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A search for the large angular scale polarization of the cosmic microwave background

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A S earch for th e L arge A n gu lar S ca le P o la r iz a tio n o f th e
C o sm ic M icrow ave B a ck g ro u n d
bv
B rian K eating
B.Sc. C ase W estern Reserve U niversity, 1993
Sc.M ., Brown U niversity, 1995
D issertation
S u b m itte d in p a rtia l fulfillment of th e requirem ents for the
Degree of D octor of Philosophy
in th e D ep artm en t of Physics a t B row n U niversity
Providence, R hode Island
M ay 2000
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
UMI Number 9987786
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© C opyright
by
B rian K eating
2000
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T h is d isse rta tio n by B rian K eating is accepted in its present form by
th e D ep artm en t of Physics as satisfying th e
d issertatio n requirem ent for th e degree of
D o cto r of Philosophy
P ete r T im bie, D irector
R ecom m ended to th e G rad u ate C ouncil
D a « e .j.
m
s .m
.
R o b ert B ran d en b erg er, Reader
D a te . TV?-.
J fj5 5
regory Tucker, R eader
A pproved by th e G ra d u a te C ouncil
D ate.
z A
m
.........................................
P ed er J . E stru p
D ean o f G ra d u a te School an d Research
ii
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A ck n o w led g m en ts
So m any people have helped me, b o th personally a n d professionally, in m y years of school­
ing. I have b een honored to know each and every one of you, an d I have benefited enor­
mously.
F irst of all to m y advisor, I am rem inded o f th e old proverb th a t states: “W hen the
stu d e n t is ready, he will find his teacher.” I am q u ite lucky to have found P e te r Tim bie.
P e te r possesses so m an y gifts - it’s difficult to list all of th e m here. He h as th e gift of
pow erful physical insight w hen th ere are problem s to solve. He has p atien ce w hen things
go w rong, a sense o f hu m o r w hen things really go w rong, an d m ost o f all: th e integrity,
com passion, an d soul needed to lead young people today. P eter, th a n k s for p u ttin g up
w ith me for these 6 y ears an d tu rn in g me into a physicist. A nd th a n k s for “le ttin g ” me
b e a t you in b ask etb all so m any times.
I ’ve been privileged to have knowntwo m en w ho have been w onderful m en tors th ro u g h ­
o u t my b rief career. Prof. A lex P olnarev ta u g h t m e m o st o f w h at I know a b o u t theoretical
physics an d cosmology'. C o n tinuing the trad itio n s p assed on by his m en to r Y .A . Zeldovich,
he also ta u g h t me th a t “N ine women can n o t have a b ab y in one m onth!” Alex, I’m hon­
ored to be y o u r “Cosm ological Son” . Dr. Lucio P iccirillo was th ere to solve th e hard
problem s th a t were essential to th e success o f P O L A R . In th e lab he is a w izard - “T h e
M aster” takes th e “cry” o u t o f cryogenics, an d p u ts it in to chess - th a n k s for never let­
tin g me win. T h an k s, also, for giving me th e tools I needed to get to th e n ex t level as an
ex p erim en talist.
N ext we com e to th e folks who helped sh ap e P O L A R . F irst of all is m y little buddy,
N ate (Dogg) S teb o r. N ate, th an k s for w orking so h a rd , for so long. T h an k s for catching
m y m istakes before th ey becam e disasters, doing a n d re-doing so m any th in g s w ithout
co m plaint. T h an k s for listening to me sing off-key for hours in th e lab, a n d th a n k s for
p u ttin g up w ith m y (m o stly unintentional) a tte m p ts to electro cu te you. C hris (Codfish)
O ’Dell has b u ilt a g re a t m any things th a t have im proved th e quality of P O L A R ’s life
im m ensely. S orry I m ad e you re-build the g ro u n d screen te n tim es ju s t to increase its
len g th by a few m icrons. I leave PO L A R in yro u r skilled hands.
Several u n d erg rad u ates have helped in th e d evelopm ent o f PO L A R . A t U W , K ip H y att
designed an d b u ilt th e o u te r ground screens - a n en g in eerin g feat th a t w ould m ake the
ancient E g y p tian s trem b le. Jessie Sincher m ade several o f th e figures in th is thesis, and has
been a good sp o rt in th e lab. B ack a t Brown, th e re w ere tw o u n d erg rad u ates w ho helped
define th e directio n P O L A R would later take. B ren d an C rill helped co n stru ct th e “P ro to P o larim eter” which allowed num erous system atics to be diagnosed, an d perform ed the
calcu latio n s which e stim ated th e polarization of th e atm o sp h ere. M elvin P h u a continued
in B re n d a n ’s shoes, refining th e d a ta acquisition tech n iq u e, building m any com ponents of
th e original p o larim eter, an d in general, was a g re a t help to m e d u rin g in-lab testing.
I have also b en efitted from innum erable conversations w ith th e “N ew G u a rd ” T h e­
orists: W ayne Hu, A rth u r Kosowsky, and M atias Z ald arriag a. C osm ology is in good
h an d s w ith you th ree gentlem en. Josh G undersE n, cosm ology’s only R abid-H acker PackerBacker b u ilt th e Rfa B an d feedhom , and p artic ip a te d in countless discussions on th e s ta tu s
an d fu tu re o f P O L A R . E d W ollack - The H ard est W orking M an in M icrowaves - bailed
m e o u t on so m any occasions. His energy, insight, a n d skill m otivate me. E d, I hope
som eday I can rep ay you!
iii
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G ra n t (W in g m an ) W ilson is a n in terestin g ch aracter. He is a th ro w b ack to the oldschool of e x p erim en tal cosm ologists. Well versed in ex p erim en tal techniques, and theory,
he has done a g reat deal for P O L A R an d for m e personally. B y ask in g th e h ard questions
such as “A re you sure you w ant to do th a t? ? ? ” an d “You can’t b e serious!?” , he has helped
define th e sh a p e of P O L A R from th e very begin n in g.
Back a t U W th e re w ere two people w ho m ad e m y life as a g rad s tu d e n t much more
enjoyable. D o n n a G arcia is a w izard - she keeps o u r group to g e th e r w ith h er knowledge,
org an izatio n , an d cheerful a ttitu d e . Lee P o tra tz an d en tire th e S taff o f th e Physics In­
stru m e n t S h o p a t U W tu r n ideas in to m etal. Lee was essential to th e success of POLAR.
He ta u g h t m e a g re a t d eal a b o u t m achining, m otorcycles, an d life in th e B adger S tate.
Now to m y friends w ho have s u p p o rte d m e m uch m ore th a n th e y know. My oldest
friend, S hane D iekm an, w as always th e re for me w hen I needed him . E rica C ushin has been
the su n sh in e th a t has alw ays brig h ten ed my sp irits. M y friends from C ase W estern: Joe
S ch arp f an d A d am G reen b erg m ake m e proud. Jeff (Jefe) T sen g is m y k in d red (Taiwanese)
sp irit. Back a t hom e in P u rd y s, Tom G ustinis a n d th e entire F reem an fam ily have always
tre a te d me rig h t. A t B row n two guys kept me e n te rta in e d for so m any years. M att P arry
is p roof th a t th e o re tic al cosm ology is in great sh ap e. A nd, finally, to m y Soul B rother.
S tep h o n A lexander: yo u r passion, insight, stre n g th , an d w isdom inspire m e to be a b etter
physicist.
“T h in k w here m a n ’s glory m o st begins an d ends, an d say m y glory was I had such
friends.” -W .B . Y eats
iv
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D e d ic a tio n
T h is thesis is ded icated to m y first an d best teachers: m y family, w ith o u t whom I’d be
lo s t...
M y D ad, R aym ond, is a tru e S tan d -U p Guy. His sense o f hum or and intellect have
always been g reat resources for me. T h ro u g h o u t m y life he has injected precision doses of
w isdom an d w it, ju s t w hen I needed th em m ost.
I’d be n o th in g w ith o u t m y little b ro th ers Nick an d C hris. T h ey m ake me laugh (m ostly)
an d cry (rarely), b u t I am so proud o f them . J u s t know ing th ey are in th e world fills m y
life w ith joy.
M y older b ro th e r K evin is my hero; m y rock. He is my b est friend, my inspiration, m y
g u ard ian . He m ade m e w h at I am.
M y m o th er B a rb a ra ta u g h t me ev ery th in g I know a b o u t life. W ords cannot ca p tu re
w h at she m eans to m e - I owe her everything. “If I could find a tw inkling sta r, one half
so w ondrous as you are, th a t s ta r would be, like m y h eart an d m e - dedicated to you.’’
v
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Contents
A ck n o w led g m en ts
iii
D e d ic a tio n
v
I n tr o d u c tio n
1
1
7
P r e lim in a rie s
1.1
R adio A stronom y F u n d am en tals
................................................................................
7
1.2
R ad io m etrv B a s i c s ............................................................................................................
7
1.3
E m ission and A bsorption o f R ad iatio n by M a t t e r .................................................
10
1.4
T h o m so n Scattering, P olarizatio n , and the Stokes P a r a m e t e r s .........................
12
1.5
T im e an d Frequency-D om ain R e la tio n s ......................................................................
16
1.5.1
A utocorrelation F unction: A C F ......................................................................
16
1.5.2
Frequency D om ain R e l a t i o n s .........................................................................
18
U nification: PSD <=>■ A C F ..............................................................................................
18
1 .6 . 1
19
1.6
2
Linear F i l t e r s .........................................................................................................
