m o re th a n th e 5a level.
I f we in c lu d e a w eak p r io r on th e H u b b le C o n s ta n t. H 0 > 5 0 k m /s /M p c .
th e n th is is s u ffic ie n t to c o n s tra in 0.98 < Q tot < 1 08 (95% co n fid e n ce in te rv a l).
C o m b in in g th e W M A P e x td a ta w ith supe rn o va m e a su re m e n ts o f th e a n g u la r
d ia m e te r d is ta n c e re la tio n s h ip (see fig u re 3.13) we o b ta in 0.98 < Q tot < 1 0 6 . T h is
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C h a p te r 3:
C o s m o lo g ic a l P a ra m e te rs
100
co nfid e nce in te rv a l does n o t re q u ire a p r io r on h.
I f we fu r th e r in c lu d e th e H S T
K e y P ro je c t m e a surem e n t o f H t) as a p rio r, then th e lim it s on O 0 im p ro v e s lig h tly :
i},„t < 1.02 ± 0.02 F ig u re 3.13 shows th e tw o d im e n s io n a l lik e lih o o d su rface fo r
v a rio u s c o m b in a tio n s o f th e d a ta .
6 .3 .
M assive N eu trin o s
C o p io u s n u m b e rs o f n e u trin o s wen* p ro du ce d in th e e a rly universe.
I f these
n e u trin o s have ru m -n e g lig ib le mass th e y can m ake a n o n - tr iv ia l c o n tr ib u tio n to th e
to ta l e nergy d e n s ity o f th e universe d u r in g b o th m a tte r a nd r a d ia tio n d o m in a tio n .
D u rin g m a tte r d o m in a tio n , th e m assive n e u trin o s c lu s te r o ti v e ry large scales b u t
fre e -stre a m o u t o f s m a lle r scale flu c tu a tio n s . T h is fre e -s tre a m in g changes th e shape
o f th e m a tte r p ow er s p e c tru m (H u et a l.. 1998) a nd m ost im p o r ta n tly , suppresses
th e a m p litu d e o f flu c tu a tio n s . Since we can n o rm a liz e th e a m p litu d e o f flu c tu a tio n s
to th e W M A P d a ta , th e a m p litu d e o f flu c tu a tio n s in th e 2dFC IR S d a ta places
s ig n ific a n t lim it s on n e u trin o p ro p e rtie s .
T h e c o n tr ib u tio n o f n e u trin o s to th e e nergy d e n s ity o f th e u n ive rse depends
u p o n th e sum o f th e mass o f th e lig h t n e u trin o species:
=
s
S
p t
(3- ° '
N o te th a t th e sum o n ly in clu d e s n e u trin o species lig h t e no u gh to d e co u p le w h ile s t ill
re la tiv is t ic.
E x p e rim e n ts th a t p ro b e n e u trin o p ro p a g a tio n fro m source to d e te c to r are
s e n s itiv e n o t to th e n e u trin o mass b u t to th e square mass d iffe re n c e betw een d iffe re n t
n e u trin o mass e ig in s ta te s .
S o la r n e u trin o e x p e rim e n ts (B a h c a ll et a l.. 2003a)
im p ly th a t th e square mass d iffe re n c e betw een th e e le c tro n a n d m u o n n e u trin o is
~ 7 x 1 0 " ’ e V T T h e d e fic it o f m u o n n e u trin o s in a tm o s p h e ric show ers im p ly th a t
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C h a p te r 3:
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th e mass d iffe re n ce betw een m u o n and ta ti n e u trin o s is
1 0
“ * e \ ( K e a r n s . '2002).
I f th e e le ctro n n e u trin o is m uch lig h te r th a n th e ra n n e u trin o , th e n th e c o m b in a tio n
o f these re su lts im p ly th a t m
<
0.1
eY : s t ill below th e d e te c tio n lim its fo r o u r
d a ta -s e t. O n th e o th e r h a n d , i f m u, ~ m „_ . th e n th e th re e n e u trin o species can leave
an o bservable im p r in t on th e C M B a n g u la r p ow er s p e c tru m and th e g a la x y large
scale s tru c tu re p ow er s p e c tru m .
In o u r a n a lysis, we co n sid e r th is la tte r case and
assum e th a t th e re arc* th re e d eg enerate s ta b le lig h t n e u trin o species.
F ig u re 3.14 shows th e c u m u la tiv e lik e lih o o d o f th e c o m b in a tio n o f W M A P . C'BI.
AC’ B A R . and 2 d F G R S d a ta as a fu n c tio n o f th e energy d e n s ity in n e u trin o s . Based
on th is a n a lysis, we c o n c lu d e th a t 12,,/r < 0.0067 ( 9 5 '/ co nfid e nce lim it ) . I f we add
th e L y m a n o d a ta , th e n flu* lim it s lig h tly weakens to i l tJ r < 0.0076.
For th re e
degenerate n e u trin o species, th is im p lie s th a t rnu < 0.23 eV . T h is lim it is ro u g h ly a
fa c to r o f tw o im p ro v e m e n t over p re v io u s analyses (e.g.. E lg a ro y et al. (2 0 0 2 )) th a t
h ad to assume s tro n g p rio rs on il„, a nd H {).
T a b le 3.9.
p rio r
no p r io r
d n s/ d In k =
n, < 1
0
9 5 'A C o n fid e n ce L im its on T e n s o r/S c a la r R a tio
WMAP
\\M A P e x t- t- 2 d F G R S
W M A P e x t-F 2 d F G R S +
Lym an o
1.28
0.81
0 .47
1.14
0.53
0.37
0.90
0.43
0.29
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C h a p te r 3:
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102
6.4.
T ensors
M a n y m odels o f in fla tio n predic t a s ig n ific a n t g ra v ity wave* b a c kg ro u n d . These
ten so r flu c tu a tio n s were1 g e n e ra te d d u rin g in fla tio n .
T ensor flu c tu a tio n s have
th e ir largest effects on large' a n g u la r scale's \vhe>re th e y add in q u a d ra tu re to the
flu c tu a tio n s gene'rate'd by sca la r mode's.
He>re'. we place* lim its on the* amplituele* o f teuisor moele*s. We de*fine* th e tensor
am plitude* u sin g the* same* c o n v e n tio n as Leach e*t al. (2002):
P !rri.w/r(A.)
r ~
_
PsrulaAk.)'
whe*re* Pt,„-.„r etnel P^nhir fire the* p rim o rd ia l am plituele* o f te*nse>r and sca la r
H u c tn a tio n s and A-. = 0.002 M p c ' 1. Since* we* se*c* no e*vide*nce fo r teuisor mode's in
o u r fit. we* s im p lify the* a n a ly s is by a ssum ing th a t the* teuisor spe'ctral inde*x satisfies
the* sin g le fielel in fla tio n a ry e-onsistency c o n d itio n :
n, = - r /
8
.
(3-8)
T h is c o n s tra in t reduces th e n u m b e r o f p a ra m e te rs in th is m o d el te)
8
: .4. lh , / r .
h. ns. d n s/ d \ n k . r a n d r. W e ig n o re th e ru n n in g o f n f. T h e a d d itio n o f th is
new p a ra m e te r does n o t im p ro v e
th e fit as fig u re (3 .15 ) shows th e c o m b in a tio n o f
W M A P e x t-f- 2 d F G R S -I- L y m a n o is able to place a lim it on th e te n so r amplituele*:
r < 0.90 (9 ~CA confidence lim it ) .
A s ta b le (3.9) shows, th is lim it is m uch m ore
s trin g e n t i f we re s tric t th e p a ra m e te r space to m o d e ls
\ d n / d In A:| =
0
w ith e ith e r
iis
<
1
or
.
P e iris et a l. (2003) discuss th e im p lic a tio n s o f o u r lim its on te n so r a m p litu d e fo r
in fla tio n a ry scenarios. U s in g th e re s u lts o f th is a n a lysis. P e iris et a l. (2003) shows
th a t the in fe rre d jo in t lik e lih o o d o f ns. d n s/ d \ n k a n d r places s ig n ific a n t c o n s tra in ts
on in fla t io n a rv m odels.
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C h a p te r 3:
C o s m o lo g ic a l P a ra m e te rs
7.
103
In trigu in g D iscrep an cies
W h ile the A C D M m o d e l's success in f it t in g C M B d a ta and a host o f o th e r
a s tro n o m ic a l d a ta is t r u ly re m a rk a b le , th e re re m a in a p a ir o f in tr ig u in g discrepancies:
on b o th the largest and sm a lle st scales.
W h ile a d d in g a ru n n in g s p e c tra l in d ex
m ay resolve p ro b le m s on s m a ll scales, th e re re m a in s a possible d is c re p a n c y between
p re d ic tio n s and o b se rva tio n s on th e largest a n g u la r scales.
F ig u re 3.1G shows the m easured a n g u la r p ow er s p e c tru m a nd the p re d ic tio n s o f
o u r best fit A —C D M m o d el, w here tin* d a ta were fit to b o th C M B and large-scale
s tru c tu re d a ta . T h e fig u re also shows th e m easured a n g u la r c o rre la tio n fu n c tio n : the
lack o f any c o rre la te d sig n a l on a n g u la r scales g re a te r th a n GO degrees is n o te w o rth y .
W e q u a n tify th is lack o f p ow er on large scales by m e a su rin g a fo u r p o in t s ta tis tic :
(3-9)
T h e u p p e r c u to ff and th e fo rm o f th is s ta tis tic were b o th d e te rm in e d a po.st.eori
in response to the shape o f th e c o rre la tio n fu n c tio n .
We e va lu a te th e s ta tis tic a l
sig n ific a n ce o f these d iscre pa n cie s by d o in g M o n te -C a rlo re a liz a tio n s o f th e firs t
1 00 .00 0
m odels in th e M a rk o v chains. T h is a llo w s us to average n ot o n ly o ve r cosm ic
v a ria n ce b u t also o ver o u r u n c e rta in tie s in co sm o lo g ica l p a ra m e te rs. F o r o u r A C D M
M a rk o v chains ( fit to th e W M A P c x t + 2 d F G R S d a ta sets), we fin d th a t o n ly 0.7% o f
th e m odels have lo w e r values fo r th e q u a d ru p o le a n d o n ly 0.15% o f th e s im u la tio n s
have low er values o f 5 . For th e ru n n in g m o d e l, we fin d th a t o n ly 0 .9 ‘X o f th e m odels
have low er values fo r th e q u a d ru p o le a n d o n ly 0.3% o f th e s im u la tio n s have low er
values o f S. T h e shape o f th e a n g u la r c o rre la tio n fu n c tio n is c e rta in ly unu su al fo r
re a liz a tio n s o f th is m o d el.
Is th is d is c re p a n c y m e a n in g fu l? T h e lo w q u a d ru p o le was a lre a d y c le a rly seen
in C’O B E and was u s u a lly dism isse d as due to co sm ic va ria n ce (B o n d et a l.. 1998)
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C h a p te r 3:
C o s m o lo g ic a l P a ra m e te rs
104
o r fo re g ro u n d c o n ta m in a tio n . W h ile th e W M A P d a ta re in fo rce s th e case fo r its low
value, cosm ic va ria nce is s ig n ific a n t on these large a n g u la r scales and a n y G aussian
fie ld w ill a lw a ys have u n u su a l fea tu re s. O n th e o th e r han d , th is d iscre p a n cy co u ld
he th e s ig n a tu re o f in te re s tin g new physics.
T h e d isco ve ry o f an a c c e le ra tin g u niverse im p lie s th a t at these la rg e scale's,
th e re is new and not u n d e rs to o d physics. T h is now physics is u s u a lly in te rp re te d
to he d a rk ene rg y o r a co sm o lo g ic a l c o n s ta n t. In c ith e r case, we w o u ld e xpect th a t
th e decay o f flu c tu a tio n s at late 1 tim e s produce's a s ig n ific a n t IS W s ig n a l. B o u g h n
et al. (1998) a rgue th a t in a AC’ D M m o d e l w ith <>„, =
0
.2 o. the re sh o u ld he a
d e te c ta b le c o rre la tio n betw een th e C’ M B sig n a l and tra ce rs o f large-scale s tru c tu re :
yet th e y wen* n ot ahh* to d e te ct a sig n a l. T h e re are' a lte rn a tiv e e x p la n a tio n s o f the
a c c e le ra tin g universe, such as e x tr a d im e n s io n a l g ra v ity th e o rie s (D e ffa ye t et al..
2 0 0 2
) th a t do n o t re q u ire a co sm o lo g ic a l c o n s ta n t and sh o u ld m ake ra d ic a lly d iffe re n t
p re d ic tio n s fo r th e C M B on these a n g u la r scales. These p re d ic tio n s have n o t yet
been c a lc u la te d .
W h a t c o u ld generate th is u n u su a l shaped a n g u la r c o rre la tio n fu n c tio n ? As an
e x a m p le , we c o m p u te th e a n g u la r c o rre la tio n fu n c tio n in a to y m o d e l, w here th e
p ow er s p e c tru m has th e fo rm :
p ( k ) = £ ; f)(A' - ^ 8 , , / r i) ) .
I
n = 1
el
( 3 - io )
w here r 0 is th e c o n fo rm a l d is ta n c e to th e surface o f last s c a tte r. T h is to y m o d e l
s im u la te s b o th th e effects o f a d is c re te p o w e r s p e c tru m due to a fin ite u niverse and
th e effects o f rin g in g in th e p ow er s p e c tru m due to a fe a tu re in th e in fla to n p o te n tia l
(see P e iris et a l. (2003) fo r a d iscu ssio n o f in fla tio n a ry m o d e ls ). F ig u re 3.16 shows
th e a n g u la r c o rre la tio n fu n c tio n a n d fig u re 3.17 show th e T E p ow er s p e c tru m o f
th e m o d el. N o te th a t th e T E p o w e r s p e c tru m is p a r tic u la r ly se n sitive to fea tu re s
in th e m a tte r p ow er s p e c tru m .
In tr ig u in g ly . th is to y m o d e l is a b e tte r m a tc h to
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C h a p te r 3:
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105
th e observed c o rre la tio n fu n c tio n th a n th e A C 'D M m o d e l and p re d ic ts a d is tin c tiv e
s ig n a tu re in th e T E s p e c tru m . C o rn is h et al. (1998) show th a t i f the' universe was
fin ite and s m a lle r th a n the vo lu m e w ith in the d e c o u p lin g surface1, th e n the re sh o u ld
be a ve ry d is tin c tiv e sig n a l: m a tche d c irc les. T h e surface o f last s c a tte r is a sphere*
centered a ro u n d W M A P . I f the* unive’rse is fin ite them th is sphere* m ust in tersect
itse'lf. th is le'acls to p a irs o f m a tch e d circle's. The'se matc h circle's provide* not o n ly
the* c h 'fin itiv e signature* o f a finite* u niverse b u t also sh o u ld enable* co sm o lo g ists to
ele*termine the* to p o lo g y o f the* universe'C ornish e*t al. (1 9 9 8 b ): \Ve*e*ks (1998). S h o u ld
we* be* able* to de*te*ct circle's i f the* p ow er s p e c tru m c u to ff is due* to th e size* o f the*
large*st mode* b e in g
1 /u
i? W hile* the*re* is no rig o ro u s the*ore*m re*lating the* size* o f
the* large’st mode* to the* diarne-ter o f the* fu u d a m e u ta l d o m a in . D . a n a lysis o f b o th
n e g a tiv e ly c u rv t'd (C o rn is h
S pergel. 2000) and p o s itiv e ly curve'd ( Lehoucq e*t al..
2002) topologie*s sugge*st th a t D ~ (0.6 -
1
)A. T h u s , i f the* "p e a k " in the* powe*r
s p e c tru m at / = 5 corre*sponds to th e large*st m ode in the* d o m a in , we* sh o u ld be* able*
to ele*te*ct a patte*rn o f circle s in the sky.
D ue to the* fin ite size* o f the' p a tch o f the u niverse visible* to W M A P (o r
a n y fu tu re s a te llite ), o u r a b ilit y to d e te rm in e th e o rig in and sig n ifica n ce o f th is
d is c re p a n c y w ill be lim ite 'd by cosm ic variance*. H ow ever, fu tu re o b s e rv a tio n s ra n
o ffe r some new in s ig h t in to its o rig in . B y c o m b in in g th e W M A P d a ta w ith tra ce rs
o f la rg e scale s tru c tu re (B o u g h n et a l.. 1998: P e iris
S pergel. 2000). a stro n o m e rs
m a y be able to d ir e c tly d e te ct th e c o m p o n e n t o f th e C M B flu c tu a tio n s clue to th e
ISW ’ e ffect.
W M A P ' s o n g o in g o b se rva tio n s o f large-scale m icro w a ve b a ckg ro u n d
p o la riz a tio n flu c tu a tio n s w ill enable a d d itio n a l m e a surem e n ts o f flu c tu a tio n s a t large
a n g u la r scales. S ince th e T E o b se rva tio n s are p ro b in g d iffe re n t re g ion s o f th e sky
fro m th e T T o b s e rv a tio n s , th e y m a y e n lig h te n us on w h e th e r th e la ck o f c o rre la tio n s
on la rg e a n g u la r scales is a s ta tis tic a l flu ke o r th e s ig n a tu re o f new physics.
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C h a p te r 3:
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8.
106
C on clu sion s
C o s m o lo g y now has a s ta n d a rd m o d e l: a Hat u niverse com posed o f m a tte r,
b a ryo n s and va cu u m energy w ith a n e a rly sc a le -in v a ria n t s p e c tru m o f p r im o rd ia l
H u e tu a tio n s.
In th is co sm o lo g ica l m o d e l, tin* p ro p e rtie s o f th e u niverse are
c h a ra c te riz e d by th e d e n s ity o f b a ryo n s. m a tte r and th e e xp a n sio n ra te: i }h.
and h.
For th e a na lysis o f C 'M B re su lts, a ll o f th e effects o f s ta r fo rm a tio n can
be in c o rp o ra te d in a single n u m b e r:
th e o p tic a l d e p th due to re io n iz a tio n , r .
T h e p r im o rd ia l H u e tu a tio n s in th is m o d el are ch a ra cte rize d by a s p e c tra l in d ex.
D e sp ite its s im p lic ity , it is an a d e q ua te Ht n ot o n ly to th e W M A P te m p e ra tu re and
p o la riz a tio n d a ta b u t also to s m a ll scale C’ M B d a ta , la rg e scale s tru c tu re d a ta , and
supe rn o va d a ta . T h is m odel is co n siste n t w ith the b a r y o n /p h o to n r a tio in fe rre d fro m
o b s e rv a tio n s o f D / H in d is ta n t quasars, th e H S T K e y P ro je c t m easurem ent o f the
H u b b le c o n s ta n t, s te lla r ages a nd th e a m p litu d e o f mass H u e tu a tio n s in fe rre d fro m
clu s te rs and fro m g ra v ita tio n a l le n sing . W h e n we in c lu d e large scale s tru c tu re o r
L y m a n o forest d a ta in th e a n a lysis, th e d a ta suggest th a t we m ay need to add an
a d d itio n a l p a ra m e te r: < h g / d \ n k. Since th e best fit m o d els p re d ic t th a t th e slope o f
th e p ow er s p e c tru m is re d de r on s m a ll scales, th is m o d e l p re d ic ts la te r fo rm a tio n
tim e s fo r d w a rf galaxies. T h is m o d ific a tio n to th e p o w e r la w AC’ D M m o d e l m ay
resolve m a n y o f its p ro b le m s on th e g a la x y scale. T a b le (3.10) lis ts th e best fit
p a ra m e te rs fo r th is m o d el.
W h ile th e re have been a host o f papers on co sm o lo g ica l p a ra m e te rs. W M A P has
b ro u g h t th is p ro g ra m to a new stage: W M A P s m ore a ccu ra te d e te rm in a tio n o f th e
a n g u la r pow er s p e c tru m has s ig n ific a n tly reduced p a ra m e te r u n c e rta in tie s . W M A P
s d e te c tio n o f T E flu c tu a tio n s has c o n firm e d th e basic m o d e l and its d e te c tio n o f
re io n iz a tio n s ig n a tu re has reduced th e n , — r degeneracy. M o s t im p o r ta n tly , the
rig o ro u s p ro p a g a tio n o f e rro rs a n d u n c e rta in tie s in th e W M A P d a ta has s tre n g th e n e d
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C h a p te r 3:
T a b le 3.10.
C o s m o lo g ic a l P a ra m e te rs
107
Basic and D e rive d C o s m o lo g ic a l P a ra m e te rs: R u n n in g S p e c tra l In d e x
M o d e l ’-1
M ean a n d G8 rA
C o n fid e n ce E rro rs
,1 = 0 .8 3 :;j;^
A m p litu d e ' o f flu c tu a tio n s
S p e c tra l In d e x at k = 0.05 M p c ~ l
D e riv a tiv e o f S p e c tra l In d e x
H u b b le C o n s ta n t
B a rv o n D e n s ity
M a tte r D e n s ity
O p tic ’al D e p th
M a tte r P ow er S p e c tru m N o rm a liz a tio n
C h a ra c te ris tic .‘5292, 1V o f V e lo c ity F lu c tu a tio n s
B a rv o n D e n s ity /C r itic a l D e n s ity
M a tte r D e n s ity /C r itic a l D e n s ity
A ge o f th e U nive rse
//., = 0.93 ± 0.03
d n j d \ n k = —0.031
/. —
“ 1
h
- M
0 .<
1 *() l)l
i h j i 2 = 0.0224 ± 0.0009
O
_ o 1 ■>- -o.oos
--rn” — O. lOO _()_()()•)
r = 0.17 ± 0 . 0 6
rrH = 0.84 ± 0.04
a j l " * = 0 .3 8 :j’;;j;J
!>„ = 0.044 ± 0.004
<>,„ = 0 .2 7 ± 0.04
/o = 13.7 ± 0.2 G y r
zr = 17 ± 4
= 1089 ± I
R e io tiiz a tio ti R e d s h ift1’
D e c o u p lin g R e d sh ift
A ge o f th e U nive rse at D e c o u p lin g
T h ic k n e ss o f S urface o f L a st S c a tte r
T h ic k n e ss o f S urface o f L ast S c a tte r
f,itr = 3 797* k y r
A
= 193
U
A c o u s tic A n g u la r Scale'
C u rre n t D e n s ity o f B a ryo n s
B a r y o n /P h o to n R a tio
_
~
f*?
41
00
II
R e d s h ift o f M a tte r /R a d ia tio n E q u a lity
S ou n d H o riz o n at D e c o u p lin g
A n g u la r Size D is ta n c e to th e D e c o u p lin g S urface
± 2
o.}oq-lU-l
•JO-210
r , = 147 ± 2 M p c
~>il
d a = 1 4 .0 :";‘ G p c
t a = 301 ± 1
r>h = (2.5 ± 0 . 1 ) x
1 0
n = (6. i:8;3) x m -10
aF it to th e W M A P . C B I. A C B A R . 2 d F G R S a n d L y m a n o fo re st d a ta
bAssum es io n iz a tio n fra c tio n . r e =
1
<- l A = ~ dc / r s
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
"' cm -1
C h a p te r 3:
C o sm o lo g ica l P a ra m e te rs
108
th e s ig n ifica n ce o f th e in fe rre d p a ra m e te r value's.
In th is paper, we have* also e x a m in e d a n u m b e r o f m ore c o m p lic a te d m o dels:
n on -H a t universe's, quintessence m o d e ls, m o d e ls w ith m assive n e u trin o s , and m o d e ls
w ith ten so r g ra v ita tio n a l wave mode's. B y c o m b in in g the W M A P d a ta w ith fin e r
scale C’ M B expe'rim e'iits anel w ith othe*r a s tro n o m ic a l d a ta sens (2 d F G R S g a la x y
powe'r s p e c tru m and S X Ia o b s e rv a tio n s ), we place> s ig n ific a n t new lim its on rhe'se
p a ra m e te rs.
C o sm o lo g y is now in a s im ila r stage 1 in its in te 'lle c tu a l dewe11 ;
*nt to p a rtic le
physics three' eh'eades ago w hen particle* p h y s ic is ts converge'd on the' e u r r i'iit s ta n d a rd
moele'l. The* sta n d a rd mode'l o f p a rtic le* physics fits a wide* range* o f d a ta , b u t does n ot
answe'r m a n y fu n d a m e n ta l epie'stions: "w h a t is the* o rig in o f mass? w h y is the're* more*
th a n one fa m ily ? , e'tc." S im ila r ly , the* s ta n d a rd c o sm o lo g ica l mode'l has m a n y elee*p
open epie'stions:
"w h a t is the* d a rk e n e rg y? w h a t is th e d a rk m a tte r? w h a t is the*
p h y s ic a l m odel b e h in d in fla tio n (o r s o m e th in g like* in fla tio n )? " O v e r th e past thre*e*
de'e aeh's. pre cisio n te*sts have* c o n firm e d th e s ta n d a rd mode'l o f p a rtic le physics anel
se*arched fo r d istin ctive * signature's o f th e n a tu ra l e xte n sio n o f th e s ta n d a rd m o d e l:
s u p e rs y m m e try .
O v e r th e c o m in g years. im p ro v in g C M B . large scale s tru c tu re ,
h 'n s in g . and supe rn o va elata w ill p ro v id e e ver m o re rig o ro u s tc*sts o f the* c o s m o lo g ic a l
s ta n d a rd m o d el and search fo r new p h ysics b eyo n d th e sta n cla rd m o d e l.
A ck n o w led g em en ts
W e th a n k E d J e n k in s fo r h e lp fu l c o m m e n ts a b o u t th e [D /H ] recent m e a surem e n ts
anel th e ir in te r p r e ta tio n . W e th a n k R a u l Jim e n e z fo r u seful discussions a b o u t th e
c o s m ic ages. We th a n k A d a m Riess fo r p r o v id in g us w ith th e lik e lih o o d su rface
fo rm th e S N IA d a ta . T h e W M A P m is s io n is m ade p ossible by th e s u p p o rt o f th e
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p te r 3:
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109
O ffic e Space a t N A S A H e a d q u a rte rs and by th e h a rd and ca pa b le w o rk o f scores o f
s c ie n tis ts , engineers, m anagers, a d m in is tr a tiv e sta ff, and review ers. LY is s u p p o rte d
by N A S A th ro u g h C h a n d ra F e llo w sh ip P F 2-30 0 22 issued by the C h a n d ra X -ra y
O b s e rv a to ry ce n te r, w h ich is o p e ra te d by tin* S m ith s o n ia n A s fro p h y s ie a l O b s e rv a to ry
fo r and on b e h a lf o f N A S A u n d e r c o n tra c t N A S 8-39073.
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C h a p te r 3:
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110
Angular scale (deg)
2
0.5
90
0.2
7000
6000
5000
E 4000
SC 3000
2000
1000
10
40
100
200
Multipole moment I
400
800
F ig . 3.1.
T h is figure* c o in p a rrs th e best fit p ow er la w A C D M m o d e l to th e W M A P
te m p e ra tu re a n g u la r p o w e r s p e c tru m . T h e g ra y d o ts are th e u n b in n e d d a ta .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p te r 3:
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3
■ji
o
C o s m o lo g ic a l P a ra m e te rs
III
Reionization
-
A d ia b a tic P r e d i c t i o n
+
0
- 1
L
0
1 0 0
200
300
1 0 0
500
V lu lt ip o le i
F ig . 3.2.
T h is fig u re co m p ares th e best f it p o w e r la w A C D M m o d e l to th e W M A P
te m p e ra tu re a n g u la r p ow er s p e c tru m .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p te r 3:
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112
25
300
20
200
0 018
0 020
0 022
0 024
0 026
0 028
12
14
0 0 8 0 10 0 1 2 0 14 0 16 0 18 0 20 0 22
i^ 2
3
2
1
0
0 4
06
0 8
10
1 10
A at k=0 05Mpc
F ig . 3.3.
T h is fig u re shows th e lik e lih o o d fu n c tio n o f th e W M A P T T -+- T E d a ta
as a fu n c tio n o f th e basic p a ra m e te rs in th e p ow er la w A C 'D M W M A P m o d e l. ( f h , / r .
Q mf r . h. .4. ns and r . ) T h e p o in ts are th e b in n e d m a rg in a liz e d lik e lih o o d fro m
th e M a rk o v ch a in a n d th e s o lid cu rve is an E d g e w o rth e xpa n sio n o f th e M a rk o v
chains p o in ts . T h e m a rg in a liz e d lik e lih o o d fu n c tio n is n e a rly G aussian fo r a ll o f th e
p a ra m e te rs except fo r r . T h e dashed lin e s show th e m a x im u m lik e lih o o d values o f
th e g lo b a l s ix d im e n s io n a l f it . Since th e peak in th e lik e lih o o d . j\ u / . is n o t th e same
as th e e x p e c ta tio n va lu e o f th e lik e lih o o d fu n c tio n . < r > . the dashed lin e does n ot
lie at th e c e n te r o f th e p ro je c te d lik e lih o o d .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p te r 3:
C o s m o lo g ic a l P a ra m e te rs
113
I
Z
r r
anr
u ^
F ig . 3.4.
T h is p lo t shows th e c o n trib u tio n to 2 In C p e r m u ltip o le b in n e d a t A / = 13.
T h e excess \ 2 com es p r im a r ily fro m th re e re g ion s, one a ro u n d ( ~ 120. one a ro u n d
f ~ 200 a nd th e o th e r a ro u n d f ~ 340.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 3:
C o s m o lo g ic a l P aram ete rs
114
F ig . 3.0.
S p e c tra l In d e x C o n s tra in ts . L o ft p a n e l: th e //, - r degeneracy in th e
W M A P d a ta fo r a p o w e r-la w AC’ D M m o d e l. T h e T E o b se rva tio n s c o n s tra in the
value o f r a nd th e shape o f th e C’( 1 s p e e tru tn c o n s tra in a c o m b in a tio n o f n , and r .
R ig h t p anel: n , — M.t,h2 degeneracy. T in 1 shaded regions show t i n 'j o i n t one and tw o
s ig m a co nfid e n ce regions.
F ig . 3.6.
(L e ft) T h is fig u re com pares th e best f it A C D M m o d el o f §3 based on
W’A /.A P d a ta o n ly to th e 2 d F G R S P ow er S p e c tru m (P e rc iv a l et a l.. 2001). T h e bias
p a ra m e te r fo r th e best fit Pow er L aw A C D M m o d e l is 1.0 c o rre s p o n d in g to a best
f it va lu e o f d = 0.45. (R ig h t) T h is fig u re co m p a re s th e best fit P ow er L aw A C D M
m o d e l o f §3 to th e p o w e r s p e c tru m a t c = 3 in fe rre d fro m th e L y m a n o forest d a ta .
T h e d a ta p o in ts have been scaled d o w n w a rd s by 2 0 9 t. w h ic h is co n siste n t w ith th e 1
a c a lib ra tio n u n c e rta in ty (C r o ft et a l.. 2002).
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C h a p t e r 3:
C o s m o lo g ic a l P a ra m e te rs
10
115
1 .0 0 0 0
1
I
iQ
0.1 0 0 0
10
I
0
0
1
2
-5
.
f
JC
0 .0 1 0 0
i1
1
1
14
0
0.0001
0
10
10
z
20
25
30
z
F ig . 3.7.
(L e ft p a n e l) T h is fig u re shows th e fra c tio n o f mass in h o u n d o b je c ts as
a fu n c tio n o f re d s h ift. T h e b lack lines show th e mass in co lla pse d o b je c ts w ith mass
g re a te r th a n \ [ f,,{l-( z) . th e e ffe ctive Jeans mass in th e absence o f //_> c o o lin g fo r o u r
best fit P L A C D M m o d e l ( th in lines are fo r the fit to W M A P o n ly and th ic k lines
a re fo r th e fit to a ll d a ta sets). T h e heavy lin e uses th e best fit p a ra m e te rs based on
a ll d a ta (w h ic h has a lo w e r rtH) a n d the lig h t lin e uses th e best fit p a ra m e te rs based
on f it t in g to th e W M A P d a ta o n ly. T h e dashed lin e s show th e mass in collapsed
o b je c ts w ith masses g re a te r th a n th e Jeans mass a ssu m in g th a t th e m in im u m mass is
1 0 \ U . . M o re o b je c ts fo rm i f th e m in im u m mass is lo w e r. (R ig h t P anel) T h is fig ure
shows th e io n iz a tio n fra c tio n as a fu n c tio n o f re d s h ift. T h e s o lid lin e show s io n iz a tio n
fra c tio n fo r th e best fit P L A C D M m odel i f we assum e th a t //•_» c o o lin g is suppressed
b y p h o to -d e s tru c tio n o f H>. T h is fig u re suggests th a t H> c o o lin g m a y be necessary
fo r enough o b je c ts to fo rm e a rly enough to be co n siste n t w ith th e W M A P d e te c tio n .
T h e h eavy lin e is fo r th e best f it p a ra m e te rs fo r a ll d a ta sets and th e lig h t lin e is
fo r th e best fit p a ra m e te rs fo r th e W M A P o n ly f it . T h e dashed lines assume th a t
th e o b je c ts w ith masses g re a te r th a n 1 0 W / . can fo rm sta rs. T h e g ra y b a n d shows
th e 68% lik e lih o o d re g io n fo r zr based on th e a s s u m p tio n o f in s ta n ta n e o u s co m p le te
re io n iz a tio n (K o g u t e t a l.. 2003).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 3:
C o s m o lo g ic a l P aram ete rs
14
60
500 r
50
400 [
300
116
[
40
I
30
\
200 \
20
\
100 ^
0 Ote
0 020
0 022 0 024
I 2t>h2
0 026
0 028
0 0 8 0 10 0 12 0 14 0 16 0 18 0 2 0 0 2 2
05
0 6
0 7
08
09
10
n
8
15
5
6
4
10
3
4
2
0
04
5
2
1
o:
06
08
10
12
A at k=0 OSMpc'1
14
0 00
0 05
0 10
0.15
T
0 20
0 25
:
0 II
0 30
0 80
______....
0 85
0 90
0 95
100
105
1 10
n, at k=0 05M dc‘ t
25
20
♦0 10 -0 05
0 00
0 05
drvdink
F ig . 3.8.
T h is figure’ shows th e m a rg in a liz e d lik e lih o o d fo r v a rio u s co sm o lo g ica l
p a ra m e te rs in th e ru n n in g s p e c tra l in d e x m odel fo r o u r a n a ly s is o f th e co m b in e d
W M A P . C B I. AC’ B A R . 2 d F G R S and L y m a n r> d a ta sets. T h e dashed lines show the
m a x im u m lik e lih o o d values o f th e g lo b a l seven d im e n s io n a l fit.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 3:
C o s m o lo g ic a l P a ra m e te rs
117
WMAP
WMAP
U
a.
K
C
/D
Jc
2dFGRS
JC
£
IB
©
C
Lya
Lyct
io*-1
to-*
toT
i
W
avenumtDer k[Moc*1]
to*3
to-2
to-’
W
avertumber k(Mpc-1]
i
F ig . .3.0.
( L e ft) T h e shaded re g io n in th e fig u re s lu m s th e I — rr c o n to u rs fo r the*
a m p litu d e o f th e p ow er s p e c tru m as a fu n c tio n o f scale fo r th e ru n n in g s p e c tra l in d e x
m o d e l fit to a ll d a ta sets. The* d o tte d line's b ra cke t th e 2 -rr re g ion fo r th is m o d el.
T h e dashed lin e is th e best fit p ow er s p e c tru m fo r th e p o w e r la w A C D M m o d el.
( R ig h t) T h e shaded region in th e fig u re show s th e 1 -rr c o n to u rs fo r th e a m p litu d e ' o f
th e a m p litu d e o f mass flu c tu a tio n s .
= (k'i / ( 2 ~ 2 ) P ( k ) . as a fu n c tio n o f scale fo r
th e ru n n in g s p e c tra l in d e x m o d e l fit to a ll d a ta sets. T h e d o tte d line's bracket the
2-rr re g ion fo r th is mode'l. T h e dashed lin e is th e be'st fit fo r the' p ow er law A C D M
mode'l.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 3:
118
C o s m o lo g ic a l P a ra m e te rs
1 .0 0 0 0
0 .1 0 0 0
10
I 0. 01 00
X
10
10
10
0 .0 0 1 0
-20
-25
0 .0001
10
10
z
15
25
20
30
z
F ig . 3.10.
(L e ft) T h is fig u re shows th e frac tio n o f the* universe* in h o u n d objec ts w ith
mass g re a te r th a n M t{HI■ (d a she d ). M J = 10f,.U . ( s o lid ) a n d M J (d o tte d ) in a m odel
w ith a ru n n in g s p e c tra l in d e x . T h e curves were c o m p u te d fo r th e 1rr u p p e r lim it
p a ra m e te rs fo r th is m o d e l (see F ig u re 3 .9 ). These s h o u ld he view ed as u p p e r lim its
on th e mass fra c tio n in co lla pse d o b je c ts . (R ig h t) T h is fig u re shows th e io n iz a tio n
fra c tio n as a fu n c tio n o f re d s h ift and is based on th e a s s u m p tio n s d escribe d in §4.7.
A s in th e fig u re on th e le ft, we use th e 1a u p p e r lim it e s tim a te o f th e p ow er s p e c tru m
so th a t we o b ta in " o p tim is tic ” e stim a te s o f th e re io n iz a tio n fra c tio n . In th e c o n te x t o f
a ru n n in g s p e c tra l in d e x fit to th e d a ta , th e W M A P d e te c tio n o f re io n iz a tio n appears
to re q u ire th a t H> c o o lin g p la ye d an im p o rta n t ro le in e a rly s ta r fo rm a tio n .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 3:
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119
F ig . 3.11.
C o n s tra in ts on D a rk E n e rg y P ro p e rtie s . T h e u p p e r le ft panel shows
th e m a rg in a liz e d m a x im u m lik e lih o o d surface fo r th e W M A P e x t d a ta alone and fo r
a c o m b in a tio n o f th e W M A P e x t 4- 2dFC*RS d a ta sets. T h e s o lid lines in th e fig u re
show th e 6 8 % a n d 95% co nfid e n ce ranges fo r th e f it C su p e rn o va d a ta fro m P e rlm u tte r
et al. (1999). In th e u p p e r rig h t p a n e l, we m u lt ip ly th e su p e rn o va lik e lih o o d fu n c tio n
by th e W M A P e x t + 2 d F G R S lik e lih o o d fu n c tio n s . T h e lo w e r le ft panel shows the
m a x im u m lik e lih o o d su rfa ce fo r h a n d iv fo r th e W M A P e x t d a ta alo ne a nd fo r the
W M A P e x t - f 2 d F G R S d a ta sets. T h e s o lid lin e s in th e fig u re s are th e 6 8 % a nd 95%
co nfid e nce lim it s on H 0 fro m th e H S T K e y P ro je c t, w here we a dd th e s y s te m a tic and
s ta tis tic a l e rro rs in q u a d ra tu re . In th e lo w e r rig h t p an e l, we m u lt ip ly th e lik e lih o o d
fu n c tio n fo r th e W M A P e x t 4- 2 d F G R S d a ta b y th e lik e lih o o d surface fo r th e H S T
d a ta to d e te rm in e th e jo in t lik e lih o o d surface. T h e d a rk areas in these p lo ts are the
6 8 % lik e lih o o d re g ion s a n d th e lig h t areas are th e 95% lik e lih o o d regions.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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A
o
n
k
F ig . 3.12.
C o n s tra in ts on D a rk E n e rg y P ro p e rtie s . T h e u p p e r le ft panel shows
th e m a rg in a liz e d m a x im u m lik e lih o o d surface fo r th e W M A P e x t d a ta a lo ne and fo r a
c o m b in a tio n o f th e W M A P e x t ■+■ 2 d F G R S d a ta sets. T h e s o lid lin e s in th e fig u re show
th e 6 8 % and 95% confid e nce ranges fo r supe rn o va d a ta fro m Riess et a l. ( 2 0 0 1 ). In th e
u p p e r rig h t p anel, we m u lt ip ly the supe rn o va lik e lih o o d fu n c tio n by th e W M A P e x t
+ 2 d F G R S lik e lih o o d fu n c tio n s . T h e lo w e r le ft panel show s th e m a x im u m lik e lih o o d
su rface fo r h and w fo r th e W M A P e x t d a ta alone a n d fo r th e W M A P e x t
2dFG R S
d a ta sets. T h e s o lid lines in the fig ures are th e 6 8 % and 95% co nfid e nce lim its on
Ho fro m th e H S T K e y P ro je c t, w here we add th e s y s te m a tic and s ta tis tic a l e rro rs
in q u a d ra tu re . In th e lo w e r rig h t p anel, we m u lt ip ly th e lik e lih o o d fu n c tio n fo r th e
W M A P e x t + 2 d F G R S d a ta by th e lik e lih o o d surface fo r th e H S T d a ta to d e te rm in e
th e jo in t lik e lih o o d surface. T h e d a rk areas in these p lo ts are th e 6 8 % lik e lih o o d
regions a n d th e lig h t areas are th e 95% lik e lih o o d regions. T h e c a lc u la tio n s fo r th is
fig u re assum ed a p r io r th a t iv > — 1 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 3:
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121
<
n
n_
n_
F ig . 3.13.
C o n s tra in ts on the g e o m e try o f the u niverse : Q m —
p la ne . T h is fig u re
shows th e tw o d im e n s io n a l lik e lih o o d surface fo r v a rio u s c o m b in a tio n s o f d a ta : (u p p e r
le ft) W M A P (u p p e r r ig h t) W M A P e x t (lo w e r le ft) \ Y M A P e x t + H S T K e y P ro je c t
(s u p e rn o va d a ta (R iess et a l.. 1998. 2001) is show n b u t n o t used in th e lik e lih o o d
in th is p a rt o f th e p an e l: (lo w e r r ig h t) W M A P e x t + H S T K e y P ro je c t -+- su pe rn o va
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 3:
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C o s m o lo g ic a l P a ra m e te rs
122
0.8
_o
D
n
±cn
0.6
"a
<u
|
0 .4
E
°
0.2
.
0 .0
0 . 0 0 0
0.0 02
0 .0 0 4
0 .0 0 6
0 .0 0 8
0.010
n vh 2
F ig . 3.14.
T h is fig u re shows th e m a rg in a liz e d c u m u la tiv e p r o b a b ility o f H J r based
on a fit to th e W M A P e x t-f- 2 d F G R S d a ta sets (dashed) and th e c u m u la tiv e p ro b a b ility
based on a f it to th e W M A P e x t 4- 2 d F G R S + L y m a n n d a ta sets (s o lid ). T h e v e rtic a l
lin e s are th e 95% co nfid e nce u p p e r lim its fo r each case (0.21 and 0.23 c V ).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 3:
C o s m o lo g ic a l P aram ete rs
1 .0
c
o
y
0.8 -
123
r<0.90
(no
priors)
r<0.43
( i d n s/ d i n k l < 0 . 0 0 5 )
r<0.29
( n s< 1)
13
_Q
C/3
S
"~o
0.6 -
CL)
>
_o
-Z 3 0 . 4 -
E
13
CJ
0.2
-
0.0
0.0
0.2
0.4
0.6
0.8
t e n s o r / s c a l a r ratio
1.0
1 .2
r
F ig . 3.15.
T h is fig u re shows the c u m u la tiv e lik e lih o o d o f th e c o m b in a tio n o f the
W M A P e x t + 2 d F G R S + L y m a n n d a ta sets as a fu n c tio n o f r . th e te n s o r/s c a la r ra tio .
T h e th re e lin e s show th e lik e lih o o d fo r no p rio rs , fo r m o d e ls w ith \ d n / ( i \ n k \ < 0.005
and fo r m o d e ls w ith n s < 1 .
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C h a p t e r .3:
124
C o s m o lo g ic a l P a ra m e te rs
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200
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universe m o d e l, and W M A P d a ta on la rg e a n g u la r scales. T h e d a ta p o in ts are
c o m p u te d fro m th e te m p la te -c le a n e d V b a n d W M A P u s in g th e KpO c u t (B e n n e tt
et a l.. 2003c).
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C h a p t e r 3:
C o s m o lo g ic a l P a ra m e te rs
4
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u
+
5
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F ig . 3.17.
T E P ow er S p e c tru m . T h is fig u re co m p a re s th e d a ta to th e p re d ic te d T E
p o w e r s p e c tru m in o u r to y fin ite u niverse m o d e l a n d th e A C D M m o d e l. B o th m o dels
assum e th a t r = 0.17 a n d have id e n tic a l c o s m o lo g ic a l p a ra m e te rs. T h is fig u re shows
th a t th e T E p ow er s p e c tru m c o n ta in s a d d itio n a l in fo r m a tio n a b o u t th e flu c tu a tio n s
a t la rg e angles. W h ile th e c u rre n t d a ta can n o t d is tin g u is h between these m o dels,
fu tu re o b s e rv a tio n s c o u ld d e te c t th e d is tin c tiv e T E s ig n a tu re o f th e m o d el.
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B ib liograp h y
B acon. D .. Massey. R .. R e fre gie r. A .. A:
E llis . R. 2002.
a s tro -p h /0 2 0 3 1 3 4
B a h c a ll. I. N .. G o n z a le z -G a rc ia . M . C’..
A:P en ya -G aray. C . 2003a. .1HEP02 (2003).
009.
B a h c a ll. X . A . A' B ode. P. 2002. a s tr o - p h /0 2 12303
B alu a ll. X . A .. O s trik e r. .1. P.. P e rlm u tte r. S.. A S te in h a rd t. P. .1. 1999. Science’. 284.
1481
B a h c a ll. X . A . e*t a l. 20021). a s tro -p h /0 2 0 5 4 9 0
B a rka n a . R .. H a im a n . Z.. A O s trik e r. J. P.
2001. Ap.J.
B arnes. C . et al. 2003. A p .IS . to appe’a r
v l4 8 t i l
in
558. 482
B e n n e tt. C . L .. B ay. M .. H a lp e rn . M .. H in sh a w . G .. Jackson. C .. J a ro s ik . X .. K o g u t.
A .. L im o n . M .. M e ye r. S. S.. Page. L .. S pergel. D . X .. T u c k e r. G . S.. W ilk in s o n .
D. T .. W o t lack. E.. A W r ig h t. E. L. 2003a. Ap.J. 583. 1
B e n n e tt. C . L .. H a lp e rn . M .. H in sh a w . G .. J a ro s ik . X .. K o g u t. A .. L im o n . M .. M eyer.
S. S.. Page. L .. S pe rg e l. D . X .. T u cke r. G . S.. W o lla c k . E .. W r ig h t. E. L .. B arnes.
C .. G reason. M .. H ill. R .. K o m a ts u . E .. X o lta . M .. O d e g a rd . X .. P e iris. H .. V erde.
L .. A W e ila n d . J. 2003b. Ap.JS. to a p p e a r in v l4 8 n l
B e n n e tt. C . L. et al. 2003c. Ap.JS. to a p p e a r in v l4 8 n l
126
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
127
B IB L IO G R A P H Y
B oesgaard. A . M . A' S te ig m a n . G . 1985. A R A A A . 23. 319
B o n d . .1. R. A' E fs ta th io ii. G . 1984. Ap.J. 285. L45
B o n d . .J. R .. .la ffc . A . H.. A- K n o x . L. 1998. P hys. R p v. D . 57. 2117
B o n d . .1. R. pt a l. 2002. a s tro -p h /0 2 0 5 3 8 6
B o r^ a n i. S.. R o s a ti.
H o ld e n . B .. D e lla
P..
T o zzi. P.. S ta n fo rd . S.A .. E is rn h a rd t. P. R .. L id n ia n . C ..
C e ra . R .. N o rm a n . C .. A S quires. G . 2001. Ap.J. 5G1. 13
B o u ^ h n . S. P.. C ritte n d e n . R. G .. A T u ro k . N. G . 1998. N ew A s tro n o m y . 3. 275
B ro w n . M . L. et a l. 2002. a s tr o - p h /0 2 10213
B u rie s . S. A T y t le r .
D. 1998a. Ap.J. 499. G99.
B u rie s . S. A T y tle r .
D. 1998b. Ap.J. 507. 732.
B u rie s . S.. N o lle tt. K . M .. A T u rn e r. M . S. 2001. Ap.J. 552. L I
C a ld w e ll. R. R .. Dave. R .. A S te in h a rd t. P. J. 1998. P hys. Rev. L e tt.. 80. 1582
C a rro ll. S.. H o ffm a n . M .. A T ro d d e n . \ L 2003. a s tro -p h /0 3 0 1 2 7 3
C’a y re l. R .. H ill. \ ’ .. Beers. T . C .. B a rb u y . B .. S p ite . NL. S p ite . F.. Plez. B..
A n d e rse n . .J.. B o n ifa c io . P.. Francois. P.. M o la ro . P.. N o rd s tro m . B .. A P rim a s. F.
2001.
N a tu re . 409. G91
C en. R. 2002. Ap.J. s u b m itte d (a s tro -p h /0 2 1 0 4 7 3 )
C h a b o y e r. B. 1995. Ap.J. 444. L9
1998. P hys. R ep.. 307. 23
C h a b o y e r. B. A K ra uss. L . M . 2002. A p J . 567. L 45
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
128
B IB L IO G R A P H Y
C h a b o y e r. B. A: K ra u ss. L. M . 2003. Science. 299. G5
C la y to n . D o n a ld D . 1988. M N R A S . 234.
1
C'olless. M .. D a lto n . G .. M a d d o x . S.. S u th e rla n d . W .. X o rb e rg . P.. C'oh'. S..
B la n d -H a w th o r n . .1.. B rid g e s. T .. C a n n o n . R .. C o llin s . C .. C o u ch. \ \ \ . C ross.
N .. D ooley. K .. De P ro p ris . R.. D riv e r. S. P.. E fs ta th io u . G .. E llis . R. S.. F re n k.
C . S.. G la z e b ro o k . K .. .Jackson. C .. L a h a v . () .. Lew is. I.. L u n is d e n . S.. M a d g w ie k .
D .. Peacock. J. A .. P eterson. B. A .. P rice . I.. S eaborne. M .. A- T a y lo r. K . 2001.
M X R A S . 328. 1039
C o rn is h . X . J.. S pe rg e l.
D .. A S ta rk m a n . G .
1998.
Phys. R ev. D. 37. 3982
C o rn is h . X . .1.. S pe rg e l. D .. A S ta rk m a n . G .
1998.
P ro c. X a t. A ca d . S ci.. 93. 82
C o rn is h . X . J. A S pe rg e l. D. X . 2000. P hys. Rev. D . 62. 87304
C o w a n. J. J.. P fe iffe r. B .. K ra tz . K .-L .. T h ie le m a n n . F .-K .. Snodon. C’ .. B u rie s. S..
T y tle r . D .. A Beers. T . C. 1999. Ap.J. 321. 194
C r o ft. R. A . C .. W e in b e rg . D . H .. B o lte . M .. B u rie s. S.. H o rn q u is t. L .. K a tz . X ..
K ir k m a n . D .. A T y tle r . D . 2002. Ap.J. 381. 20
C ro ft. R. A . C .. W e in lie rg . D . H.. K a tz . X .. A H e rn q u is t. L. 1998. Ap.J. 493. 44
Dedeo. S.. C a ld w e ll. R .. A S te in h a rd t. P. 2003. a s tro -p h /0 3 0 1 2 8 4
D e ffa y e t. C’ .. D v a li. G .. A G abadadze. G . 2002. P hys. Rev. D . 63. 44023
D 'O d o ric o . S.. D essauges-Zavadsky. M .. A M o la ro . P.
E fs ta th io u . G . A B o n d . J. R. 1999. M X R A S . 304.
Eke. V . R .. X a v a rro . J.
F .. A S te in rn e tz. M .
2 0 0 1
. A A A . 368. L21
75
2001. Ap.J. 554. 114
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
B IB L IO G R A P H Y
129
E lg a ro y . ( ) .. L a lm v . () .. P e rciva l. W .
Peacock. .1. A .. M a d g w ie k . D. S.. B rid le .
S. L .. B a u g h . C. M .. B a ld rv . I. K .. B la tu l-H a w th o ru . .1.. B rid g e s. T .. C a n n o n . R..
C o le . S.. C’olless. M .. C o llin s . C’ .. C ouch. \V .. D a lto n . G .. de P ro p ris . R .. D riv e r.
S. P.. E fs ta th io u . G . P.. E llis . R. S.. F renk. C. S.. G la z e h ro o k . K .. Jackson. C’ ..
Lew is. I.. Lurnsd e n. S.. M a d d o x . S.. X o rb e rg . P.. P eterson. B. A .. S u th e rla n d . \ \ \ .
A* T a y lo r. K . 2002. Phys. Rev. L e tt.. 89. 6 1 3 0 1
E p s te in . R. I.. L a trirn e r. .1. M .. A S chram m . D. X . 197G. X a tu re . 203. 198
F ish e r. K . B .. D a vis. M .. S trauss. M . A .. Y a h il. A .. A H u c h ra . .1. 199-1. M X R A S .
2GG. 50
F re e dm an . \V . L .. M a d o re . B. F.. G ib so n . B. K .. Ferrarese. L .. K e lso n . D. D .. Sakai.
S..
M o u ld . .). R.. K e n n ic u tt. R. C .. Ford. H. C’ .. G ra h a m . .1. A .. H u c h ra . .1. P..
Hughes. S. M . G .. Illin g w o r th . G . D .. M a c ri. L. M .. A S te tso n . P. B. 2001. A p J .
55:3. 47
G a rn a v ic h . P. M .. .Iha. S.. C’h a llis . P.. C lo c c h ia tti. A .. D ie rcks. A .. F ilip p e n k o .
A.
\ '. . G illila n d . R.
L .. H ogan. C. J.. K irs h n e r. R. P.. L e ib u n d g u t. B .. P h illip s .
M . M .. R('iss. D .. Riess. A . G .. S c h m id t. B . P.. S ch o m m e r. R. A .. S m ith .
R. C’ ..
S p y ro m ilio . J.. S tu b b s . C .. S un tze ff. X . B .. T o n ry . .1.. A C a r r o ll. S. M . 1998. A p J .
509. 74
G n e d in . X . Y . A H a m ilto n . A . J. S. 2002. M X R A S . 334. 107
G r a tto n . R. G .. Fusi Pecci. F .. C a rre tta . E .. C le m e n tin i. G .. C o rs i. C . E.. A L a tta n z i.
M . 1997. A p J . 491. 749
H a irn a n . Z .. A b e l. T ..
A Rees. M . J. 2000. A p J . 534. 11
H a im a n . Z .. Rees. M .
J.. A Loeb. A . 1997. A p J . 484. 985
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
B IB L IO G R A P H Y
130
H a n n e s ta d . S.. H ansen. S. H .. V illa n te . F. L .. A’ H a m ilto n . A . .1. S. 2002. A s tro p a rtic le
P hysics. V o lu m e 17, Issue 3. p. 373-382.. 17. 373
H ansen. B. M . S.. B re w e r. I.. F a hlrn an . G . G .. G ib s o n . B. K .. Ib a ta . R.. L im o n g i.
M .. R ich . R. M .. R ich e r. H. B .. S hara. M . M .. A" S te tso n . P. B. 2002. A p .I. 374.
L 133
H ill. \ \ . Plez. B .. C a y re l. R.. Beers. T . C .. N o rd s tro m . B .. A nd e rse n . J.. S p ite . M ..
S p ite . F.. B a rb u y . B .. B o n ifa c io . P.. D epagne. E.. F ra n cois. P.. A P riin a s . F. 2002.
A A A . 387. 3G0
H in s h a w . G . F. cr al. 2003a. A p .lS . to a p p e a r in v l4 8 t i l
. 2003b. A p .lS . to a p p e a r in v l4 8 t i l
H o e k s tra . H.. v a il W aerbeke. L .. G la d d e rs . M . D .. M e llie r. Y .. A Yee. H. K . C . 2002.
A p J . 377. G04
H u . \V . 2001. in RESC’ E C : 1099: B ir t h a n d E v o lu tio n o f th e C niverse. 131
H u . \V .. E ise n ste in . D. J.. A T e g in a rk . M . 1998. Phvs. R ev. L e tt.. 80. 3233
H u . W .. F u k u g ita . M .. Z a ld a rria g a . M . A T e g rn a rk . M . 2001. A p J . 349. G69
H u e y. G .. W a n g . L.. Dave. R.. C a ld w e ll. R. R .. A S te in h a rd t. P. J. 1999.
P hys. Rev. D . 59. 63005
H u ffe n b e rg e r. K . A S e lja k. E. 2003. a s tro -p h /0 3 0 1 3 4 1
J a ro s ik . X . et a l. 2003a. A p J S . 145
— . 2003b. A p J S . to a p p e a r in v l4 8 n l
J a rv is . M . et a l. 2002. a s tr o - p h /0 2 10604
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
B IB L IO G R A P H Y
131
.Jenkins. E. B.. T rip p . T . M .. W 'ozniak. P.. Sofia. t \ .J.. A S o n n e b o rn . G . 1999. Ap.J.
320. 182
Jim e n e z. R. 1999. in D a rk M a tte r in A s tro p h y s ic s and P a rtic le P hysics. 170
Jim e n e z. R. A' P adoan. P. 1998. A p J . 498. 704
J im e n ez. R.. T h e jll. P.. .Jorgensen. I . G .. M a c D o n a ld . J.. A Pagel. B. 199G. M X R A S .
282. 926
Jones. M . E. et al. 2001. a stro -p h /0 1 0 3 0 4 G
K a lu z n y . J.. T h o m p s o n . I.. K rz e m in s k i. \ \ \ . O le ch. A .. P ych . \V .. A M ochejska.
B.
2002. in A S P C o n f. Ser. 265: O m e ga C e n ta u ri. A U n iq u e W in d o w in to
A s tro p h y s ic s . 133
K a m io n k o w s k i. M .. S pergel. D . X .. A S u g iya m a . X . 1994. A p J . 426. L37
K ea rn s. E. T . 2002. F ra sca ti Phys. Ser.. 28. 413
K irk m a n . D .. T y tle r . D .. B u rie s. S.. L u b in . D. A O 'M e a ra . J .M . 2000. A p J . 329. 633
K irk m a n . D .. T y tle r . D .. S u z u k i. X .. O 'M e a ra . J.. A L u b in . D . 2003. p re p rin t
ast ro -p h /0 3 0 2 0 0 6
K n o x . L .. C h riste n se n . X . A S ko rd is. C . 2001. A p J . 563. L93.
K o g u t. A . et a l. 2003. A p J S . to a p p e a r in v l4 8 i l l
K o m a ts u . E. A S e lja k. l \ 2002. M X R A S . 336. 1256
K o m a ts u . E. et al. 2003. A p J S . to a p p e a r in v l4 8 n l
K o s o w sky. A . A T u rn e r. M . S. 1995. P hys. R ev. D . 52. 1739
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
132
B IB L IO G R A P H Y
K u o . C'. L. et a l. 2002. A p J . a s tr o - p h /0 2 12289
Leach. S. M .. L id d le . A . R.. M a r tin . .1.. A- Schwarz. D. J. 2002. Phys. Rev. D .
6
G.
23515
Lehoucq. R.. W eeks. J.. L'zan. J.. C lausinann. E.. A L u in iu e t. .1. 2002. C lassical and
Q u a n tu m G ra v ity . 19. 4683
Levsh a kov. S .A .. A g a fo n o va . 1.1. D ’ O d o ric o . S.. W o lfe . A .M .. A Dessauges-Zavadsky.
M .. a p j. 582. 59G.
M ason. B. S.. M ye rs. S. T .. A Readhead. A . C. S. 2001. A p J . 555. L
11
M oon*. B. 1994. N a tu re . 370. G29
M oos. H. W .. S etnbach. K . R .. Y id a l- M a d ja r . A .. Y o rk . D. G .. F rie d tn a n . S. D ..
H e b ra rd . G .. K ru k . J. W .. L ehner. N .. L em oin e . M .. S4692
1
rn . G .. W o o d . B. E..
A ke. T . B .. A n d re . M .. B la ir. W . P.. C’ hayer. P.. G ry . C'.. D u p ree . A . K .. F e rle t.
R.. F e ld m a n . P. D .. G re en . .1. C’ .. H o w k. J. C .. H u tc h in g s . .1. B .. .Jenkins. E. B..
L in s k y . .J. L .. M u rp h y . E. M .. O egerle. W . R .. O liv e ira . C .. R o th . K .. S;441
D.
w.
.J.. Savage. B. D .. S h u ll. J. M .. T rip p . T . M .. W e ile r. E. J.. W elsh. B. Y ..
W ilk in s o n . E .. A W o o d g a te . B. E. 2002. A p JS . 140. 3
N a v a rro . J. F .. F renk. C . S.. A W h ite . S. D. M . 1997. A p J . 490. 493
O h . S. P. 2001. A p J . 553. 499
O 'M e a ra . J. M .. T y t le r . D .. K irk m a n . D .. S u zu ki. N .. P rochaska. J. X .. L u b in . D .. A
W o lfe . A . M . 2001. A p J . 552. 718
P aczyn ski. B . 1997. in T h e E x tra g a ia c tic D ista n ce Scale. 273 280
Page. L . et a l. 2003a. A p J S . to a p p e a r in v l4 8 n l
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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. 2003c. A p .I.
0 8 0
. in press
Peacock. .1. A .. C ole. S.. X o rb e rg . P.. B a u g h . C'. M .. B la n d -H a w th o rn . .1.. B rid g es.
T .. C a n n o n . R. D .. C’olless. M .. C o llin s . C .. C o u ch . W .. D a lto n . C>.. D eelev. K .. De
P ro p ris . R.. D riv e r. S. P.. E fs ta th io u . G .. E llis . R. S.. Frenk. C. S.. G la z e b ro o k .
K .. Jackson. C’ .. L a h a v. () .. L ew is. I.. L u m sd e n . S.. M a d d o x . S.. P e re iva l. \V . J..
Peterson. B. A .. P rice . I.. S u th e rla n d . \V .. A’ T a y lo r. K . 2001. N a tu re . 410. 169
Pearson. T . J.. M ason. B. S.. R eadhead. A . C. S.. S hepherd. M . C’.. Sievers. .1. L..
T d o m p ra s e rt. P. S.. C a r tw r ig h t. .1. K .. F a rm e r. A . J.. P a d in . S.. M yers. S. T ..
B o n d . J. R.. C 'o n ta ld i. C’. R.. Pen. L '.-L .. P ru n e t. S.. Pogosyan. D .. C a rls tro rn .
•J. E.. K ovac. J.. L e itc h . E. M .. P ryke . C’ .. H a lve rso n . N. W .. H o lz a p fe l. \V . L ..
A lta tn ira n o . P.. B ro n fm a n . L .. Casassus. S.. M a y . J.. A- Jo y. M . 2002. A p .I.
su bm it ted (ast ro -p h /0 2 0 3 3 8 8 )
Peebles. P. J. E. A R a tra . B. 1988. A p .I. 323. L 1 7
P eiris. H. et al. 2003. A p .lS . to a p p e a r in v l4 8 r i l
P eiris. H. V . A S pergel. D . N . 2000. A p .I. 340. 603
P ereival. \V . J.. B a u g h . C . M .. B la n d -H a w th o rn . .!.. B rid g es. T .. C a n n o n . R .. C ole.
S..
C'olless. M .. C o llin s . C .. C o u ch . \V .. D a lto n . G .. De P ro p ris . R .. D riv e r. S. P..
E fs ta th io u . G .. E llis . R. S.. F re n k. C. S.. G la z e b ro o k . K .. Jackson. C .. L a h a v .
( ) .. L ew is. L . L u m sd e n . S.. M a d d o x . S.. M o o d y . S.. X o rb e rg . P.. Peacock. J. A ..
P eterson. B . A .. S u th e rla n d . \Y .. A T a y lo r. K . 2001. M X R A S . 327. 1297
P e rln m tte r. S.. A ld e rin g . G .. G o ld h a b e r. G .. K n o p .
R. A .. X u g e n t. P..C a s tro .
D e u stu a . S.. F a b b ro . S.. G o o b a r. A .. G ro o m . D . E .. H o o k. I. M .. K im .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
P. G ..
A . G ..
D ID L IO G R A P m
134
K im . M . Y .. Leo. .J. C'.. N unes. X . .1.. P a in . R.. P ennypaeker. C. R.. Q u iin b y . R..
L id m a n . C.. E llis . R. S.. Ir w in . XL. M c M a h o n . R. G .. R u iz -L a p u e n to . P.. W a lto n .
V . Schaefer. B .. B oyle. B . .1.. F ilip p e n k o . A . \ ’ .. M a th e s o n . T .. F ru e h te r. A . S..
P anagia. N .. X e w b e rg . H. J. M .. C o u ch. W . .1.. A- T h e S u p e rn ova C o sm o lo g y
P ro je c t. 1999. A p .I. 317. 363
P e ttin i. XI. A B ow en. D. V . 2001. A p .I. 360. 41
P h illip s . XI. XI. 1993. A p .I. 413. L103
P ie rp a o li. E .. B o rg a n i. S.. S c o tt. D .. A W h ite 1. XI. 2002. a s tr o - p h /0 2 10367
R a tra . B. A Peebles. P. .1. E. 1988. Phys. Rev. D . 37. 3406
Reese1. E. D .. C a rls tro m . I. E .. .Joy. XI.. X lo h r. .1. J.. G re go . L .. A H o lza p fe l. W . L.
2002. A p .I. 381. 33
R e fre gie r. A .. Rhocles. .1.. A G ro th . E. .1. 2002. A p .I. 372. L131
R e ip ric h . T . H. A B o h rin g e r. H. 2002. A p .I. 367. 716
R e n z in i. A .. B ra g a g lia . A .. F e rra ro . F. R .. G ilm o z z i. R .. O r to la n i. S.. H o lb e rg . J. B..
L ie b e rt. J.. W esem ael. F .. A B o h lin . R. C . 1996. A p .I. 463. L 2 3 +
R ic h e r. H. B .. B re w e r. .1.. F a h lm a n . G . G .. G ib s o n . B. K .. Hansen. B. X I.. Ib a ta . R ..
K a lir a i. J. S.. L irn o n g i. X I.. R ic h . R. X I.. S aviane. I.. S h a ra . XI. X I.. A S tetson.