T h e S ta n d a rd C osm ological M o d e l and th e P o la riz a tio n o f th e C M B
2.1
21
A nisotropy of th e CM B an d th e generation of P o la r iz a tio n ....................................21
vi
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3
2.2
Ionized E p o ch s in th e E volution of th e U n iv e rs e .......................................................... 26
2.3
P o larizatio n P ro d u ced by C osm ological P e rtu rb a tio n s
2.4
T h e F irs t Ionized Epoch: R ecom bination
2.5
T h e Second Ionized Epoch: R e io n iz a tio n ........................................................................ 40
2.6
P o larizatio n Pow er S p e c t r u m ..............................................................................................44
.....................................................................35
A n In tr o d u c tio n to R a d io m etry
3.1
3.2
51
T o tal Pow er R a d io m e te r ........................................................................................................51
3.1.1
M inim um D etectable S i g n a l ................................................................................... 52
3.1.2
L im itatio n s of the T otal Pow er R ad io m eter T e c h n iq u e .............................. 53
T h e C o rrela tio n R adiom eter T e c h n iq u e ............................................................................54
3.2.1
4
............................................ 30
M inim um D etectable S i g n a l ................................................................................... 57
P O L A R : E x p e r im e n ta l D e sc rip tio n
59
4.1
P O L A R : E x p erim en tal O v e r v ie w .......................................................................................60
4.2
T h e P O L A R R ad io m eter
4.3
C old R eceiver C om ponents
4.4
4.5
.................................................................................................... 63
.................................................................................................63
4.3.1
D e w a r .............................................................................................................................63
4.3.2
V acuum System
4.3.3
C T I C ry o c o o le r............................................................................................................65
........................................................................................................64
O p t i c s ........................................................................................................................................... 6 6
4.4.1
C o rru g a te d Scalar Feed h o r n ............................................................................... 6 6
4.4.2
O p tic al C ross P o la riz a tio n .......................................................................................73
4.4.3
O rth o m o d e Transducer: O M T
H E M T A m plifiers
............................................................................ 74
.................................................................................................................. 79
vii
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4.6
5
R oom T em p eratu re R a d io m e te r Box: RTRB
............................................................ 81
4.6.1
S u p erh etero d y n e C o m p o n e n ts ............................................................................. 82
4.6.2
T he C o r r e l a t o r .......................................................................................................... 84
4.6.3
M u ltip lic a tio n ............................................................................................................. 87
4.6.4
E lectronics Box a n d H o u s e k e e p in g ................................................................... 96
4.6.5
P o st-D etectio n E lectronics: P D E ...................................................................... 97
4.6.6
DAQ H ard w are a n d S o f t w a r e ............................................................................. 98
4.7
R o ta tio n M ount a n d D rive S y s t e m ................................................................................ 99
4.8
In stru m en t B a n d p a s s e s ........................................................................................................101
C a lib ra tio n
104
5.1
C alib ratio n d e s id e r a t a ........................................................................................................104
5.2
T w isted-C old L oad (T C L ) C alib ratio n s
5.2.1
5.3
5.4
5.5
....................................................................106
System Noise T e m p e r a t u r e ...............................................................................108
W ire G rid C alib rato r: W G C
.........................................................................................112
5.3.1
G ain M a tr ic e s ........................................................................................................... 120
5.3.2
C alib ratio n U sing th e W G C ...............................................................................121
5.3.3
R esults o f W G C : G a in M atrices and S ystem atic E f f e c t s ...................... 122
Noise A nalyses an d N oise E qu iv alen t T em peratures: N E T s
............................. 124
R esults of Noise A n a l y s e s ................................................................................................. 128
6
S y s te m a tic E ffects
132
7
L arge A n gu lar S c a le F o r e g ro u n d s in th e K & b an d
137
7.1
S ynchrotron E m i s s i o n ....................................................................................................... 139
7.2
P o larizatio n P ro d u ced by In te rste lla r D u s t ................................................................ 141
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7.2.1
T h e rm a l D ust E m is s i o n ...................................................................................... 142
7.2.2
Diffuse Em ission from R o ta tin g In te rste lla r D ust G r a i n s ........................ 142
7.3
B rem sstrah lu n g E m i s s i o n ................................................................................................ 143
7.4
E x tra g a la c tic P o in t S o u r c e s .............................................................................................146
7.5
A tm ospheric C o n ta m in a tio n .............................................................................................146
7.5.1
7.6
8
S u m m ary of A strophysical Foregrounds
....................................................................149
O b serv a tio n s
150
8.1
S ite
8.2
A tm ospheric e f f e c ts ..............................................................................................................151
8.3
O b serv atio n S trategy' and Sky C o v e r a g e ....................................................................153
8.3.1
9
P o larized Em ission from th e E a r th ’s A tm osphere in th e K & B and . . 147
........................................................................................................................................... 150
S en sitiv ity to the Power S p e c t r u m .................................................................154
D a ta R e d u c tio n and A n alysis
157
9.1
D a ta A nalysis M e th o d o lo g y ............................................................................................ 157
9.2
B inning of D a t a .................................................................................................................... 158
9.3
9.2.1
In ter-b in C orrelations caused by ro ta tio n of
th e p o la r im e te r ........ 161
9.2.2
Is O u r B inning S trategy O p t i m a l ? ................................................................ 165
C o rrelatio n B etw een Sky P i x e l s ..................................................................................... 168
10 R e su lts
172
10.1 E x p ected Long-Term Perform ance o f P O L A R .......................................................... 173
10.2 Long-T erm In teg ra tio n Tests
.......................................................................................... 174
10.3 E stim a te d P olarized Signal Level an d U n c erta in ty
10.3.1
................................................175
E s tim a te d T otal Signal Level in M odels w ith E arly R eionization
ix
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. . 175
10.4 T em perature-P olarization C ross-C orrelation an d C O B E .................................... 180
10.4.1 C orrelations Between T em p eratu re an d Stokes P a r a m e te r s ....................180
10.4.2 T he C O B E D M R I n s t r u m e n t ........................................................................... 183
10.4.3 M odel an d D a ta I n p u t ......................................................................................... 184
11 C o n clu sio n s
186
x
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List of Figures
1 .1
T hom son S catterin g G e o m e t r y ......................................................................................... 14
2.1
O rigin of th e S calar Q u a d ru p o la r D is trib u tio n ............................................................. 24
2.2
T h e T h ree Q u a d ru p o lar D i s t r i b u t i o n s ........................................................................... 25
2.3
T h e O bserved P o larizatio n P a tte r n P roduced By a Single S calar P ertu rb atio n 27
2.4
Effect of D u ratio n o f R eco m b in atio n on the P o larizatio n o f th e C M B . . . .
2.5
P olarization Power S p e c tra w ith an d W ith o u t E arly R e io n iz a tio n ...................... 43
3.1
T o tal Power R ad io m eter S c h e m a tic ..................................................................................52
3.2
C o rrelatio n R ad io m eter S c h e m a t i c ..................................................................................55
4.1
C om ponents o f th e P O L A R K*. B and R adiom eter
4.2
P O L A R D e w a r .........................................................................................................................64
4.3
P O L A R K &B and Feed h o rn El-plane Beam P a tte rn a t 29 G H z: C om parison
40
.................................................. 62
W ith T h e o r y ........................................................................................................................... 70
4.4
P O L A R K a B and 31 G H z H -plane Beam P a t t e r n ...................................................... 72
4.5
29 G Hz C ross-P olarization B eam M a p ...........................................................................74
4.6
G roundscreens
4.7
H E M T S t r u c t u r e ..................................................................................................................... 79
........................................................................................................................ 77
xi
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4.8
E lectrical S chem atic of M u ltip lie r ......................................................................................85
4.9
P ream plifier S c h e m a tic ........................................................................................................... 98
4.10 R o ta tio n M o u n t S chem atic
.............................................................................................. 100
4.11 P ream plifier S c h e m a tic .........................................................................................................102
4.12 C o rrela to r B a n d p a s s e s .........................................................................................................102
5.1
T w isted C old Load C a l ib r a to r ...........................................................................................107
5.2
Y -F actor M e a s u r e m e n ts ..................................................................................................... 113
5.3
W ire G rid C a lib ra to r In P l a c e .......................................................................................... 114
5.4
W ire G rid C a l ib r a to r ............................................................................................................ 115
5.5
C alib ratio n R u n for C o rre la to r C hannel J2
5.6
Pow er S p e c tra o f All C o rrela to r C hannels a n d T o tal Power D etecto rs . . . .
5.7
Low frequency P S D o f All C o rrelato r C hannels: In-phase Lock-ins an d
.............................................................. 119
129
Q u a d ra tu re P h ase lo c k - in s ..................................................................................................130
7.1
B rightness T em p eratu re S p e c tra o f E xpected Polarized A strophysical Fore­
g rounds
..................................................................................................................................... 138
7.2
E x p ected P olarized S y n ch ro tro n Em ission a t 31 G H z .............................................141
7.3
C O B E Free-Free A n te n n a T em p eratu re M ap
8.1
S p ectru m o f A tm ospheric A n ten n a T em p eratu re in the K a b an d vs. P W V . 152
8.2
In teg ra ted A tm ospheric A n te n n a T em p eratu re in th e RTa b an d vs. P W V . . 152
8.3
1998 P W V vs. D a y ............................................................................................................... 153
9.1
C o rrela to r J l : Ten R o ta tio n s of D a ta O v erp lo tted
9.2
C o rrela to r J l : Ten R o ta tio n s o f D a ta C o-A dded
.......................................................... 146
................................................159
................................................... 160
xii
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10.1
A Q r m s and A U r m s
vs.
In te g ra tio n T im e
.............................................................174
10.2 T em p eratu re A nisotropy Sky R ealization: No R e io n iz a tio n .............................175
10.3 T em p eratu re A nisotropy Sky R ealization: T o tal R eionization a t z = 50 . . . 176
10.4 P olarized CM B Sky R ealization: No R e io n iz a tio n .............................................. 176
10.5
P olarized CMB Sky R ealization: T o tal R eionization a t z = 50........................177
10.6 S im ulation of CM B A nisotropy a n d P olarization C orresponding to P O L A R ’s O bserving S t r a t e g y ............................................................................................. 178
10.7 E x p ec ted O bservable RMS P o larizatio n vs. R edshift o f R eionization . . . .
179
10.8
C O B E T em p eratu re A nisotropy C en tered on P O L A R ’s O bserving Fields . . 185
10.9
T em p eratu re-P o larizatio n C ro ss-C o rrelatio n Sky R e a liz a tio n ......................... 185
xiii
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List of Tables
1
E x p erim en tal L im its on Linear Polarization (95% C onfidence Level). . . .
4.1
N orm alized Pow er Coefficients for G auss-L aguerre M o d e s ...................................69
4.2
P O L A R ATa-B an d M easured and Modeled FW H M B eam W id th s ........................ 72
4.3
P ro p erties of P O L A R ’s O M T: A tlantic M icrowave M odel 2800........................
75
4.4
Tolerances on F requency Response V ariations for a 2.5% R eduction in SN R
90
4.5
R adiom eter C en tro id s, B andw idths, and O bserving Sensitivities (T.4 nt~ 15K) 101
4.6
P O L A R ATa B an d R ad io m eter C o m p o n e n ts .................................................. 103
4.7
P O L A R O b serv in g P a r a m e t e r s .......................................................................... 103
5.1
System Noise T em p eratu re O btained From C o rrela to r C hannels Using..Lin­
6
ear In tercep t M e th o d ................................................................................................112
5.2
E stim ated P ro p erties of G rid and Loads Used For C a lib r a tio n ............... 118
5.3
N E T E stim ates from PSD an d RMS C om pared W ith P red icted N ET . . . . 131
6.1
E xpected S y stem atic E f f e c ts ................................................................................. 133
7.1
P ro p erties of H II R e g i o n s .....................................................................................145
xiv
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Introduction
T h e 2.7K Cosm ic M icrowave B ackground (C M B ) radiation is a v ital p ro b e o f all m odern
cosm ological theories. T h is rad iatio n provides a “sn ap sh o t” of th e epoch a t w hich radiation
an d m a tte r decoupled, approxim ately 300,000 years after th e Big B ang, a n d carries the
im p rin t of th e ionization history of th e universe. T his inform ation has been used to tightly
co n strain theories o f cosmological s tru c tu re form ation, and has ushered in th e era when
“cosm ological accu racy ” is no longer a p ejo rativ e term .
T h e CM B was definitively identified in 1965 by Penzias a n d W ilson [lj. T h e three
defining ch aracteristics of this relict ra d ia tio n are: its spectrum , sp a tia l anisotropy, and
polarizatio n . Since th a t tim e, num erous ex p erim en ts have characterized its sp e c tru m and
s p a tia l anisotropy:
S p e c tr a l M ea su r e m en ts
T h e C O B E F ar-In frared A bsolute S p ectro p h o to m er (FIRA S) has d eterm in e d the th e r­
m odynam ic blackbody te m p eratu re of th e C M B to be 2.725 ± 0 .0 0 2 K [2]. However, since
th is in stru m en t p robes th e CM B over a lim ited sp ectral range (v ~ 10 to 600 G H z), to
pro b e th e b lackbody n a tu re of th e CM B over th e w idest possible ran g e o f frequencies we
m u st also consult su p p lem en tal ex p erim en tal evidence from lower frequency m easurem ents
[3]. T h ese lower frequency m easurem ents d o m ore th a n simply confirm th e resu lts of FI-
1
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RAS. they serve to co n stra in d ev iatio n s from th e P lanck sp e c tru m to (nearly) negligible
levels. M ost notably, th ese low -frequency m easurem ents co n stra in th e so-called “chem ical
p o te n tial” d isto rtio n , w hich resu lts in a decrem ent of th e P lan c k sp e c tru m a t low frequen­
cies. P h o to n s, being bosons, have a num ber d en sity in eq u ilib riu m o f n = —„/fcbr. 1-^,,—[•
T h e P lanck sp ectru m se ts fi0 = 0, b u t th e existence of a non-zero p 0 can not be ruled-out
a t present. T h e best 95% confidence u p p er-lim its suggest th a t \p.a\ < 4 x 10-
*1 [4 ].