P. B. 2002. A p .I. 574. L151
Riess. A . G .. F ilip p e n k o . A . V .. C h a llis . P.. C lo c c h ia tti. A .. D ie rcks. A .. G a rn a v ic h .
P. XI.. G illila n d . R . L .. H o g a n . C . .J.. .Jha. S.. K irs h n e r. R. P.. L e ib u n d g u t. B ..
P h illip s . XI. X I.. Reiss. D .. S c h m id t. B. P.. S ch o m m e r. R. A .. S m ith . R. C ..
S p y ro m ilio . J.. S tu b b s. C .. S u n tz e ff. N . B .. A T o n ry . J. 1998. A.J. 116. 1009
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
B IB L IO G R A P H Y
Riess. A . G .. N u g e n t. P. E .. G illila n d . R. L .. S c h m id t. B. P.. T o n rv . .J.. D ic k in s o n .
M .. T h o m p s o n . R. I.. B u d a v a ri. T . . S.. Evans. A . S.. F ilip p e n k o . A . V .. L iv io . M ..
S anders. D . B .. S ha p le y. A . E .. S p in ra d . H .. S te id e l. C . C’ .. S te rn . D .. Surace. .1..
A V e ille u x . S. 2001. A p .I. 500. 49
Ricss. A . G .. Press.
\V . H .. A K irs h n e r. R. P.1995. A p .I. 438. L17
S chatz. H .. T o en je s. R .. P fe iffe r. B .. Beers. T . C .. C o w a n. .1. .1.. H ill. \ ’ .. A* K ra tz . K .
2002. A p .I. 579. (520
Schueeker.P.. B o h rin g e r.H .. C o llin s . C .A .. A G u zzo . L .. a. 398. 807
Schuecker. P.. C’a h lw e ll. R .R .. B o h rin g e r. H .. C o llin s . C .A .. G u zzo . L . A W e in b e rg .
N .N .. a. 402. 53.
S e lja k . U. A Z a ld a rria g a . M . 1990. A p .I. 409. 437
S h e th . R. K . A T o rm e n . G . 1999. M X R A S . 308. 119
S o n n e b o rn . G .. T r ip p . T . M .. F e rle t. R.. .le n kin s. E. B .. S ofia. l_\ I.. V id a l- M a d ja r .
A .. A W o z n ia k . P. R.
2 0 0 0
. A p .I. "545. 277
S pergel. D . N. A S te itd ia rd t. P. .1. 2000. P hys. R ev. L e tt.. 84. 3700
T e g rn a rk. M . A S ilk . .1. 1995. A p .I. 441. 458
T h ie le m a n n . F .-K ..
T .. K r a tz . K .- L ..
H auser. P.. K o lb e . E.. M a rtin e z -P in e d o . G .. P an o v. I.. R auscher
P fe iffe r. B .. Rosswog. S.. L ie h e n d o rfe r. M .. A M ezza cap p a. A .
2002. Space Science R e view s. 100. 277
T h o m p s o n . I. B .. K a lu z n v . .1.. P ych . \ \ \ . B u rle y . G .. K rz e m in s k i. W .. P a czyh sk i. B.
Persson. S. E .. A P re s to n . G . \V .
2 0 0 1
. A .I. 121. 3089
T o th . C,. A O s trik e r. J. P. 1992. A p .I. 389. 5
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
136
B IB L IO G R A P H Y
T u rn e r. M . S. .V: Riess. A . G . 2002. A p .I. 569. 18
V a n d e n B e rg . D. A .. R ic h a rd . ().. M ic h a u d . G .. A" R ich e r. I. 2002. A p .I. 571. 487
V an W aerbeke. L. T e re no . I.. M e llie r. Y .. A B erna d ea u . F. 2002. a s tro -p h /0 2 1 2 1 5 0 .
V an W aerbeke. L .. M e llie r. Y .. P ello. R.. Pen. L’ .-L .. M cC ra cke n . H ..I. A .Jain. B.
2002.
a. 393. 369.
V enkatesan. A .. G iro u x . M . L.. A S h u ll. .J. M . 2001. Ap.J. 563. 1
V erde. L .. Heavens. A . F.. P ereival. W . .!.. M a ta rre se . S.. B au g h. C’. M .. B la n d H a w th o rn . .1.. B rid g e s. T .. C a n n o n . R.. C o le . S.. C’olless. M .. C o llin s . C .. C ouch.
W .. D a lto n . G .. De P ro p ris . R .. D riv e r. S. P.. E fs ta th io u . G .. E llis . R. S.. Frenk.
C.
S.. G la z e b ro o k . K .. .Jackson. C .. L a h a v. ( ) .. L ew is. I.. L u m sd e n . S.. M a d d o x .
S..
M a d g w ic k . D .. N o rb e rg . P.. Peacock. J. A .. P eterson. B. A .. S u th e rla n d . W .. A
T a y lo r. K . 2002. M X R A S . 335. 432
V erde. L. et al. 2003. A p JS . to a p p e a r in v l4 8 n l
W echsler. R. H .. B u llo c k . J. S.. P ritn a c k . J. R .. K ra v ts o v . A . V .. A D ekel. A . 2002.
A p J . 568. 52
W eeks. J. R. 1998. C lass. Q u a n t. G ra v . 15. 2599.
W e tte ric h . C'. 1988. N u c le a r P hysics. B 302.
6 6 8
W illic k . J. A . A S trau ss. M . A . 1998. A p J . 507. 64
Z e n tn e r. A . R. A B u llo c k . J. S. 2002. P hys. R ev. D .
6 6
. 43003
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C h ap ter 4
Im p lica tion s for In flation
Abstract
We c o n fro n t p re d ic tio n s o f in fla tio n a ry scenarios w ith th e \ Y \ l . \ P d a ta , in
c o m b in a tio n w ith c o m p le m e n ta ry sm a ll-sca le C M B m e a surem e n ts and large-scale
s tru c tu re d a ta .
T h e W M A P d e te c tio n o f a la rg e -a n g le a n ti- c o r re la tio n in tin*
te m p e ra tu re p o la riz a tio n cross-pow er s p e c tru m is th e s ig n a tu re o f a d ia b a tic
s u p e rh o riz o n flu c tu a tio n s at th e tim e o f d e c o u p lin g . T h e W M A P d a ta are d escribed
by p u re a d ia b a tic flu c tu a tio n s :
is o c u rv a tu re c o m p o n e n t.
we place an u p p e r lim it on a c o rre la te d C 'D M
U sin g W M A P c o n s tra in ts on th e shape o f th e sca la r
p ow er s p e c tru m and th e a m p litu d e o f g ra v ity waves, we e x p lo re th e p a ra m e te r
space o f in fla tio n a r y m odels th a t is co nsiste n t w ith th e d a ta .
W e place lim it s
on in fla tio n a r y m o d els: fo r e xa m p le , a m in im a lly -c o u p le d Ao ' is d isfa vo re d at
m o re th a n 3-rr u sin g W M A P d a ta in c o m b in a tio n w it h s m a lle r scale C M B and
la rg e scale s tru c tu re su rve y d a ta . T h e lim its on th e p r im o r d ia l p a ra m e te rs u sin g
W M A P d a ta a lo ne are: « , ( * „ = 0.002 M p c - 1 ) = 1 .2 0 :g ;l'f. d n j d \ n k = - 0 . 0 7 7 : ^ 5 ° .4 ( ^ 0 = 0.002 M p c _ l ) = 0 .7 irS :l? (687c C L ), a n d r ( k 0 = 0.002 M p c - 1 ) < 1.28 (9 5 7
C L ).
137
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C h a p t e r 4:
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I m p lic a t io n s f o r In f la t io n
1.
In tro d u ctio n
A n epoch o f accelera te d e xpa n sio n in th e e a rly u niverse, in fla tio n , d y n a m ic a lly
resolves c o sm o lo g ica l puzzles such as h o m o g e n e ity, is o tro p y , and flatness o f the
u niverse (G u rh . 1981: L in d e . 1982: A lb re c h t A' S te in h a rd t. 1982: S ato. 1981). and
generates s u p e rh o riz o n flu c tu a tio n s w ith o u t a p p e a lin g to fin e -tu n e d in it ia l setups
(M u k h a n o v A C h ib is o v . 1981: H a w k in g . 1982: G u th A P i. 1982: S ta ro b in s k y . 1982:
B ardeen et a l.. 1983: M u k h a n o v et a l.. 1992).
D u rin g tin* accelera te d e xpa n sio n
phase, g e n e ra tio n a n d a m p lific a tio n o f q u a n tu m flu c tu a tio n s in sca la r fields are
u n a v o id a b le (P a rk e r. 19G9: B ir r e il A D avies. 1982).
These flu c tu a tio n s become
classical a fte r cro ssin g the event h o rizo n . L a te r d u r in g th e d e c e le ra tio n phase' th e y
re -e n te r th e h o riz o n , and seed the m a tte r and th e ra d ia tio n flu c tu a tio n s observed in
th e universe.
T h e m a jo r ity o f in fla tio n m odels p re d ic t G a u s s ia n , a d ia b a tic , n e a rly scalein v a ria n t p rim o rd ia l flu c tu a tio n s .
These p ro p e rtie s are g e n e ric p re d ic tio n s o f
in fla tio n a r y m odels. T h e co sm ic m icrow ave b a c k g ro u n d ( C M B ) ra d ia tio n a n is o tro p y
is a p ro m is in g to o l fo r te s tin g these p ro p e rtie s , as th e lin e a r ity o f th e C M B a n is o tro p y
preserves basic p ro p e rtie s o f th e p rim o rd ia l flu c tu a tio n s .
In c o m p a n io n papers.
S pergel et al. (2003) fin d th a t a d ia b a tic s c a le -in v a ria n t p r im o r d ia l flu c tu a tio n s fit
th e W M A P C M B d a ta as w e ll as a host o f o th e r a s tro n o m ic a l d a ta sets in c lu d in g
th e g a la x y and th e L y m a n -o p ow er sp e ctra : K o m a ts u et a l. (2003) fin d th a t the
W 'A /.A P C M B d a ta is co n siste n t w ith G au ssia n p r im o r d ia l flu c tu a tio n s . These re su lts
in d ic a te th a t p re d ic tio n s o f th e m ost basic in fla tio n a r y m o d e ls are in g o o d agreem ent
w ith th e d a ta .
W h ile th e in fla tio n p a ra d ig m has been v e ry successful, ra d ic a lly d iffe re n t
in fla tio n a r y m odels y ie ld s im ila r p re d ic tio n s fo r th e p ro p e rtie s o f flu c tu a tio n s :
G a u s s ia n ity . a d ia b a tic ity . a n d n e a r-sca le -in va ria n ce . T o b re a k th e degeneracy a m o n g
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C h a p t e r 4:
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I m p lic a t io n s f o r I n f la t io n
th e m o d els, we need to m easure tin* p rim o rd ia l flu c tu a tio n s precisely. Even a slig h t
d e v ia tio n fro m G aussian, a d ia b a tic , n e a r-s c a le -in v a ria n t flu c tu a tio n s can place s tro n g
c o n s tra in ts on the m o d els (L id d le A" L y th . 2000). T h e C M B a n is o tro p y a ris in g fro m
p rim o rd ia l g ra v ita tio n a l waves can also be a p o w e rfu l m e th o d fo r m o d el te s tin g . In
th is p a p e r, we c o n fro n t p re d ic tio n s o f va rio u s in fla tio n a r y m odels w ith th e C M B
d a ta fro m th e W M A P . C 'B I (P earson et a l.. 2 002). and AC’ B A R (K u o et a l.. 2002)
e x p e rim e n ts , as w ell as th e 2 d F G R S (P e re iv a l et a l..
2 0 0 1
) and L y m a n -o pow er
s p e c tra (C ro ft et al.. 2002: G n e d in A" H a m ilto n . 2002).
T h is p a p e r is o rg a n ize d as fo llo w s. In (j
2
. we show th a t the W M A P d e te c tio n
o f an a n ti-c o rre la tio n betw een th e te m p e ra tu re and th e p o la riz a tio n flu c tu a tio n s
at I ~~ 130 is th e d is tin c tiv e s ig n a tu re o f a d ia b a tic s u p e rh o riz o n flu c tu a tio n s . We
co m p a re th e d a ta w ith sp ecific p re d ic tio n s o f in fla tio n a r y m odels: s in g le -fie ld m odels
in (( 3. a n ti d o u b le -fie ld m o dels in § 4. We e x a m in e th e evidence fo r features in the
in fla to ti p o te n tia l in (j o. F in a lly , we s u m m a riz e o u r re su lts and d ra w conclusio n s in
§
0.
2.
Im p lica tio n s o f W M A P “T E ” D e te c tio n for th e Inflationary P aradigm
A fu n d a m e n ta l fe a tu re o f in fla tio n a ry m o d e ls is a p e rio d o f accelerated
e x p a n s io n in th e ve ry e a rly u niverse.
D u rin g th is tim e , q u a n tu m flu c tu a tio n s
are h ig h ly a m p lifie d , a n d th e ir w a ve len g ths are s tre tc h e d to o u ts id e th e H u b b le
h o riz o n . T h u s , th e g e n e ra tio n o f large-scale flu c tu a tio n s is an in e v ita b le fe a tu re o f
in fla tio n . These flu c tu a tio n s are co he re n t on w h a t a p p e a r to be s u p e rh o riz o n scales
at d e c o u p lin g . W ith o u t a ccelerated e x p a n sio n , th e causal h o riz o n a t d e c o u p lin g is
~ 2 degrees. C a u s a lity im p lie s th a t th e c o rre la tio n le n g th scale fo r flu c tu a tio n s can
be no la rg e r th a n th is scale. T h u s , th e d e te c tio n o f s u p e rh o riz o n flu c tu a tio n s is a
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C h a p t e r 4:
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I m p lic a t io n s f o r I n f la tio n
d is tin c tiv e s ig n a tu re o f th is e a rly epoch o f a c c e le ra tio n .
T ilt ' C O D E D M R d e te c tio n o f large scale flu c tu a tio n s has been som etim es
d escribe d as a d e te c tio n o f su p e rh o riz o n scale flu c tu a tio n s . W h ile th is is th e m ost
lik e ly in te r p r e ta tio n o f th e C O D E re su lts, it is n o t u niqu e . T h e n * an* several possible
m echanism s fo r g e n e ra tin g large-scale te m p e ra tu re flu c tu a tio n s .
F o r e xam ple ,
te x tu re m o d els p re d ic t a n e a rly sc a le -in v a ria n t s p e c tru m o f te m p e ra tu re flu c tu a tio n s
on large a n g u la r scales (Pen et a l.. 1994). T h e C ' O D E d e te c tio n sounded th e d e a th
kn e ll fo r these p a r tic u la r m o d els not th ro u g h its d e te c tio n o f flu c tu a tio n s , b u t
due to th e lo w a m p litu d e o f tin* observed flu c tu a tio n s . The* d e te c tio n o f a co u stic
te m p e ra tu re flu c tu a tio n s is also som etim es evoked as th e d e fin itiv e s ig n a tu re o f
.superhorizon scale flu c tu a tio n s (H u A W h ite . 1997). S trin g and defect m odels do not
p ro d u ce sh a rp a c o u s tic peaks (A lb re c h t et a l.. 199G: T u ro k et a l.. 1998). However,
th e d e te c tio n o f a co u stic peaks in the te m p e ra tu re a n g u la r p ow er s p e c tru m does not
prove th a t th e flu c tu a tio n s are s u p e rh o riz o n , as causal sources a c tin g p u re ly th ro u g h
g ra v ity can e x a c tly m im ic tin * observed peak p a tte rn (T u ro k . 1996a.b). T h e recent
s tu d y o f causal seed m odels by D u rre r et al. (2 0 02 ) shows th a t th e y can re p ro du ce
m uch o f th e observer! peak s tru c tu re and p ro v id e a p la u s ib le fit to th e \ m ' - W M A P
C M B d a ta .
T h e la rg e -a n g le (50
^
150) te m p e ra tu r e - p o la ri/a tio n a n ti-c o rre la tio n
d e te cte d by W M A P (K o g u t et a l.. 2003) is a d is tin c tiv e ' s ig n a tu re o f su p e rh o riz o n
a d ia b a tic flu c tu a tio n s (S pergel
Z a ld a rria g a . 1997). T h e reason fo r th is co n clu sio n
is e x p la in e d as fo llo w s. T h ro u g h o u t th is se ctio n , we co n sid e r o n ly scales la rg e r th a n
th e sound h o riz o n a t th e d e c o u p lin g epoch. Z a ld a rria g a
H a ra ri (1995) show th a t,
in th e tig h t c o u p lin g a p p ro x im a tio n , th e p o la riz a tio n sig n a l arises fro m th e g ra d ie n t
o f th e p e c u lia r v e lo c ity o f th e p h o to n flu id .
0
i-
A E — -0 .1 7 (1 - /r ) A t /dt.r^ 0 i(t/dt,r ).
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(4-1)
C h a p t e r 4:
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I m p lic a t io n s fo r In f la t io n
w here A /. is th e E -m o d e (p a rity -e v e n ) p o la riz a tio n flu c tu a tio n . //,/,,• is th e c o n fo rm a l
tim e at d e c o u p lin g . A t/,/,, is th e thickness o f tin ' su rface o f last s c a tte rin g in
c o n fo rm a l tim e , and // = cos (A- • //). T h e v e lo c ity g ra d ie n t generates a q u a d ru p o le
te m p e ra tu re a n is o tro p y p a tte rn a ro u n d e le ctron s w h ic h , in t u r n , pro du ce s the
E -m o d e p o la riz a tio n . N o te th a t w h ile re io n iz a tio n v io la te s th e a s su m p tio n s o f tig h t
c o u p lin g , the* e xiste n ce o f c le a r a co u stic o s c illa tio n s in the' te m p e ra tu re -p o la riz a tio n
( T E ) and te m p e ra tu re -te m p e ra tu re ( T T ) a n g u la r p o w e r s p e c tra im p ly th a t m ost
( 8 5 ' / ) C M B p h o to n s d e te cte d by W M A P d id in d ee d com e fro m c = 1089 where'
th e tig h t c o u p lin g a p p ro x im a tio n is v a lid . The' v e lo c ity <—
) i is redated to the* p h o to n
d e n s ity H uctuatiem s. (-),). th ro u g h the* c o n tin u ity e'epiation. A(-)t = - 3 ((-)() -r ‘h ). whe're
<t> is B arde-ens c u rv a tu re ' p e r tu rb a tio n . The> observable' te'inpe'rature' flu c tu a tio n s
on large' sc ale's are a p p ro x im a te ly given by A / = B 0 (//,/,.,.) -*- 4M//,/,, ). w here 4* is
the* N e w to n ia n p o te n tia l, w h ich equals —<f> in th e absence o f a n is o tro p ic stress.
Therefore*, ro u g h ly sp e a kin g , th e p h o to n d e n s ity flu c tu a tio n s gemerate te m p e ra tu re
flu c tu a tio n s , w h ile th e v e lo c ity g ra d ie n t generates p o la riz a tio n flu c tu a tio n s .
T h e tig h t c o u p lin g a p p ro x im a tio n im p lie s th a t th e b a ry o n p h o to n flu id is
governed by a sin g le se co n d -o rd e r d iffe re n tia l e q u a tio n w h ic h y ie ld s a se'ries o f
a c o u s tic peaks (P eebles A’ Y u. 1970: H u A S u g iy a m a . 1995):
(B o + * ) + (i 1 + «
o+
+ A’V ' ( 0 o + * ) =
w here th e so un d speed c, is g ive n by c~ = [3(1 -I- /?)]
f\ c <l> "
i J
•
(4" 2)
*. T h e large-scale s o lu tio n to
th is e q u a tio n is (H u A S u g iya m a . 1995)
B 0(t/) + <(>(//) = [ B 0(0 ) + <t>(0)jcos(AT,r/) + A t, J
r / / / '[<!>(//) -
# ( / / ') ] sin [Ac, (// - r f ) J.
(4-3)
a nd th e c o n tin u ity e q u a tio n gives th e s o lu tio n fo r th e p e c u lia r v e lo c ity .
•3c,
= [ B o(0) + < W ) \ s in (A t.,//) - A t, f d r f [ $ ( / /') - ^ { r f ) \ c o s [A r,(r/ - //')].
Jo
(4-4)
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C h a p t e r 4:
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I m p lic a t io n s fo r In fla tio n
These s o lu tio n s (e q u a tio n s (4 -1 ). (4 -3 ). and (4 -4 )) are v a lid regardless o f th e n a tu re
o f th e source o f flu c tu a tio n s .
In in fla tio n a r y m odels, a p e rio d o f a ccelerated e xp a n sio n generates
s u p e rh o riz o n a d ia b a tic flu c tu a tio n s , so th a t tin* firs t te rm in e q u a tio n (4-3)
and (4 -4 ) is n on-zero.
Since 'I' ~
-<I> and B 0 (0 )
on s u p e rh o riz o n scale's, one' o b ta in s A /
~
‘H O ) =
.7 *H 0 ) =
^ ‘H ty /rr)
— ^ ‘Ht/</.r) cos(Av,//</,.,.). and
A/,- ~ 0.17(1 - / r ) H y A//,/,.r ‘H '/ (/rr ) s in (A -rv /,/,,) (see* H u A* S u g iy a m a (1993)
and Z a le la rria g a A f la r a r i (1993) fo r e ie riv a tio n ). The're'fore. the* cross c o rre la tio n is
fo u n d to be*
( A / A /,)
-0 .0 3 (1 - //-)(A -e \A t/,/,,.)P + (/,-) s in ( 2 A-e-,//,/,,.).
w here P,p(k) is the* pe»\ver spe*e-truni o f <1>(?/,/,, ).
(4-3)
The 1 observable* c o rre la tio n
fu n c tio n is e*stimateel as A - '( A /A /..) . C le a rly . th e re is an anti-e-e>rre*latiem pe*ak ne*ar
k<\tUu-r ~~ 3 7T/ 4 . w h ic h co rre sp on d s to I ~ 130: th is is the* d is tin c tiv e s ig n a tu re o f
p rim o rd ia l a d ia b a tic flu c tu a tio n s . In o th e r worels. the* a n ti- c o r re la tio n appe*ars on
s u p e rh o riz o n scales at d e c o u p lin g . because o f the* tnoelulatiem bet\ve*en th e d e n s ity
m ode. cos(k(\ti,trr). anel th e v e lo c ity moele. sin (At , t),l r r). y ie ld in g sin (2 At .,
w hieh
has a pe*ak on scales la rg e r th a n the h o riz o n size. r//,/,,. ~ \/3 r ,r /,/f.r .
C o s m ic s trin g s anel te x tu re s are exam ples o f active* m o dels. In these m odels,
causal fielel d y n a m ic s e*e>ntinuously generate s p a tia l v a ria tio n s in th e e*nergy elensity o f
a fielel. M a g u e ijo et a l. (1996) elescribe th e g en e ra l d y n a m ic s o f a c tiv e moelels. These
m o dels elo n o t have th e firs t te rm in e q u a tio n (4 -3 ) a n d (4 -4 ). b u t th e flu c tu a tio n s
are p ro d u ce d b y th e seconel te rm , th e g ro w th o f <I> anel
T h e same a p p lie s to
p rirn o re lia l is o c u rv a tu re flu c tu a tio n s , w here th e n o n -a e lia b a tic pressure causes ‘t> anel
to g ro w . W h ile th e p ro b le m is m ore c o m p lic a te d , these m o d e ls give a p o s itiv e
c o rre la tio n betw een te m p e ra tu re a n d p o la riz a tio n flu c tu a tio n s on la rg e scales. T h is
p o s itiv e c o rre la tio n is p re d ic te d n ot ju s t fo r te x tu re (S e lja k et a l.. 1997) a n d sca lin g
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C h a p t e r -I:
I m p lic a t io n s f o r I n f la t io n
seed m o d els ( D u r r e r et al.. 2 002). h u t is th e gen e ric s ig n a tu re o f any causal m odels
(H u i'c W h ite . 1 9 9 7 )1 th a t la ck a p e rio d o f accelera te d e xp a n sio n .
F ig u re 4.1 shows the p re d ic tio n s o f th e T E la rg e angle c o rre la tio n p re d icte d in
ty p ic a l p r im o r d ia l a d ia b a tic , is o c u rv a tu re , and causal sca lin g seed m o d e ls com pared
w ith the W M A P d a ta . T h e causal sca lin g seed m o d e l show n is a Hat F a m ily 1
m o d el in th e cla ssiH ca rio n o f D u rre r et al. (2002) th a t p ro v id e d a g o o d fit to the
p re -W ’A /.A P te m p e ra tu re d a ta .
T h e \ \ M A P d e te c tio n o f a T E a n ti-c o rre la tio n at I
eorres,34
1
o() — 130. scales th a t
to s u p e rh o riz o n scales at the epoch o f d e c o u p lin g , ru le s o u t a b ro ad class
o f a c tiv e m odels. It im p lie s th e existence o f s u p e rh o riz o n . a d ia b a tic flu c tu a tio n s at
d e c o u p lin g . I f these flu c tu a tio n s were generated d y n a m ic a lly ra th e r th a n by s e ttin g
sp ecial in it ia l c o n d itio n s th e n th e T E d e te c tio n re q uires th a t th e universe had a
p e rio d o f accelera te d e xp a n sio n . In a d d itio n to in fla tio n , th e p re -B ig -B a n g scenario
(G a s p e rin i .k: V eneziano. 1993) and the E k p y r o tic scenario ( K h o u r y et a l.. 2001.
2 0 0 2
) p re d ic t t in 1 e xistence o f s u p e rh o riz o n flu c tu a tio n s .
3.
S in g le F ield Inflation M o d els
In th is se ctio n we e xp lo re how p re d ic tio n s o f sp e cific m o d e ls th a t im p le m e n t
in fla tio n (see L y th .k: R io tto (1999) fo r a su rve y) c o m p a re w ith c u rre n t obse rva tio ns.
1
H u «k: W h ite (1997) use an o p p o s ite sign c o n v e n tio n fo r th e T E cross pow er
s p e c tru m .
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C h a p t e r 4:
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I m p lic a t io n s f o r I n f la tio n
3.1.
In tro d u ctio n
T h e d e fin itio n o f "s in g le -fie ld in fla tio n " encom passes th e class o f m odels in
w h ie h th e in fla tio n a r y epoch is d escribe d by a sin g le sca la r fie ld , th e in fla to n fie ld .
W e also in c lu d e a class o f m o d els ca lle d " h y b r id " in fla tio n m o d els as s in g le -fie ld
m odels. W h ile h y b rid in fla tio n re q uires a second fie ld to end in fla tio n (L in d e . 1994).
th e second fie ld does n ot c o n trib u te to th e d y n a m ic s o f in fla tio n o r th e observed
flu c tu a tio n s . T h u s, th e p re d ic tio n s o f h y b rid in fla tio n m o d e ls can be s tu d ie d in the’
c o n te x t o f s in g le -fie ld m odels.
D u rin g in fla tio n the p o te n tia l ene rg y o f th e in fla to n fielel I
over the’ k in e tic energy.
d o m in a te s
T h e Frie'elmann e q u a tio n them tedls us th a t the*
e’xpansiem rate1. H . is ne*arly co n s ta n t in time*: H = h / n ~
A /pi = ( 8 ~G’ ) ~ l , “ - t o , , | / \ / 8 rr = 2.4 x
1 ()1H
A /p| ' ( \ / 3
) 1
where 1
G e \ ‘ is the* re d uce d P la n c k energy. The 1
universe 1 th u s undergoes an a ccelerated e xp a n sio n phase1, empaneling e x p o n e n tia lly
as a( t ) x e x p ( / Hrft) ~ e x p ( H t ) . O ne u s u a lly use’s the- e -fo ld s reu na in in g at a givem
tim e . ,V(.M. as a m easure o f how m uch the 1 universe 1 expanels fro m t te» the 1 enel o f
in fla tio n .
-V (/) = ln [n (f,.r„ i )] - ln [r/(f)] =
H(t)dt.
It is kn o w n th a t flatness
and h o m o g e n e ity o f th e u nive rse re q u ire -V (/Stiirt) > of), w h e re fstart is th e tim e at the
onset o f in fla tio n (i.e .. th e universe ne’eds to be e xp a n d e d to a t least e ,0 ~ o x 10JI
tim e s la rg e r by f,.n(i). T h e a ccelerated e xp a n sio n o f th is a m o u n t d ilu te s any in it ia l
in h o m o g e n e ity and s p a tia l c u rv a tu re u n t il th e y becom e n e g lig ib le in th e observable
universe to d a y.
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C h a p t e r 4:
I m p lic a t io n s f o r I n f la t io n
3 .2 .
Fram ew ork for d a ta a n a ly sis
Par amet eri zi ng the pr i mor di al poi rer spectra
Tht> p ow er s p e c tru m o f th e C M B a n is o tro p y is d e te rm in e d by th e pow er
s p e c tra o f th e c u rv a tu re and te n s o r p e rtu rb a tio n s .
M o s t in fla tio n a r y m odels
p re d ic t sca la r a n d te n s o r p ow er sp e ctra th a t a p p ro x im a te ly fo llo w p ow er laws:
Here. 7Z is th e c u rv a tu re p e r tu rb a tio n in the c o rn o v in g gauge. a n d / / . and h . are the
tw o p o la riz a tio n sta te s o f the p rim o rd ia l te n so r p e r tu rb a tio n . T h e s p e c tra l indices
tt.s and i>t v a ry s lo w ly w ith scale, o r not at a ll. As s p e c tra l in d ice s d e v ia te m ore and
m o re fro m scale in v a ria n c e (i.e ..
= I and n, = 0 ). th e p o w e r-la w a p p ro x im a tio n
u s u a lly becom es less a n d less a ccu ra te .
T h u s, in g en e ra l, one m u st co n s id e r the
scale d e p e nd e nt " r u n n in g " o f th e s p e c tra l indices, d n j d \ u k a n d d n , / d \ n k.
We
p a ra m e te riz e these p o w e r sp e ctra by
« « ( Ar<) i - 1 - - { t i n , t l U i k ) \ t u k
k ti)
(4-6)
rtt (ko)~ t (dnt
d \ n k ) I r i ( Ar k t) )
(4-7)
w h e re A ^ A 'o ) is a n o r m a liz a tio n c o n s ta n t, a n d kQ is som e p iv o t w a ve n u m b e r. T h e
ru n n in g . d n / d \ n k . is d efin e d b v th e second d e riv a tiv e o f th e p ow er s p e c tru m .
dn/dhik =
( P S i / d In k'~. fo r b o th th e sca la r a n d th e te n s o r m odes, and is
in d e p e n d e n t o f k. T h is p a ra m e te riz a tio n gives th e d e fin itio n o f th e s p e c tra l in d e x .
d In A
(4-8)
d In k
fo r th e sca la r m odes, a n d
d In A i
d In k
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(4-9)
C h a p t e r 4:
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I m p lic a t io n s f o r In f la t io n
fo r th e ten so r m odes. Iri a d d irio ti. we re -p a ra m e te riz e th e te n so r p o w e r s p e c tru m
a m p litu d e . Aj;(A-0 ). by tin * "te n s o r/s c a la r ra tio r " . th e re la tiv e a m p litu d e o f the*
te n s o r to sca la r m odes, give n b y-
r=IKT h e r a tio o f th e te n so r q u a d rtip o le to th e scalar q u a d ru p o le . r-,. is o fte n q u o te d when
re fe rrin g to th e te n s o r/s c a la r ra tio . T h e re la tio n betw een r 2 and th e d e fin itio n o f the
te n s o r/s c a la r r a tio above is so m e w ha t co sm o lo g y-d e p e n d e n t. For an S C 'D M universe
w ith no re io n iz a tio n , it is:
r , = 0.8625 r.
( 1- 1 1 )
F or c o m p a ris o n , fo r th e m a x im u m lik e lih o o d sin g le fie ld in fla tio n m o d e l fo r the
U '. \ M P e x t - r 2 d F G R S d a ta sets presented in tlie* ta b le notes o f T a b le 4.1 in §3.3. th is
re la tio n is r> = 0.6332 r.
F o llo w in g n o ta tio n a l c o n ve n tio n s in Spergel et al. (2 0 03 ). we use .4(A „) fo r tin*
s c a la r p ow er s p e c tru m a m p litu d e , w here A( h {)) and A jj(A -„) a rt' re la te d th ro u g h
A 'i(A -„)
Here. T c m h
=
2.725 x
10
=
8 0 0 --
zz
2.95 x 1 0 ' 9 .4(Ao).
h( / t / \ ). T h is
O ne can usee q u a tio n s (4 -6 ).
= A — .-M o )
leva
\-5J
(4-12)
(4-13)
re la tio n is d e riv e d in
V erde et al.
(4 -8 ). a n d (4-9) to e va lu a te .4.tts. a n d
ntat
(2 0 03 ).
a d iffe re n t
w a ve n u m b e r fro m A0. re sp e ctive ly. Hence.
/,
-4(A-i) = .4 (A -0)
\ n,(fc0 ) - 1- ((r/n,
rtln k)
I n(fci Ar(>)
'
(4-14)
CAMB code (L e w is
et a l.. 2 0 0 0 ) a n d r in Leach e t a l. (2 0 02 ). W e have m o d ifie d CMBFAST (S e lja k
- T h is d e fin itio n o f r agrees w ith th e d e fin itio n o f T / S in th e
Z a ld a rria g a . 1996) a c c o rd in g ly to m a tc h th e same c o n v e n tio n .
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C h a p t e r 4:
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I m p lic a t io n s f o r In f la t io n
We have 6 observables (.4. r. ri.. n,. dtts/ d \ n k. d n , / d \ n k ) . each o f w h ich can
be co m p ared to p re d ic tio n s o f an in fla tio n a ry m odel.
T h e c o m p le m e n ta ry a p p ro a ch (w h ic h we do not in v e s tig a te in th is w o rk ) is
to p a ra m e te riz e t in ' p rim o rd ia l p ow er s p e c tru m in a m o d e l-in d e p e n d e n t w ay (see.
fo r e xam ple . W a n g et a l. (1 9 9 9 )). These a u th o rs a n tic ip a te d th a t W M A P has the
p o te n tia l a b ilit y to reveal d e v ia tio n s fro m sca le -in va ria n ce w hen co m b in e d w ith large
scale s tru c tu re d a ta . M u k h e rje e .k: W a n g (2003a.b) e x te n d th is a p p ro a ch and use it
to p u t m o d e l-in d e p e n d e n t c o n s tra in ts on t in 1 p rim o rd ia l p ow er s p e c tru m usin g flu*
pre- W M A P C M B d a ta .
Sl ow roll parameters
In the c o n te x t o f slow ro ll in fla tio n a ry m odels, o n ly th re e " s lo w -ro ll p a ra m e te rs ” ,
plu s th e a m p litu d e o f th e p o te n tia l, d e te rm in e the s ix observables (.4. r. //„. nt.
d n j d h i k . d i i f / d In k). T h u s , one can use th e re la tio n s a m o ng th e observables to
e ith e r reduce th e n u m b e r o f p a ra m e te rs to fo u r, o r cross-check i f tin* slow ro ll
in fla tio n p a ra d ig m is co n siste n t w ith th e d a ta . T h e s lo w -ro ll p a ra m e te rs are defined
by ( L id d le .k: L y th . 1992. 1993):
\ f-
( V’\ 1
■ "
‘ v )
>h
=
A /J I —
ScC
=
=
-Ao/n
p' , (
.
(4-15)
).
V'Vm
—j . . ,-
(4-16)
.
w here p rim e denotes d e riv a tiv e s w ith respect to th e fie ld o.
(4 -r
Here.
q u a n tifie s
"steepness” o f th e slope o f th e p o te n tia l w h ic h is p o s itiv e -d e fin ite . rp q u a n tifie s
"c u rv a tu re ” o f th e p o te n tia l, a n d
u n fo rtu n a te ly o fte n d e n o te d
. (w h ic h is not p o s itiv e -d e fin ite , b u t is
in th e lite ra tu r e because it is a second o rd e r
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C h a p t e r 4:
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I m p lic a t io n s f o r In f la t io n
p a ra m e te r). q u a n tifie s th e t h ir d d e riv a tiv e o f th e p o te n tia l, o r " je r k " . A ll p a ra m e te rs
m ust he sm a lle r th a n one fo r in fla tio n to o e ru r.
We denote1 these "p o te n tia l slow
r o ll" p a ra m e te rs w ith a s u b s c rip t \ ' to d is tin g u is h th e m fro m t in 1 "H u b b le slow
r o ll" p a ra m e te rs o f A p p e n d ix A . G r a tfo n et al. (2003) discuss th e e q u iva le n t set o f
p a ra m e te rs fo r the E k p y ro tie scenario.
P a ra m e te riz a tio n o f slow r o ll m o d els by f x . t/v . and G a vo id s re ly in g on specific
m odels, and enables one to e x p lo re a large* m o d e l space* w ith o u t assum ing a specific
m o d e l.
Each in fla tio n m o d e l p re d ic ts tin * s lo w -ro ll p a ra m e te rs , and hence the*
observable's. A s ta n d a rd slow ro ll a n a lysis give*s o b se rva b le q iia n titic *s in te*rms o f the*
slow ro ll paratne*te*rs te> firs t o rd e r as (se*e* L idelle A- L y th (2000) fo r a re*vie*w).
_
'V' A• L*[>!
*
r,!i
"
24~-Vv
r
=
lG r, .
//, - 1
=
- 6 e v -h 2/a =
n,
=
- 2 r v = —g .
=
._
,
3
_
lG fx r/x - 2 4 f- - 2G = r /a -- —
/•
'
v
32
\
>
<7//,
r/InA-
=
(l n <
(I In k
=
(4 -1 8 )
(4-19)
O
-f- 2/a .
(4-20)
(4 -2 1 )
-
oc
-SI
[ ( / « ., - D - - 4 / / : ] - 2 G -
(4-22)
.
o ’ r \,
4 f v /A - 8 fj. = - ( n , - 1) + ~ '1
8 1 /" '
"
8J
(4’ 23)
3
T h e te n so r t i l t in in fla tio n is a lw a ys red. n t < 0. T h e e q u a tio n rit = —r / 8 is kn o w n as
th e co nsiste n cy re la tio n fo r s in g le -fie ld in fla tio n m o d els ( it weakens to an in e q u a lity
fo r m u lti-fie ld in fla tio n m o d e ls).
We use th e re la tio n to reduce th e n u m b e r o f
p a ra m e te rs . W h ile we have a lso c a rrie d o u t th e a n a ly s is in c lu d in g nt as a p a ra m e te r,
and v e rifie d th a t th e re is a p a ra m e te r space s a tis fy in g th e co n siste n cy re la tio n ,
in c lu d in g n t o b v io u s ly weakens th e c o n s tra in ts on th e o th e r observables. G iv e n th a t
we fin d r is co nsiste n t w ith zero (§ 3 .3 ). th e ru n n in g te n s o r in d e x d n t/ d \ i \ k is p o o rly
c o n s tra in e d w ith o u r d a ta set: thu s, we ig n o re it a nd c o n s tra in o u r m o d els u sin g th e
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C h a p t e r 4:
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I m p lic a t io n s fo r In f la t io n
o th e r fo u r observables (.4. r. n s. d n s/ d In A-) as free p a ra m e te rs.
3.3.
D eterm in in g th e pow er sp e ctru m p aram eters
W e use a M a rk o v C h a in M o n te C a rlo ( M C M C ) te c h n iq u e to e x p lo re the
lik e lih o o d surface.
Verde et al. (2003) d escribe o u r m e th o d o lo g y .
We use the
W M A P T T (H in s h a w et a l.. 2003) and T E (K o g u t et a l.. 2003) a n g u la r pow er
s p e c tra . To m easure th e shape o f th e s p e c tru m (i.e .. n s and d n s/ d \ n k ) a ccu ra te ly ,
we w ant to p ro b e th e p r im o rd ia l p ow er s p e c tru m over as w id e a range o f scales as
possible. T h e re fo re , we also in c lu d e th e C B I (P earson et a l.. 2002) a n d AC’ B A R
(K u o et a l.. 2002) C M B d a ta . L y m a n o forest d a ta (C ro ft et a l.. 2002: G n e d in V
H a m ilto n . 2002). and the 2 d F G R S large-scali* s tru c tu re d a ta (P e re iv a l et a l.. 2001)
in o u r lik e lih o o d ana lysis. We re fe r to th e co m b in e d U '.M .V P + C B I-t-A C ’ B A R d a ta as
WMAPcxt.
In to ta l, th e sin g le fie ld in fla tio n m o d e l is d e scrib e d by an 8 -p a ra m e te r m o d el:
4 p a ra m e te rs fo r c h a ra c te riz in g a F rie d rn a n n -R o b e rts o n -W a lk e r u niverse (b a ry o n ic
d e n s ity SI*,/?*, m a tte r d e n s ity Q mh~. H u b b le c o n s ta n t in u n its o f 100 km s ’ M p W 1
/). o p tic a l d e p th r ) . and 4 p a ra m e te rs fo r th e p rim o rd ia l p ow er s p e c tra (.1. r. n„.
d n s/ d InA-). W h e n we add 2 d F G R S d a ta , we need tw o fu r th e r la rge-scale s tru c tu re
p a ra m e te rs , .i and rrp. to m a rg in a liz e o ver th e shape and th e a m p litu d e o f the
2 d F G R S p ow er s p e c tru m (V e rd e et a l.. 2003). W e ru n M C M C w ith these e ig ht
( W M A P o n ly m o d e l) o r ten ( U ’.\/.4 P e x t-f-2 d F G R S . l \ ’.\/.4 P e x t+ 2 d F G R S -t-L y o
m o d e ls ) p a ra m e te rs in o rd e r to get o u r c o n s tra in ts .
T h e p rio rs on th e m o d e l are: a fla t u niverse, a c o s m o lo g ica l c o n s ta n t e q u a tio n
o f s ta te fo r th e d a rk energy, a n d a re s tric tio n o f r < 0.3.
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Table -1.1.
Param eters For P rim o rd ia l Power Spectra: Single Field ln lla tio ii M odel
Param eter
\ n/.-w*
//,(A „ = 0.002 Mpc ')
r(k „ = 0.002 Mpc
l - y> o il
i hi . Jd lit A'
/!(*•„ = 0.002 Mpc ')
ihh*
1
h
T
Oh
U ’A M P e xt ± 2 d F (!H S a
-i»
< 1.28/0.81/0.-I<
_0
0 7 7
1
* » i » r. n
0.0.V2
0 71 ,0n il10
0.021 ± 0.002
0.127 ± 0 .0 1 7
0.78 ± 0.07
0.22 ± 0.00
0.82 HM.’I
0.12
/ *e xt-f2 d F (!H S ± l.vm an n a
1.18 ± 0.08
1.18’ °0.11
,J
Hi
< l . i i/o.88/o.87
< 0 .0 0 /0 .I8/0.291'
,1 M T , t 0 . 0 I I
0 .0 1.) oniri
0.78 ± 0.00
0.028 ± 0.001
0.181 ± 0.000
n
m o:i
0.01
0.20 ± 0.00
0.88 ± 0.08
I I n - r t 0 .02 H
0.0.)i>
11 - - i II.IIK
0.00
0.021
±
0.001
0.181
0.78
0.18
0.88
±
±
±
±
0.000
0.08
0.00
0.08
aT h c (|iioted values are the mean am i the 0 8 '/ p ro b a b ility level o f the 1 <1 m arginalized like lih o o d . For
b o th \ \ ’A /.\/V x t± 2 d F C H S and \ l ’A /.-\/V x t± 2 d F (IH S ± L y m a n a data sets, the 10 d m a xim u m like liho o d
p oint in tin* M arkov C h a in (1.8 x 10*' steps) for th is m odel is ji hJ i ’ — 0.021, U „ J r - 0.182, h = 0.77,
"(^'n.nii') = 11-8, c(An.i),)i) = 0.12, dn. J i l hi k — -0 .0 8 2 , .1(A|,nii2) = 0.78, r - 0.21, a H - 0.87]. Mere,
A'o.oiu >s A’o = 0.002 M pc '. The m a xim u m like lih o o d m odel in the M C M C using M’A/.-W* data alone
is [»,,/»* = 0.028, i l j r
= 0.122, h = 0.70, n( ku.i m ) =
1.27, / ( A , -- (l.8(i, <l nj <t \ uk = -0 .1 0 ,
• MA'ii.iio') = 0.7-1, t = 0.29]. (Ire a l care must be taken in in te rp re tin g th is p oint.
It is given here for
completeness only, and we do not recommend it for use in any analysis. There is a long, Hal degeneracy
between ii and r , as described in jj.9 Spergel et al. (2008), and th is point happened to lie at the very
bint* edge o f th is degeneracy right at tin* edge o f o n r upper lim it p rio r on r.
This M arkov chain had
e x tra freedom because we are adding three param eters over the m odel discussed in .Spergel et al. (2008),
thereby in tro d u c in g significant new degeneracies (see F igure 1.8).
'’T he 98% upper lim its for the tensor-scalar ra tio are quoted fo r various priors in the fo llo w in g order:
[no p rio r on r l n J d U i k o r //,] / [d n ,/» /ln A = 0] / [//, < 1], The p rio rs were applied to the o u tp u t o f the
MCMC.
C h a p t e r 4:
I m p lic a t io n s fo r In fla tio n
151
T a b le 4.1 shows results o f o u r a n a lysis fo r th e W M A P . W M A P c x i 4 - 2 d F G R S
and W M A P c x x -t- 2 d F G R S 4- L y m a n o d a ta sets. We e v a lu a te / / s. .4. and r in the
fit a t
= 0.002 M p c - 1 . T h us, th is ta b le a n d th e fig ures to fo llo w re p o rt th e re su lts
fo r .4 a n d ris a t A'<, = 0.002 M p c -1 . N o te th a t S pergel et al. (2003) re p o rt these
q u a n titie s e v a lu a te d at A0 = 0.05 M p c -1 (u s in g e q u a tio n s (4-14) and (4 -8 )). T h e re
are 3.2 r fo ld s betw een A() = 0.002 M p c -1 a n d A0 = 0.05 M p c - 1 .
We d id n o t fin d a n y ten so r m odes. T a b le 4.1 show s 9 5 '/ u p p e r lim its fo r the
te n s o r-s c a la r r a tio r at A- = 0.002 M p c - 1 . fo r v a rio u s c o m b in a tio n s o f the* d a ta sets.
As we w ill see la te r, th e re are s tro n g degeneracies present betw een the p a ra m e te rs
r and d n s/ d In A-. For e xam ple, one can add p ow er at lo w m u ltip o le s by in cre a sin g
/• and th e n rem ove it w ith a b lu e r n , w h ile ke e p in g th e lo w I a m p litu d e c o n s ta n t.
T h u s , one can o b ta in stro n g e r c o n s tra in ts on r by a ssu m in g d iffe re n t p rio rs on u ,
and d n s/ d In A-. In th e ta b le , we lis t th e 9 5 '/ C L c o n s tra in ts on r th a t w o u ld be
o b ta in e d i f (1 ) th e re were no p rio rs on
o r r / n , / r / I n A \ (2) i f one o n ly considers
m o d e ls w ith no ru n n in g o f th e scalar s p e c tra l in d e x , and (3) i f o n ly m o dels w ith red
s p e c tra l in d ice s are considered ( n o n -h y b rid -in tta tio n m o d e ls p re d ic t red in d ices in
g e n e ra l).
T h e n o - p r io r r lim it r < 0.9. a lo n g w ith th e 2 a u p p e r lim it on th e a m p litu d e
.4 (A- = 0.002 M p c - 1 ) < 0.75 4- 0.08 x 2. im p lie s th a t th e energy scale o f in fla tio n
\ ’ 1 1 < 3.3 x 10lfi G e V at th e 9 5 '/ co nfid e n ce level. W e can also p u t a lim it on
th e e n e rg y d e n s ity o f p r im o rd ia l g ra v ity waves fro m in fla tio n . T h is is c a lc u la te d in
A p p e n d ix B.
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C h a p t e r 4:
I m p lic a t io n s fo r I n f la t io n
N o te th a t in tin* case o f the W M A P -o n ly M a rk o v ch a in , the degeneracy between
n s. r a n d r i n j d In A' is cu t o ff by the p r io r r < 0.3 ( r is denegerate w ith n s). T h u s , a
b e tte r u p p e r lim it on r w ill s ig n ific a n tly tig h te n th e c o n s tra in ts on th is m o d el fro m
t in ' W M A P d a ta alone.
A ll c o s m o lo g ica l p a ra m e te rs are co n siste n t w it h the b e s t-fit ru n n in g m o d el o f
S pergel et al. (2 0 0 3 ). w h ic h was o b ta in e d fo r a A C D M m o d e l w ith no tensors and a
ru n n in g s p e c tra l in d e x . A d d in g the e x tra p a ra m e te r r does not im p ro v e th e fit.
O u r c o n s tra in t on r/s shows th a t th e sca la r p ow er s p e c tru m is n e a rly scalein v a ria n t. O ne im p lic a tio n o f th is re su lt is th a t flu c tu a tio n s were generated d u rin g
a ccelera te d e xp a n sio n in n e a rly d e -S itte r space (M u k h a n o v A- C h ib is o v . 1981:
H a w k in g . 1982: G u rh A P i. 1982: S ta ro b in s k y . 1982: B ardeen et a l.. 1983: M u k h a n o v
et a l.. 1992). w here th e e q u a tio n o f s ta te o f th e sca la r fie ld is ic ~ —1. R ecently.
G r a tto n et al. (2003) have show n th a t th e re is o n ly one o th e r p o s s ib ility fo r ro b u s tly
o b ta in in g a d ia b a tic flu c tu a tio n s w ith n e a rly s c a le -in v a ria n t sp e ctra : ir 2> I. T h e
E k p y r o tic /C y c lic scenarios co rre sp o n d to th is case. N o te, how ever, th a t p re d ic tio n s
fo r th e p rim o rd ia l p e r tu rb a tio n s p e c tru m re s u ltin g fro m th e E k p y ro tie scenario are
c o n tro v e rs ia l (see. fo r e xa m p le . T s u jik a w a et a l. (2 0 0 2 )).
W e fin d a m a rg in a l 2rx preference fo r a ru n n in g s p e c tra l in d e x in a ll th re e d a ta
sets: d n s/ ( l \ n k = —O.OoolS.o-ra ( W M A P c x t -f-2 d F G R S -r L y m a n o d a ta se t). T h is
same preference was seen in th e a n a lysis w ith o u t ten so rs c a rrie d o u t in Spergel et al.
(2 0 03 ).
F ig u re 4.2 shows o u r c o n s tra in t on n , as a fu n c tio n o f k fo r the W M A P .
W’A /.A P ext-t- 2 d F G R S and W M A P e x t + 2 d F G R S + L y m a n a d a ta sets. A t each
w a v e n u m b e r k. we use e q u a tio n (4-8) to c o n v e rt ris(ko = 0.002 M p c - 1 ) to n* ( k) a t
each w a ve n u m b e r. T h e n , we e va lu a te th e m ean (s o lid lin e ). 68% in te rv a l (shaded
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 4:
153
I m p lic a t io n s fo r In fla tio n
a rea), a n d 95 ‘X in te rv a l (dashed lines) fro n t th e M C’ M C s. T h is shows a h in t th a t
th e s p e c tra l in d e x is ru n n in g fro m blue (r ;s > 1) on la rg e scales to red ( / / s < 1) on
s m a ll scales. In o u r MC’ M C s. fo r tin* U ’A /.A P d a ta set alone. 91' / o f m o dels e xp lo re d
by th e c h a in have a sca la r s p e c tra l in d e x ru n n in g fro m b lu e a t k = 0.0007 M p c ' 1
(/ ~
10) to red at k = 2 M p c -1 . F o r th e \\W /A P e x t-t-2 d F G R S d a ta set. 95VA o f
m o dels go fro m a b lu e in d e x at large scales to a red in d e x at s m a ll scales, a n d when
L y m a n ft forest d a ta is added, th e fra c tio n ru n n in g fro m b lu e to red becomes 96' A.
O n e -lo o p c o rre c tio n and re n o rm a liz a tio n u s u a lly p re d ic t ru n n in g mass a n d /o r
ru n n in g c o u p lin g c o n s ta n t, g iv in g some d n ^ / d In k. D e te c tio n o f it im p lie s in te re s tin g
q u a n tu m p he n om en a d u rin g in fla tio n (see L y th A' R io tto (1999) fo r a re v ie w ). For
th e ru n n in g o f tin* sca la r s p e c tra l in d e x (e q u a tio n 4 -2 2).
(4-24)
Since th e d a ta re q uires n s ~~ I (see T a b le 4 .1 ). (n , — 1)*’
w hen tt, — 1 ~ 2 t/v . (sec Case A and Case D in
0.01. It is e sp e cia lly s m a ll
3 .4 ). T h e re fo re , i f d n s/ d \ n k is large
eno u gh to d e te c t. d n j d \ w k > 10~“ . th e n d n „ / d \ u k m u st be d o m in a te d b y 2£v . a
p ro d u c t o f th e firs t and th e th ir d d e riv a tiv e s o f th e p o te n tia l (e q u a tio n (4 -1 7 )). T h e
h in t o f d n s/ d In k in o u r d a ta can be in te rp re te d as £ v ~ — \ d n j d In k = 0.028 ± 0 .0 1 5 .
H ow ever, o b ta in in g the ru n n in g fro m b lu e to red. w h ic h is suggested by th e d a ta ,
m a y re q u ire fin e -tu n e d p ro p e rtie s in th e shape o f th e p o te n tia l.
M o re d a ta are
re q u ire d to d e te rm in e w h e th e r th e h in ts o f a ru n n in g in d e x are real.
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C h a p t e r -I:
154
I m p lic a t io n s f o r In fla tio n
3.4.
S in gle field m o d els con fron t th e d ata
Tcst.mii a specific i nflation model: A o l
A s a p re lu d e to sh o w in g c o n s tra in ts on b ro a d classes o f in fla tio n a r y m o d els, we
firs t illu s tr a te th e p ow er o f the d a ta u sin g th e e x a m p le o f th e m in im a lly -c o u p le d
\ ' = A o ' / l m o d e l, w h ic h is o fte n used as an in tr o d u c tio n to in fla tio n a r y m o d e ls
(L in d e . 1990). W e show th a t th is te x tb o o k e x a m p le is u n lik e ly .
T h e F rie d m a n n and c o n tin u ity e q u a tio n s fo r a hom ogeneous sca la r fie ld lead to
th e s lo w -ro ll p a ra m e te rs, w h ich one can use in c o n ju n c tio n w ith th e e q u a tio n s o f § 3.2
in o rd e r to o b ta in p re d ic tio n s fo r th e observables. For th e p o te n tia l 1 (o ) = A o '/ A
one o b ta in s th e p o te n tia l slow ro ll p a ra m e te rs as:
\f-
f> = S '-£ .
o-
\t-
r/v = 12- —
o-
W1
G = 9 6 — fi.
o'
(4 -2 5 )
T h e n u m b e r o f i -fo ld in g s re m a in in g t i l l th e end o f in fla tio n is d efin e d by
w here f v (o,.„a) =
I defines the end o f in fla tio n .
h o riz o n e x it scale as o —
n /8 .V .\/pi
and
A
A s s u m in g o rnd
= 50. one o b ta in s
O. ta k in g th e
= 0.94 a nd r
=
0.32
u s in g e q u a tio n s (4 -1 9 ) a n d (4 -2 0 ). As d n s/ d \ n k is n e g lig ib le fo r th is m o d e l, we use
d n s/ i l In A- = 0.
W e m a x im iz e th e lik e lih o o d fo r th is m o d e l b y ru n n in g a s im u la te d a n n e a lin g
code. W e fit to W *.\/.4 P e x t+ 2 d F G R S d a ta , v a ry in g th e fo llo w in g p a ra m e te rs: Q bI r .
{}m/ r . h. r . .4*. .1 a n d <rp. w h ile keeping n.,. i l n ^ / d h i k . a n d r fixe d a t th e A o ‘
values. T h e m a x iin u rn lik e lih o o d m o d e l o b ta in e d has [ f V r = 0.022. Q mh 2 = 0.135.
'W h ile .4 is an in fla tio n a r y p a ra m e te r, it is d ir e c tly re la te d to th e s e lf-c o u p lin g A
w h ic h we d o n o t kn o w : th u s, we tre a t it as a p a ra m e te r.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 4:
I m p lic a t io n s f o r In fla tio n
r = 0.07. .4 = 0.G7. h = 0.G9. rrH = 0.7G], T h is b e s t-fit m o d e l is co m p a re d in
T a b le 4.2 to th e c o rre s p o n d in g m o d el w ith the fu ll sot o f single' fie ld in fla tio n a r y
p a ra m e te rs. T h e A o ' m o d e l is d isp lace d fro m th e m a x im u m lik e lih o o d g eneric single*
fie ld m o d e l by S \ ; f f = IG [ ± \ j f f ( \ Y U A P ) = 14. A y ^ C B I + A C B A R - J e lF G R S )
= 2], whe*re*
= —2 In £ and C is the> lik e lih o o d (see* V erde e>t al. (2 0 0 3 )). Sinc e*
th e re la tiv e lik e lih o o d be*twee*n the* m o dels is e x p ( —8 ). and the* n u m b e r o f degrees o f
fre'c'dom is a p p ro x im a te ly th re e . Ao* is d isfa vo re d a t more* th a n Her. The* ta b le shows
th a t a d d in g e x te rn a l d a ta sets does n ot make* a s ig n ific a n t differene-e to the* A \ 7 / y
be*twe*en the* m o d e ls, and the* c o n s tra in t is p r im a r ily c o m in g fro m W .M .4P d a ta .
T h is re su lt h o ld s o n ly fo r E in s te in g ra v ity .
Whe*n a ru m -m in im a l c o u p lin g o f
the* fo rm £o~ R ( f = 1/G is the* e o n fo rm a l c o u p lin g ) is added to the* L a g ra n g ia n . the*
c o u p lin g change's the* d y n a m ic s o f o. T h is m o d e l pre'die ts o n ly a t in y a m o u n t o f
re*nsor mode's (K o m a ts u V Futam ase. 1999: H w a n g V N o li. 1998) in agreeme*nt w ith
the* d a ta .
One* can pe*rform a s im ila r a n a lysis on any g ive n in fla tio n a r y m o d e l to se*e* w h a t
c o n s tra in ts the* d a ta p u t on it.
R a th e r th a n a tte m p t th is He*rcule*an ta s k , in th e
fo llo w in g se ctio n we s im p ly use o u r c o n s tra in ts on //,. rl ns/ r l \ i \ k . and r a nd th e
predic tio n s o f v a rio u s classes o f sin g le fie ld in fla tio n a r y m o d e ls fo r these p a ra m e te rs
T a b le 4.2.
M odel
B e s t-fit in fla tio n
A o 1 m odel
Goodness-•o f-fit C o m p a ris o n fo r A o 1 M o d e l
\tff
( WMAP)
1428
1442
\i//
(e x t+ 2 d F G R S )
T o ta l \ ; f f / u
(V \.\/A P e x t+ 2 d F G R S )
36
38
1 4 6 4 /1 3 79
148 0 /1 3 82
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Cha pt e r 4:
I mpl i cati ons for Inflation
in o rd e r to p u t b ro a d c o n s tra in ts on th e m .
Test mi/ a broad class of infl ati on models
N a ive ly, tin* p a ra m e te r space in observables spanned by th e slow ro ll p a ra m e te rs
appears to be large. W e sh a ll show below th a t “ v ia b le " slow ro ll in fla tio n m o dels (i.e.
those th a t can su s ta in in fla tio n fo r a su ffic ie n t n u m b e r o f <-fo ld s to solve co sm o lo g ica l
p ro b le m s) a c tu a lly o ccu p y s ig n ific a n tly s m a lle r regions in th e p a ra m e te r space.
H o ffm a n A' T u rn e r (2001): K in n e y (2002a): E a sth e r A: K in n e y (2002): Hansen A'
K u n z (2 0 02 ): C’a p r in i et al. (2003) have in v e s tig a te d gen e ric p re d ic tio n s o f slow ro ll
in fla tio n m o d e ls by u sin g a set o f in fla tio n a ry How e q u a tio n s (see A p p e n d ix A fo r
a d e ta ile d d e s c rip tio n a nd d e H n itio n o f c o n v e n tio n s ). In p a r tic u la r K in n e y (2002a)
and E a sth e r A K in n e y (2002) use M o n te C a rlo s im u la tio n s to e x te n d th e slow ro ll
a p p ro x im a tio n s to ttfth o rd e r. These a u th o rs fin d " a ttr a c to r s " c o rre s p o n d in g to fixe d
p o in ts (w h e re a ll d e riv a tiv e s o f th e flo w p a ra m e te rs v a n is h ): m o d e ls c lu s te r s tro n g ly
near th e p o w e r-la w in fla tio n p re d ic tio n s , r = 8( 1 — n s) (see § 3 .4 ). and on th e zero
ten so r m odes, r = 0.
F o llo w in g th e m e th o d o f K in n e y (2002a) and E a s th e r A K in n e y (2 0 02 ). we
c o m p u te a m illio n re a liz a tio n s o f th e in fla tio n a ry flo w e q u a tio n s n u m e ric a lly ,
tr u n c a tin g th e flo w e q u a tio n h ie ra rc h y a t e ig h th o rd e r a n d e v a lu a tin g th e observables
to second o rd e r in slo w ro ll u sin g e q u a tio n s (A 1 5 ) (A 1 7 ). W e m a rg in a liz e over th e
a m b ig u ity o f c o n v e rtin g between o and k. in tro d u c e d by th e d e ta ils o f re h e a tin g and
th e ene rg y d e n s ity d u r in g in fla tio n by a d o p tin g th e M o n te C a rlo a p p ro a ch o f th e
above a u th o rs . T h e o bservable q u a n titie s o f a g ive n re a liz a tio n o f th e flo w e q u a tio n s
are e v a lu a te d at a sp e cific value o f e -fo ld in g . .V. H ow ever, o b se rva b le q u a n titie s are
m easured a t a sp e cific value o f k. T h e re fo re , we need to re la te A ' to k. T h is re q uires
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 4:
I m p lic a t io n s f o r In fla tio n
d e ta ile d m o d e lin g o f re h e a tin g . w h ich ca rrie s an in h e re n t u n c e rta in ty . We a tte m p t
to m a rg in a lize ' over th is by ra n d o m ly d ra w in g .V value's From a u n ifo rm d is tr ib u tio n
.V = [40. 70].
F ig u re 4.3 shows p a rt o f th e p a ra m e te r space o f v ia b le slow ro ll in fla tio n m o dels,
w ith th e W M A P 9aVi confid e nce region show n in b lue. Each p o in t on these panels is
a d iffe re n t M onte1 C a rlo re a liz a tio n o f the flo w e q u a tio n s, a n d corre'sponds to a v ia b le
sle>w ro ll m o d e l. N ot a ll p o in ts th a t are viable* slow redl m o d e ls co rre sp o n d te> spe*cific
p h y s ic a l m o d e ls c o n s tru c te d in th e lite ra ture*. M o st o f the* me>ele*ls cluste*r ne*ar the*
a ttra c to rs . sparse'ly p o p u la tin g the* re*st o f th e large* p a ra m e te r space* alle>we*d by the*
slow ro ll c la s s ific a tio n . It m u st be* e'tuphasixed th a t the*se* s c a tte r p lo ts shotllel n ot be*
inte*rpre*te*el in a s ta tis tic a l se*nse* since* we elo n o t kn o w how the* in it ia l c o n d itio n s fo r
the* unive*rse* are* selected. Even if a g ive n re a liz a tio n o f the* flo w e q u a tio n s eloe*s not
s it on the* a ttr a c to r , th is eloe*s not m ean th a t it is n ot favore*d. Each p o in t e>n th is
p lo t carrie*s e qu a l we'ight. and e*ach is a viable* m o d e l o f in fla tio n . Notice* th a t th e
M ’A /.A P d a ta do not lie* p a r tic u la r ly close to th e r = 8( I — //.,) " a ttr a c to r " s o lu tio n ,
at the* 2-rr le ve l, b u t is q u ite co nsiste n t w ith th e r — 0 a ttr a c to r .
O ne m a y ca te g o rize slo w m il m o dels iiite) seweral e-lasses e x p e n d in g upem w here
th e p re 'd ic tio n s lie on th e p a ra m e te r space spanned by n„. d n s/ d In A\ and r (D o d e ls o n
et a l.. 1997: K in n e y . 1998: H a n n e sta d et ah. 2 00 1 ). Each class sh o u ld co rre sp o n d to
s p e cific p h y s ic a l m o dels erf in fla tio n . H e re a fte r, we d ro p th e s u b s c rip t I " unless th e re
is an a m b ig u ity
it sh o u ld o th e rw is e be im p lic it ly assum ed th a t we are re fe rrin g
to th e s ta n d a rd slow rcdl p a ra m e te rs. W e c a te g o rize th e m exlels em the basis o f th e
c u rv a tu re o f th e p o te n tia l r/. as it is the o n ly p a ra m e te r th a t e nte rs in to th e re la tio n
betw een ns a n d r (e q u a tio n (4 -2 0 )). and betw een n s a n d d n ^ / d l n k + 2£ (e q u a tio n
(4 -2 2 )). T h u s , r/ is th e m o st im p o rta n t p a ra m e te r fo r c la s s ify in g th e o b s e rv a tio n a l
p re d ic tio n s o f th e slow r o ll m odels. T h e classes are d e fin e d by
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C h a p t e r 4:
158
I m p lic a t io n s f o r I n f la tio n
(A ) n e g a tive curvature* m o dels. // < 0.
(B ) s m a ll p o s itiv e (o r zero) c u rv a tu re m odels. 0 < '/ < 2 '(C ) in term e dia te * positive* c u rv a tu re m odels. 2 f < rj < 3 f. a n d
(D ) la rg e positive* c u rv a tu re m odels, rj > 3e.
Each class o ccup ie s a c e rta in region in th e parame*te*r space.