At frequencies n ear th e peak of th e Planck sp ectru m a C o m p to n -y d istortion m ight
be expected from C o m p to n sc a tte rin g of p h otons by electro n s h eated by a hypothesized
energy release p rio r to deco u p lin g a t a redshift o f z ~ 105. T his ty p e o f distortion has
a characteristic sp e c tra l sig n a tu re in th a t th e h o t electro n s increase th e frequency of
scattered p h otons while conserving photon n um ber density. T h e y -facto r param eterizes
th e te m p e ra tu re difference betw een th e hot electrons an d cooler photons: y = / drfc&(Te —
Tcmij)f m ec2,where Te is th e k in etic te m p e ra tu re of th e electro n s. T h e resu lt of a C om ptony disto rtio n is a sp ectra l d ecrem en t below th e p eak frequency of th e C M B , and a spectral
increm ent above th e p eak . T h e b est u p p er lim its on th e C o m p to n -y factor suggest \y\ <
1 X l ( r 3 [ 5 ] ,[ 6 ] .
A n iso tro p y M e a su r e m e n ts
Large Scale A n iso tro p y
T here are a t present a p le th o ra of detections o f an iso tro p y of th e CM B, the largest of
which is th e dipole an iso tro p y resu ltin g from th e e a r th ’s p ro p e r m otion w ith respect to
th e nearly isotropic P lan c k ian sp e c tru m m entioned above. T h is effect is non-cosmological,
so it is never included in m odels w hich predict th e an iso tro p y o f th e CM B. E xpanding
th e observed te m p e ra tu re p a tte r n on th e celestial sphere in to spherical harm onics we
have:
7
^ (
0
,<£) = T ,e,m aL Y^m , . T h e first cosm ologically significant anisotropy is the
2
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quadrupole, w ith a n ex p ec tatio n value o f C 2 = r ^ 7l6 ( |a 2 ,m|2) = y /^ E m = - 2 la 2 ,m|2- T h e
RM S quad ru p o le is used to norm alize th e CM B pow er sp ectru m . T he am p litu d e of th e
pow er sp ectru m is defined to be:
Since th e angle su b ten d ed on the celestial sp h ere by a spherical harm onic m ultipole P.
scales as 6 ~ ~ j l , th e q u ad ru p o le anisotropy is a large-scale anisotropy ( 6 ~ 90°. T he
best lim its on Q r m s com e from th e C O B E D M R ex p erim en t, which a t 95% confidence
are 4p K < Q r m s < 28p K , where th e relative im precision is a ttrib u te d prim arily to
co n tam in atio n by g alactic em ission [7], In th e a n g u la r range 90° < 9 < 10°, th e D M R also
provides th e m o st precise m easurem ent o f te m p e ra tu re anisotropy. O n 10° scales, C O B E
detects A T
30^xK [8 ].
Interm ediate a n d S m a ll Scale A nisotropy
T here has been a n intense effort to m easure th e an iso tro p y of th e CMB a t sm aller scales
because the flu ctu atio n s on these scales are exp ected to b ear th e im print of m icrophvsical
processes occuring in th e early universe prior to an d d u rin g decoupling.
All o f these
experim ents are e ith e r ground o r balloon based. VVe refer th e reader to th e recent reviews
[7] a n d [9] for c u rre n t results.
W h a t A b o u t P o la riza tio n ?
P olarization of th e C M B has received co m p arativ ely little experim ental atte n tio n , de­
sp ite its fu n d am en tal n atu re . T h e anisotropy an d p o la rizatio n depend on th e prim ordial
pow er sp ectru m of flu ctu atio n s as well as th e io n izatio n h isto ry o f th e universe in different
ways. As I will d e m o n stra te in C h ap ter 2, a d etectio n o f polarization would com plem ent
3
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th e detectio n s of anisotropy by facilitating th e reco n stru ctio n o f th e in itial spectrum of
p e rtu rb a tio n s as well as th e ionization history* of th e universe.
T h e m ag n itu d e an d sp atial d istrib u tio n o f p o larizatio n is d eterm in ed by factors such
as: th e source o f th e CM B anisotropy, th e d en sity p a ra m e te r fi0, th e baryon content
of th e universe f ig , th e H ubble co n stan t H 0, and th e ionization histo ry o f the universe.
CM B p o larizatio n is p aticu larly sensitive to th e ionization h isto ry of th e universe, which
includes th e d u ra tio n of recom bination and th e epoch of reionization. T h e detection of,
or a fu rth er co n strain t on, th e po larizatio n o f th e C M B has th e p o te n tia l to dram atically
enhance o u r u n d erstan d in g of th e pre-galactic evolution o f th e universe.
Sim ilar to th e CM B anisotropy power sp ectru m , th e p o larizatio n pow er spectrum
contains in fo rm atio n on all angular scales. Large an g u lar scales (larger th a n ~ 1°) corre­
spond to regions on th e last scatterin g surface which were larger th a n th e causal horizon
a t : ~ 1000. In th e absence o f reionization, these scales were affected only by the long
w avelength m odes o f th e prim ordial pow er sp ectru m . T h is region of th e anisotropy power
sp ectru m was m easured by th e C O B E D M R , an d establishes th e n o rm alization for models
of large scale s tru c tu re form ation. Similarly, m easurem ents of po larizatio n a t large angu­
lar scales will norm alize th e entire po larizatio n power sp ectru m . Because th e anticipated
signal size is sm all a t all angular scales, p o larization m easurem ents face m ore form idable
challenges th a n an iso tro p y m easurem ents.
T h e ex p erim en t described in this thesis m easures p o larizatio n signals on large angular
scales. W hile th ese signals may be weaker th a n signals on sm all scales, th e design of a large
an g u lar scale m easurem ent is com paratively sim ple an d co m p act, w ith p o ten tially lower
suscep tib ility to sources o f system atic error. A d etection, o r im proved u p p er lim it, a t large
an g u lar scales is a n a tu ra l first step tow ards probing th e p o larizatio n pow er spectrum on
4
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all an g u lar scales. In th is reg ard , th e present sta te o f th e p o larizatio n field is rem iniscent
of the an iso tro p y field a decade ago.
In th is thesis we w ill review theoretical arg u m en ts w hich suggest th a t th e ratio of
polarization to a n iso tro p y should be in th e range
0 .1 %
to
1 0 %,
a t large a n g u lar scales.
The best cu rren t u p p e r lim its o n polarization are co m p arab le to th e m easured anisotropy
level itself (see T ab le
1 ).
the level o f A T /T cmb ~
M easurem ents of anisotropy, by C O B E , an d o th e r ex perim ents on
1
x 10 - 5 indicate th e required level of sen sitiv ity to polarization
m ust be a t least A 7 y T cmb < 1 x 10- 6 . T hus, to o b ta in new non -triv ial inform ation,
either a positive d e te c tio n , o r a n im proved u p p er lim it cap a b le o f d iscrim in atin g betw een
different cosm ological scenarios, necessitates extrem ely precise m easurem ents.
C u rren t d e te c to r technology is capable of achieving th e required level o f sensitivity.
However, in a d d itio n to achieving high sensitivity it is essen tial to d iscrim in ate th e p o lar­
ization from sy ste m a tic effects, such as non-cosm ological astro p h y sical sources of polarized
radiation. S pace-based m issions, such as MAP[10] an d P lan ck Surveyor [1 1 ] will produce
full-sky an iso tro p y m ap s, an d are expected to achieve th e req u ired sen sitiv ity level to m ea­
sure p o larization as well. T h e pro jected sensitivity levels w ill allow for per-pixel detections
of anisotropy w ith sig n al-to -n o ise ratios (SNR) > 1. T h e p o larizatio n m ap s from these
missions, however, a re ex p ected to have S N R < 1 for each beam -sized pixel, and will be
of lower resolution th a n th e aniso tro p y m aps. F o rtu n ately , p o larizatio n observations are
also possible from th e g ro u n d ; as we will d em o n strate, po larized atm o sp h eric em ission is
expected to be negligible.
T his thesis describ es a n ongoing, ground-based p o la riz a tio n experim ent, P olarization
O bservations o f L arg e A n g u lar Regions (P O L A R ), o p tim ized to m easure C M B polariza­
tion a t 7° scales, for ~ 36 pixels. T h e design in co rp o rates m an y techniques developed for
5
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previous an iso tro p y an d p o larizatio n experim ents, from th e g ro u n d , balloons, an d space.
In C h a p te r 2 we review th e th e o ry o f CM B polarization, w hich m o tiv ates P O L A R ’s
ex p erim en tal design. We describe th e in stru m en t in C h a p te r 4, an d th e calibration of
P O L A R in C h a p te r 5. In C h a p te r 7 we focus ou r a tte n tio n on a significant challenge
to the d etectio n o f th e p o larizatio n of th e CM B: th e discrim in atio n of C M B polarization
from p o larized foreground sources. C h a p te r
presents an overview of o u r observing strat-
8
egy. which is designed to m inim ize th e tim e required to d e te c t a cosm ological signal. In
C h ap ter 9 we presen t th e p relim in ary analysis of the initial observing ru n of PO LA R.
C h a p te r 10 sum m arizes th e resu lts o b tain ed to d ate, along w ith a form alism for com par­
ing fu tu re d a ta sets w ith te m p e ra tu re aniso tro p y d ata, an d calcu latin g e stim ates of the
epoch of reionization. Finally, we sp ecu late on th e conclusions w hich could be draw n from
a d etectio n of C M B po larizatio n , as well as fu tu re directions an d goals o f th e PO L A R
cam paign.
T able
1:
E x p erim en tal L im its on L inear Polarization (95% C onfidence Level)
Reference
P e n z ia s & W ils o n 1965 [1]
C a d e rn i e t a l. 1978 [12]
N a n o s 1979 [13]
Sm oot
L u b in 1979[14]
L u b in &£ S m o o t 1981 [15]
P a r tr id g e e t a l. 1997 [16]
A n g u la r S c a le
scattered
n e a r g a la c tic c e n te r
6
6
=
=
+40°
3 8 ° ,5 3 ° ,6 3 °
6 - - 3 7 ° to + 6 3 °
40™ a t
8 =
+43.5°
L im it
---0 .5 °
< 0 < 40°
15°
7°
7°
1'
~ 10° C a p a ro u n d N C P
w
o
N 'e ttc rfie ld e t a l. 1996 [18]
4.0
1 0 0 -6 0 0
9.3
33
33
8.4
26 - 36
26 - 46
S k y C o v e ra g e D e c = £
o
W o llack e t a l. 1993 [17]
F re q u e n c y (G H z )
~ 10° C a p a ro u n d N C P
0.5°
6
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0 .1
0 .0 0 1
- 0 .0 1
6 x 1 0 -4
3 x 10 " 4
6 x 10~5
1 .1 x 1 0 " b
9 x 10_b
6 x 10- b
Chapter 1
Preliminaries
This ch ap ter is in ten d ed to serve as a reference for several ch ap ters in th is thesis. T h e
results qu o ted here are, in general, not derived, an d th e in ten t is sim ply to compile a sm all
repository of in fo rm atio n which will be quoted th ro u g h o u t th is thesis. T h e reader m ay
feel free to refer to th is c h ap ter only by necessity, or skip it altogether.
1.1
R a d io A str o n o m y F u n d am en tals
We first sum m arize som e results which outline ou r observables and th e ir connection to
our m easurem ent technique.
1.2
R a d io m e tr y B a sics
A radio source observed in direction 9 can be characterized by its Brightness, with units
of [W /m 2 /H z /sr]. T h e brightness, B v {6), of a source is related to its detected power in an
area elem ent d A via:
( 1. 1)
7
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w here dCl = sin Od0d<t> , d A , du a re infinitesim al solid angle, area, a n d frequency ele­
m ents, respectively, an d 6 is th e angle betw een th e n orm al to th e area elem ent and the
line-of-sight.
T h e source’s F lu x D ensity is defined by:
Su
=
j
B„{0)dD.
T h e unit of S„ is th e Jansky = J v = 10
26
(1.2)
W /m 2 /H z.
For a blackbody, th e brightness is a function of th erm o d y n am ic te m p e ra tu re T :
2 h i?
1
B „ (T ) = - j - — -------,
c
(1.3)
evr - i
which has a m axim um brightness a t a frequency
= 5 8 .7 8 9 (^ ).
T he num ber
den sity of p h o to n s in a blackbody ra d ia tio n field is 2.03 x 107T 3m - 3 [19]. A t frequencies
m uch g reater th a n umax, we get th e fam ed W ien Law: B „ (T ) = ^ r - e ~ hu/ kT . In the
Rayleigh-Jeans region o f th e sp ectru m we have
hu <C k T => B r j ( u,T ) = ^ 4 - k T .
From this e q u atio n we o b ta in th e definition of B rightness Temperature:
rp
c2B „
T e = 2h S -
(L4)
From th e definition o f th e brightness te m p e ra tu re , which is in d ep en d en t o f th e receiver,
we o b ta in th e A n te n n a Temperature w hich is d ep en d en t on th e beam of th e rad io telescope.