U sin g
// = (/;, — l ) n / [ 2 ( o — 3 )]. \vhe*re* t/ = of . one* fin d s
( A ) n, <
1. 0
<
r < !j( I —//.,). - = ( 1 -
n , ) ' < d n j t ! In k — 2£ < 0.
( B)
n, <
1. ^(1 - //,) < r < 8( 1 - n, ) . - = ( I - n J 2 < d n , / d In k 4 - 2£ < 2( 1 - n , )*’ .
(C)
n, <
1. r > 8( 1 — ns). tin , / d In A* -t- 2^ > 2(1 — //,)~ . and
(D )
ns>
I. r > 0. d n , / d \ u k 4- 2£ > 0.
T o firs t o rd e r in slo w ro ll, th e subspace* ( ns. r) is u n iq u e ly divide*d in to the fo u r
classe's. and th e w h o le space spanned by the*se pararne*ters is <le*fine*d by th is
c la s s ific a tio n . T h e d iv is io n o f th e o th e r subs pace ( n, . d n s/ \ n k ) is le*ss u n iq u e . and
d n s/ d \ n k < —2£ — ^(1 — n s)~ is n o t covered by th is c la s s ific a tio n . T o h ig h e r o rd e r in
slow ro ll, these b o u n d a rie s o n ly h o ld a p p ro x im a te ly - fo r in sta n ce . Case C can have
a s lig h tly b lu e sc a la r in d e x. and Case D can have a s lig h tly red one.
W e s u m m a riz e basic p re d ic tio n s o f th e above m o d e l classes to firs t o rd e r in slow
r o ll u sin g th e re la tio n betw een r a nd
h,
(e q u a tio n (4 -2 0 )) re w r itte n as
8
16
r = ^ ( i - fis) + y ' i T h is im p lie s :
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(4‘27)
C h a p t e r 4:
I m p lic a t io n s fo r In fla tio n
159
(A ) n e g a tive c u rv a tu re m o dels p re d ic t t\ < 0 a n d 1 — n s > 0 : th e second te rm
n e a rly cancels th e firs t to g ive r to o s m a ll to d e te c t.
(B ) s m a ll p o s itiv e c u rv a tu re m odels p re d ic t I - ns > 0 and // > 0: a la rg e r is
p ro d u ce d .
(C ) in te rm e d ia te p o s itiv e c u rv a tu re m odels p re d ic t 1 - //, > 0 and r/ > 0: a large r
is p ro d u c e d , a nd
(D ) large p o s itiv e c u rv a tu re m odels p re d ic t 1 — n , < 0 and r/ > 0: th e firs t te rm
n e a rly cancels tin* second to give r to o s m a ll to d e te ct.
T h e c a n c e lla tio n o f th e te rm s in Case A and Case D im p lie s r/„ -
1 ~ 2//: the
steepness o f th e p o te n tia l in Case A and Cast* D is in s ig n ific a n t c o m p a re d to the
c u rv a tu re , r <SC b/j.
O n the' o th e r hand,
in Case B a nd Case C th e steepness is
la rg e r th a n o r c o m p a ra b le to the c iirv a tu n *. by d e fin itio n : th u s, n o n -d e te c tio n o f
r can e xclu d e m a n y m o d els in Case B a m i Case C’ . A s we have show n in § 3.4.
a m in im a lly -c o u p le d A o 1 m o d el, w h ich fa lls in to Case B. is e x clu d e d at high
s ig n ific a n ce , la rg e ly because o f o u r n o n -d e te c tio n o f r (see also § 3.4).
F o r an o v e rv ie w .
F ig u re 4.4 shows
re a liz a tio n s c o rre s p o n d in g to th e m o d el
th e M o n te C a rlo How e q u a tio n
classes A D above on th e ( n ,. r).
( n s. d n j d h i k ) . and ( r . d n s/ d \ n k ) planes, fo r th e W M A P . U ’A /.4 P e x t-i-2 d F G R S
and U A /A P e x t -K 2 d F G R S + L y m a n o d a ta sets.
In T a b le 4.3. we show th e ranges take n by th e o bservables n s. r a n d d n s/ d In k
in th e M o n te C a rlo re a liz a tio n s th a t re m a in a fte r th r o w in g o u t a ll th e p o in ts w h ich
are o u ts id e a t least one o f th e jo in t- 9 5 l/c co nfid e nce levels. These p o in ts have been
se pa ra ted in to th e m o d e l classes A -D v ia th e ir qv . T hese c o n s tra in ts were c a lc u la te d
as fo llo w s. F ir s t, we fin d th e M o n te C a rlo re a liz a tio n s o f th e flo w e q u a tio n s fro m each
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C h a p t e r 4:
1G0
I m p lic a t io n s fo r I n fla tio n
m o d e l class th a t fa ll inside all the jo in t-O o 'X co n fid e n ce levels fo r a given d a ta set.
s e p a ra te ly fo r th e W M A P . l\\\/.4 P e x t+ 2 d F G R S a n d l\\\/.4 P e x t+ 2 d F G R S + L y m a n
o d a ta sets (i.e. tin * m odels show n on F ig u re 4.-1). T h e n we fin d fo r each m o d el
class th e m a x im u m and m in im u m values p re d ic te d fo r each o f th e observables w ith in
these subsets. These c o n s tra in ts mean th a t o n ly those m o d e ls ( w ith in each class)
p re d ic tin g values fo r th e observables th a t lie o u ts id e these lim it s an* e xclu de d by
these d a ta sets at 9 o ‘X C'L. N o te th a t th e b e s t-fit m o d e l w it h in th is p a ra m e te r space
has a
— 14G4/1379. Here, re ca ll a g a in th a t th e observables were e va lu a te d
to second o rd e r in slow ro ll in those c a lc u la tio n s . T h is is th e reason th a t th e Class
C’ range in n , goes s lig h tly blu e and th e Class D range in n , goes s lig h tly red: the
d iv is io n s o f th e i^ c la s s ific a tio n are o n ly exact to firs t o rd e r in slow ro ll.
In th e fo llo w in g subsections we w ill discuss in m o re d e ta il th e c o n s tra in ts on
sp e c ific p h y s ic a l m o d els th a t fa ll in to th e classes A
D. F o r a given class, we w ill
p lo t only th e How e q u a tio n re a liz a tio n s fa llin g in to th a t c a te g o ry th a t are co nsiste n t
w ith th e Oo'X co nfid e nce regions o f all th e planes ( n ,. r ) . ( n s. dn^/ d In I') and ( r .
d n s/ d In k-).
N o te th a t v e ry few m odels p re d ic t a "b a d p o w e r la w " , o r \ d n j d In A'| > O.Oo.
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Chapter
P roperties o f In fla tio n a ry M odels Present W ith in the .lo iu t-9 5 1/
\ YM. \ l >
Model
A
H
C
P
(1 x 10
r < 0.1-1
U ’A /A /V xt + i d l ’CHS
(2 x 10
r < 0.19
WA/ A/ Vxt t-id P C H S + l.vm a n o
(1 X 10 ,i)l,< /• < 0.10
0.9-1 < //, < 1.00
-0 .0 2 < d i i j d l i i k < 0.02
0.95 < n, < 1.00
0.91 < n, < 1.00
-0 .0 1 < t i n, / t l \ u k < 0.02
(7 x 10 :’ )(,< r < O.d")
0.91 < n, < 1.01
-0 .0 2 < t i n, / t l ltd - < 0.02
(7 x 10
/• < 0.52
0.95 < n, < 1.01
-0 .0 1 < dn, / t l hi k < 0.02
0.02 < d n , / d hi k < 0.001
(7 x 10
r < 0.20
(O.OOd)'^ r < 0.50
0.95 < n, < 1.02
(0.005)l,< /• < 0.52
0.90 < n, < 1.02
-0 .0 1 < d n j d ltd - < 0 .0 1
- 0 . 0 1 < d n j d h i k < 0.01
0.0 < r <
0.0 < r < 0.89
1.10
0.99 < /i., < 1.28
-0 .0 9 < t i n, / tl In A- < 0.05
1.00 < n, < 1.28
-0 .0 9 < d n J d U i k < 0.01
0.91 < n, < 1.01
Implications for Inflation
Confidence Hegion'1
4:
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Table -1.5.
-0 .0 2 < # /n ,/r/ In*- <0. 01
(0.05)t,< r < 0,10
0.97 < n, < 1.02
-0 .0 1 < d n , / d i n k < 0.001
(8 x 10 " ) '’< r < 0.89
1.00 < n, < 1.28
-0.0 !) < t l n, / t l \ u k < 0 .0 0 1
aT h e ranges taken by the p re dictt'd observables o f slow ro ll models (to second order in slow ro ll)
w ith in the jo in t f)5(/
C Ls front the specified data sets. T h e model classes are: Case A (// < 0), Case
U (0 < // < 2<), Case C (2f < // < 5 (), Case 1) (// > .’ it).
''T h e lower value o f /• does not represent a detection, but ra th e r the m in im a l level o f tensors predicted
bv any point in the M onte C a rlo th a t falls w ith in in th is class and is consistent w ith the data. We
include tin* lower lim it to help sc! goals fo r fu tu re C.MB p o la riza tio n missions.
5
C h a p t e r -I:
162
I m p lic a t io n s f o r I n fla tio n
Case .4: nei/tit.ire cumi t i i r c models i/ < 0
T h (' to p ro w o f F ig u re 4.6 shows th e M o n te C a rlo p o in ts b e lo n g in g to Case A
w h ic h are co n s is te n t w ith a ll th e jo in t - 9 6 '/ co n fid e n ce re g ion s o f tin* observables
show n in th e fig u re , fo r th e W ".\L 4 P e x t-t-2 d F G R S -l-L y m a n o d a ta set.
T h e n e g a tiv e // m o d els o fte n arise fro m a p o te n tia l o f sp o n ta n e o u s s y m m e try
b re a k in g (e.g.. new in fla tio n - A lb re c h t A' S te in h a rd t (1 9 8 2 ): L in d e (1 9 8 2 )).
W e c o n s id e r n e g a tiv e -c u rv a tu re p o te n tia ls in th e fo rm o f \ ’ = A ' f l — ( o / p ) p\
w here /> > 2. W e re q u ire O < p fo r th e fo rm o f th e p o te n tia l to be v a lid , and A
d e te rm in e s th e ene rg y scale o f in fla tio n , o r th e e ne rg y s to n 'd in a false va cuu m . O ne
fin d s th a t th is m o d e l a lw a ys gives a red t i l t tts <
n, — I =
I to firs t o rd e r in slow ro ll, as
—6 f — 2 |/ / j < 0.
For p =
g ive n b y .V ~
2. th e n u m b e r o f f-fo ld s a t o b e fo re th e end o f in fla tio n is
( / / - / 2 . \ / “, ) ln ( / / / o ) . w here we have a p p ro x im a te d o,.n,i — P-
B y u sin g th e sam e a p p ro x im a tio n , one fin d s n , — 1 ~
—4 ( M p i / p ) 2. and
r ~ 3 2 ( o “ A / p | / / i1) ^ 8(1 — n ,) r ~ - N(1-" * ). In th is class o f m o d e ls, n , ca n n o t be ve ry
close to 1 w ith o u t p b e c o m in g la rg e r th a n m p|.
p ~
lO A/pi ~ 2 m p|.
F o r e x a m p le , n , = 0.96 im p lie s
F o r th is class o f m o d e ls, r has a peak value o f r ~ 0.06 a t
n s = 0.98 (a ssu m in g .V = 60).
Even th is pea k v a lu e is to o s m a ll fo r W M A P to
d e te c t. W e see fro m T a b le 4.3 th a t th is m o d e l is c o n s is te n t w ith th e c u rre n t d a ta ,
b u t re q u ire s p > rnpi to be v a lid .
F o r p > 3. h s -
1 ~ —( 2 / . Y )(p -
1 ) / ( p - 2) o r 0.92 < n„ < 0.96 fo r .V = 60
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C h a p te r T
1G3
I m p lic a t io n s f o r I n fla tio n
regardless o f a value o f //. and r ~ 4
/ pi / / / ) “ (o / / / )~(
11 is n e g lig ib le as o
//.
T hose m o d e ls lie in th e jo in t 2 <r c o n to u r.
T h e n e g a tive
=
a
/
m o d e l also arises fro m th e p o te n tia l in th e fo rm o f
A ' [ l -r- o l n ( o / / / ) ] . a o n e -lo o p c o rre c tio n in a s p o n ta n e o u s ly broken
s u p e rs y m m e tric th e o ry (D v a li et a l.. 199-1). Here the c o u p lin g c o n s ta n t o sh o u ld be
s m a lle r th a n o f o rd e r 1. In th is m o d el o ro lls d o w n to w a rd s the* o rig in . O ne fin d s
—1 = —(1 + -,o )/.V w h ic li im p lie s 0.95 <
< 0.98 fo r I > o > 0 (th is fo rm u la is not
v a lid w hen o = 0 o r o = ft)- Sine*’ r = S aj/.Y = 8o( 1 -r .^aj ) ~ 1( 1 — aj„ ) = 0 .0 lG (o /(). 1).
th e te n so r m o d e is to o s m a ll fo r W’A/.AP to d e te c t, unless the* <95 t '' g a takes its
m a x im a l value, o ~~ 1. T h is ty p e o f m o d el is co n siste n t w ith th e d a ta .
Case D: smal l positive curvature models 0 <
a,
< 2e
T tu 1 second row o f F ig u re T o shows th e M o n te C a rlo [jo in ts b e lo n g in g to Case
B w h ic h are co n siste n t w ith a ll th e jo in t- 9 o ‘X co nfid e nce regions o f th e observables
sh ow n in th e fig u re .
T h e 's m a ll" p o s itiv e tf m odels co rre sp o n d to m o n o m ia l p o te n tia ls fo r 0 < n < 2 f
a n d e x p o n e n tia l p o te n tia ls fo r r/ = 2f. T h e m o n o m ia l p o te n tia ls ta ke th e fo rm o f
\ ' = \ l ( o / p ) p w here p > 2. a n ti th e e x p o n e n tia l p o te n tia ls
= A 1e x p ( o / / / ) . T h e
zero a/ m o d e l is \ ' = A x{ o / p ) . T o firs t o rd e r in slow r o ll, th e sca la r s p e c tra l in d e x
is a lw a ys red. as n s — I = —6 f 4- 2 1) < —4 f < 0. T h e zero rj m o d e l m a rks a b o rd e r
betw een th e n e g a tive ij m o dels and th e p o s itiv e i/ m o d e ls, g iv in g r = |( 1 — n ,) .
T h e m o n o m ia l p o te n tia ls o fte n a p p e a r in c h a o tic in fla tio n m o d e ls (L in d e .
1983). w h ic h re q u ire th a t o be in it ia lly d isp la ce d fro m th e o rig in by a la rg e
a m o u n t. ~ m pi. in o rd e r to a vo id fin e -tu n e d in it ia l values fo r o. T h e m o n o m ia l
p o te n tia ls can have a p e rio d o f in fla tio n a t o ^
rnpi. a n d in fla tio n ends w h e n o
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C h a p t e r 4:
164
I m p lic a t io n s f o r I n f la t io n
ro lls d o w n to n ea r t lit ' o rig in .
F o r p = 2. in fla tio n is d riv e n by tin* mass te rm ,
w h ic h gives o = 2 \ / X M p \ . » , = 1 — 2 /.Y = 0.96. r
- S /.Y = 4(1 — n s) = 0.16.
and d n s/ d \ i \ k = - 2 / S 1 = —( I — n s) - / 2 = - 0 . 8 x 1 0 " '.
F o r p = 4. in fla tio n is
d riv e n by th e s e lf-c o u p lin g , w h ic h gives o = 2 \ / 2 S A /P|. t/, =
r = 1 6 /.Y = f ( l - n j
I — 3 /.Y = 0.94.
= 0.32. and <lns/ d In A- = - 3 / . Y - = - ( l - n s)J/ 3 = - 1 . 2 x 1 0 " '.
T h e m ost s tr ik in g fe a tu re o f the s m a ll p o s itiv e p m o d e ls is th a t th e g ra v ita tio n a l
wave a m p litu d e can be la rg e, r > 0.16.
O u r d a ta suggest th a t, fo r m o n o m ia l
p o te n tia ls to lit* w ith in tin * jo in t 9-VX c o n to u r, r < 0.26 (T a b le 4.3). A A o 1 m o d e l is
e xclu d e d at
3 n (jj 3 .4 ). and a ny m o n o m ia l p o te n tia ls w ith p > 4 are also e xclu de d
at h ig h sig n ifca n ce . M o d e ls w ith p — 2 (m ass te rm in fla tio n ) are co n siste n t w ith the
d a ta .
T h e e x p o n e n tia l p o te n tia ls a p p e a r in th e B ra n s D icke th e o ry o f g ra v ity (B ra n s
lY D icke . 1961: D icke . 1962) c o n fo rm a lly tra n s fo rm e d to th e E in s te in fra m e (th e
e x te n d e d in fla tio n m o d els) (L a A' S te in h a rd t. 1989). O ne fin d s n s = 1 — (/ / / . \ / pi)~ .
r = 8(1 — n ,) . and d n s/ d In A- = 0. T h u s , th e e x p o n e n tia l p o te n tia ls p re d ic t an
exact p o w e r-la w s p e c tru m a nd s ig n ific a n t g r a v ita tio n a l waves fo r s ig n ific a n tly tilte d
sp e c tra . Since // = .Y .\/f; , / ( o - cv n,f)- n * = 1 ~ [.V .\/pi / ( o - C W ) p . T h e 9o'X range
fo r n s in T a b le 4.3 im p lie s th a t o — cy,1(/ > 4 .Y .\/pl ~ 2 0 0 .\/pi ~ 4 0 //tp(.
T h e e x p o n e n tia l p o te n tia ls m a rk a b o rd e r betw een th e s m a ll p o s itiv e p m o dels
and th e p o s itiv e in te rm e d ia te r/ m o d els d e scrib e d b e lo w .
Case D : large positive curvature models p > 3 f
B efo re d e s c rib in g Case C . it is usefu l to d e scrib e Case D firs t. T h e fo u rth ro w o f
F ig u re 4.5 show s th e M o n te C a rlo p o in ts b e lo n g in g to Case D w h ic h are co n sis te n t
w ith a ll th e jo in t - 9 5 1/? co nfid e nce regions o f th e o bservables show n in th e fig u re .
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C h a p t e r 4:
I m p lic a t io n s fo r In f la t io n
165
T in * "la rg e " p o s itiv e c u rv a tu re m o d els c o rre sp o n d to h y b r id in fla tio n m odels
(L in d e . 1994). w h ic h have re c e n tly a ttra c te d m uch a tte n tio n as an /? -in v a ria n t
s u p e rs y m m e tric th e o ry n a tu ra lly realizes h y b rid in fla tio n (C o p e la n d et a l.. 1994:
D v a li et a l.. 1994). W h ile it is p o in te d o u t th a t s u p e rg ra v ity effects add to o large
an e ffe ctive mass to th e in H a ton fie ld to m a in ta in in fla tio n , th e m in im a l K a h le r
s u p e rg ra v ity does n ot have such a la rg e mass p ro b le m (C o p e la n d et a l.. 1994: L in d e
A- R io tfo . 1997). T h e d is tin c tiv e fe a tu re o f th is class o f m odels w ith if >
is th a t
th e s p e c tru m has a b lu e t i l t , u , — I = —Gf -r- l i f > 0. to firs t o rd e r in slow ro ll.
A ty p ic a l p o te n tia l is a m o n o m ia l p o te n tia l p lu s a c o n s ta n t te rm .
C = A * [1 ■+■ ( o / f i ) 1']. w h ich enables in fla tio n to o c c u r fo r a s m a ll value o f o.
o < m ,,|. A t firs t s ig h t, in fla tio n never ends fo r th is p o te n tia l, as th e c o n s ta n t te rm
su sta in s th e ex,4604
C
1 e xp a n sio n forever.
second fie ld n w h ic h c66
H y b rid in fla tio n m o d els p o s tu la te a
' 's to o. W h e n o ro lls s lo w ly on th e p o te n tia l, n stays
at tin* o rig in and has no effect on th e d y n a m ic s . F o r a s m a ll va lu e o f o in fla tio n is
d o m in a te d by a false va cu u m te rm . \ ’ (o. n = 0) 2: A '. W h e n o ro lls d o w n to some
c r itic a l va lu e , a s ta rts m o v in g to w a rd a tru e v a cu u m state*, l ’ ( o . a ) = 0. a n d in fla tio n
ends. A n u m e ric a l c a lc u la tio n (L in d e . 1994) suggests th a t th e p o te n tia l is described
b v o o n ly u n t il o reaches a c r itic a l value. W h e n o reaches th e c r itic a l value*, in fla tio n
sueleienly ends anel a neeel n ot be cemsidered. T h u s , we incluele* th e h y b rid m odels in
o u r eliscussiem o f s in g le -fie ld m o dels.
F o r p = 2. enie finels th a t .V ~
im p lie s // ~
^ ( /e /4 /pi) J ln ( o /o , . n,i) ^
50. w h ic h , in tu rn ,
1 0 .\/Pi ~ 2 m pi fo r ln ( o / o emi) ~ 1- T h e s p e c tra l sletpe is e s tim a te d as
ns ~ 1 + 4 ( A /Pi j p ) 1 ~
1.04. a n d th e tense>r/se-alar ratie>. r ~ 3 2 ( o / / e ) * ( . l/ pi//e ) “ =
8 { o / p ) 2{ ns — 1). is n e g lig ib le as in fla tio n o ccurs a t o <
/e. T h e ru n n in g is also
n e g lig ib le , as d n s/ d l n k ~ 6 4 ( o / p ) ~ ( M pi / p ) 1 = 4 ( o / p ) 2{ ns — 1)J
10- 2 . T h is ty p e
o f m o d e l lies w ith in th e jo in t 955c c o n to u rs.
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C h a p te r 4:
Im p lic a t io n s fo r I n f la t io n
1GG
O n e -lo o p c o rre c tio n in a s o ftly broken s u p e rs y m m e tric th e o ry induces a
lo g a rith m ic a lly ru n n in g mass. \ ' = A 1 {1 -f ( o / p ) 2 { 1 -f- o l n ( o / Q ) ] }. w h e n ' o is
a c o u p lin g co n s ta n t and Q is a re n o rm a liz a tio n p o in t.
d e te rm in e d by
Since //, is p ra c tic a lly
th is p o te n tia l gives rise to a lo g a rith m ic ru n n in g o f n , ( L y th A’
R io tto . 1999). These m o d els w o u ld 1it* in th e re g ion o ccu p ie d by th e M o n te C a rlo
p o in ts th a t have a large, n e g a tive d o s/ d h \ k. T h is ty p e o f m o d e l is consistent w ith
the d a ta .
Cast' C: mtennedi nt e positive r ur ni t nr e models 2t < r / < Hr
T h e th ir d row o f F ig u re 4.5 shows th e M o n te C’<3478
1 ,
s b e lo n g in g to Case
C w h ich are co nsiste n t w ith a ll th e jo in t - 9 5 '/ confid e nce regions o f the observables
show n in th e fig u re .
T h e " in te rm e d ia te " p o s itiv e c u rv a tu re m o d els are d efin e d, to firs t o rd e r in slow
ro ll, as h a v in g a red t i l t , n H — 1 = —Ge + 'hj < 0. o r th e e x a c tly s c a le -in v a ria n t
s p e c tru m , tt, — 1 = 0 .
w h ile n o t b eing d e scrib e d by m o n o m ia l o r e x p o n e n tia l
p o te n tia ls . These c o n d itio n s lead to a p a ra m e te r space where 2f < tj < 3 f. Here
we discuss o n ly e xam ple s o f p h ysica l m o d els th a t do n o t so le ly liv e in Case C . b u t
b rie fly pass th ro u g h it as th e y tr a n s itio n fro m Case D to Case B o r Case A.
T h e tra n s itio n fro m Case D to Case B m a y co rre sp o n d to a sp ecial case o f h y b rid
in fla tio n m o dels d e scrib e d in th e p re vio u s su b se ctio n (Case D ). \ ’ = A 1[ 1 + ( o / p ) p].
W h e n o 3> p. th e p o te n tia l becomes Case B p o te n tia l. I ’ —> A l ( o / p ) p. and the
s p e c tru m is red. rt, < 1. W h e n o <SC p. th e p o te n tia l d riv e s h y b r id in fla tio n , and
th e s p e c tru m is b lu e. n„ > 1. O n th e o th e r h a n d , w hen o ~ p. th e p o te n tia l takes
a p a ra m e te r space som ew here betw een Case B a n d Case D . w h ic h corre sp on d s to
Case C. O ne m a y argue th a t th is m o d e l re q u ire s fin e -tu n e d p ro p e rtie s in th a t we
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C h a p te r 4:
Im p lic a t io n s fo r I n fla tio n
1G7
ju s t tra n s itio n fro m one regim e to the o th e r. H ow ever, th e Case C' regim e has an
in te re s tin g p ro p e rty : th e s p e c tra l in d e x n s ru n s fro m red on large scales to blue on
s m a ll scales, as o undergoes tin* tra n s itio n fro m Case B to Case D. T h is e xam ple has
th e w ro n g sign fo r tin* ru n n in g o f the in d e x co m p a re d to the* d a ta a t th e ~ 2-rr level.
L in d e
R io tto ( 1997) is one e xa m p le o f a tr a n s itio n fro m Cast' D to Case A.
T h e y c o n s id e r a s u p e rg ra v ity -m o tiv a te d h y b rid p o te n tia l w ith a o n e -lo o p c o rre c tio n ,
w h ich can be a p p ro x im a te d d u rin g in fla tio n as
I ' r= A ' [ l
o lt i( o / Q ) •+■ A (o / / / ) 1] .
(-1-28)
S uppose th a t th e o n e -lo o p c o rre c tio n is n e g lig ib le in some e a rly tim e . i.e.. o — Q.
T h e s p e c tru m is b lue. (T h e th ir d te rm is p ra c tic a lly u n im p o r ta n t, as in fla tio n is
d riv e n b y th e firs t te rm at th is stage.)
I f th e lo o p c o rre c tio n becom es im p o rta n t
a fte r several f- fo ld s , th e n » , changes fro m blue to red. as th e lo o p c o rre c tio n gives a
red t i l t as we saw in
.‘3.4. T h is e xa m p le is co n siste n t w it h th e d a ta . T h e tra n s itio n
(fro m Case D to Case A ) is possible o n ly w hen n and Q c o n s p ire to balance th e first
te rm a m i th e second te rm rig h t a t the scale accessible to o u r o b se rva tio n s.
4.
M u ltip le F ield In flation M o d els
4.1.
Fram ew ork
In g en e ra l, a c a n d id a te fu n d a m e n ta l th e o ry o f p a r tic le p hysics such as a
s u p e rs y m m e tric th e o ry requires n o t o n ly one. b u t m a n y o th e r sca la r fields. It is thus
n a tu r a lly e xp e cte d th a t d u rin g in fla tio n th e re m a y e x is t m o re th a n one scalar fie ld
th a t c o n trib u te s to th e d y n a m ic s o f in fla tio n .
In m o st s in g le -fie ld in fla tio n m odels, th e flu c tu a tio n s p ro d u c e d have an a lm o s t
s c a le -in v a ria n t. G a u ssia n , p u re ly a d ia b a tic p o w e r s p e c tru m whose a m p litu d e is
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C h a p t e r 4:
I m p lic a t io n s f o r In f la t io n
1G8
c h a ra c te riz e d by th e e o m o v in g c u rv a tu re p e r tu rb a tio n . R . w h ic h re m a in s co n s ta n t
on s u p e rh o riz o n scales. T h e y also p re d ic t ten so r p e r tu rb a tio n s w ith th e co nsiste n cy
c o n d itio n in e q u a tio n (4 -2 1 ).
W ith th e a d d itio n o f m u ltip le Helds, th e space o f p ossib le p re d ic tio n s
w idens c o n s id e ra b ly .
T h e m o st d is tin c tiv e fe a tu re is th e g e n e ra tio n o f e n tro p y ,
o r is o c u rv a tu re , p e r tu rb a tio n s betw een one Held a nd tin* o th e r.
T in * e n tro p y
p e r tu rb a tio n . S . can v io la te th e co n se rva tio n o f R on s u p e rh o riz o n scales, p ro v id in g
a source fo r th e la te -tim e e v o lu tio n o f R w h ich weakens tin* sin g le Held co nsiste n cy
c o n d itio n in to an u p p e r b o u n d on tin* te n s o r/s c a la r r a tio (P o la rs k i <k; S ta ro b in s k y .
199-3: Sasaki <!c S te w a rt. 199G: C ia re ia -B e llid o A; W ands. 199G). L im its on th e possible
level o f th e e n tro p y p e r tu r b a tio n thus d is c rim in a te betw een th e m u ltip le Held m o dels
and th e sin g le Held m o d els. In th is se ction , we co n s id e r th e m in im a l e xte n sio n to
single-H eld in fla tio n
4.2.
a m o d e l c o n s is tin g o f tw o m in im a lly -c o u p le d sca la r Helds.
C o rrela ted A d ia b a tic /Iso c u r v a tu r e F lu c tu a tio n s from D o u b le-F ield
Inflation
T h e \ \ A L A P d a ta c o n firm th a t p ure is o e u rv a tu re flu c tu a tio n s do n o t d o m in a te
th e observed C’ M B a n is o tro p y . T h e y p re d ic t large-scale te m p e ra tu re a n is o tro p ie s
th a t are to o la rg e w ith respect to th e m easured d e n s ity flu c tu a tio n s , a n d have th e
w ro n g p e a k /tro u g h p o s itio n s in th e te m p e ra tu re a n d p o la riz a tio n p o w e r sp e c tra
(H u
W h ite . 1996: Page et a l.. 2003). T h e W M A P o b s e rv a tio n s lim it b u t do
n o t p re clu d e th e p o s s ib ility o f c o rre la te d m ix tu re s o f is o c u rv a tu re a n d a d ia b a tic
p e rtu rb a tio n s , w h ic h is a g e n e ric p re d ic tio n o f tw o -fie ld in fla tio n m o d els.
B o th
is o c u rv a tu re a n d a d ia b a tic p e rtu rb a tio n s receive s ig n ific a n t c o n trib u tio n s fro m a t
least one o f th e s c a la r fields to p ro d u ce th e c o rre la tio n (L a n g lo is . 1999: P ie rp a o li
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C h a p t e r 4:
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I m p lic a t io n s f o r I n f la t io n
et a l.. 1999: L a n g lo is <k: R ia zu elo. 2000: G o rd o n et a l.. 2001: B a r to lo f t al.. 2001.
2002: A m e n d o la f t a l.. 2002: W a n d s f t a l.. 2002). W e focus on th fS f m ix e d m odels
in th is se ctio n .
Let itrmt ‘»nd S ra,i be th e c u rv a tu re and e n tro p y p e rtu rb a tio n s deep in tin*
ra d ia tio n era. re sp e ctive ly. A t la rg e scales, th e te m p e ra tu re a n is o tro p y is given by
( L a n g lo is . 1999):
(4-29)
in a d d itio n to th e in te g ra te d Sachs W’olfe e ffect.
T h e e n tro p y p e r tu rb a tio n .
S ni,t = ' V v , / m — ( 3 / 1)<ip-. //)-.. re m a in s co n s ta n t on large scales u n til re -e n try in to
th e h o riz o n . I f 1Zril,t and S rn,i have tin* same sign (c o rre la te d ), th e n th e large scale
te m p e ra tu re a n is o tro p y is reduced.
I f th e y have o p p o s ite signs (a n ti-c o rre la te d ),
th e n th e te m p e ra tu re a n is o tro p y is increased. S pergel et al. (2003) fin d th a t there is
an a p p a re n t lack o f p ow er at th e v e ry largest scales in tin ' W’.'MAP d a ta : thu s, one o f
th e m o tiv a tio n s o f th is s tu d y is to see w h e th e r a c o rre la te d S rn,t can p ro v id e a b e tte r
f it to th e W M A P lo w -/ d a ta th a n a p u re ly a d ia b a tic m o d e l.
T h e ('v o lu tio n o f th e c u rv a tu re /e n tro p y p e r tu rb a tio n s fro m h o rizo n -cro ssin g to
th e ra d ia tio n -d o m in a te d era can be p a ra m e te riz e d b y a tra n s fe r m a t r ix (A m e n d o la
et a l.. 2002).
(4-30)
Here. T rr = 1 a n d T sr = 0 because o f th e p h y s ic a l re q u ire m e n t th a t R is conserved
fo r p u re ly a d ia b a tic p e rtu rb a tio n s , and th a t R c a n n o t source S . A ll th e q u a n titie s
in e q u a tio n (4 -3 0 ) are w e a k ly sca le -de p en d e nt, a n d m a y be p a ra m e te riz e d by
p ow er-Iaw s. H ence, we w r ite th is e q u a tio n as
Rrad
—
A r k nidr -r A sk n' u s.
(4-31)
(4-32)
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C h a p t e r 4:
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I m p lic a t io n s fo r I n f la t io n
w hore >ir and ris are in d e p e n d e n t G aussian ra n d o m va ria b le s w ith u n it
va ria nce .
A 2is-(k)
(drn \ )
=
rt'r.„-
T h e c ro s s -c o rre la tio n s p e c tru m
is given
by
- (I "' /' 2~2)('Rm,tSra,i) — .4, Z J A - " ' . O ne m a y d efin e th e c o rre la tio n
co e ffic ie n t u sin g an angle A as
('RrudSr.ui)
sig n ( B ) A sk ’,A
COS A = ---:-------------- :----------- =
:— ■ -= = .
(1Z; Hll) 1 - ( S 2ld) 1
^ A 2k 2" ‘ + .4;’ A--’»*
w here — I < cos A
<
1.
(4-5d)
T h u s, in general, six p a ra m e te rs (.4 r . .4,. cos A . n |.
n 2. n :!) are needed to c h a ra c te riz e th e d o u b le -in fla tio n m o d e l w ith c o rre la te d
a d ia b a tic /is o c u rv a tu r e p e rtu rb a tio n s , w h ile cos A
is sca le -de p en d e nt.
In
o rd e r to s im p lify o u r a n a lysis, we neglect the scale-dependeneo o f cos A :
thu s. //,
=
/>.{ ^
n > and cos A
w r itte n as A ^ ( A )
=
=
signf B ) A J A .
( k 2/ 2 - 2 ) ( H 2
rad) =
T h e powc'r s p e ctra are
( A ; + .4;)A -’" '
=
A-’ A * " - '- 1. and
A jdA -) = { k " ' / 2 - 2) ( S ; nit) = B 2k 2,‘ - = A 2f 2Hk n‘"’ ~ x. W e have d efin e d
-
1 = 2n,
a nd n lso — 1 = 2n> to c o in c id e w ith th e s ta n d a rd n o ta tio n fo r th e sca la r s p e c tra l
in d e x .
T h e "is o c u rv a tu re fra c tio n " defined by / „ „
re la tiv e a m p litu d e o f 3 to i t .
=
13/A d e te rm in e s the
T h e c ro s s -c o rre la tio n s p e c tru m is th e n w r itte n as
A ^ .(A -) = c o s A s j A 2rJ k ) ^ 2 (k) = A'2f , xo cos A k {,,f‘,l~n'
T h e te m p e ra tu re a nd p o la riz a tio n a n is o tro p ie s are g iven by these pow er sp ectra :
_
It,
(T
/
I, \
C’?J
x
-4' I
c r r
*
A 2f ' , a cos A I
rt'iU
[-J7J
1
[//“ d(A-)]‘ .
y
( 0
(4-34)
[^ (A -^ r tA -) ] •
(4-36)
a n d th e to ta l a n is o tro p y is C \ot = C’[‘d -F G '/’ " + 2 C f orr. Here. ry,{k ) is the ra d ia tio n
tra n s fe r fu n c tio n a p p ro p ria te to a d ia b a tic o r is o c u rv a tu re p e rtu rb a tio n s o f e ith e r
te m p e ra tu re o r p o la riz a tio n a n is o tro p ie s . N o te t h a t th e q u a n titie s n ad. n tso. a n d f , so
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C h a p t e r 4:
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I m p lic a t io n s fo r I n f la t io n
aro defin e d a t a sp e cific w a ve n u m b e r k{). w h ich we ta ke to be A0 = 0.05 M p c
1 in rfie
M C’M C . T o tra n s la te th e c o n s tra in t on f ts„ to a n y o th e r w a ve n u tn b e r. one* uses
(4-37)
W e can re s tric t
> 0 w ith o u t loss o f g e n e ra lity .
S ince we can rem ove .4 by
n o rm a liz in g to th e o v e ra ll a m p litu d e o f flu c tu a tio n s in th e W M A P d a ta , we are
le ft w ith 4 p a ra m e te rs . n,llt.
and cos A .
W e neglect th e c o n trib u tio n o f
te n s o r m odes, as th e a d d itio n o f tensors goes in th e o p p o s ite d ire c tio n in te rm s o f
e x p la in in g th e low a m p litu d e o f th e lo w -/ T T p ow er s p e c tru m . We also neglect the
scale-dependence o f n,,,/ and n,stl. as th e y are not w e ll c o n s tra in e d by o u r d a ta sets.
We fit to th e W M A P e x t -r-2dFG R S and W '.\f.4 P e x t-f-2 d F C iR S -‘-L y m a n o d a ta
sets w ith th e 11 p a ra m e te r m o d e l (f2 ft/r. V.mh~. h. r . n,t,{. t),*,,.
cos A . .4. .1
(7p). T h e re su lts o f th e fit fo r th e d o u b le in fla tio n m o d e l p a ra m e te rs are shown
in T a b le 4.4.
F ig u re
4.G shows th e c u m u la tiv e d is t r ib u t io n o f
T h e b e s t-fit
n o n -p rim o r d ia l c o s m o lo g ica l p a ra m e te r c o n s tra in ts are ve ry s im ila r to th e single fie ld
case.
W h ile th e fit trie s to reduce th e large-scale a n is o tro p y w it h an a d m ix tu re o f
correlated is o c u rv a tu re m odes as e xpected (n o te th a t cos A
1 Z r n ii
and
S
r a ,i
< 0 co rre sp o n d s to
h a v in g th e sam e sign, fro m th e d e fin itio n o f in it ia l c o n d itio n s in th e
C M B F A S T co d e ), th is o n ly leads to a s m a ll re d u c tio n in a m p litu d e a t th e q u a d ru p o le .
T a b le 4.5 co m p a re s th e g o o d n e s s -o f-fit fo r th is m o d e l a lo n g w it h th e m a x im u m
lik e lih o o d m o d e ls fo r th e A C D M a nd sin g le fie ld in fla tio n cases. Because \ i f f / v is
n o t im p ro v e d b y th e a d d itio n o f th re e new p a ra m e te rs a n d c o n s id e ra b le p h ysica l
c o m p le x ity , we c o n c lu d e th a t th e d a ta d o n o t re q u ire th is m o d e l. T h is im p lie s th a t
th e in it ia l c o n d itio n s are c o n siste n t w ith b e in g f u lly a d ia b a tic .
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C h a p t e r 4:
I m p lic a t io n s fo r I n f la t io n
T a b le 4.4.
172
C o s m o lo g ic a l P aram ete rs: A d ia b a tic 4- Is o c u rv a tu re M o d e l
P a ra m e te r
U ’A /A P e x t-F 2 d F G R S -f-L y m a n o
< 0.32“
0.9 7 ± 0.03
< 0 .3 3 “
0.95 ± 0.03
-0 .7 6 :;> ;i*
0.82 ± 0.10
0.023 ± 0.001
0.133 ± 0.007
0.072 ± 0.04
0.1G ± 0.0G
-o .7 G :!!;ir;
0.78 ± 0.08
0.023 ± 0 .0 0 1
0.131 ± 0.00G
i
■o
q
II
U ’A /A P e x f+ 2 d F G R S
cos A
A ( k 0 = 0-05 M p c ” 1)
n hi r
ih J r
h
r
0.072 ± 0.04
0.14 ± 0.0G
0.81 ± 0.04
0.84 ± 0.0G
rr.s
“ T h e c o n s tra in t on th e is o c u rv a tu re fra c tio n .
T a b le 4.5.
is a 9 5 '/ u p p e r lim it.
G o o d n e s s -o f-F it C o m p a ris o n fo r A d ia b a tie /Is o e u rv a tu r e M o d e l
M odel
ACDM
S in g le fie ld in fla tio n
A d ia b a tic /Is o c u r v a tu r e
“ These
values
1 4 6 8 /1 3 81
1 4 6 4 /1 3 7 9
1 4 6 8 /1 3 7 8
are
fo r
th e
U '.\/A P e x t+ 2 d F G R S d a ta set. Here
we d o n o t g ive
fo r th e L y m a n
n d a ta , as th e co va ria n ce betw een th e
d a ta p o in ts is n o t k n o w n (V e rd e e t a l..
2003).
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C h a p t e r 4:
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I m p lic a t io n s f o r In f la t io n
5.
S m o o th n ess o f th e Inflaton P o te n tia l
S pergel of al. (2003) p o in t o u t th a t th o rn arc sovoral s h a rp foa tu ro s in tho
W M A P T T a n g u la r p ow er spoor ru m th a t c o n trib u te to th o re d u c e d -\~ y j fo r tho
b e s t-fit m o d e l b e in g
1.09. T h e large \ ; f j m ay re s u lt fro m n e g le c tin g 0.3 V'A
c o n trib u tio n s to th e W'.W.AP T T p ow er s p e c tru m co va ria n ce m a tr ix : fo r e xam ple ,
g r a v ita tio n a l lo u sing o f th e C M B . beam a s y m m e try , a n d n o n -G a u s s ia n itv in noise
m aps.
W hen in c lu d e d , these effects w ill lik e ly im p ro v e th e re d u c e d -\,Jyy o f the
b e s t-fit AC’ D M m o d e l. A t t in 1 m o m e n t we ca n n o t a tta c h a ny a s tro p h y s ie a l re a lity to
these fea tu re s. S im ila r features a p p e a r in M o n te C a rlo s im u la tio n s .
While ire do not claim these i/litehes are rosrnoloijieallij sn/mfieant. it is
in tr ig u in g to co n sid e r w h a t th e y m ig h t im p ly i/ ’ t hoy tu r n o u t to be s ig n ific a n t a fte r
fu r th e r s c ru tin y .
In th is se ction we in ve s tig a te w h e th e r th e re d u c e d -\;!y y is im p ro v e d by fr y in g to
fit one o r m o re o f these "g litc h e s " w ith a fe a tu re in th e in fla tio n a r y p o te n tia l. A d a m s
et al. (1997) show th a t a class o f m o d els d e rive d fro m s u p e rg ra v ity th e o rie s n a tu ra lly
gives rise to in fla to n p o te n tia ls w ith a la rg e n u m b e r o f sudden d o w n w a rd steps. Each
step co rre sp o n d s to a s y m m e try -b re a k in g phase tr a n s itio n in a fie ld co u p le d to the
in fla to n . since th e mass changes s u d d e n ly w hen each tr a n s itio n occurs. I f in fla tio n
o c c u rre d in th e m a n n e r suggested by these a u th o rs , a s p e c tra l fe a tu re is expected
e very 10-15 e -folds. T h e re fo re , one o f these fea tu re s m a y be v is ib le in th e C M B o r
large-scale s tru c tu re sp e ctra .
W e use th e fo rm a lis m a d o p te d by A d a m s e t a l. (2 0 0 1 ). a n d m o d e l th e step by
th e p o te n tia l
(4-38)
w here o is th e in fla to n fie ld , a n d th e p o te n tia l has a ste p s ta r tin g a t o., w ith
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C h a p t e r 4:
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174
am p litud e * and g ra d ie n t d e te rm in e d by c and <1 re sp e ctive ly. In p h y s ic a lly re a lis tic
m odels, th e pres<*nce o f th e ste p does n ot in te r r u p t in fla tio n , b ut a ffe cts d e n s ity
p e rtu rb a tio n s by in tro d u c in g scale-dependent o s c illa tio n s .
A d a m s et al. (2001)
describe tin* p h e n o m e n o lo g y o f these m odels: th e s h a rp e r th e step, th e la rg e r the
am plitude* a nd lo n g e v ity o f th e "rin g in g ." For o u r c a lc u la tio n s o f th e p ow er s p e c tru m
in these* tnoeiels. we n u m e ric a lly integrate* the* K le in G o rd o n e q u a tio n u sin g the*
p re s c rip t io n o f A d a m s et al. (2001).
We* also p h c u o m e u o lo g ic a lly tnode*l a d ip in th e in fla to n p o te n tia l u sin g a to y
tnoele*! o f a G au ssia n d ip cente*reel at o , w ith h eigh t c anel w id th d:
W (O ) =
(4-59)
I 1 - r ‘ *xp
We* fix the* n o il- p r im o r d ia l co sm o log ica l parame*te*rs at the* m a x im u m lik e lih o o d
value's fo r th e A C D M
me>ele>| firte*el to the* W M A P c x t d a ta . [S2h/r = 0.022.
o rnf,- — 0.13. r = 0.11. .4 = 0.74. h = 0.72]. We the*n ru n s im u la te d anne*aling coeles
fo r o n ly the* three* param ete*rs: o s. c. anel e/. fo r e*ach p o te *n tia l. f it t in g to th e W M A P
T T a n d T E d a ta o n ly . F o r th is sectiem. since th is mealed p re d ic ts sh a rp fe*ature*s in
th e a n g u la r pe»wer s p e c tru m . we hael te) m o d ify th e stanelarel C M B F A S T s p lin in g
re s o lu tio n . s p lin in g a t A / = 1 fo r 2 < / < 50 anel A / = 5 fo r I > 50.
T h e b e s t-fit p a ra m e te rs fo u n d fo r e*ach p o te n tia l are g iv e n in Table* 4.6. a lo n g
w ith th e
fo r th e W M A P T T and T E elata.
F ig u re 4.7 shews the*se m o dels
plotte'el a lo n g w ith th e W M A P T T d a ta . T h e b e s t-fit m o d e ls preelie-t fea tu re s in
th e T E s p e c tru m a t sp ecific m u ltip o le s . w h ic h are w e ll b e lo w d e te c tio n , give n th e
e-urrent u nce rta in tie*s. T h e step m o d el d iffe rs fro m th e A C D M m o d e l b y A
th e d ip m o d e l b y A
= 10.
= 6. We are n o t c la im in g th a t these are th e best p ossible
m odels in th is p a ra m e te r space, o n ly th a t these are th e b e s t-fit m o d e ls fo u n d in 8
s im u la te d a n n e a lin g ru n s. N o te th a t th e m odels w ith fe a tu re s were n o t a llo w e d th e
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C h a p t e r 4:
I m p lic a t io n s f o r I n f la t io n
175
freedom to im p ro v e the lit by a d ju s tin g th e c o s m o lo g ic a l p a ra m e te rs.
A ve ry s m a ll fra c tio n a l change in th e in fla to n p o te n tia l a m p litu d e . <•
O .l'X . is
su ffic ie n t to cause sh a rp features in the a n g u la r p o w e r s p e c tru m . M o d e ls w ith m uch
la rg e r c w o u ld have d ra m a tic effects th a t are n ot seen in the W M A P a n g u la r p ow er
s p e c tru m .
These m o d els also p re d ic t sh a rp fea tu re s in th e large-scale s tru c tu re pow er
s p e c tru m .
F ig u re -1.8 shows th e m a tte r p o w e r s p e c tra fo r tin * b e s t-fit s te p /d ip
m odels. F o rth c o m in g large-scale s tru c tu re surve ys m a y lo o k fo r th e presence o f such
features, and test th e v ia b ilit y o f these m o dels.
6.
C o n clu sio n s
W’.\/.-\P has m ade s ix key o b se rva tio n s th a t are o f im p o rta n c e in c o n s tra in in g
in fla tio n a ry m odels.
(a) T h e u niverse is c o n siste n t w ith b e in g Hat (S p e rg e l et a l.. 2003).
(b ) T h e p r im o r d ia l flu c tn a tio n s are d e scrib e d by ra n d o m G au ssia n fields (K o m a ts u
T a b le 4.G.
M odel
S tep
D ip
ACDM
B e s t-F it M o d e ls w ith P o te n tia l F eatures"
o , ( . \ / pl)
c
d (A /p i)
W M AP \ i f f /»
15.5379
0.00091
0.00041
0.01418
0.00847
1 4 2 2 /1 3 39
15.51757
X /A
X /A
X /A
142 6 /1 3 39
143 2 /1 3 42
"W e g ive as m a n y s ig n ific a n t figures as are needed in o rd e r to re p ro d u ce o u r re su lts.
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C h a p t e r -i:
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I m p lic a t io n s f o r I n f la t io n
et a l.. 2003).
(c) We have show n th a t rho W M A P d e te c tio n o f an a n ti- c o r ro la tio n botwoon C M B
to m p o ra tiiro a n d p o la riz a tio n flu c tu a tio n s at 0 > 2° is a d is tin c tiv e s ig n a tu re o f
a d ia b a tic flu c tu a tio n s on s u p e rh o riz o n scales at th o epoch o f d e c o u p lin g . T h is
d e te c tio n agrees w it h a fu n d a m e n ta l p re d ic tio n o f th e in fla tio n a r y p a ra d ig m .
(d) In c o m b in a tio n w ith c o m p le m e n ta ry C M B d a ta (th e C’ B I and th e A C 'B A R
d a ta ), th e 2 d F G R S large-scale s tru c tu re d a ta , a n d L y m a n o forest d a ta .
W M A P d a ta c o n s tra in th e p rim o rd ia l sca la r and te n so r p ow er sp ectra
p re d ic te d by s in g le -fie ld in fla tio n a ry m o dels.
F o r th e sca la r m odes, the
m ean and th o G8‘X e rro r level o f th e 1 d m a rg in a liz e d lik e lih o o d fo r th e
pow er s p e c tru m slo p e and th e ru n n in g o f th e s p e c tra l in d e x are. re sp ective ly.
n„(ko = 0.002 M p e - 1 ) - 1.13 ± 0 . 0 8 and rfns/ d In A- - -O .O oolIJ Jj.];'). T h is value
is in agreem ent w ith d n s/ r l In A- = —0 . 0 3 1 of S pergel et a l. (2003). w h ic h
was o b ta in e d fo r a A C D M m o d e l w ith no tensors and a ru n n in g s p e c tra l in d e x.
T h e d a ta suggest a t th e 2-rr level, b u t do not re q u ire th a t, th o scalar s p e c tra l
in d e x ru n s fro m
> 1 on la rg e scales to //, < 1 on s m a ll scales. I f tru e , the
t h ir d d e riv a tiv e o f th e in fla to n p o te n tia l w o u ld be im p o rta n t in d e s c rib in g its
d y n a m ic s .
(e) T h e U A /A P e x t± 2 d F G R S c o n s tra in ts on n v <Ins/ d \ n k . and r p u t lim its on
s in g le -fie ld in fla tio n a r y m o d e ls th a t give rise to a la rg e te n so r c o n tr ib u tio n and
a red (n , < 1) t i l t . A m in im a lly -c o u p le d A o 1 m o d e l lies m o re th a n 3-rr aw ay
fro m th e m a x im u m lik e lih o o d p o in t. T h e c o n tr ib u tio n to th e
tw o p o in ts fro m W M A P
betw een the
a lone is 14.
( f) W e test tw o -fie ld in fla tio n a r y m odels w it h an a d m ix tu r e o f a d ia b a tic and
C D M is o c u rv a tu re co m p o n e n ts. T h e d a ta d o n o t ju s t if y a d d in g th e a d d itio n a l
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C h a p t e r -I:
I m p lic a t io n s f o r In fla tio n
177
p a ra m e te rs needed fo r th is m o d el. and flu* in it ia l c o n d itio n s a rc consistent
w ith b e in g p u re ly a d ia b a tic .
W M A P b o th c o n firm s tin* basic tenets o f tin* in fla tio n a r y p a ra d ig m and begins to
q u a n tita tiv e ly test in fla tio n a ry m odels. H ow ever, we ca n n o t ye t d is tin g u is h between
b ro a d classes o f in fla tio n a ry the o rie s w h ich have d iffe re n t p h y s ic a l m o tiv a tio n s . In
o rd e r to go b eyo n d m o d e l b u ild in g and le a rn s o m e th in g a b o u t th e physics o f the e a rly
universe, it is im p o rta n t to be able to m ake such d is tin c tio n s at h ig h significance.
T o a c c o m p lis h th is , one re q u ire m e n t is a b e tte r m e a surem e n t o f th e flu c tu a tio n s at
h ig h /. a n d a b e tte r m easurem ent o f r . in o rd e r to bre ak th e degeneracy between
and r.
We n o te th a t an exact sca le -in v a ria n t s p e c tru m (n , = I a nd f/n ,/r /ln A - = 0) is
n ot yet e x c lu d e d at m ore th a n 2n level. E x c lu d in g th is p o in t w o u ld have p ro fo u n d
im p lic a tio n s in s u p p o rt o f in fla tio n , as p h ysica l sin g le fie ld in fla tio n a r y m odels
p re d ic t n o n -ze ro d e v ia tio n fro m exact sca le -in va ria n ce .
We co n c lu d e by sh o w in g th e ten so r te m p e ra tu re a n d p o la riz a tio n pow er
sp e c tra fo r th e m a x im u m
lik e lih o o d s in g le -fie ld in fla tio n m o d e l fo r th e
W M A P o x t + 2 d F G R S + L y m a n n d a ta set. w h ich has te n s o r/s c a la r r a tio r — 0.42
(F ig u re 4 .9 ). T h e d e te c tio n and m easurem ent o f th e g ra v ity -w a v e p ow er sp e c tru m
w o u ld p ro v id e th e n e xt im p o rta n t key test o f in fla tio n .
A ck n ow led gem en ts
T h e W M A P m issio n is m ade possible by th e s u p p o rt o f th e O ffice o f Space
Sciences a t N A S A H e a d q u a rte rs and by th e h a rd a n d ca p a b le w o rk o f scores
o f s c ie n tis ts , engineers, te ch n icia n s, m a c h in is ts , d a ta a n a ly s ts , b u d g e t a na lysts.
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C h a p t e r I:
178
I m p lic a t io n s f o r I n f la t io n
m anagers, a d m in is tr a tiv e s ta ff, and review ers. W e th a n k .Janet W e ila n d and M ich a el
X o lta fo r th e ir assistance w ith d a ta a n a lysis and fig ures.
W e th a n k I'ro s S eljak
fo r his h e lp w ith m o d ific a tio n s to C M B F A S T . H Y P ackno w le d ge s th e s u p p o rt o f
a D o d d s F e llo w s h ip g ra n te d by P rin c e to n U n iv e rs ity .
L Y is s u p p o rte d by N A S A
th ro u g h C h a n d ra F e llo w s h ip P F 2-30 0 22 issued by th e C’h a n d ra X -ra y O b s e rv a to ry
ce n te r, w h ic h is o p e ra te d by th e S m ith s o n ia n A s tro p h y s ic a l O b s e rv a to ry fo r and on
b e h a lf o f N A S A u n d e r c o n tra c t X A S 8 -3 9 0 7 3 . We th a n k M a r tin K u n z fo r p ro v id in g
the causal seed s im u la tio n re su lts fo r Figure' I and W ill K in n e y fo r useful discussions
a b o u t M o n te C a rlo s im u la tio n s o f How e q u a tio n s.
A.
In flation ary F low E q u ation s
We b egin by d e s c rib in g th e h ie ra rc h y o f in fla tio n a r y flo w e q u a tio n s described
by the* gen e ra lize d
"H u b b le S low R o ll" (H S R ) p a ra m e te rs. In th e H a m ilto n -.J a c o b i
fo r m u la tio n o f in fla tio n a r y d y n a m ic s , one expresses th e H u b b le p a ra m e te r d ire c tly
as a fu n c tio n o f th e fie ld o ra th e r th a n a fu n c tio n o f tim e . H = H ( o ) . u n d e r the
a s s u m p tio n th a t o is tn o n o to n ie in tim e . T h e n th e e q u a tio n s o f m o tio n fo r th e fie ld
a nd b a c k g ro u n d a rt' g ive n by:
( A l)
Here, p rim e denotes d e riv a tiv e s w ith respect to o.
E q u a tio n (A 2 ). re ferre d to as
th e Hami lt on- Jac obi Equation, a llo w s us to co n sid e r in fla tio n in te rm s o f H ( o )
ra th e r th a n U ( o ) . T h e fo rm e r, b e in g a g e o m e tric q u a n tity , describe s in fla tio n m ore
n a tu ra lly . G iv e n H ( o ) . e q u a tio n (A 2 ) im m e d ia te ly gives l ( o ). a n d one o b ta in s H ( t )
b y u s in g e q u a tio n ( A l ) to c o n v e rt betw een H ' a n d H . T h is can th e n be in te g ra te d
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C h a p t e r 4:
179
I m p lic a t io n s f o r I n f la t io n
to give a ( t ) i f d esired . since H ( t ) = d / a . R e w ritin g e q u a tio n (A 2 ) as
H-(o)
>-
(A 3 )
3.1/$
we o b ta in
= 3^'r,0)-^
i d
so t iia t th e c o n d itio n fo r in fla tio n ( d / a ) > 0 is s im p ly g ive n by f „ < 1.
T h u s , m u ’ can d efine a set o f HSR p a ra m e te rs in a n a lo g y to th e PSR p aram ete rs
of
3.2. th o u g h th e re is no a s s u m p tio n o f s lo w -ro ll im p lic it in th is d e fin itio n :
{m£)~
s
,
^ 2A/* ( ^ )
A
4
>
IA5)
C
•s / /
'A „ ^
(2 A /pI) ' ^
P
^
^
.
(A 7 )
W e need one m o re in g re d ie n t: th e n u m b e r o f f- fo ld s b efo re th e end o f in fla tio n .
.V is
d efin e d by.
X =
f tr
f°' H
1
do
,
■
f
H dt = I — do = - 7 =--------- I
Jt
Jo O
v ‘2 .\/p| Jo. ^ j f „ ( o )
(A 8 )
w here tr and o r are th e tim e and fie ld value a t th e end o f in fla tio n , a nd .V increases
th e e a rlie r one goes b a rk in tim e (/ > 0 => d X < 0 ). T h e d e riv a tiv e w ith respect to
.V is th e re fo re .
=
<A9)
T h e n , an in fin it e h ie ra rc h y o f in fla tio n a ry "flo w ” e q u a tio n s can be d e fin e d by
d iffe r e n tia tin g e q u a tio n s ( A 4 ) - ( A 7 ) w ith respect to .V:
~
d .\
=
2 ( h ( t]h — ( h )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(A 1 0 )
C h a p t e r 4:
I m p lic a t io n s fo r In f la t io n
180
(All)
T lit* d e fin itio n o f th e sca la r and ten so r p ow er sp e ctra are:
■)
* := « //
Since d e riv a tiv e s w ith respect to w a ve nu m b er k can be expressed w ith respect to .V
as:
_d_
(All)
Is
the observables are given in te rm s o f the HSR p a ra m e te rs to second o rd e r as (S te w a rt
L y th . 1093: L id d le et a l.. 1904).
nH-
r
=
l G r „ [1 + 2 C ( f „ -
1
=
( 2 rhl - 4 r „ ) [ l -
w here C = 4 ( ln 2 - f * ) — 5 and *
2;
(A
( A 15)
//„)]
i ( 3 - 5C > „
- (3 - 5C ) ' i -r ^ ( 3 - C )S „ (A 1 6 )
0.577 is E u le r's c o n s ta n t. N o te th a t, as p o in te d
o u t in K in n e v (2 0 0 2 b ). th e re is a ty p o g ra p h ic a l e rro r in d e fin in g C in L id d le et al.
(1994) th a t was in h e rite d by K in n e y (2002a). We have used th e co rre ct value fro m
S te w a rt
L y th (1993).
F in a lly , th e P SR p a ra m e te rs are g ive n in te rm s o f th e H S R p a ra m e te rs to firs t
o rd e r in slow r o ll as:
(A 18)
t/v - f I=
£1
— 3ev t/v + 3 f
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(A 19)
(A 20)
C h a p t e r t:
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I m p lic a t io n s fo r I n f la t io n
B.
A n e stim a te o f i l (;w
T ensor p ( 'rtu rl) a tio !is generated by in fla tio n are s to c h a s tic in n a tu re , so the
g ra v ity wave p e r tu rb a tio n can be e xpanded in p la ne waves
M ' X ) =
/
k > V ~ , kX + / ' - ( ' k K / - , k x ] .
(Bl)
when* f" is th e p o la riz a tio n te n so r, and « = -*-. x are th e tw o p o la riz a tio n s in the
Transverse Traceless ( T T ) gauge (in w h ich htm = hUj 7 0). T h e stress-energy te n s o r
fo r g ra v ity waves is d efined as
and in th e T T gauge, we have
T*> = •j
(B 3 )
Cf
Thus.
(IH-
f d*k'
<v-> - / ^ / i S —
c
[< /»_ (- k ) M r . k ' ) ) v ^ ,J +
< / » , ( r . k ) / » , ( r . k ') K / " ']
(B 4 )
T h e va ria n ce o f th e p e rtu rb a tio n s in th e h fie ld s can be w r itt e n as
< /,„(- k ) / t a( r . k ' ) ) = |/ia(r. k ) |“ (2 rr );,P;$( k - k ') .
and since
(B o )
= 2. we o b ta in
( h tJh'J) = I
d3k
0 - 2
(2~y
[ | / u ( r . k ) | 2 + |/t <( r . k ) | “ ] .
(B 6 )
W r itin g
h a( T . k ) = ha ( 0 . k ) T ( r . k ) .
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(B 7 )
C h a p t e r -I:
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I m p lic a t io n s fo r I n f la t io n
w here T is th e tra n s fe r fu n c tio n , we have
=
J
k ) |“ + | / i . (0. k ) i “] T~( r. k)
=
[!''-((>• k) |- + |/| - (0. k)!-] t-(-.k-).
(B8)
F ro m the d e fin itio n o f th e p r im o r d ia l te n so r p ow er s p e c tru m .
A j (A’) =
[ |/|.((). k ) | ' -r !//. (0. k)j-] .
(B9)
we o b ta in
(h.j h' J) = j (I In A-A/, ( k)T~( r . k ) .
( BI O)
N ow
Tm = prav =
(B11>
th u s we have
,1 In k
( B 12)
3 2 -6 ’
R e m e m b e rin g th a t i l = p x (8~G ’/ 3 / / (f ). we o b ta in
r /» o n
= ± j ( k ) t - ( r . k)
d\n k
12H $
( B 13)
T h e re fo re .
/r•- X- f)
ftra r — /
, a H k ) T 2( T . k )
d in A-
77-
--------•
,nitv
( B l- l)
O n large scales, we can ta k e th e m a tte r d o m in a te d tra n s fe r fu n c tio n fo r te n s o r
m odes
T (r.k) = 3 ^ 4 ^ At
(B l 5 >
w here j \ is th e s p h e ric a l Bessel fu n c tio n o f o rd e r 1. Hence
7 ( r . A) = -3 A -^ 4 ^ .
KT
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(B 1 6 )
C h a p t e r 4:
183
Im p lic a t io n s fo r I n f la t io n
w here f lic second o rd e r s p h e ric a l Bessel fu n c tio n is given by
(3 /;■ ’ -
1 / r ) sin c — 3 cos : / z2.
F u rth e r, we have th e fo llo w in g d e fin itio n s :
A-
A / i (A’ )
—
A jj(A 'o )
(B 1 7 )
.
I
A'o
A);(A-„j
( B 18)
( A'd )
where
(B 1 0 )
T o e lim in a te //,. we use th e in fla tio n a r y s in g le -fie ld co n siste n cy re la tio n , n, -= —r / 8 .
C o m b in in g tlie se e q u a tio n s, a nd e v a lu a tin g th e m at th e present. 7(). we are left
w ith
- r
ih;w
=
S
3
P ' /A'
7 m l,
J>( A'm)
P
(B20)
A'm)
'■(i) l"*- dk I k
jo
m
A*
-r s
J ( A' M) )
A-'
A'm)
\ A*o
(B 2 1 )
w here A- and r0 are to be e v a lu a te d in u n its o f 1 /M p c a nd A() = 0.002 M p c '. We
can now change to th e d irn e n sio n le ss v a ria b le s = k r 0 a nd o b ta in
2.21 x H C '1 r A ( k 0) s<,r/H r
<Tnc =
Jo
(Is X
l-r
8
j-A-r)
x
(B 2 2 )
w here x 0 = k0r0. Now
f (I-yj/1—- n—m+ S lrn/
A/on. —
7(1
(B 2 3 )
a n d / / 0r 0 = 3.35 fo r f>,„ = 0.29. T h e n we o b ta in
Or.vc = 1-97 x I Q ' 10 r A ( k 0 )
ds -r ‘ _r "
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(B 2 4 )
C h a p te r 4:
184
I m p lic a t io n s f o r In f la t io n
T h e in te g ra n d in Eq. '24. / = . r 1~r ,H[j-»(.r)/s\~. is show n in F ig . 4.10 fo r r = 0.9.
T h is in te g ra l is q u ite in s e n s itiv e to the valiu* o f r: its value changes by UYA fro m
0.0822 a t r = 0.1 to 0.0734 at r = 0.9. T h e in te g ra n d peaks at .r — 3. T h is im p lie s
th a t th e m a x im u m c o n trib u tio n to
is c o m in g fro m A- = 3 / / ()/3 .3 o . w h ic h is 1.1
tim e s th e h o riz o n size.
F in a lly , using h = 0.73 (a g a in , th e q u a n tity f r r H is re la tiv e ly in s e n s itiv e to the
value o f r and ft), we o b ta in
O f;u - 2.09 x 10 11 (,-.4)
(B 2 3)
T h e lim it fro m th e U W /.-V P ext-f-'idF G R S -^-Lyo d a ta is g ive n by r.-l = 0.29 ± 0 .1 8 .
Hence, ra k in g th e m ean value. Ih vu
— 6.07 x 10“ u .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 4:
I m p lic a t io n s fo r I n f la t io n
180
F ig .
4.1.
T e m p e ra tu re -P o la riz a tio n a n g u la r p o w e r s p e c tru m .