T h e a n ten n a has a peak-value norm alized beam pattern, P (9 ) w hich results in an acceptance
8
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P (0 )d Q . T he m a in beam is defined as th e solid angle, fI m , su b ten d ed
solid angle o f
by th e p o rtio n o f P (0 ) betw een its first nulls. T h is resu lts in a m a in beam efficiency factor
of Ub =
T h e n th e a n te n n a te m p eratu re is defined to be:
(1.5)
If a ra d io source fills th e entire beam o f a rad io telescope an d is optically th ic k 1
th e n we see th a t th e a n te n n a te m p eratu re will eq u al th e th erm odynam ic te m p e ra tu re
of th e source. T h is fact is qu ite rem arkable, for it suggests th a t if we could couple th e
rad ia tio n received from th is d ista n t source to a b lack b o d y in o u r lab, th e lab b lackbody’s
te m p e ra tu re w ould equal th e tem p eratu re o f th e d is ta n t source. In th is way we can literally
“take th e te m p e ra tu re ” o f extrem ely d ista n t o b jects.
O ne final co ncept is useful here; th a t of th e a n te n n a ’s effective area. A radio telescope
a n te n n a m ay have zero physical area, as in th e case o f a n ideal dipole antenna, yet it is
defined to have an effective area of:
( 1 .6 )
w here, n is th e n u m b er of sp atial m odes, A is th e w avelength o f th e rad iation received
by th e a n te n n a , an d Q \r is th e m ain-beam solid angle[20][19]. For (diffraction-lim ited)
m easu rem en ts, n =
1
a n d A eQ = A2 = £7 . T h e p ro d u c t A eQ m is, in astronom ical circles,
known as th e throughput o r etendue. T h e co ncept o f a n te n n a te m p e ra tu re is useful since
it im plies a source of flux density S u will p ro d u ce an a n te n n a te m p eratu re of:
‘M eaning th a t it is in bu lk therm al equilibrium if it is a diffuse source, such as a cloud, and has an
em issivity o f unity.
9
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S vA e
~ 2 kb ’
(1.7)
which is co n sisten t w ith th e conventional relatio n sh ip between flux an d area.
1.3
E m issio n an d A b so r p tio n o f R a d ia tio n b y M a tte r
Here we present a brief, phenom enological, d escription of emission and ab so rp tio n as it
p erta in s to o b je cts frequently observed in rad io astronom ical applications [2 0 ].
Emission
We first consider an em ittin g m edium , w ith num ber density of em ittin g particles p,
and emission coefficient j u (which is th e energy e m itted by a volume elem ent d V = dr d A
in th e intervals du,dCl,dt), which produces an infinitesim al brightness:
dBv =
47r
j„ p d r e ~T,
(1.8)
p er u n it thickness o f th e medium, dr. T he optical depth, r, is defined to be:
t
= r R ctpdr,
Jo
w here a p is know n as th e absorption coefficient, w ith u n its of m - 1 , an d r is o p tical depth
which is dim ensionless Absorption
Now consider a collim ated beam of ra d ia tio n traveling a distance dr th ro u g h an ab ­
sorbing m edium . T h e brightness o f th e ra d ia tio n afte r traversing a d ista n ce R through
th e m edium is re la ted to the in itial brightness, B 0 by:
10
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B = B 0e~
apdr = B ae~ T.
(1.9)
C om bining b o th em ission an d absorption by a m edium , we have th a t:
B v = B 0e~T +
-^ -(1
7ra p
- e~T).
( 1 . 10 )
K irchofFs Law sta te s th a t, in equilibrium , we have:
dB = 0
( 1. 11 )
which relates brightness of an em ittin g an d absorbing cloud to its bulk properties in
therm odynam ic equilibrium .
A b so rp tio n and S catterin g : E x tin c tio n
T he B oltzm ann eq u atio n gives th e relation betw een th e in p u t an d o u tp u t intensities of a
collim ated beam of rad ia tio n traveling an infinitesim al distance dr th ro u g h an absorbing,
e m ittin g m edium :
dB„
( 1 . 12 )
w here th e em ission coefficient, j„, is th e energy em itted by a volume elem ent d V =
ds d A in th e intervals du, dCl, dt, an d k uB u is th e energy absorbed from a beam of specific
inten sity B u.
11
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E x tin ctio n , d en o ted by Q e, is th e com bination of ab so rp tio n and scatterin g o u t of
a collim ated beam .
E x tin ctio n is p aram eterized by Q e =
where s„ is th e o p tical
sc a tte rin g cross-section, an d er^ is th e geom etric cross section of th e scatterer. Q e varies
w ith index of refractio n o f th e scatterer, ap p ro ach in g u n ity as th e w avelength decreases to
zero, an d can becom e »
1 .4
1
a t wavelengths co m p arab le to th e dim ension of th e s c a tte re r.
T h o m so n S c a tte r in g , P o la r iz a tio n , a n d th e S to k es P a ­
r a m e te r s
T h e descrip tio n o f p o larized rad iatio n appears q u ite frequently in this thesis, b o th in th e
th eo retical d escrip tio n o f po larizatio n of th e C M B, an d th e ex p erim en tal description of its
detectio n . We choose to present a unified discussion prior to em barking on eith er course.
We s ta r t by considering a generally 2 polarized electrom agnetic wave w ith an g u lar
frequency, uj:
E = E yo sin(u;t - Sy) y -F E x 0 sin(u,'< — c5x )x.
(1-13)
T h e p o larizatio n s ta te of th e wave is com pletely ch aracterized by th e Stokes p aram e­
ters: I, Q, U, an d V .
Iy
=
{E y20)
(1-14)
Ix
=
{ E l o)
(1.15)
2i.e ., e ith e r circularly o r linearly (or both)
12
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I
= Iy + I X
(1.16)
Q
= Iy-Ix
(1.17)
U
= 2 E yo E xo cos(<5y — 8X)
(1-18)
V
= 2EyoExosin(6y — 8X).
(1-19)
Physically, I is the to ta l in ten sity o f th e rad iatio n , an d is always p o sitiv e. T h e param ­
eters Q an d U quantify th e linear p o larizatio n of th e wave, and V q u an tifies the degree
of circu lar p o larization (w hen V = 0, th e rad iatio n is linearly p o larized o r unpolarized).
T h e level of p o larization is defined as II = Y®
^ +v : and th e p o la rized intensity is
/poi = n x / .
N ote th a t th e Stokes p a ra m e te rs a re defined in intensity = brightness u nits, whereas
ou r ex p erim en t m easures th e ante nn a temperature of th e incident ra d ia tio n field. If we
need to convert between th e two we ap p ly eq u atio n s 1.4 and 1.5.
We now tu r n from th e phenom enological description of p o la rizatio n to its generation
via p h o to n -electro n scatterin g . T h o m so n scatterin g is th e low -energy lim it of C om pton
scatterin g , differing due to th e fact th a t a T h om son-scattered p h o to n will have th e same
frequency before and a fte r sc a tte rin g .
We will o nly discuss s c a tte rin g of photons by
electrons, n o t o th e r charged p articles.
T h e differential scatterin g cross-section for T hom son scatterin g gives th e intensity of
ra d ia tio n sc a tte re d into solid angle dCl :
^
=
(1-20)
w here o t is the T hom son cross-section w ith u n its o f m 2, an d th e vectors e' and e
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rep resen t th e incom ing a n d outgoing polarizations, respectively.
T h e Thom son cross-
section a ? oc r~, where r e is th e classical electron radius. We refer th e reader to figure 1.1
for definitions of coordinates used in th e scatterin g problem .
i
F ig u re 1.1:
Thomson Scattering Geometry
Following Kosowsky [21], we consider an incident unpolarized plane wave of intensity I '
which is subsequently sc a tte re d along th e z-axis. From th e sc a tte rin g cross-section defined
above we find th a t th e Stokes p aram eters of th e outgoing ra d ia tio n are:
I
^ - I ’{ 1 + cos 2 0 )
( 1 .2 1 )
Q
3tfT j, „. 2
■/ sin 6
( 1 .2 2 )
U
0,
(1.23)
87T
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where 6 is th e scatterin g angle. To d eterm in e th e n et polarization produced by an
incident d istrib u tio n I ' (6, <p), we m ust in teg rate th e above eq uations over all incom ing an d
outgoing angles. We find, as expected, th a t th ere will now be a non-vanishing U:
f = ^ L [ d n I ’( 6 , 0)(1 + cos 2 0)
167T J
q
_
if =
f
lOTT J
I*(Q ^ ((>) s in2 0cos2<£
f dCt I ' ( 0 ,
1 6 tt J
0
) sin 2
0
sin
20
(1.24)
.
At this p o in t, it is cu sto m ary to expand I'(0,4>) into spherical harmonics, viz:
I \ B , 4>) =
J 2 a t , r n Y t , m ( 0 , <t>)-
(1-25)
1,171
By the o rthogonality of th e spherical harm onics, we find from equations 1.24 and 1.25
th a t th e resulting polarization is:
(L 26 )
u =
( l 27>
which shows th a t it is th e q u ad ru p o le content o f th e incident radiation field which
determ ines th e polarization produced.
15
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1.5
T im e and F r e q u e n c y -D o m a in R e la tio n s
B o th of th e following subsections d raw heavily from th e excellent discussion given in Me
D onough a n d W halen [22]. R esults from these sections are used heavily th ro u g h o u t this
thesis in m an y seem ingly d isp arate discussions. A th o ro u g h u n d ersta n d in g is required in
ord er to co nnect the predictions o f th e o ry to th e exp erim en tal observables.
1 .5 .1
A u t o c o r r e la t io n F u n c tio n : A C F
We begin by considering a tim estream , rep resen tin g th e voltage o u t o f a d etecto r, denoted
by x(t ). T h e expectation value of x{ t) is: E [x(t)] =
x(t)p [x t]d x t, w here p[x£] is the
p ro b ab ility th a t x(t) = x t a t tim e t.
T h e n th e au to -correlation fun ctio n (A C F) of x{t) is defined to be:
= E [ x ( t i ) ( x ( t 2 )]
(1.28)
/•O O
I
X\x%p(xi, X2, t \ , ^2) d x |dx%
J — OC
If th e process which g enerates th e tim estream is a stationary ra n d o m process then
E [x (t)] = m , th e mean is a co n stan t, as is th e variance a \ . In this case we can disregard
th e ab so lu te tim es t\ an d t i in e q u atio n 1.29 an d consider in stead only th e tim e difference
betw een th em , denoted by r .
T h en eq u atio n 1.29 becomes:
Rx(t\,ti - t ) =
J
x ( t i ) x ( t i - r ) p [ x ( t i ) , x ( t i - T)]dxtld x t l -T = R i ( r ) .
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(1.30)
If R x { t i , t 2 ) = R x ( t i — t 2 ) th e n th e process is called xvide-sense statio n ary .
T hese
definitions resu lt in:
o \ — /2x(0) — m 2.
(1-31)
For a com plex function x ( t ) w hich is w ide-sense s ta tio n a ry we have th a t:
(1-32)
N ote th a t from th e definition o f th e A C F, R x ( t ) = R x {—t ). Sim ilar definitions hold
for th e cross-correlation fu n ctio n ( = Rxy) of two tim estream s x( t) and y(t), though we
will n o t present these g eneralizations of eq u atio n 1.30 here.
We wish to em phasize here th a t, alth o u g h we have been d ealin g in th e tim e-dom ain,
these results im m ediately generalize, for exam ple, to functions o f angle. As a toy exam ple,
let x { 9 ) be a scalar fu n ctio n of angle, o n a sm all-enough p o rtio n of th e sky which can
essentially be tre a te d as p la n a r, o r in th is case, one-dim ensional. C ontinuing w ith th e toy
exam ple, let x(9) = cos kO, th e n th e A C F of x{9) is:
R x { 9 , 9 — t ) = R x (t ) = i cos(A:t).
(1.33)
We see th a t a sinusoidal fu n ctio n on th e sky produces a sinusoidal correlation function.
T h is generic resu lt will be useful w hen we discuss th e au to an d cross-correlation functions
of cosm ological observables, in C h a p te r 2, w hich are often assum ed to be described by
G au ssian ran d o m fields.
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1 .5 .2
F r e q u e n c y D o m a in R e la t io n s
A tim estream x( t ) has a Fourier Transform :
X{iu) =
as long as
J — OO
x ( t ) e ~ 2*il/tdt
(1.34)
\x(t)\dt < oo. O f course th is inequality does not hold for a tim estream
rep resen tin g a physical process, such a s th e o u tp u t from our detectors. In stead we relax
th is m o re-strict criterion, an d instead require:
. 1
lim ( - )
r T /2
/
T —oo ' • / / J —T / 2
\x(t)\2d t < oo,
which can usually be satisfied. In th is case, we see th a t x ( t ) / y / T has a F ourier tra n s ­
form d enoted by X ( v , d v ) , since in th e lim it th a t T —* oo, we o b tain an infinitesim al
\f d v . Usually, we are interested in th e infinitesim al power contained in a n infinitesim al
frequency b an d , du. This becomes:
S x ( v ) d v = E[ \X (v, d v ) \ 2}.