T h e la rg e -a n g le
T E p o w e r s p e c tru m p re d ic te d in p rim o rd ia l a d ia b a tic m o d e ls (s o lid ), p rim o rd ia l
is o c u rv a tu re m o d e ls (d a she d ), a n d in causal sc a lin g seed m o d e ls (d o tte d ).
The
H W /.4 P T E d a ta (K o g u t et a l.. 2003) is show n fo r c o m p a ris o n , in b in s o f A l = 10.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C T ia p f^r 4:
2. 0
>
-
186
I m p lic a t io n s fo r I n f la t io n
20
WMAP
WMAP
WMAP
2dFGRS
o
1 .5
Lva
♦
.0
M
M
7
7
I
r
M
e
2dPGRS
0 .5
0 .5
I
0 .0 .
to-* 10-* 10** 10“'
wovenumber m [Mpc~( ]
1
o.o
10-* 10"* 10"* 10“ *
0.0
I
10 *4
10” S
1 0 "*
1 0 -'
1
•ov«number ii [Mpc~*]
■Mnuf.Br . [M
p
c~
*]
F ig .
4.2.
T h is fig u re shows n , as a fu n c tio n o f A- fo r th e W M A P (le ft).
U '.\/.4 P e x t+ 2 d F G R S (m id d le ) a nd U A /.A P e x t + 2 d F G R S -i-L y m a n o (r ig h t) d a ta s e ts .
T h e m ean (s o lid lin e ) and th e G8% (shaded area) a n d 95% (dashe'd lines) in te rv a ls
are show n. T h e scales p robed by W' M A P. 2 d F G R S and L y m a n n
are in d ic a te d on
th e fig ure.
-o.oo
05
C - 0 .0 5
-
S'
0.4 0.6 0 .8 1.0 1.2
0.0 02 M pc-
0.2 0.4 0 .6 0 .8 1.0 1.2
n , a t k - 0 .0 0 2 M pc"
0.10
-0 .1 5
0.5 1.0 1.5 2.0 2.5
te n s o r/s c a la r rotio r
F ig . 4.3. — T h is set o f figures show s p a rt o f th e p a ra m e te r space spanned by v ia b le
slow ro ll in fla tio n m o dels, w ith th e W ' M A P 68% co n fid e n ce re g io n show n in d a rk blue
a n d th e 95% co nfid e n ce re g ion show n in lig h t b lue.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r -I:
187
Im p lic a t io n s f o r I n fla tio n
n
o>
0 .8
1.0
1.2
1 .4
n , o l k -0 .0 0 2 M p c "'
1.
-0 .2 5
0 .8
- 0 .2 5
0 .5
1.0
1.5
te n s o r/s e o la r rotio r
1.0
1.2
1.4
n , a t k -0 .0 0 2 Mpc
v _
E -0 .1 5
-
0.20
-0 .2 5
0 .8
1.0
1.2
1 .4
nt ot k -0 .0 0 2 M p c "'
1.6
0 .8
0 .5
1.0
1.5
te n s o r/s c a lo r rotio r
1.0
1.2
1.4
n , a t k -0 .0 0 2 Mpc'
-0.10
o 0.5
-
0.20
- 0 .2 5
0 .8
1.0
1.2
1.4
n , at k -0 .0 0 2 M p c "'
1.6
1.0
1.2
1.4
n . a t k -0 .0 0 2 M pc"
0.0
0 .5
1.0
1.5
te n s o r/s c a lo r ro tio r
F ig .
4.4. — T h is set o f fig ures co m p ares th e fits fro m th e W M A P (to p ro w ).
H ‘.\/.4 P e x t-t-2 d F G R S (m id d le ro w ) a n d W M A P e x l - j - 2 d F G R S + L y o d a ta (b o tto m
ro w ) to th e p re d ic tio n s o f specific classes o f p h y s ic a lly m o tiv a te d in fla tio n m odels.
T h e c o lo r c o d in g shows m o d e l classes re ferre d to in th e te x t: (A ) red. (B ) green.
(C ) m a g e n ta . (D ) b la ck. T h e d a rk a n d lig h t b lu e re g ion s are th e j o i n t 1 <J and 2 o
re g ion s fo r th e sp ecified d a ta sets (c o n tra s t th is w it h th e 1-d m arginalized 1 a e rro rs
g ive n in T a b le 4.1). We show o n ly M o n te C a rlo m o d e ls th a t are co n s is te n t w ith a ll
th re e 2 - a re g ion s in each d a ta set. T h is fig u re does n o t im p ly th a t th e m odels n ot
p lo tte d are ru le d o u t.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r -I:
188
I m p lic a t io n s fo r I n fla tio n
>0.09
-<U»
- 0 .1 0
-0.19
-0.19
0.09
-0-10
-0.19
-0.09
-O.OB
-0.10
-0.10
-ai9h
% « ft*<L003 MK
F ig . 4 .5 .— T h is set o f figures com p ares th e fits fro m th e U '.\/A P e x t+ 2 d F G R S + L y o
d a ta to th e p re d ic tio n s o f a ll fo u r classes o f in fla tio n m o d els. T h e to p ro w is C lass A
[red d o ts j. T h e second ro w is C lass B [green d o ts ]. T h e t h ir d ro w is C la ss C [m a ge n ta
d o ts ]. T h e b o tto m ro w is C lass D [b la c k d o ts]. T h e d a rk a n d lig h t b lu e regions are
th e jo in t I cr a n d 2 - a regions fo r th e \V .\fA P e x t+ 2 d F G R S + L y a d a ta . W e show o n ly
M o n te C a rlo m o d e ls th a t are c o n siste n t w ith 2 - a re g ion s in a ll p anels. T h is fig u re
does n o t im p ly t h a t th e m o d els n o t p lo tte d are ru le d o u t.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 4:
189
I m p lic a t io n s f o r I n f la t io n
f;so< 0 . 3 3
( 9 5 % CL
c
o
VVMAP e x t + 2 d FC R S
D
_Q
+ Lya)
ui
'td
CD
>
o
0 .4
Z3
E
D
O
0 .0
0.0
0.1
0.2
0 .3
0 .4
0 .5
Lso
F ig .
4.6.
T h e c u m u la tiv e d is t r ib u t io n o f th e is o c u rv a tu re fra c tio n . f ls„. fo r th e
U '.\/.4 P e x t-t-2 d F G R S -fL y m a n a d a ta set.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 4:
190
I m p lic a t io n s f o r In fla tio n
c
/
/
F ig . 4.7.
B e s t-fit m o d els (s o lid ) w ith a step (le ft) and a d ip ( r ig h t) in tin* in fia to n
p o te n tia l, w ith th e W ' M A P T T d a ta . T h e b e s t-fit AC’D M m o d e l to W A /.A P ext d a ta
is show n (d o tte d ) fo r co m p a ris o n .
F ig . 4 .8 .— T h e la rge-scale s tru c tu re p o w e r sp e c tra fo r th e b e s t-fit p o te n tia l s te p ( le ft)
a n d d ip (r ig h t) m odels.
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 4:
191
I m p lic a t io n s fo r I n f la t io n
103
10"3
.
. . .
I
.
3• s-
,____ 1_LI -
10
a
100
1000
l
F ig . 4.9.
T h e te n s o r pow er s p e c tru m fo r th e m a x im u m lik e lih o o d m o d e l fro m a fit
to U '.\/.4 P e x t+ 2 d F G R S d a ta sets. T h e p lo t shows th e T T (s o lid ). E E (d o ts ). B B
(s h o rt dashes) a nd th e a b so lu te value o f T E n e g a tive (d o ts a n d dashes) and p o s itiv e
(lo n g dashes) te n s o r sp ectra .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r -/:
I m p lic a t io n s fo r I n f la t io n
I
0.06
0.05
0.04
0.03
0.02
0.01
2
F ig . 4.10.
4
6
8
T h e in te g ra n d o f Eq. B24
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
10
B ib liograp h y
A d a m s . .1.. C re ssw e ll. B .. Ac East her. R. 2001. Phys. Rev.. D 64. 123514
A d a m s . .1. A .. Ross. G . G .. A: S arka r. S. 1997. N u cl. Phys. B 503. 10")
A lb re c h t. A .. C o u lso n . D .. F e rre ira . P.. X: M a g tie ijo . .1. 1996. Phys. Rev. L e tt.. 7G.
1113
A lb re c h t. A . Ac S te in h a rd t. P. .J. 1982. P hys. Rev. L e tt.. 18. 1220
A m e jid o la . L .. G o rd o n . C .. W ands. D .. A: Sasaki. M . 2002. Phys. Rev. L e tt.. 88.
211302
B arde e n. .1. M .. S te in h a rd t. P. .!.. A- T u rn e r. M . S. 1983. P hys. Rev. D . 28. G79
B a rto lo . N .. M a ta rre s e . S.. A: R io tto . A . 2001. Phys. R ev.. D 64. 123304
. 2002. P hys. R ev.. D 65. 103505
B ir r e ll. N. D . & D avies. P. C. \V . 1982. Q u a n tu m fie ld s in cu rv e d space (C a m b rid g e
U n iv e rs ity Press)
B ra n s . C . Ac D icke . R. H. 1961. P h y s ic a l R e vie w . 124. 925
C a p rin i. C .. Hansen. S. H .. Ac K u n z . M . 2003. M X R A S . 339. 212
C o p e la n d . E. J.. L id d le . A . R .. L y th . D . H .. S te w a rt. E. D .. Ac W ands. D . 1994.
P hys. R e v.. D 49. 6410
193
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
194
B IB L IO G R A P H Y
C ro ft. R. A . C \. W e in b e rg . D. H .. B o lte . M .. B urie s. S.. H e rn q u is t. L.. Ka t z . N ..
K irk iu a n . D .. A’ T y t le r . D. 2002. A p .I.
0 8
1. 20
D icke. R. H. 1962. P h y s ic a l R e vie w , vol. 125. Issue
6
. pp. 2163-2167. 125. 2163
D o d elso n . S.. K in n e y . W . H .. A K o lb . E. \V . 1997. Phys. Rev.. D 56. 3207
D u rre r. R.. K u n z . M .. A M e lc h io rri. A . 2002. Phys. R e p t.. 364.
1
D v a li. C . R .. S hafi. Q-. A- Schaefer. R. 1994. Phys. Rev. L e tt.. 73. 1886
E asrher. R. A K in n e y . \V . H. 2002. Phys. Rev. D. s u b m itte d ( a s tr o - p h /0 2 10345)
G a rc ia -B e llid o . .1. A" W ands. D. 1996. Phys. Rev.. D53. 5437
G a s p e rin i. M . A V eneziano. G . 1993. A s tro p a rt. P hys..
1
. 317
G n e d in . N. 5 ’ . A H a m ilto n . A . .1. S. 2002. M N R A S . 334. 107
G o rd o n . C .. W ands. D .. B asse tt. B. A .. A M a a rte n s . R. 2001. P hys. Rev.. D63.
023506
G r a tto n . S.. K h o u ry . .1.. S te in h a rd t. P.. A T u ro k . N. 2003. p re p rin t (a s tro p h /0 3 0 1 3 9 5 )
G u th . A . H. 1981. P hys. Rev. D . 23. 347
G u th . A . H. A P i. S. V . 1982. P hys. Rev. L e tt.. 49. 1110
H a n n e sta d . S.. H ansen. S. H .. A V illa n te . F. L.
2001. A s tro p a rtic le P hysics. 16. 137
Hansen. S. H. A K u n z . M .
1007
2 0 0 2
. M N R A S . 336.
H a w k in g . S. W . 1982. P hys. L e tt.. B 115. 295
H in s h a w . G . F. et a l. 2003. A p .IS . to a p p e a r in v l4 8 n l
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
B IB L IO G R A P H Y
H o ffm a n . M . B. A T u rn e r. M . S. 2001. P hys. R ev.. DG4. 02330G
H u . \V . A S u g iya m a . N . 1995. A p .I. 4 14. 489
H u . W . A- W h ite . M . 199G. Ap.J. 471. 30
. 1997. P hys. Rev. D . 36. 69G
H w ang. .1. A' N o li. H. 1998. P h y s ic a l R e vie w L e tte rs , V o lu m e 81. Issue 24. D ecem ber
14. 1998. p p .3274-3277. 81. 3274
K h o u ry . .1.. O v r u t. B. A .. S eibe rg . X .. S te in h a rd t. P. .J.. A T u ro k . X . 2002. Phys.
Rev.. DG3. 08G007
K h o u ry . .).. O v r u t. B. A .. S te in h a rd t. P. .1.. A T u ro k . X . 2001. Phys. Rev.. DG4.
123322
K in n e y . W . H. 1998. P hys. R e v.. D 38. 12330G
. 2002a. Phys. R ev.. DGG. 083308
. 2002b. p re p rin t (a s tro -p h /0 2 0 6 0 3 2 )
K o g u t. A . et al. 2003. A p .IS . to a p p e a r in v l4 8 n l
K o m a ts u . E. A F u fam a se . T . 1999. Phys. R ev.. D 59. 0G4029
K o m a ts u . E. et a l. 2003. A p .IS . to a p p e a r in v l4 8 n l
K u o . C. L . et a l. 2002. A p .I. a s tr o - p h /0 2 12289
L a . D. A S te in h a rd t. P. J. 1989. P hys. R ev. L e tt.. G2. 376
L a n g lo is . D . 1999. P hys. R ev.. D 5 9 . 123512
L a n g lo is . D . A R ia z u e lo . A . 2000. P hys. R e v.. D 6 2 . 043504
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
B IB L IO G R A P H Y
196
Loach. S. NL. L id d le . A . R .. M a rtin . .1.. A' Schw arz. D . .1. '2002. P hys. Rev. D. 66.
23515
Lew is. A .. C’h a llin o r . A ., A Lasenby. A . 2000. A p .I. 538. 473
L id d le . A . R. A’ L y th . D . H. 1992. Phys. L e tt.. B 291. 391
. 1993. Phys. R e p t.. 231. I
. 2000. C o s m o lo g ic a l in fla tio n a m i large-scale s tru c tu re (C a m b rid g e U n iv e rs ity
Press)
L id d le . A . R .. P arsons. P.. A B a rro w . .1. D. 1994. Phys. R ev.. D 50. 7222
L in d e . A . D. 1982. Phys. L e tt.. B 108. 389
. 1983. Phys. L e tt.. B 129. 177
. 1990. P a rtic le physics and in fla tio n a ry co sm o lo g y (C h u r. S w itz e rla n d : H a rw o o d )
. 1994. Phys. R ev.. D 49. 748
L in d e . A . D . A R io tto . A . 1997. P hys. R ev.. D 56. 1841
L y th . D. H. A R io tto . A . 1999. P hys. R e p t.. 314. 1
M a g u e ijo . .J.. A lb re c h t. A .. C o u lso n . D .. A F e rre ira . P. 1996. P hys. Rev. L e tt.. 76.
2617
M u k h a n o v . V . F. A C h ib is o v . G . V . 1981. .JE T P L e tte rs . 33. 532
M u k h a n o v . V . F .. F e ld m a n . H. A .. A B ra n d e n b e rg e r. R. H. 1992. Phys. R e p t.. 215.
203
M u k h e rje e . P. A W a n g. V . 2003a. 1562. A p .I. s u b m itte d (a s tro -p h /0 3 0 1 5 6 2 )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
B IB L IO G R A P H Y
197
. 2003b. 1058. A p .I. s u b m itte d (a s tro -p h /0 3 0 1 0 5 8 )
Page. L . et al. 2003. A p.IS . to a p p e a r in vl-18 i l l
P arke r. L. 19G9. Phys. R ev.. 183. 1057
Pearson. T . .1.. et al. 2002. A p .I. s u b m itte d (a s tro -p h /0 2 0 5 3 8 8 )
Peebles. P. I. E. A" Y u. .1. T . 1970. A p .I. 162. 815
Pen. I ’ .-L .. S pe rg e l. D. N.. A- T u ro k . N. 1994. P hys. R ev.. D 49. G92
P e rriv a l. \Y . .1.. et al. 2001. M N R A S . 327. 1297
P ie rp a o li. E .. C la re ia -B e llid o . .1.. A B o rg a n i. S. 1999. .Journal o f H ig h E ne rg y Physics.
10. 15
P o la rs k i. D . A S ta ro b in s k v . A . A . 1995. Phys. L e tt.. B3-5G. 19G
S asaki. M . A S te w a rt. E. D. 199G. P rog. T h e o r. P hys.. 95. 71
S ato . K . 1981. M N R A S . 195. 4G7
S e lja k. L’ .. Pen. I ’ .-L .. A T u ro k . N. 1997. P hys. R ev. L e tt.. 79. 1615
S e lja k . L". A Z a ld a rria g a . M . 199G. A p .I. 469. 437
S pergel. D. N . A Z a ld a rria g a . M . 1997. Phys. R ev. L e tt.. 79. 2180
S pergel. D . N . et al. 2003. A p.IS . to a pp e ar in v l4 8 n l
S ta ro b in s k v . A . A . 1982. P hys. L e tt.. B 117. 175
S te w a rt. E. D . A L y th . D. H. 1993. Phys. L e tt.. B 30 2 . 171
T s u jik a w a . S.. B ra n d e n b e rg e r. R .. A F in e lli. F. 2002. P hys. R ev. D . 66. 83513
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
B IB L IO G R A P H Y
198
T u ro k . N. 1996a. P hys. Rev. L e tt.. 77. 4138
. 199Gb. A p .I. 473. L3
T u ro k . N'.. Pen. t '. - L . . A: S eljak. I '. 1998. P hys. R e v.. D 3 8 . 02330G
V erde. L. f t a l. 2003. A p .IS . to a p p e a r in v l4 8 n l
W ands. D .. B a rto lo . X .. M a ta rre se . S.. A’ R io tto . A . 2002. P hys. R ev.. DGG. 043o20
W a n g . Y .. S pe rg e l. D. X .. A S trauss. M . A . 1999. A p .I. o lO . 20
Z a ld a rria g a . M . A H a ra ri. D. D. 199"). Phys. R ev.. D-V2. 3276
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C h ap ter 5
F u tu re P r o sp e c ts w ith
W
M
A
P
There is a theory which states that i f ever anybody diseorers exactly what the
V ni rerse is f o r and why it is here, it will instantly disappear and be replaced by
somethin;/ even more bizarre and inexplicable.
There is another theory which states
that this has already happened.
D o u g la s A d a m s
The aim of science is not to open the door to infinite wisdom, but to set, a limit to
infinite eiwor.
B e r to lt B re cht
1.
M o tiv a tio n
W h ile W M A P has revealed the em ergence o f a s ta n d a rd co sm o log y, d e te rm in in g
key c o sm o lo g ic a l p a ra m e te rs w ith u np re ce de n te d p re c is io n , it also leaves m a n y
u nansw ered q u e s tio n s th a t have dogged c o s m o lo g ists fo r decades.
T h e universe
app e ars to co n sist o f a b iz a rre m ix tu re o f co ld d a r k m a tte r (2 4 % ) a n d d a rk energy
(7 1 % ). w ith th e fa m ilia r b a ry o n ic m a tte r th a t co m p rise s e v e ry th in g we k n o w b eing
re le g a te d to a m ere 5% . W h a t is d a rk m a tte r? W h a t is d a r k energy? W h a t is the
199
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C 'h n pte r -3:
F u t u r e P ro sp e cts w it h W' M A P
in fla to n '.’ T h e answ ers to these classic (jn c s tio n s a rc u n kn o w n .
200
W M A P a fte r one
y e a r p ro vid e s ta n ta liz in g h in ts th a t the la rg e angle p ow er s p e c tru m m ay c o n ta in
su rp rise s, and th a t th e shape o f th e p r im o rd ia l p ow er sp e c tru m m a y n ot co n fo rm to
e x p e c ta tio n s , h u t none o f these h in ts p ro v id e d e fin itiv e answers.
Some o f these q u e stio n s m ay have to a w a it advances in th e o re tic a l cosm o log y
a n d / o r p a rtic le physics.
H ow ever, the new W’A/.AP re su lts have shed lig h t on a
d iffe re n t ty p e o f u n c e rta in ty . W h ile we can now c le a rly ru le o ut cosm ologies w hich
p re d ic t g ro ssly d iffe re n t C’ M B p ow er sp e ctra to the s ta n d a rd fla t A C 'D M m o d el w ith
a d ia b a tic in it ia l c o n d itio n s (e.g. defect m o d e ls), it is s t ill d iffic u lt to d is tin g u is h
betw een m o d e ls w h ic h are p e rtu rb a tio n s a ro u n d th is s ta n d a rd m o d e l. These m odels
c o n ta in one o r tw o e x tra p a ra m e te rs w h ich seem o n ly s lig h tly favo re d a t th e < 2rr
level. It is im p o rta n t to d eve lo p q u a n tita tiv e s ta tis tic a l tech n iq ue s w h ich take in to
account th e ro le o f p rio rs on o rd e r to d is tin g u is h betw een m odels w h ich have ro u g h ly
th e same lik e lih o o d b u t d iffe r by one o r tw o e x tra p a ra m e te rs. O n e o fte n resorts to
O c c a m 's R a zo r, w h ic h says th a t e n titie s sh o u ld not be m u ltip lie d w ith o u t necessity,
th a t is. th e sim p le s t e x p la n a tio n is p re fe rre d t i l l th e d a ta dem an d s a m ore co m p le x
m o d e l. R ecent w o rk seems to im p ly th a t th is h e u ris tic p rin c ip le h olds up u n d e r
q u a n tita tiv e s c ru tin y : s im p le m o d e ls seem indeed to be p re fe rre d by p ro b a b ility
th e o ry . W e w ill b rie fly re vie w tw o approaches to th is p ro b le m th a t have appeared in
th e recent lite ra tu r e .
1.1.
B ayesian E v id en ce
W h ile th e peak o f th e lik e lih o o d p ro vid e s a m e ch an ism to choose between
d iffe re n t m o d e ls w ith th e same p a ra m e te riz a tio n , it is n o t a d e q ua te fo r ch oo sin g
betw een m o d e ls w h ic h have d iffe re n t v a ria b le s o r d iffe re n t n um be rs o f va ria b le s. T h e
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p te r ~r.
F u t u r e P ro sp e c ts w it h W M A P
201
concept o f "B a ye sian E v id e n c e " ( D ro ll et a l.. 2000: S a in i or a l.. 2003) oharacrorizos
th is ohoioo in te rm s o f th e r a tio o f p o s te rio r B ayesian p ro b a b ilitie s o f th e tw o m odels.
A m o d el is a set o f ru le s to p re d ic t d a ta fro m a given set o f p a ra m e te rs and
a p rio r w h ich q u a n tifie s th e p ro b a b ilitie s o f th e d iffe re n t p a ra m e te r values in the
absence o f d a ta . i.e. one s a s s u m p tio n s , w h ich m a y be based on th e o ry ( tin ' universe
is H a t), lo g ic ({}[, > 0 ). o r based on know ledge o f o th e r d ata -se ts (0.3 < h < 1.0). Let
a p a ir o f m odels be sp e cifie d by tin* hypotheses .1 and D. w h ic h arc1 p a ra m e te riz e d
by th e sets { f t , ' }
and { o ^ } and w here the n u m b e r o f p a ra m e te rs in
be d iffe re n t. O ne wishes
th e sets m ay
to kn o w w h ic h o f tin ' hypotheses are m ost co nsiste n t w ith
the d a ta (D ) and th e p rio rs (I). T h e n , using Bayes's T h e o re m , th e r a tio o f th e ir
p o s te rio r p ro b a b ilitie s is g iv e n l»y
= P (A \D .I) _ P(A\I)
Ui
P(D\D. I)
f P( D \ A . / )
P( D\ I ) * P(D\D. I)
where P ( A . I ) is th e p r o b a b ility o f h yp o th e sis A given the p rio rs , and P ( D \ A . l ) is
the lik e lih o o d o f th e d a ta m a rg in a liz e d over a ll possible values o f th e p a ra m e te rs a , 1
given th e p rio rs . T in * la t te r te rm can be c a lc u la te d d ir e c tly fro m th e o u tp u t o f the
M C M C : it is e ffe c tiv e ly g ive n b y tin* ra tio o f lik e lih o o d m a x im a C maA A ) / C max( B ) .
W h e n r , ta is m uch la rg e r th a n one. h yp o th e sis .4 is p re fe rre d , w h ic h h yp o th e s is B is
p re fe rred i f T
is m uch s m a lle r th a n one. I f th e ra tio is o f o rd e r 1. th e n th e re is no
c le a r preference.
1.2.
T h e R azor
W o rk by B a la s u b ra m a n ia n ( 1996a.b): M y u n g et a l. (2000) shows th a t th e
s im p lic ity o f a m o d e l fa m ily can be tie d to g e o m e tric p ro p e rtie s o f th e m o d e l seen as
a subspace o f th e space o f d is tr ib u tio n s . F ro m a g e o m e tric p e rs p e c tiv e , a p a ra m e tric
m o d el fa m ily o f p r o b a b ility d is tr ib u tio n s fo rm s a R ie m a n n ia n m a n ifo ld em bedded
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C h a p te r o:
202
F u f i i r r P r n s p r c ts w it h W M A P
in the space o f a ll d is trib u tio n s , d escribe d by a m e tric w h ic h can be take n to be
th e F ish e r In fo rm a tio n m a trix . These a u th o rs d efine a m easure o f th e c o m p le x ity
o f a p a ra m e tric d is tr ib u tio n re la tiv e to a g ive n tru e d is tr ib u tio n , ca lle d th e r a ; o r o f
a m o d el fa m ily . B ayesian in feren ce and tin* M in im u m D e s c rip tio n L e n g th ( M D L )
p rin c ip le (w h ic h state s th a t th e best m o d e l fo r d e s c rib in g a d a ta set is th e one th a t
p e rm its th e g re ate st co m pression o f th e d a ta d e s c rip tio n ) are show n to give e m p iric a l
a p p ro x im a tio n s o f th e ra zo r. T h e ra z o r re q uires th a t th e p r io r d is tr ib u tio n m ust give
equal w eight to each dist.imjuishablc d is tr ib u tio n in d exe d by the m o d e l p a ra m e te rs.
Because the level o f new n o ta tio n and concepts th a t need to be in tro d u c e d fo r a
d e ta ile d d e s c rip tio n o f tin* ra zo r is not a p p ro p ria te fo r th is d iscu ssio n , th e reader is
referred to the papers above fo r fu r th e r d e ta ils .
F u rth e r w o rk needs to be done to a p p ly th is te c h n iq u e to lik e lih o o d surfaces
o b ta in e d by M C M C . since in its c u rre n t fo r m u la tio n th e re is an a m b ig u ity in
d e c id in g w h a t p a rt o f th e phase* v o lu m e to co n sid e r in th e case* where* a {trio r em a
give'n param ete*r is ne*ve*r h it by th e M a rk o v ch a in .
Le*aving th is to p ic fo r fu tu re w o rk , we* tm w co n sid e r how in tro d u c in g e x tra
p a ra m e te rs b eyond th e s im p le s t m o d e l can bias p a ra m e te r e*stimatie>n.
It is
in tu itiv e ly cle*ar th a t w hen lo n g , fla t (i.e . moelels a lo n g th e d egeneracy have ro u g h ly
rlie same lik e lih o o e l) degeneracies in p a ra m e te r space are present, e sp e cia lly fo r
p a ra m e te rs w here th e ch'generacy is n o t s y m m e tric a b o u t th e n u ll o r n o m in a l
value o f th e p a ra m e te r (such as r. r. and ir). th e m ean values o f o th e r p a ra m e te rs
(even those th a t are n o t p a rt o f th e degeneracy) can be* " p u lle d " . T h is is because
m a rg in a liz in g o ve r th e degeneracy su rface w ill y ie ld a m ean va lu e th a t is a t its
"c e n te r o f mass” , a n d i f th e re is m o re phase on one side o f a n o m in a l va lu e th a n th e
o th e r, the m a rg in a liz e d m ean va lu e w ill be on th e side w it h g re a te r phase space.
F o r e xa m p le , th e re is m o re phase space fo r Q tot > 1 th a n Q tot < 1 (spp F ig- 3.13).
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203
and so th e m ean i l , ot y ie ld s a closed m o d el, even th o u g h th is drive's m eans o f some
p a ra m e te rs away fro m th e ir Hat A C 'D M values (fo r exam ple', th e m ean value o f h
is m o re th a n Irr low er fro m the AC’ D M m e a n). S im ila rly , th e re is a lo t m ore phase'
space fo r ec < — t th a n ir > — I (see F ig . 3 .1 1 ). a gain re s u ltin g in a m ean value th a t
has ir < - 1 and p u llin g some* o th e r p a ra m e te rs (the* m ean o f /; is me>re' th a n 1a
h ig h e r th a n its AC’ D M va lu e ). For th e single1 Held in H a tio n case1, one* can e x p lic itly
see* fro m T a b le 4.1 how fa r the1 o the'r p a ra m e te rs have be>en p u lle d by the1 ( u ,. r)
d egeneracy o ff the* AC’ D M m ean values in Table* 3.7.
T h u s . we> can se*e th a t m in im iz in g o r bre'aking de'ge'ne>racie's is h ig h ly desirab le
fo r c o s m o lo g ica l p a ra m e te r e s tim a tio n . Even s im p ly n a rro w in g the1 w ie lth o f a lo n g
degene'racy d ire c tio n can im p ro v e the1 consiste'ncy betw een the1 m eans o f param ete rs
th a t are net p a rt o f th e de'ge'ne'racv. w hen c o m p a rin g twe> m o d els whete1 one1 co n ta in s
th e degeneracy and th e e th e t d o e s n 't (fo r exam ple1. AC’ D M and ir + C ’ D M . where* th e
la t te r c o n ta in s an e x tra degeneracy betwee'n [ic. h j a nd [w. f?r„j ) .
W hile* one can
b re a k degemeracies by a d d in g d a ta se'ts w h ich have* d iffe re n t degeneracy elirectiems
fro m the- C M B d a ta set (e.g. la rg e scale s tru c tu re a n d su p e rn o va su rve ys), it is
also tru e , a t le'ast fo r th e foreseeable fu tu re , th a t the>se non-C ’ .MB analyses are
c o m p lic a te d by s y s te m a tic u n c e rta in tie s th a t are s ig n ific a n tly la rg e r th a n fo r the
C M B case, w here s y s te m a tic ^ are w e ll-u n d e rs to o d a n d can be rig o ro u s ly c o n tro lle d
by e x p e rim e n ta l design and a n a lysis techniques.
T h u s , it is in te re s tin g to see
w h a t can be don e to m in im iz e degeneracies usin g th e C’ M B a lone, by u sin g a ll the
in fo r m a tio n present in th e te m p e ra tu re . E- a n d B -m o d e p o la riz a tio n a u to - and
c ro s s -c o rre la tio n p ow er sp e ctra , as a fu n c tio n o f th e n u m b e r o f years U W /.4 P co u ld
p o te n tia lly be ke p t o p e ra tin g .
T h is c h a p te r in ve stig a te s th re e base cosm ologies: (1) th e s ta n d a rd A C D M
m o d e l and a p a ir o f m o d e ls w h ic h p e r tu rb a ro u n d it : (2 ) A C D M + r u n n in g . and (3)
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C h a p t e r o:
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204
th e "s in g le -fie ld -in fla tio n " A C D M + ru rm in g -H e n s o rs . T in * selected p a ra m e te rs fo r
th e hast* cosm ologies are close to th e ir re sp ective m a x im u m lik e lih o o d m odels fou n d
in the a n a lysis o f C h a p te rs 3 and 4. T h e goal o f th is c h a p te r is to e valua te the*
lik e lih o o d surfaces fo r these m o d els a ssu m in g th a t W M A P co ntin u es o p e ra tin g fo r 4.
G and 8 years and o b ta in reasonable e stim a te s fo r tin* e rro r bars one can expect fo r
th e ir re sp ective p a ra m e te rs in each o f th e cases. As a b y -p ro d u c t, we seek to discover
effects (such as o ff-d ia g o n a l te rm s ) th a t need to be in c lu d e d in the covarian ce m a tr ix ,
and m e th o d o lo g y th a t m ig h t reduce biases in p a ra m e te r e s tim a tio n in fu tu re W M A P
analyst's.
2.
C rea tin g T est D ata
T h t' co sm o lo g ica l p a ra m e te rs fo r th e th re e bast' cosm ologies are lis te d in
T a b le j . l .
F ig u re o . l shows th e bast' m o d e l fo r th e p ow er law AC’ D M case w ith the
co sm ic va ria n ce -s noise e rro rs a fte r 4. G a n ti 8 years fo r th e T T . T E . E E and D D
p ow er sp e ctra .
F o r each case, s im u la te d M o n te C a rlo re a liz a tio n s o f fu ll sky C M B m aps w ith
w h ite noise fo r Q . V a n ti \Y channels were cre a te d , ta k in g th e noise bias n{ r .
n f E = 2 n f r = n BB at th e c o rre c t levels fo r th e re q u is ite n u m b e r o f years o f
o p e ra tio n . B a sica lly, n / v fo r y years o f o p e ra tio n is g ive n by [ u / x fo r 1 ye ar o f
o p e ra tio n ]///. A d iffe re n t M o n te C a rlo re a liz a tio n was used fo r each tim e -fra m e o f
o p e ra tio n , fo r each m o d e l. T h e m aps were c o m b in e d w ith A’JJ w e ig h tin g (w h ic h is
n o t o p tim a l in a ll ( regim es, b u t is u n b ia se d ), th e K p 2 sky cu t was a p p lie d to th e
m a p . a n d th e p ow er s p e c tra fo r C' J1 . C ' f h . C f h . C BB. C’[ B were e s tim a te d fro m
th e s im u la te d m aps. Since th e la st sh o u ld be zero, it serves as a co n siste n cy check.