(1.35)
S x {l>) is know n as the Power S p ectral D ensity (P SD ).
1.6
U n ifica tio n : P S D <=*» A C F
It is well-known th a t the statistic al p ro p erties o f a G aussian random v ariab le are com ­
pletely d eterm in ed by its first two m om ents: its m ean and variance. It is also well-known
th a t such a process can be com pletely described by its PSD . T his leads us to hypothesize
th a t th e tim e do m ain is no m ore fu n d am en tal th a n th e frequency dom ain, a n d we expect
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th a t th e inform ation co n tain ed in th e A C F should be re la ted to th a t in the PSD.
Let us look once again a t equation 1.30. Using th e F ourier representation of x(t) we
have:
R x { t ) = E [x(t)x*(t - r)]
(1.36)
= f - o c f - o c E[X(i/, d u ) X m(i/, d i/)e " 2,ri[^ - ‘/( t - r)J]dt
(1.37)
=
E[|X(z/, dv)\2} e ~ ™ ^ }
= /* o
(1.38)
(1.39)
w here we have assum ed th a t th e phases of th e F ourier tran sfo rm s are uncorrelated
to get th e final equality. T hus, from equation 1.39 we have th e fam ed W iener-K hintchin
T heorem which s ta te s th a t th e A C F, Rx and th e P S D , Sx are Fourier transform pairs.
We also have th a t:
crx + m 2 = R x ( 0) =
f
S x (u)di/,
(1-40)
J —OO
which is used extensively in th is thesis. This discussion will also be relevant in C h ap ter
2 w hen we relate th e observable correlation functions o f th e C M B te m p eratu re/p o la rizatio n
fields to th eir pow er sp ectra , provided by theory.
1 .6 .1
L in e a r F ilte r s
We close this section w ith a b rief discussion on th e response o f linear filters. Again we
follow Me D onough an d W h alen in sp irit and n o tation.
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C onsider a sig n a l x ( t ) w hich is passed th ro u g h a lin ear filter. T h e o u tp u t from th e
filter is a convolution of th e filter’s response function h(t) w ith th e signal, viz:
y(t) = [
h( t — t' ) x ( t ' ) d t' .
J —oo
(1-41)
T h e convolution th eo rem sta te s th a t th e Fourier tran sfo rm s of t/(£), h(t), x ( t ) are re­
lated via:
Y(u)
= H (u)X(u).
(1.42)
If th e filter’s tra n s fe r function, H ( u ) is known, th e n eq u atio n 1.42 lets us determ ine the
signal o u t of th e filter by a sim ple inverse-Fourier tran sfo rm . T h is fact allows us to analyze
num erous p h en o m en a w hich are found th ro u g h o u t th is th esis in topics
as (seem ingly)
un related as P O L A R ’s sen sitiv ity to th e p o larizatio n ’s a n g u lar pow er sp e c tru m an d noisesp ectru m o ut of P O L A R ’s lock-in detectors!
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Chapter 2
The Standard Cosmological Model
and the Polarization of the CMB
2.1
A n is o tr o p y o f th e C M B and th e g e n e r a tio n o f P olariza­
tio n
T h e aniso tro p y o f th e C M B is in tim a te ly related to its p o larizatio n . If o u r universe were
isotropic there w ould b e no m etric p e rtu rb a tio n s, an d space-tim e w ould be com pletely
characterized by a F ried m an -R o b ertso n -W alk er m etric. M etric p e rtu rb a tio n s generate the
an iso tro p y of th e C M B , which, as we will d em onstrate, g enerates th e polarization. In an
u n p e rtu rb e d universe (indeed, even in a universe w ith only first o rd e r dipole anisotropy
caused by th e e a r th ’s p ecu liar velocity w ith respect to th e last s c a tte rin g surface) there
would be no p o la rizatio n o f th e m icrow ave background. As shown in C h a p te r 1, polarized
ra d ia tio n is p ro d u ced b y s c a tte rin g o f unpolarized rad ia tio n which possesses a quadrupole
m om ent. M onopole ra d ia tio n produces no polarization up o n sc a tte rin g . D ipole radiation
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is really ju s t a special case o f m onopole rad ia tio n as m easured by an observer in a mov­
ing reference fram e. Since th e velocity o f th e observer can be transform ed to zero by an
a p p ro p ria te boost, a n o th er observer would observe a m onopole field incident on th e elec­
tro n . Since th e to ta l polarized in ten sity by eith er observer is fram e-independent, th e only
reconciliation o f this (ap p aren t) parad o x is th a t dipole rad ia tio n can produce no scattered
po larizatio n . T his is not tru e of a quad ru p o le rad iatio n field - it can n o t be transform ed
aw ay by any L orentz tran sfo rm atio n , an d this fact coupled w ith the o rthogonality of the
spherical harm onics, can be used to show th a t it is only th e quadrupole content of the
incident field which can produce p o larization via T hom son scatterin g .
T h ere are th ree classes o f p e rtu rb a tio n th a t g en erate an iso tro p y of th e CM B: scalar
co n trib u tio n s, g en erated by m a tte r an d rad iatio n density inhom ogeneities, vector p e rtu r­
batio n s g en erated by vortical flows in th e photon-baryon fluid, and tensor contributions,
associated w ith g rav itatio n al waves.
All th ree p e rtu rb a tio n s give rise to te m p eratu re
aniso tro p y in th e CM B via th e Sachs-W olfe effect [23].
We have m o tiv ated th e fact th a t T hom son scatterin g of anisotropic rad iatio n by free
electrons generates po larizatio n [24].
From equation 1.27 we see th a t scatterin g by a
single electron produces polarized rad iatio n w ith an in ten sity approxim ately
10%
of the
an iso tro p y q u ad ru p o le am p litu d e w hen averaged over all directions of ph o to n incidence
and scatterin g . In th e case o f CM B polarization th e exact p o larizatio n level, as well as the
an g u lar scale o f th e d istrib u tio n of p o larization on th e sky dep en d on th e optical d ep th
along th e observer’s line o f sight, and on th e p articu la r sources of m etric p e rtu rb a tio n
([25]; [26]; [27]; [28]). For a recent review see: [29] and [30].
W e see th a t th ere are tw o p rim ary ingredients in o rd er for p o larization of th e CM B
to arise:
free-electrons an d an incident, anisotropic, ra d ia tio n field possessing a non-
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vanishing q u ad ru p o le m om ent. T h e C M B polarization observed to d a y roughly scales as
~ /o° n e(O a 2 , o w h e r e n e(f) is free-electron density a t tim e t,
0 2 ,0
is th e quadrupole
com ponent of th e CM B te m p e ra tu re rad ia tio n a t t, and th e in teg ral is to be taken along
the line of sig h t from decoupling to today. D ue to the ap p earan ce of s in 2 term s in equation
1.24, a n d th e orthogonality o f th e sp h erical harm onics, only th e q u a d ru p o le m om ent on
the incident p h o to n field co n trib u tes to th e Thom son scatterin g . H owever, th e quadrupole
condition is easily satisfied for alm o st any class of p ertu rb a tio n .
Recall th a t prior to recom b in atio n , m a tte r (in the form o f p ro to n s an d free-electrons)
and p h o to n s were tightly coupled, i.e. th e spatial d istrib u tio n a n d te m p e ra tu re o f one
m irrored th a t of th e o th er. T h e “lin ear regim e” is th e epoch w hen th e d ensity p ertu r­
bations (due to m a tte r anisotropy) were sm all w ith respect to th e global d istrib u tio n of
m a tter. In th is case we can ex p an d th e p ertu rb atio n s induced in th e m etric of space-tim e
into a sp ectru m of plane-waves. A n individual plane wave has a m u ltip o le expansion in
cylindrical coordinates which is azim u th allv sym m etric w ith resp ect to its wave-vector.
C hoosing an explicit rep resen tatio n for an individual plane wave, we have:
OO
eizcos 0 = Y / (2e + l ) i ej t (z)Pe(cos0),
(2.1)
0
w here 9 is defined w ith respect to th e z-axis of the cylindrical co o rd in a te system , j i is
a Bessel function and Pi is a L egendre polynom ial.
A gain, we stress th a t th e p h o to n anisotropy prior to decoupling will have a m atching
spatial d istrib u tio n . T h e d ensity p e rtu rb a tio n produced by this p la n e wave will affect
the su b seq u en t evolution of th e d is trib u tio n of m a tter an d p h o to n s. A t th e peaks of the
density p lane wave we are a t a relativ e over-density in th e m a tte r d istrib u tio n , and more
23
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Figure 2.1: M atter falling into potential wells drags radiation with it. This radiation is blue
shifted and thus appears hotter than radiation coming to the electron along the troughs of the
plane-wave perturbation. The result produces a radiation field with an intensity given by the
quadrupolar pattern Y o,qm a tte r w ill ten d to accu m u late here, dragging “w arm er” p h o to n s w ith it. A peak in th e
density field is equivalent to a trough in th e g ra v ita tio n a l p o te n tia l field. As a result,
looking in d irectio n s p arallel to th e plane w ave’s w ave-vector ± z we see w arm er photons,
while in d irectio n s alo n g th e troughs we see cooler p h o to n s (see figure
2 . 1 ).
T his d istrib u tio n of p h o to n s seen by th e electro n a t rest varies as cos 2 6 , o r as th e
spherical harm onic oc y^o • F ig u re 2.2 shows this sph erical h arm onic along w ith th e o th er
two spherical h arm onics w ith £ = 2.
Now consider a “toy-u n iv erse” w ith only one d en sity p e rtu rb a tio n prior to decoupling 1 .
An observer a t th e p re se n t who is looking a t th e surface o f last sc a tte rin g will see th e results
of T hom son s c a tte rin g from electrons “seeing” th e ir ow n local q u a d ru p o la r rad ia tio n fields.
Due to th e q u a d ru p o la r an iso tro p y as seen by each electro n , th e scattered rad iatio n will
be polarized an d w ill trav el to th e observer a t p resen t, encoding th e m agnitude of th e
'S u ch a p e rtu rb a tio n is also known as a “scalar” p e rtu rb a tio n since, unlike “vector” or “ten so r” p er­
tu rb atio n s, it has no handedness, i.e., it is invariant under a p a rity transform ation.
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F ig u re 2.2:
The three quadrupolar radiation fields as seen in the rest frame of an electron
which can produce polarization via Thomson scattering. For each angular index £ there are 2 ^ + 1
values of the magnetic index m. Shown here, from top to bottom, is the radiation field produced
by tensor (gravitational wave) perturbations, ^ 1^ 2 , - 2 12 + |V2 .-I-2 l2i the radiation field produced
by scalar (density) perturbations |V2 ,o|2; and the radiation field produced by vector (vorticity)
perturbations >/|V 2 , - i l 2 + |Vb.-t-i I2- In this figure, th e darker colors represent cooler, less intense
photons, and lighter colors represent warmer, more intense photons. The polarization axis is always
oriented along the direction of the more intense photons.
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quadrupole of th e ra d ia tio n field surrounding th a t electron p rio r to decoupling. Once
generated, th e m ag n itu d e o f th e observed polarization d ep en d s only on th e orientation
of th e hot-lobe o f th e q u ad ru p o le w ith respect to th e line o f sig h t of th e observer. T he
to ta l polarization seen by th is observer in this “toy-universe” will vary as oc V^.o sin 2 9,
(see figure 2.3). To o b ta in th e com plete polarization p a tte rn resu ltin g from all density
(scalar) p ertu rb a tio n s w hich are observed a t th e present tim e, we m ust integrate the
con trib u tio n of a d is trib u tio n o f plane waves w ith a rb itra ry w ave-vectors. We note th a t
th e superposition of an y n u m b er o f scalar p ertu rb atio n s p roduces a polarization p attern
on th e celestial sphere w hich is curl-free. We refer th e read e r to [31] for an excellent
discussion of th e g eo m etry o f CM B polarization as p roduced on th e sky by vector and
ten so r p ertu rb atio n s.