In c lu d in g C BB in m o d e ls (1 ) a nd (2 ). w h ic h c o n ta in no te n so r m odes, also serves as
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C h a p t e r 3:
F u t u r e P rospects w it h W M A P
T a b le 3.1.
P a ra m e te r
203
C o s m o lo g ic a l P a ra m e te rs fo r F id u c ia l M o d els
(1) P ow er Law
(2) R u n n in g In d e x
AC’ D M
A C 'D M
0.0238
0.144
0.0224
0.133
« ,a
7
0.99
0.16G
0.93
0.168
.1“
h
0.88
0.72
0.83
0.71
<>„//-’
d u s/ d In k
rb
“ D e fin ed a t k = 0 .0 3 M p c
fo r m o d e l (3 ).
-0 .0 3 1
(3 ) S in g le F ie ld
In fla tio n
0.0233
0.134
1.13
0.184
0.73
0.73
- 0 .0 3 3
0.41
1 fo r m odels (1 ) and (2 ). a n d k = 0 .0 0 2 M p c
b D e fin ed at k = 0 .0 0 2 M p c -1 .
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1
C h a p t e r 5:
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206
a c o n s iste n cy check and lets us a scerta in w h e th e r the co va ria n ce m a tr ix requires the
in c lu s io n o f a d d itio n a l effects.
F or lo n g e r o p e ra tio n o f W M A P . tin ' covarian ce s betw een th e T T . T E . E E and
D D becom e im p o r ta n t. T in * d ia g o n a l p a rr o f th e co va ria n ce m a tr ix can be m odeled
as D ‘(J w here { i \ = { T T . T E . E E . D D . T D . E D ) . T h e co va ria n ce m a tr ix is g ive n by
zy,. = ia y o y . p* -
(o-l)
it’■
w here th e d e fin itio n s o f r (f a n d f f f are th e sam e as in S e ctio n 2.2. For > 4 years
W M A P o p e ra tio n , wo set tin* beam u n c e rta in tie s and p o in t source s u b tra c tio n erro rs
to zero (i.e . f tr = 0 ). We o n ly in c lu d e o ff-d ia g o n a l te rm s due to t in 1 sky cut th a t
co u p le d iffe re n t /-m o d e s fo r t in ’ d ia g o n a l te rm s E " . T h e c u rv a tu re m a tr ix is g ive n by
:o-2)
Q '/r = B y - ' * *
\J D /‘ D J
‘
For t in ’ T T T T te rm , we use th e lik e lih o o d a p p ro x im a tio n g ive n in Eqs. 2-11 and
2-12. and fo r th e o th e r te rm s we use th e G a u ssia n lik e lih o o d given bv
--> In £ = £
,J
Z
(5-3)
( C " - C’r ) Q ’/ r (Cl-"' - Cl) .
ff’
In th is a n a ly s is we have e x p lic itly set to ze ro te rm s such as D EBEB. D EEBB.
D f rBB e tc . w h ic h are s m a ll, ta k in g
/ D rrrr
d
D ‘J ~
t t t e
D j TEE
\
d
t t t e
d
j t e e
()
()
d
t e t e
d
t e e e
0
()
D EEEE
0
0
D j EEE
0
0
0
dbbbb
()
0
0
0
0
D j BTB
\
■
(5-4)
b u t such te rm s can e a s ily be in c lu d e d in fu tu re analyses. T h e v a rio u s d ia g o n a l te rm s
(in ( space) are g ive n by
Q T TT T
=
(T T T T ) - (TT)2
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C 'h a prer 7:
F iit u r v P r o s p r r t s w it h W M A P
207
2(C’/ / ',h - A ' 77 )D ( h:rh:
=
(T E T E ) - { T E ) 2
(c /Trh +
D j " i:
=
y rr)(cf'-'h+ a 7^ )
(-^ - n/;,,
+ (C'7--"')-
( T T T E ) - {T T ) ( T E )
(c j r ' h h- A 7 7 ') e ' 7 -,h
(2/ + i)/;i,.v
D l :i:i:l
=
(2/ - i ) / ; i , v
=
w
=
-
=
(•7-9)
n
(T E E E ) - ( T E ) ( E E )
( c ; ;A'-, h + A 7 -7-') c r K-ih
D w««w
(7-8)
( E E E E ) - {T T ) ( E E )
(c Y /';',h )J
D ! i :,: h
O- I
( E E E E ) - (E E )'2
2{C?:h:Ah+ X h:h:)'2
D n /•:/•:
(7-G)
(7-10)
(2^ + 1)/Av
(DDDD) _ (i3Z?>J
7 - 11;
(2 / + l ) / & ,
D,r /# r fl
=
( T D T B ) - (TB)'2
( C j r ' h + A 7 7 )(C /,H th -r A *
( 2 f+ l) /^
w here
7-12)
= 0 . 8 7 / \/ l- 1 4 . th o n u m e ra to r c o m in g fro m th e K p 2 sky c u t and the
d e n o m in a to r fro m th e
w e ig h tin g .
3.
P a ram eter E stim a tio n R e su lts
W e n ow c a rry o u t p a ra m e te r e s tim a tio n u sin g M C M C fo r each o f th e m odels
(1 ). (2) a n d (3 ). T h e re su lts are presented below .
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C h a p t e r 5:
F u tu r e P ro spe cts w it h W M A P
3.1.
208
A C D M M od el
T a b le 5 .2 gives tin* re su lts o f th e p a ra m e te r e s tim a tio n fo r th e A C D M m o d el
fo r 4. C and 8 years, re sp ective ly.
F ig u re 3.2 shows tin ' 1-d im e n s io n a l lik e lih o o d
p ro file s fo r the m o d e l p a ra m e te rs for 8 years o f o p e ra tio n - 4 and 6 ye ar lik e lih o o d
p ro file s lo o k ve ry s im ila r. T h e mean recovered p a ra m e te rs flu c tu a te up and d o w n by
lrr ro u g h ly a t h ir d o f th e tim e , as expected. B u t th e ta b le shows th a t r and .4 are
c o n s is te n tly biased h ig h e r th a n the fid u c ia l values in p a ra m e te r e s tim a tio n fro m a ll
three M o n te C a rlo re a liz a tio n s . F ig u re 3.3 shows th e reason: by I years o f o p e ra tio n ,
the r vs n s d egeneracy is broken, and th e d o m in a n t degeneracy is betw een r and .1.
T h is degeneracy is im p o r ta n t even a fte r 8 years o f o p e ra tio n , and is dep e nd e nt on
the p o s itio n o f th e p iv o t p o in t o f rlit* p rim o rd ia l p ow er s p e c tru m (see F ig u re 3.4).
T h is suggests th a t th e p iv o t p o in t sh o u ld be chosen w ith care to m in im iz e th is
degeneracy, p e rha p s a ro u n d A- = 0.02 M p c " 1.
3.2 .
R u n n in g Ind ex A C D M M o d el
T a b le 5.3 gives th e re su lts o f the p a ra m e te r e s tim a tio n fo r th e ru n n in g sca la r
in d e x m o d e l fo r 4. 6 a n d 8 years, re sp ective ly.
W h ile m o st o f th e recovered
p a ra m e te rs show co nsiste n cy, n ,. r / n , / r / ln A \ r a n d h show som e biases in th e 4 a n d 6
ye ar a n a lysis. T h is can a g a in be a ttr ib u te d to th e effect o f degeneracies. B y 8 years
o f o p e ra tio n , th e degeneracies are s m a lle r and th e in p u t p a ra m e te rs are recovered
w e ll. F ig u re 5.5 show s th e degeneracies o f r . n , a n d d n j d \ n k w h ic h s t ill e xist a fte r
8 years. T h e e x tra degeneracy o f n„ vs. d n s/ d In k o ver th e A C D M m o d e l a ccou n ts
fo r th e fa c t th a t th e e rro r on ns fo r th e ru n n in g in d e x m o d e l does n o t im p ro v e to th e
level it does in th e A C D M m o d el.
I t a p p e ars th a t i f one is a tte m p tin g to c o n s tra in m o d e ls w ith a ru n n in g s p e c tra l
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 5:
F u tu r e P ro spe cts w it h W M A P
T a b le 0.2.
E s tim a te d P a ra m e te rs fo r A C D M M o d e l
P a ra m e te r
4 W a rs
n hf r
0.0 2 37 x 0.0007
0.138 x 0.007
0.98 x 0.02
H ,J r
r>P
i
G W a rs
0.202 ± 0.020
0.92 ± 0.04
0.74 ± 0.03
.1"
h
0.0248
0.139
1.02
0.198
0.93
0.7G
0.2G x 0.03
Hu
"D e fin e d at k = 0 .0 5 M p c
T a b le o.3.
±
x
±
±
±
x
8 W a rs
0.0007
0.00G
0.02
0.019
0.04
0.03
0.0234
0.144
0.988
0.19G
0.94
0.G9
0.24 ± 0.03
±
x
x
±
x
x
F ra c tio n a l
E rro r
0.000G
0 .0 0 5
0.01G
0.01G
0.03
0.02
2..VZ
3 ..V /
27,'
0.30 x 0.03
1 0 '/
3‘Z
3 '/
I VX
'.
E s tim a te d P a ra m e te rs fo r R u n n in g In d e x A C D M M o d e l
P a ra m e te r
n bh2
Urnh2
T
(lns/ i l In k
.4"
h
209
4 \'e a rs
0.0236
0.121
1.00
0.199
0.025
6 W a rs
± 0.0008
± 0 .0 0 7
± 0.04
± 0 .0 2 1
± 0.023
0.85 ± 0.04
0.79 ± 0.04
"D e fin e d a t k = 0 .0 5 M p c
0.0248
0.122
1.03
0.188
0.030
± 0.0009
± 0.007
± 0 .0 4
± 0 .0 1 9
± 0.0 2 7
0.84 ± 0.04
0.81 ± 0.04
8 W a rs
0.0223 ± 0.0007
0.133 ± 0 .0 0 6
0.94 ± 0.03
0.1 9 9 ± 0.015
- 0 .0 1 9 ± 0 .0 2 1
0.88 ± 0.03
0.70 ± 0.03
l.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 5:
F u t u r e P ro spe cts w ith W M A P
210
in d e x, e s p e c ia lly i f one does nor have the* lu x u ry o f ru n n in g th e U '.\/.4 P e x p e rim e n t
fo r 8 years, it is im p e ra tiv e to add d a ta at s m a ll scales so as n ot to Idas th e recovered
p a ra m e te rs.
H a v in g a w e ll-c o n s tra in e d a n ch o r p o in t at s m a ll scales e lim in a te s
th e e x tra degeneracies caused by d n , / d In k. For in sta n ce , one can use d a ta fro m
g ro u n d -b a se d C’M B e x p e rim e n ts , c a lib ra te d to th e H W /.A P m a p . fo r th is purpose.
Even a t 8 ye ars' W M A P o p e ra tio n , w h ile o th e r p a ra m e te rs are n ot biased, one
does n ot o b ta in a s ig n ific a n t c o n s tra in t on d u s/ d \ w k unless it is m uch la rg e r th a n
the value sp ecified here. - 0 .0 3 . B ut it is q u ire lik e ly th a t a ru n n in g in d e x o f th is
m a g n itu d e can be d e te cte d at 3rr by a d d in g s m a lle r scale d a ta .
3.3.
“S ingle F ield In flation ” M od el
T a b le o.4 gives th e re su lts o f th e p a ra m e te r e s tim a tio n fo r th e "sin g le fie ld
in fla tio n " m o d e l fo r 4 and 8 years, re sp ective ly.
T h e 4 -ye a r o p e ra tio n case s t ill c o n ta in s degeneracies w h ic h bias th e recovered
p a ra m e te rs s ig n ific a n tly . F ig u re 5.6 shows th a t th e degeneracy surfaces in th e ( r . n ,.
d n s/ d In k) planes are s t ill s ig n ific a n t (co m p a re w ith F ig u re 4.3).
T h e 8 -ye a r o p e ra tio n case recovers th e in p u t p a ra m e te rs q u ite w e ll. F ig u re 5.7
shows how m uch th e degeneracy surfaces in th e (r. n s. d n s/ d \ n k ) planes have
tig h te n e d (c o m p a re w ith F ig u re 4.3). I t appears th a t i f th e level o f te n so r m odes is
as h ig h as in th is m o d e l ( — 0 .4 ). a 3a d e te c tio n is p ossible a fte r 8 years o f W M A P
o p e ra tio n .
A g a in , a d d in g s m a lle r scale C’ M B d a ta c a lib ra te d to W M A P sh o u ld
tig h te n c o n s tra in ts s ig n ific a n tly .
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C h a p t e r 5:
F u tu r e P ro spe cts w it h W M A P
3 .4 .
211
R e s u lt s in t h e (»•, U , „) a n d (» •, h ) P la n e s
W’o e x p lo re th e (i r. <>,„) and (ir. h) degeneracies u sin g th e s im u la te d d a ta fro m
th e A C D M m o d e l. M o d e l (1) above.
We c a rry o u t p a ra m e te r e s tim a tio n using
M C M C fo r a m o d el w ith a co n sta n t e q u a tio n o f state*, u sin g p a ra m e te rs [U ft/r.
D
n s. r. .4. h. </•]. We assume 8 years o f W M A P o p e ra tio n in th is c a lc u la tio n .
F ig u re o.8 shows th e c o n s tra in ts o b ta in e d on these planes, w h ic h , w hen co m p ared
w ith Figure* 3.11. shows th a t the* degeneracy has tiglite*iie*< 1 e s p e c ia lly fo r if > — I.
The* degeneracy surfae e* is bi-m o e la l. The* r vs. U , „ f r jo in t like-lihooel surface is also
b i-tn o d a l fo r th is m o d e l. w h ich suggests th a t the* d eg e ne ra cy is be*ing p a r tia lly broke*n
by the* improve*me*nt in the* p o la riz a tio n d a ta .
3 .5 .
R e s u lt s in t h e ( i i \ , D , „ ) P la n e
F in a lly , we* e x p lo re the* (S>A. <>,„) degeneracy. a g a in u sin g the* s im u la te d d a ta
freun the* Hat A C D M m o d e l. We c a rry o u t parame*te*r e*stimatie>n usin g MC’ M C fe>r a
n o n -fla t m o d e l w ith th e p a ra m e te rs [ lh , / r .
D \ - "*■ r - -T h]. We assum e 4 and
8 years o f U ’A/.AP o p e ra tio n in th is c a lc u la tio n .
F ig u re 5.9 shews how fa r the c o n s tra in t has tig h te n e d ce)tnpare*el te) th e c o n s tra in t
fremi th e firs t ye ar d a ta (F ig u re 3.13). T a b le 5.5 show s t h a t th e c o n s tra in t o b ta in e d
fremi 8 years is n ot a s ig n ific a n t im p ro v e m e n t over 4 years.
4.
C on clu sion s
T h e firs t ye a r W M A P a n a lysis was c a rrie d o u t w it h a level o f rig o r and
care in o rd e r to d o ju s tic e to th e p re cisio n o f th e d a ta .
H ow ever, it is th e firs t
w o rd on W M A P . n o t th e la s t. I t goes w ith o u t s a y in g t h a t one m u st c o n tin u e to
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 5:
212
F u tu r e P ro spe cts w it h W M A P
T a b le 5.4.
E s tim a te d P aram ete rs fo r "S in g le -F ie ld In fla tio n " M o d e l
P a ra m e te r
4 W a rs
8 5 ears
<Vr
i l jr
n A
r
0.0256 ± 0.0011
0.110 ± 0.008
0.0238 x 0.0007
t h y / ( I In A.4“
h
0.75 ± 0.23
0.75 ± 0.00
0.120 ± 0.00G
1.12 ± 0.05
0.210 x 0.014
-0 .0 4 1 x 0.022
0.50 x 0.15
0.70 x 0.05
0.85 ± 0.05
0.75 ± 0.03
1.00 ± 0.08
0.215 ± 0.019
-
0 .0 0 0
± o.o:i8
'‘ D e fin ed at k= 0 .0 ()2 M p e
T a b le 5.5.
D ,„,
aF la t
E s tim a te d C o n s tra in ts fo r
4 W a rs
8 W a rs
0.981 ± 0 .0 0 9
0.993 ± 0.014
ACDM
fid u c ia l
co sm o lo g y
fro m T a b le 5.1
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p to r 5:
F u tn ro P r o s p e rs w it h W M A P
213
im p ro v e th e level o f a ccu ra cy o f b o th th e covariance* m a tr ix and th e p a ra m e te r
e s tim a tio n /lik e lih o o d a n a lysis. In th is c h a p te r we have* in v e s tig a te d how p a ra m e te r
e s tim a tio n can be* biased by th e pre*sence o f degeneracy surface's in the* lik e lih o o d .
It is c le a r th a t th a t th e m in im iz a tie m o f de*ge*neracy dire*ctions is re*e|uire*d in oreler
to tru s t the* me*an value's o f parame*te*rs. give*n th e c o n tin u a lly im p ro v in g q u a lity o f
th e elata. R e c o m m e n d a tio n s fe>r future* U A / A P analyses base*d on the*se* sim ulate'd
p aram e'ter e stim a tie)ns are* as follo w s:
1. L’se the* fu ll covariance* m a tr ix be*twe*en the* possil)le* c o m b in a tio n s o f pe>we*r
sp e ctra . The* E E B D and E D E B te*rms. w h ie h we*re* ne*glecte*d in th is a na lysis,
sh o u ld alse* be* inclueh'el. since* th e y in flu e n ce the* covariance* o f r w hieh s t ill
appe*ars to be* de*ge*ne*rate* w ith a n o th e r parame*te*r to a s m a ll e*xte*nt e*ve*n in the*
8 ye a r elata.
2. The* p iv o t p o in t o f the* p rim o rd ia l powe*r spe*ctrum in p aram e'ter e*stimatiem
sh o u ld be* se*le*cte*el w ith gre*at care* to m inim ize* de*gene*racie*s. A selection o f
p iv o t p o in ts sh o u ld be trie*d to d e'te rm ine th e be*st p o s itio n to use*.
T h e re is no s ig n ific a n t impre>ve*ment in d e te rm in in g th e paramete*rs e>f a A C D M
u niverse by e x te n d in g th e o p e ra tio n o f W M A P to G o r 8 years.
How ever, the
im p ro v e m e n t in lo n g e r te rm o p e ra tio n is s ig n ific a n t in c o n s tra in in g m o d els w ith
one o r twe> e x tra p a ra m e te rs. W e fin d th a t c e rta in p a ra m e te rs o f m ore c o m p lic a te d
m odels are biased because o f th e existence o f degeneracies. These c o u ld p o te n tia lly
be m in im iz e d o r e lim in a te d b y o p e ra tin g W M A P fo r up to 8 years.
In th e case
o f a m o d e l w ith a ru n n in g in d e x , th e degeneracies m a in ly o c c u r because e rro rs
on sm a ll-sca le flu c tu a tio n s are to o large: th e re fo re it is lik e ly th a t th e same effect
as o p e ra tin g W M A P fo r lo n g e r can be achieved b y c o m b in in g W M A P d a ta w ith
g ro u n d -b a se d C M B e x p e rim e n ts th a t m easure h ig h - f m o d e ls a c c u ra te ly , c a lib ra te d
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a pt e r 5:
214
Future Prospects with W M A P
to W M A P . In th e case o f a m o d el w ith a h ig h level o f te n s o r m odes (we have
in v e s tig a te d r ~ 0 .4 ). a 3<r d e te c tio n o f te n so r m odes is p ossible a fte r 8 years. T h e
c o n s tra in ts on flatness do n o t im p ro v e s ig n ific a n tly w hen ru n n in g W M A P fo r 4 vs.
8 years. T h e W M A P 8 year c o n s tra in t on »•. w hen c o m b in e d w ith re su lts fro m
ne xt g e n e ra tio n supernovae e x p e rim e n ts (e.g. S X A P . h ttp ://s n a p .lb l.g o v /) can y ie ld
tig h t lim its on th e e q u a tio n o f s ta te o f d a rk energy, a ssu m in g th a t tin* s y s te m a tic
u n c e rta in tie s in tin ' fu tu re su pe rn o va d a ta can be c o n tro lle d to be o f th e same level
as th e ir s ta tis tic a l u n c e rta in tie s .
W e have m e n tio n e d th a t t in ’ c o m b in a tio n o f \ \ . \ / . \ P w ith s m a lle r scale C’ M B
e x p e rim e n ts (fo r e xa m p le , th e n e x t-g e n e ra tio n version o f th e D O O M E R u t i G
e x p e rim e n t, h ttp ://o b e r o n .r o m a l.in fn .it/b o o m e r a n g /b 2 k /. a nd fu tu re g ro un d -b ase d
e x p e rim e n ts ) can s ig n ific a n tly im p ro v e the science re tu rn o f a lo n g e r-liv e d \ \ ’.\/.4 P
e x p e rim e n t fo r n o n -m in im a l-A C D M cosm ologies.
C a lib r a tin g a s m a lle r scale
e x p e rim e n t to W M A P can s ig n ific a n tly im prove' c a lib ra tio n u n c e rta in tie s in the
sm a ll-sca le e x p e rim e n t and hence improve* th e parame'te*r c o n s tra in ts . The* c a lib ra tie m
u n c e rta in ty is m in im iz e d the' closer W M A P is to b e in g c o s m ic -v a ria n c e -Iim ife d in
th e region o f o v e rla p betw een the' tw o e x p e rim e n ts . The' co sm ic va ria n ce lim it o f
W M A P is f ~ 495 (4 ye ars). f ~ 545 (6 ye a rs). a n d I ~ 570 (8 years). T h u s , the
lo n g e r W M A P ru n s, the b e tte r th e science re tu r n w ill be frem i c o m b in in g its elata
w ith sm a ll-sca le e x p e rim e n ts .
We have show n th a t it is v e ry im p o rta n t to m in im iz e deg e ne ra cy d ire c tio n s in a
give n m o d e l fo r th e unbiased re*covery o f co sm o lo g ic a l p a ra m e te rs . T h e im p ro v e m e n ts
in th e d a ta (e s p e c ia lly in th e p o la riz a tio n d a ta ) g a in e d b y ru n n in g th e W M A P
d a ta fo r lo n g e r w ill s ig n ific a n tly increase its p o w e r to d is tin g u is h a m o n g m odels
w h ic h are p e rtu rb a tio n s a b o u t th e s ta n d a rd " m in im a l" A C D M m o d e l. O ne can
im p ro v e c o n s tra in ts fu r th e r by a d d in g sm a lle r-sca le g ro u n d -b a s e d C M B e x p e rim e n ts
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r o:
F u tu r e P ro spe cts w ith W M A P
c a lib ra te d to th e H ’A /.A P m aps, and th is c a lib ra tio n u n c e rta in ty decreases w ith the
le n g th o f tim e W M A P c o n tin u e s o p e ra tin g .
W e co n c lu d e by r e ite ra tin g th a t g re at care m u st be ta ke n w hen f it t in g m o d els
w ith p a ra m e te rs w h ic h have one o r m o re fla t dege ne ra cy d ire c tio n s fo r a given
d a ta -s e t.
F o r e xa m p le , i f one a tte m p ts to s im u lta n e o u s ly c o n s tra in th e usual
F R W -(-p rim o rd ia l co sm o lo g ic a l p a ra m e te rs [12*//-. V.mh~. r . h. //,. .Aj p lu s U \ . ir.
( I n^/ i l \ nk- a n d r. w ith p ro je c te d C 'M B d a ta , one is lik e ly to recover co sm o lo g ica l
p a ra m e te rs w h ic h are biased due to th e m a n y degeneracies in h e re n t between these
p a ra m e te rs . A n in d e p e n d e n t
VZ d e te rm in a tio n o f th e H u b b le p a ra m e te r w ith
m in im a l s y s re m a tic s is lik e ly to do m uch m ore fo r o u r a b ilit y to s im u lta n e o u s ly
c o n s tra in th is p a ra m e te r set th a n la rg e im p ro v e m e n ts to C’ M B d a ta over w h a t w ill
be achieved in th e n e xt few years. U n fo rtu n a te ly , th is m a y n ot be a re a lis tic g oal,
at least u sin g present m e th o d s. H ow ever, given tin * fo rm id a b le a rra y o f e x p e rim e n ts
p la n n e d fo r t in 1 n e xt co u p le o f decades, u sin g a w id e range o f m e th o d s (C 'M B
p o la rim e try . space-based in te rfe ro m e try . g ra v ity wave1 d e te c to rs , th e successor to the
H S T . p a rtic le a cce le ra to rs, n e u trin o e x p e rim e n ts , la rg e scale s tru c tu re surveys, and
s u p e rn o va su rve ys) as w e ll as advances in n u m e ric a l and s ta tis tic a l tech n iq ue s, we
m a y lo o k fo rw a rd to s ig n ific a n t advances in o u r k n o w le d g e o f th e cosmos.
I a ckno w le d ge th e h o s p ita lity o f th e A spe n C e n te r fo r P hysics, w here m uch o f
th is w o rk was c a rrie d o u t. and A n to n y L e w is. .M anoj K a p lin g h a t. L lo y d K n o x . O liv ie r
D o re . A n n W a n g . P ia M u k h e rje e a n d R o b e rt C ritte n d e n fo r usefu l co n ve rsa tio n s on
p a ra m e te r e s tim a tio n .
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r o:
F u t u r e P ro spe cts w it h W M A P
216
7000
6000
R 5000r
t
4000 [
~
3000
(5
K
200
It
io o
\
„L
i -ioo(
C. 2000 [
-2 0 0
200
400
600
800
i
100
10
1000
1
10
10
o
S'
-5
-1 0
-10
10
100
10
100
F ig . o . l.
T h e base m o d e l (b la c k lin e ) fo r th e p ow er la w A C D M case w it h th e cosm ic
v a ria n ce -I- noise e rro rs a fte r 4 ( lig h t g ra y b a n d ) . 6 (m e d iu m g ra y b a n d ) and 8 (d a rk
g ra y b a n d ) years fo r th e T T ( to p le ft) . T E (to p r ig h t) . E E (b o tto m le ft) and D B
(b o tto m r ig h t) p o w e r sp e ctra .
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 3:
■o
o
o
£
e
M
F u t u r e P ro spe cts w it h W M A P
800
80
600
60
217
20
'O
o
o
£
400
jt
V
Jt
20
200
2.20
2.JO
2.40
2.50
0.65
Qm*2
CV>2 [« t 0 0 ]
0.70
0.75
30
20
10
20
g
§
£
jt
— y
>»—
■
0.80 0.85 0.90 0.95 1.00 1.05 1.10
Amplitude
F ig .
5.2.
0.16 0.18 0.20 0.22 0.24 0.26
0.94 0.96 0.98 1.00 1.02
nt ot h-O.Ob M pc"’
O ne d im e n s io n a l m a rg in a liz e d lik e lih o o d p ro file s fo r th e A C D M m odel
a fte r 8 years o f W M A P o p e ra tio n . T h e lik e lih o o d p ro file s fo r 4 and G years do not
d iffe r s ig n ific a n tly fro m th is p lo t.
0.28
0.28 r
0 .2 6 ;
0 .2 4 :
0.26
0.24
|
0.22
0.20
o
0.2 2 :
0.20 r
0.18
0.18
0.16
0 .1 6 :
0 .94 0.96 0 .98 1.00 1.02 1.04
n
0.80 0.85 0 .90 0.95 1.00 1.05
A
F ig . 5.3. — n a vs r and A vs r jo in t lik e lih o o d surfaces fo r A C D M m o d e l a fte r 4
years o f W M A P o p e ra tio n . T h e lik e lih o o d surfaces fo r 6 and 8 years do n o t d iffe r
s ig n ific a n tly fro m th is p lo t.
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r j:
218
F u t u r e P ro spe cts w it h W M A P
<
4
3
2
1
w a v e n u m b e r k [ M p c “ 1]
F ig . 5.4.
T h e a m p litu d e p a ra m e te r .4 as a fu n c tio n o f k fo r 4 years o f W M A P
o p e ra tio n fo r th e A C D M m o d e l. T h e s o lid lin e gives th e m ean va lu e. T h e shaded
area show s 689c e rro rs and th e d o tte d lines g ive th e 959£ e rro rs . T h e c o n s tra in ts fo r
6 a n d 8 years d o n o t d iffe r s ig n ific a n tly fro m th is p lo t.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 5:
219
F u tu re P rospects w it h W M A P
-0 .0 8
0 .8 5
0 .9 0
0 .9 5
n
1.00
0 .8 5
0 .9 0 0 .9 5 1.00
dn/<Jlnk
- 0 . 1 0 - 0 . 0 5 0 .0 0
d n /d ln k
0 .0 5
F ig . 5.5.
T h e 2 -d im c n s io n a l jo in t lik e lih o o d planes fo r r vs. (.4. //,. du^/ d InA-) fo r
th e ru n n in g in d e x m o d e l. T h e shaded area shows 68% e rro rs a n d th e d o tte d lines
give the 95% e rro rs. T h e degeneracy surfaces fo r 1 and 6 years are so m e w h a t worse
th a n in th is p lo t.
0 .8 0 0 .9 0 1.00 1.10 1 .20
* , a t k « 0 .0 0 2 M p c * '
1.30
0 .8 0 0 .9 0 1.00 1.10 1.20
n , a t k « 0 .0 0 2 M p c * '
1.30
0 .0 0 .2 0 .4 0 .6 0 .8 1.0 1.2 1.4
t« n s o r /s c a la r r a tio r
F ig . 5.6.
T h e 2 -d im e n s io n a l jo in t lik e lih o o d planes fo r ( r . n,s. d n ,08
fo r the
"s in g le fie ld in fla tio n '" m o d e l: th e 68% confidence re g io n (d a rk b lu e ) a n d th e 95%
confid e nce re g ion ( lig h t b lu e ) are show n fo r 8 years o f W M A P o p e ra tio n .
0 .9
1.0
1.1
1.2
n t o t k « 0 .0 0 2 M p c * '
0 .9
1.0
1.1
1.2
n , 0 t k - 0 .0 0 2 M p c -'
0 .0
0 .2
0 .4
0 .6
0 .8
t a n s o r /K O io r ra tio r
1.0
F ig . 5 .7 .— T h e 2 -d im e n s io n a l jo in t lik e lih o o d planes fo r ( r . n s. d n s/ d l n k ) fo r th e
"s in g le fie ld in fla tio n ” m o d e l: th e 68% co nfidence re g io n (d a rk b lu e ) a n d th e 95%
c o nfid e nce re g io n ( lig h t b lu e ) are show n fo r 8 years o f W M A P o p e ra tio n .
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C h a p t e r 5:
*
220
F u t u r e P rospects w it h W M A P
\
-1 .5
-
2.0
-3 .0 t
0 .0
0 .2
0 .4
0 .6
0 .8
0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1.0 1.2 1.4
1.0
h
flm
F ig . 5.8.
T h e 2 -d im e n s io n a l jo in t lik e lih o o d planes fo r (12,,,. ir) a nd (//. »•) p la in's,
using th e A C D M m o d e l as the base cosm ology: t in 1 08% confid e nce re g ion (d a rk blue)
a nd th e 95% co n fid e n ce region (lig h t b lu e ) are show n fo r 8 years o f W M A P o p e ra tio n .
1.0
1 .0 :...........................................
0 .8 :
0 .8 1
0 .6 :
C
°-6 ^
0 .4 :
0 .4 :
0.2 *:
0.2 ;
0 .0
0 .0
0 .0
0 .2
0 .4
0 .6
o.
0 .8
1.0
-v
0 .0
................
0.2
0 .4
0 .6
0 .8
1
n.
F ig . 5.9.
T h e 2 -d im e n s io n a l jo in t lik e lih o o d p lane fo r th e (f2rn. Q \ ) p lane, using
th e A C D M m o d e l as th e base co sm o lo g y: th e 68% confid e nce re g io n (d a rk b lu e ) and
th e 95% co n fid e n ce re g io n (lig h t b lu e ) are show n fo r -1 (le ft p a n e l) a n d 8 (r ig h t panel)
years o f W M A P o p e ra tio n .
R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
B ib liograp h y
B a la s u b ra m a n ia n . V . 199G. ada p -o rg /9 G 01 0 01
B a la s u b ra m a n ia n . \ ’ . 199G. co n d -rn a t/9 G 0 1 0 3 0
D re ll. P. S.. L o re d o . T . I.. W assorm an. I. 2009. Ap.J. -">30. 593
M v u n g . I. .1.. B ala.su brarna n ia n. \ \ . A: P it t . M . 2000. P roceedings o f th e N a tio n a l
A c a d e m y o f Sciences. 97. 11170
S a in i. T . D .. W e lle r. I..
B rid le . S. L. 2003. a s tro -p h /0 3 0 o o 2 6
221
R eproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
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