2 .2
Io n ized E p o c h s in th e E v o lu tio n o f t h e U n iv erse
As m entioned earlier, th e tw o key ingredients necessary to p ro d u ce polarization of the
CM B are an anisotropic ra d ia tio n field, and a supply of free-electrons to Thom son scatter
this radiation. To satisfy th e free-electron condition we need to identify epochs in the
evolution of th e universe w hen a plasm a existed. We will now describe two such epochs:
recom bination, an d reionization. A ccording to th e s ta n d a rd m odel o f th e evolution o f the
pre-galactic m edium a fte r recom bination, the previously ionized hydrogen combined 2 to
form neu tral hydrogen w hich was tran sp aren t to th e C M B. How'ever, th e universe has
undergone a secondary ionization o f this recom bined hydrogen. G u n n
Peterson [32]
form ulate a m easure of th e ionization fraction of th e in terg alactic m edium using the lack
2Showing th a t th e term re-com bination is a aggregious misnomer!
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F ig u r e 2 .3 : S een h e re is a sin g le plane-w ave (sc a la r) p e r tu r b a tio n , w ith w av ev ccto r k in d ic a te d
by th e arro w , a t la s t s c a tte r in g p ro d u c in g a local q u a d ru p o le fo r e a c h e le c tro n lo c a te d a t th e
tro u g h s o f th e p o te n tia l p e r tu r b a tio n (h o riz o n ta l lin es). T h e q u a d ru p o la r p a tte r n Y 2.0 seen by
each electro n a t th e s u rfa c e o f la s t s c a tte rin g p ro d u c e s the s a m e am ou n t o f s c a tte r e d ra d ia tio n .
T h e o b serv er a t th e c e n te r o f th e d ia g ra m sees e a c h in d iv id u a l s c a tte re d r a d ia tio n field w ith a n
in te n sity w hich v a rie s a s s in 2 8, (8 = an gle b etw e e n k a n d th e lin e -o f sig h t) c o rre s p o n d in g to th e
s u b te n d e d an g le o f t h e h o t lo b e o f th e in d iv id u a l q u a d ru p o le s .
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o f a L v m a n -a tro u g h in th e observed s p e c tra o f d is ta n t quasars. R ecen t resu lts show th a t
th e m a jo rity o f in terg alactic hydrogen is h ighly ionized to a red sh ift o f a t least z ~
5
,
in d icatin g th a t th e universe m ust have reionized a t a n earlier ep o ch [33]. Several models
p red ic t reio n izatio n occurred in th e redshift ran g e approxim ately 30 < z„ < 70 [34], [35],
[36], [37], [38], [39].
In general, m odels o f reionization often rely on stru ctu re s such as a n early generation
o f sta rs (P o p u latio n III), or energetic proto-galaxies to provide e ith e r ionizing radiation or
collisional h e a tin g m echanism s. Thus, for ev ery m odel of reionization th e re corresponds a
s tru c tu re fo rm atio n scenario, as well as a co m m en su rate set of cosm ological param eters to
be co n fro n ted w ith o b serv atio n al evidence. W e will n o t speculate here on th e plausibility of
specific m odels o f reionization. As noted above, th e G un n -P eterso n te s t provides definitive
evidence for an ionized intergalactic m edium o u t to a redshift o f a t le a st z = 5. In fact, the
u p p er lim it o n th e redshift of reionization is se t only by th e p a u c ity o f observed quasars
beyond z = 5 an d , in principle, could be m uch higher th an this. T h e C O B E FIRAS limit
on th e C om pton-?/ p a ra m e te r
V
=J
dTkb(Te - Tcmh) / m ec2 < 2.5 x
10-5
[40], severely restric ts th e energy in p u t allow ed in m odels o f reio n izatio n , b u t does not
tig h tly co n stra in th e epoch o f reionization o r th e ionized fractio n o f th e intergalactic
m edium . T h e lim it is com p atib le w ith m an y early reionization scenarios.
As far as th e sm all a n d interm ediate-scale te m p eratu re an iso tro p y m easurem ents are
concerned, we refer th e read er to [41], w ho p erfo rm a m u lti-p a ra m eter fit to th e published
te m p e ra tu re an iso tro p y results in o rd er to e x tra c t th e earliest p erm issab le redshift of
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reionization.
T h e y conclude th a t for th e c u rren tly fashionable “Cosmic C oncordance”
model (h = 0.65, Q0 = 0.3, an d A ^ O ) , reionization is p e rm itte d as early as Zretontzation ~
35.
In c o n tra st to th e s ta n d a rd model of recom bination, no n -stan d ard m odels invoke ad ­
ditional ionized epochs. T hese models predict a prolonged, or even non-existent, recom ­
bination a n d /o r su b seq u en t ionization of th e recom bined plasm a. Since p olarization is
generated by s c a tte rin g o f photons on free electrons, its m ag n itu d e and sp atial d istrib u ­
tion could be used to d iscrim in ate between n o n -stan d ard m odels and th e stan d ard m odel
[42],
[43] , [26] ,
[44] , ngandng96,
[45] , [46],
[47], A n early reionization effectively
introduces an a d d itio n a l ‘la st’ scatterin g surface. T h is h as two effects, b o th of which, in
principle, can e n h an c e th e m agnitude of th e p o larizatio n on large angular scales. P rim a r­
ily. th e a d d itio n a l s c a tte rin g of photons d u rin g reio n izatio n can create new, or am plify
existing, p o larized ra d ia tio n v ia the T hom son m echanism discussed above. A dditionally,
the second ‘la s t’ s c a tte rin g surface occurs a t a m uch lower redshift, im plying th a t the
causal horizon on th is rescatterin g surface is larger, a n d th u s, will subtend a larger angle
on the sky today.
As we will d e m o n stra te , for reasonable n o n -stan d ard m odels, the am plitude o f po­
larization on
10°
a n g u la r scales is on th e level o f
10%
o f th e anisotropy, while for the
stan d ard m odel o f recom bination th e corresponding po larizatio n level does not exceed
1%. It is w o rth m e n tio n in g th a t all of these m odels p red ict approxim ately th e sam e level
of aniso tro p y a t
10°
scales, an d hence all of th em a re co m patible with the results o f th e
C O B E D M R ex p erim en t.
In th e re m a in d e r o f th is section we will illu stra te th e im p o rta n t theoretical features of
th e p o larizatio n o f th e C M B . We will first describe a n an aly tic trea tm e n t which predicts
29
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th e level o f polarization for b o th s ta n d a rd an d non-standard reionization histories. Next
we will describe a num erical sim u latio n of th e effect of a n o n -stan d ard reionization historyon th e p o larizatio n of th e CM B. W e will find th a t the m ore q u alitativ e analytic results
agree q u ite well w ith th e q u a n tita tiv e results of our num erical sim ulations.
2.3
P o la r iz a tio n P r o d u c e d b y C osm ological P ertu rb a tio n s
We begin o u r discussion by developing a m athem atical form alism w hich will allow us to
describe th e polarization o f th e C M B in a consistent fashion. W ith th ese tools, we will
su b sequently determ ine th e p o larizatio n signal we expect to observe, using two differ­
en t techniques. T h e first m eth o d is a n an aly tic approach which will provide a physical
fram ew ork for u n d erstan d in g th e p o larizatio n of the CM B. T h e second approach is more
q u a n tita tiv e , an d will allow us to o b ta in num erical estim ates of th e p o larizatio n signal. In
o rd er to describe th e polarization o f th e CM B, we will first introduce a param eterization
which describes th e polarization s ta te o f a rb itra ry rad iatio n fields. We th en apply this
form alism to th e polarization s ta te o f th e cosmological signal which we are seeking to
d etect.
An a lte rn a te representation for th e Stokes param eters will be of use in the following
sections.
We introduce a sym bolic vector for the distrib u tio n fun ctio n of occupation
0 -
num bers of polarized rad iatio n : n = ^ r l , where I is the symbolic v ector introduced in
C h an d rasek h a r (1960) an d is related to th e Stokes param eters in th e following way:
30
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(
\
It
I =
( 2 .2 )
U
v /
Since T hom son s c a tte rin g can n o t produce circu lar po larizatio n , V = 0, we will only
consider th e 3-vector:
(
\
I
t
I =
\
U J
An u npolarized d istrib u tio n in zero-th o rd er ap p ro x im atio n is given by:
f
O o
—-
1
\
TXq
\ 0 /
As shown in [26], an d fu rth e r discussed in [48]; [45], polarized radiation in th e presence
of cosmological p e rtu rb a tio n s can be represented as:
(
1
A = nQ
\
+ Ai
1
^
0
(2.3)
)
where fii = fiA + f in is th e correction to th e uniform , isotropic, a n d unpolarized
rad iatio n described by fiQ- T h e P lanck sp ectru m , fio, dep en d s only on frequency, and
31
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+ n n axe th e anisotropic an d polarized co m p o n en ts, respectively, which a re functions
of th e conform al tim e,
77,
com oving sp a tia l co o rd in ates, x a , p h o to n frequency, u, and
p h o to n p ro p a g a tio n direction specified by th e u n it v ector e{ 0 , 0 ) w ith p o lar angle,
azim u th a l angle,
0
0
, and
, given in an a rb itra rily o rien ted spherical co o rd in ate sy stem .
T h e eq u atio n o f radiative tran sfer in te rm s o f
6
(77, x a , u, p, 0 ), w here p = cos 0, is:
dh
a dh
dhdu
dr7 + e ‘ d x Q ~ ~ d u d q ~ 9 (n ~
(
^
w here
J = ^ / i
J
P ( p , 0 , / x ', 0 /) n ( 77, x Q,i/ ,/ i " , 0 " )d ^ /d 0 ' ,
(2.5)
w here q = crTNea. e “ are th e basis vectors, an d th e Einstein su m m atio n convention
is im plied.
In th ese expressions, a is th e cosm ological scale factor, P is th e scatterin g
m a trix describ ed by C h an d rasek h ar [24], u t is th e T hom son cross section, a n d N e is th e
com oving n u m b er d en sity of free electrons. In general, th e effects o f a p a rtic u la r choice of
m etric p e rtu rb a tio n are manifest in th e first te rm on th e right h an d side of (2.4). In th e
synchronous gauge we have th at:
du
1 d h a@ Q n
-x— = - _ e e r u
dq
2 dq
[23]. A fter re ta in in g term s up to first o rd e r in m etric p ertu rb atio n s, h a0 , a n d since ^
is o f th e first ord er, we can replace ^
in th e source term (uQ is th e u n p e rtu rb e d
by
frequency). T h is im plies th a t th e factor
7
=
= duJll gives a u n iversal frequency
d ep en d en ce for an iso tro p y and p o larizatio n effects, independent of th e ty p e of m etric
p e rtu rb a tio n s [26].
32
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The angular dependence of P is such that
4t
t / i
I o
, < t > ) n o d y 'd < p '= Q ,
( 2 -6 )
(w here 0 is th e sym bolic zero-vector), so we conclude in th e zero -th order approxim a­
tion, J = 0. For th e first o rd er approxim ation, in th e following, we will u n derstand by
“J " a ctu ally J i , in which n is replaced by n iA fter linearization a n d sp atial Fourier tran sfo rm atio n , th e eq u atio n of transfer takes
th e following form (w ith u0 replaced by u):
dti ■*
- ^ + ifcAm l f = 7 t f f c - g ( n l £ - J c ).
Here, Hg = —^ h a^ e a e ^ , an d li ‘ ” =
(2.7)
We have specified spherical coordinates in such
a way th a t y = cos 6, w here 6 is th e angle betw een a vector along th e line of sight, e, and
th e wave vector k and <z> is th e azim u th al angle of th e vector e, in th e plane perpendicular
to th e v ector k.
For a given k, h a^ can be represented as a sup erp o sitio n of scalar waves (below we will
use su b sc rip t liS” ) and ten so r g rav itatio n al waves (subscript “T ” ). Taking into account
th e ten so rial stru c tu re o f th e waves, and restrictin g o u r consideration to perturbations
w ith w avelengths longer th a n th e cosm ological horizon a t th e m om ent of equipartion (i.e.
a t th e m o m en t w hen th e energy d en sity of m a tte r equals th a t of rad ia tio n , see for exam ple
[49]), we can w rite
H k = ^T ik2n 2Ks(k) - ^ 3 (1 - y 2) cos2(i>—
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Here, th e ^/|/C5j’(^ )l2 are the am plitudes o f th e corresponding m etric p e rtu rb a tio n s a t
th e m om ent when th e ir wavelengths are equal to th e cosmological horizon.
For krj <g; 1 we have:
1
3
H r ~ — Tjk2fj.2Ks(k) - —(1 - fj.2) cos 24>Kr{k)
15
(2.9)
Z
while for kq S> 1,
1
3
H r = — q k 2ti2Ks{k) ■+• -—r-(l — fi2) cos 2(j>cos kr]K.T(k)
K
15
kq
( 2 . 10 )
For a plane wave p e rtu rb a tio n w ith w avevector k, th e anisotropy an d polarization can
be described [26] as:
/
1
\
/
*»a = ctsi/J2 - i )
\
cos 20
f d - ^ 2)
/
1
\
(1 + fi2) cos 20
^
+ 0T
-1
= 0 s ( 1 - M 2)
(2 .11)
\ 0/
\ 0 /
n n
1
\ ° /
( 2 . 12 )
—(1 -f- fj.2) cos 20
4^rsin20
\
/
S u b stitu tin g (2.11) an d (2.12) into th e integro-differential E quation of R adiative T rans­
fer, (2.4). we o b ta in th e following system of coupled o rd in ary differential equations for Q s.r
and 0s,T-
3
0 s,T +
Y
q
1
Q 0 S ,T =
- J
q
Q ^ S ,T
34
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(2.13)
£s,r + q£s,r = Fs ,t ,
(2.14)
w here £s,T — a S,T 4- /3s, T- a n d i* s,r is th e ap p ro p riate source fu n ctio n for scalar or
te n so r p ertu rb a tio n s. T his sy stem o f coupled equations illu strates th e in tim ate relation be­
tw een anisotropy an d th e g en era tio n o f polarization. In teg ra tin g th is system of equations
we o b ta in th e following gen eral solution for fis,T:
(2.15)
w here t {t], t]') =
2 .4
q (x a ) d x a is th e o p tical dep th w ith resp ect to T hom son scattering.
T h e F irst Io n iz e d E p och : R e c o m b in a tio n
We are now in a position to in v estig ate th e effects of a prolonged recom bination of the
pre-galactic plasm a. Recall th a t th e co n trib u tio n to th e p e rtu rb a tio n s p e c tra for scalars
and tensors is p aram eterized by F s ,t - B y specifying th e form o f F s ,t we are effectively
enforcing a p a rticu la r choice for o u r m odel. We will see th a t th e p o la rizatio n we observe
to d a y depends only w eakly on th e effect o f th e details of reco m b in atio n , and is m ore
sensitive to i\s,rFor w avelengths large in co m p ariso n w ith th e cosmological h o rizo n a t th e m om ent of
decoupling, t]d ,
1), th e source function, Fs ,t a t th is m o m en t can be approxim ated
by [48]:
(2.16)
35
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It can be show n t h a t th e source functions are r a th e r insensitive to th e e x a c t functional
form of th e variatio n o f th e o p tical dep th w ith resp ect to tim e [26], [44], T h ese functions
are p rim arily ch aracterized by th e epoch a n d d u ra tio n o f decoupling. Follow ing [45], we
a d o p t the following ap p ro x im atio n for th e tim e v aria tio n of th e o p tical d ep th:
ar = — d r1 r
A 77d
(see also [26] , [44] for a m ore detailed discussion). H ere A tjd is th e ch aracteristic tim e
scale of the d u ra tio n o f decoupling. A p proxim ating th e source functions u n d er th e integral
(2.15). by th e ir values a t th e m om ent of decoupling t)d , which gives th e m ain co n trib u tio n
to polarization, we have
Ps.T — -z {F s ,t )\ d A t}D f
7
Jo
\e~T — e “ ioT| —
L
(2.17)
J r
1 , 10. _
.. .
= j hi — (F s ,t )\ d A t]D
Hence.
(
n = —- 1 - In I p i
105
3
^ K S ( k ) ( l - fj.2)
1
(
- |« r ( f c )
-1
I0
<
\
y
o
(1 -F fjr) cos 20
\
—(1 + fj.2) cos 20
^
4/i sin 2 0
y
(2.18)
C om paring e q u a tio n (2.18) w ith eq u atio n (2.12), we find th a t th e p o larizatio n gener­
a ted by a single p e rtu rb a tio n m ode w ith w avevector k is given by:
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2
10
f
n^r = ~ Y o g ln y f a D f c H A z j f c b |k s ( * : ) ( 1 -
fj.2 )
3
r
- - KT ( k )
[(1+ /j-2) c o s 2d> +
1
2 f j . s in 20
(2.19)
Now we can calculate th e ro o t m ean sq u are (RM S) p o larization m easured by an an­
te n n a w ith a n full-w idth-at-half-m ax (F W H M ) 0 ^ . T he m ain co n trib u tio n to the RMS
polarization, n ( 0 / t ) , is co n trib u ted by m odes w ith k < fcmax(©A) % ?ir- =
A
9 A
n ( © ^ ) = J ( n 2)k>^
= 105 ln Y ^ d ^
dIV
(2.20)
Q s Bs + Q t Bt ,
(2.21)
where:
BS
=
J \l -
Bt
=
g — y j (1 +
Here Q s ,t =
9 f rl
n 2)2d n
=
fJ.2)2 j
( 2 . 22 )
r27T
cos“ 24>d<j) +
J
rl
4fi2dfj. J
y*27T
sin2 2<p = y .
(2.23)
fc4|/c s ,r(^ )|2x > w ith lK5,r(A:)|2 = KQsTk ns-T, a n d y/|«os,r l2 ^
th e am p litu d es of p ertu rb a tio n s w ith w avelengths equal to th e cosm ological horizon a t the
present m om ent (n = 0 corresponds to a scale invariant H arrison-Zeldovich spectrum ).
T hese am p litu d es are norm alized to th e CO B E D M R anisotropy q u ad ru p o le detection
which is approxim ately equal to 2 x 10- 5 . A ssum ing th a t n s = n r = n, we o b tain
37
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j .
n< e-4) = l o f T f l 1,1
f nD*m7rn! ( f ? )
+t <-
+T
<2'24)
T aking into account th e relatio n sh ip betw een redshift an d conform al tim e, z ~ 4 y, we
have th a t -D ~ 2^*idD ,7 hence
2
1-
VD&VD - V d —
TjD
1 A z£»
= n — 2~ = n ~
z
ZD
*
ZD
1
( zsd \
---- ( —
ZS D
\
ZD J
)
w here z s d is th e redshift o f decoupling p redicted by the standard m odel of recom bination,
and z d is th e redshift o f decoupling we are considering, i.e ., it is th e “n o n -standard”
decoupling redshift. Finally:
n (e . ) = 4 x i o - 7^
( S £ ) ( £ ) 2+f 7 n „ ,
(2.25)
where
2 x 10-2 In 4^ ^ 7°
105n/I5
1360°;
\ / Kl r + Kos 103 /
2 x 1 0 - 5 zSD I
n \ ~ 1/ 2 / 360° \ * 11 -1+ 4>/
\ 7° ) V I + 92
(2.26)
and
Kor
9 = —~
is th e ratio of th e tensor p e rtu rb a tio n am plitudes to th e scalar am p litu d es.
T h e factor
inco rp o rates th e p e rtu rb a tio n am plitudes, norm alized to th e anisotropy
quad ru p o le m easured by th e C O B E D M R . It contains all in fo rm atio n a b o u t the type of
m etric p ertu rb a tio n , allow ing us to isolate factors which d ep en d u p o n th e n atu re of the
38
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p e rtu rb a tio n s , an d th o se w hich do not. For n = 0, an d g = 0 (i.e., no ten sor p e rtu rb a ­
tions),
g =
1
~
1.
W h en g = oo (i.e., no scalar p e rtu rb a tio n s), N0iff ~ 1.84. Finally, w hen
(i.e.. equal te n so r an d scalar co n trib u tio n s), we find N0i3 ~ 1.47. From th is we
observe th a t # n,g is ra th e r insensitive to th e ra tio of te n so r to scalar am plitudes, g.
We now em phasize th e an g u lar regions to w hich th e preceding discussion is relevant.
E q u atio n s (2.24) - (2.26) ( w hich are based on asy m p to tic form ula (2.9), an d the ap p ro x ­
im ations used in (2.17)), are valid for m odes w hich satisfy: k A r jo < 1. In term s o f angle
on th e sky,
360° A tjd
— VD <
© ,4
I-
VD
We can ap p ly eq u atio n s (2.24) - (2.26) to an observation which has an an g u lar resolu­
tion ©.4 . as long as:
^
^
180 f A zd \ f zSD\ 1/2
e , > e , „ 1„ = 3 6 0 I - ^
= - ^ ( —
j
coA zd
= 6 —
f zSD\ 1/2 f A zS d \ ~ 1/2
(— )
(-n jr)
■
S p ecific M o d e ls o f R e c o m b in a tio n
As an exam ple, th e s ta n d a rd m odel of recom bination pred icts
~ o .l, which im plies
th a t ©,4 m,„ — 0.6°. For p u re scalar p e rtu rb a tio n s (n = 0), th e expected level of p o lariza­
tion a t th is an g u lar scale is: 11(0.6°) ~
6
x 10 - 6 . For an observation w ith
~
6
°, th e
p o larizatio n is 11(6°) ~ 5 x 10- 8 . T h e observed p o larizatio n is suppressed by a facto r of
~
100
w ith th is lower resolution beam .
C onsider a n o th e r exam ple, a n o n -stan d ard m odel for w hich
~ ]_t an d Zq ~ z s d »
th e an g u lar scale is: © /tmtri — 6 °. T h e p o larizatio n pred icted in this scenario is: 11(6°) ~
5 x 10- 7 . Finally, for
0 ,4
<
0 .4 mi„,
th e p o larizatio n is suppressed, and its d ependence on
39
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~ 6 x 1 O '1
.
0
F i g u r e 2 .4 :
A s c h e m a tic d ia g ra m sh o w in g th e effe cts o f s ta n d a r d , in s ta n ta n e o u s re c o m b in a ­
tio n , vs. n o n - s ta n d a r d , p ro lo n g e d re c o m b in a tio n o n th e a n g u la r sc a le a n d m a g n itu d e o f C M B
p o la riz a tio n .
is d eterm in e d by th e d etails of th e ionization h isto ry [50] , [48] , [44] , [42] , [43].
© .4
To sum m arize, for a given 0,4, th e p o larizatio n level is p ro p o rtio n al to
(see
eq u atio n (2.25)) and is sm allest for th e s ta n d a rd m odel of in stan tan eo u s recom bination.
A lternatively, th is analytic ap p ro x im atio n applies to sm aller angles in th e s ta n d a rd model,
as opposed to th e larger angles p red ic ted by n o n -stan d ard m odels, w ith prolonged recom ­
bin atio n , (see [44] for po larizatio n in s ta n d a rd a n d n o n -stan d ard m odels.
2.5
T h e S eco n d Io n ized E p o ch : R e io n iz a tio n
For P O L A R to d etect a non-zero p o la rizatio n signal a t th e large an g u lar scales which it
probes would require th a t th e ionization h isto ry o f th e universe differ significantly from th e
40
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so-called “s ta n d a rd m odel” . As m entioned earlier, n o n -stan d ard m odels o f the ionization
history are characterized by p ro tra c te d decoupling a n d /o r non-zero o p tical d ep th along
CM B photon trajecto ries. W e have discussed the effect a of n o n -sta n d a rd recom bination
using the an aly tic m e th o d tre a te d above. We now wish to exam ine th e effect of reionization
on th e details o f th e p o larizatio n o f th e CM B. This investigation lends itself particularly
well to th e num erical ev alu atio n o f th e polarization power sp e c tru m , calculated using
num erical routines such as C M B FA ST [51].
T he effect o f reionization can be param eterized in two equivalent form s. O ne m ethod
is specified by th e o p tical d e p th , r r;, for photons due to T h o m so n scatterin g along a
line of sight to th e last sc a tte rin g surface.
The second m eth o d specifies the redshift
o f reionization, zri, an d th e fractional ionization x (electro n -to -p ro to n ratio ). T he two
param eterizatio n s are re la te d as follows [33]:
- 0 0015 ( f ) 5^
( ? ) " /2 ( 4
) (1 +
<227)
w here h is th e H ubble p a ra m e te r, Q is th e to ta l density p a ra m e te r o f th e universe, and Q q
is th e density p a ra m e te r o f baryonic m a tte r. Equation (2.27) shows th e effect of curvature
of th e universe on th e o p tic a l d e p th . For reionization occurring a t th e sam e redshift and
ionization fraction, in a n o p en universe (fl < 1), the optical d e p th will be g reater th a n in
a flat or closed universe. W e also n o te th a t th e physical size of regions w hich are in causal
co n tact (H ubble radius) a t th e epoch of reionization,
is o f o rd er ~ c
t We expect
th a t regions sm aller th a n th is size will produce coherent p o la rizatio n o f th e CM B, and
affect the observed p o la rizatio n pow er sp ectru m at angular scales w hich correspond to th e
an g u lar scale su b ten d ed by th e horizon size a t th e epoch of reionization. T h is argum ent is
41
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sim ilar to those w hich p red ic t a coherence scale in th e CM B aniso tro p y pow er spectrum .
For exam ple, th e acoustic p eak s in th e CM B an isotropy pow er sp ectru m arise from causal
m echanism s (i.e., sound waves pro p ag atin g in th e photo n -b ary o n fluid) actin g on scales
o f o rd er th e horizon size a t th e epoch o f decoupling. A sim ilar effect occurs for th e CM B
po larization power sp ectru m , th o u g h in this case it is th e horizon size of th e re-scattering
surface, not th e ‘p rim a ry ’ sc a tte rin g surface, which is im p rin ted in th e observed power
sp ectru m .
Following Peebles [33], we ex p ect th a t th e observed CM B p o larizatio n an g ular correla­
tio n scale will be: @ri ~ O .lC ftsfth )1/ 3 rad. For ft = 0.1, f I b = 0.1 , h = 1 we find 0 rj ~ 1°,
an d for ft = l , f t g = 0.05, h = 0.65 we find 0 h ~ 2°. T his new an g u lar scale, ab sen t
in non-reionized m odels, is m anifested in th e sp atial p o larizatio n correlation function and
creates a peak in th e reionized p o larization pow er sp ectra a t a n g u lar scales 9 > 0.5°.
R eferring to subsection 2.6, a m ore q u a n titativ e p red ictio n of th e angular d istri­
b u tio n of po larizatio n on th e sky is o b tain ed from th e a n g u lar power sp ectrum .
Us­
ing a publicly available softw are ro u tin e (C M B F A S T -elab o rated on below) to calculate
th e power sp ectra, we have generated po larizatio n sp ectra c re a te d by scalar p e rtu rb a ­
tions in a Cold D ark M a tte r (CD M ) d o m in ated , com pletely reionized, universe w ith
x = l . f t = l , f t s = 0.05, h = 0.65. By varying th e redshift o f reionization in the range
0 < zri < 100, we co m p u te m ultiple p o larization power sp ectra , which are displayed in
figure 2.5. T h e power s p e c tra illu strate th e m ain features ex p ected from th e theoretical
principles d etailed above.
Large angular scales correspond to m odes w ith wavelengths
g re a te r th a n th e w id th of th e last scatterin g surface. P rio r to reco m b in atio n photons and
baryons were tig h tly coupled an d th e relatively sh o rt tim escale for acoustic oscillations
prevented th e form ation of long-w avelength p ertu rb a tio n s. T h ese effects are p articu larly
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12
100
10
1. 0
8
0.8
6
0.6
4
0 .4
1c
=
o
X
cJ
50
0.2
2
0
0
20
40
60
80
100
I
F igure 2.5: Polarization Power Spectra with and W ithout Early Reionization. The solid curve
is the POLAR window function, Wi, and the theoretical model predictions are indexed by the
redshift at which reionization occurcd.
evident in m odels w ith o u t reionization.
In m odels w ith early reionization, polarization a t large a n g u la r scales is enhanced due
to m u ltip le p h o to n sc a tte rin g following reionization. A t sm aller a n g u la r scales (£ ~ 100), in
m odels w ith a n d w ith o u t reionization, th e polarization pow er s p e c tra ex h ib it oscillatory
behavior, caused by th e sam e ty p e of acoustic oscillations w hich g en erate th e D oppler
peaks in th e an iso tro p y pow er sp e c tra [47] , [50]. T h ough n o t relev an t for th e large angular
scale co n sid eratio n s discussed here, for £ 2> 100 th e p o larizatio n is highly suppressed due
to Silk D am p in g [29].
T h e pow er s p e c tra are, effectively, predictions of th e p o la riz a tio n w hich should be
observable given a p a rtic u la r observing strategy.
We will show in C h a p te r 8 th a t th e
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R M S p o la riz a tio n ex p ec ted from th e s p e c tra show n in F igure 2 , w ith
= 7°, is in the
ran g e 0.05/uK < / po[ < l.O ^K , w here th e lower Umit is s ta n d a rd recom bination w ith no
reio n izatio n , an d th e u p p e r lim it is for to ta l reionization s ta rtin g a t z = 105. T hese limits
agree well w ith th e a n a ly tic estim ates for n o n -stan d ard ionization histories discussed a t
th e beg in n in g o f th is c h a p te r. For a 6° exp erim en t an d a n o n -stan d ard ionization history,
figure 2.5 p re d ic ts a p o la rizatio n level o f 5 x 10-7 ~ l^ K w hich agrees well w ith our
n u m erical sim u latio n s o f early reionization (e.g., for
2 .6
2 re io n iz a tio n
= 105).
P o la r iz a tio n P o w er S p e c tr u m
T h e a n a ly tic tre a tm e n t above describes th e essential physics responsible for th e genera­
tio n o f C M B p o la rizatio n . W e have discussed th e aspects of n o n -stan d ard recom bination
w hich a re relev an t to th e large scale p o larizatio n of th e CM B . In o rd er to estim ate the
o b serv ab le p o la rizatio n sig n atu re, we now d etail a m ore q u a n tita tiv e ap p ro ach based on
th e p o la riz a tio n pow er sp e c tru m . T his ap p ro ach also allows us to analyze th e effect of an
early reio n izatio n on th e observed p olarization.
F o r q u a n tita tiv e e stim ates, th e p o larizatio n an d aniso tro p y source term s which appear
in th e e q u a tio n of tra n sfe r can be decom posed into Legendre series. T he individual modes
are th e n evolved to th e p resen t w here th e sp a tia l stru c tu re of th e CM B can be com puted
[42] .
[43] ,
[51]. B ecause th e flu ctu atio n s in th e CM B are im p rin t d u rin g th e epoch
of lin ear evo lu tio n of p e rtu rb a tio n s, th e individual m odes evolve independently.
This
tre a tm e n t lends itself p a rtic u la rly well to num erical analysis [51].
S eljak an d Z ald arriag a have created a highly accu rate code to co m pute th e evolution of
th e m u ltip o le m o m en ts [51]. We s ta r t by considering th e te m p e ra tu re o f th e CM B which,
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being a scalar valued function, can be expanded in a spherical harm onic series on th e sky,
a t a p a rtic u la r point on th e sky, x:
T (x ) = $ >
l,m
7 y my W
x)
(2.28)
where th e V/m (x) are th e spherical harm onics a t x , and
aJim) = Y q ] d n T ( n ) Y { im )(n)
(2.29)
are th e te m p e ra tu re m ultipole coefficients and To is th e m ean CM B te m p eratu re.
T h e te m p e ra tu re tw o-point correlation function is given by:
C r t = 2£~+ 1
(2.30)
T h e variance of th e a-r.tm is given by th e C t,{ , since V ar[a 7 '^m] = (|ar,£m |2) —(|ar,£m |)2 =
(|ar./fm|2) = Cy,£ if th e ax,fm axe G aussian d istrib u ted w ith zero m ean, an d ( ...) denotes
a w hole-sky average followed by an average over all observational positions.
T h e p o larizatio n of th e CM B is a tensor-valued function, w ith a sy m m etry group dif­
ferent from th a t of th e anisotropy. T h e m ain com plication arises since th e polarization
observables are co o rd in ate d ep en d en t w hereas an y th eo retical m odel w orth its salt will
pro d u ce fram e-independent predictions. T hus we are led to consider fram e-independent
estim ato rs of th e power sp ectra . A h in t a t how to proceed is provided by th e single hereto­
fore neglected Stokes p aram eter: V. Recall th a t V is associated w ith circu lar polarization
and hence is in variant u n d er ro tatio n s a b o u t th e line o f sight. So if we can construct
a “Stokes P a ra m e te r” w hich looks like V, b u t param eterizes linear po larization it too
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may be ro ta tio n a lly independent.
T h e new p aram eter is actu ally two complex, linear
com binations o f Q an d U :
(Q ± iU ) " ( x ) = e
x
p
± iU ) ( x ) .
(2.31)
N ote th a t since th e ex p o n en tial is essentially a phase factor it will vanish w hen we
con stru ct th e pow er sp ectru m of (Q ± i U ) since th en we will calcu late ((Q ± i U ) ' ( Q ± i U )*'),
which kills off th e fram e-dependent factor.
P o larizatio n is usually described as e ith e r a vector field on th e 2-sphere, or by th e
(fram e d ep en d e n t) Stokes p aram eters w ritten as if th e celestial sp h ere were actually a
plane. N eith er o f which is correct, and b o th tak e away essential geom etric inform ation
contained in th e po larizatio n field. T h e p o larization is actu ally a spinor field on the 2sphere - a m o st unfam iliar concept. To characterize it com pletely requires an expansion
of th e p o la rizatio n field into a su itab le basis for spinors on th e 2-sphere.
Z aldarriaga
and Seljak [52] have m ade initial w ork in th is direction based on th e formalism originally
developed by R oger Penrose an d o th ers [53] for g rav itatio n al wave applications.
An equivalent, th o u g h m ore geom etric approach, was developed by Kamionkowski,
Kosowskv, an d S teb b in s [54]; h ereafter referred to as KKS. T h e K KS technique involves
expanding th e p o larizatio n field on th e sky as a tensor field : Vabin)- T h e polarization
tensor is a 2 x 2 sym m etric (V ab = ‘Pba) a n d trace-free (gabV ab = 0) tensor, param eterized
by two real q u an tities. Given th e Stokes p aram eters Q an d U m easured in any coordinate
system , we can co n stru ct V ab• For exam ple, in spherical p o lar coordinates, (0,4)), th e
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m etric is gab = d ia g ( l, sin 2 0) and
/
Q(n)
—£ /( n ) s in 0
Vabin) = \
(2.32)
^ —£ /(n )s in #
—Q ( n ) s in 2 0
As pointed o u t by K K S , th e factors of sin 6 m u s t be included since th e coordinate basis
for (6, <p) is a n o rth o g o n al, b u t n o t a n o rth o n o rm al basis.
We now seek th e eq u atio n s corresponding to 2.28 for th e expansion of th e polarization
ten so r in term s o f a com p lete set o f o rth o n o rm al basis fu n ctio n s for sym m etric trace-free
(S T F ) 2 x 2 te n so rs o n th e 2-sphere,
(2.33)
1=2 m=—l
w here now we have two sets of expansion coefficients:
f d n V a b (n )Y l$ -(n )
(2.34)
N ote th a t, since th e scalar Spherical H arm onics o b ey o rth o n o rm ality conditions, th e
basis functions ^ ^ n)ai ( n ) and ^ / ^ ^ ( n ) will as well. T hese functions are given in term s
of covariant deriv ativ es o f th e scalar spherical harm o n ics by [54]:
Y(lm)ab ~~ N l ( Y(lm):ab
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(2.35)
an d
(2.36)
(lm)ab
w here ea&is th e com pletely an tisy m m etric tensor, th e “:” den o tes covariant differentiation
on th e 2-sphere, an d
(2.37)
is a n o rm alizatio n factor. T h e new o rth o n o rm ality p ro p erties are:
(2.38)
(2.39)
T h ese functions are th e rank-2 generalizations o f th e “V ector S pherical H arm onics”
used to F ourier ex p an d th e electrom agnetic field [55]. As po in ted o u t by K KS, the existence
of tw o sets of basis functions, “G ” an d “C ” , is due to th e fact th a t a n S T F 2 x 2 tensor
is specified by tw o in d ep en d en t p aram eters, which shows, n o t surprisingly, th a t the linear
p o larizatio n o f a region on th e celestial sp h ere is com pletely specified by tw o param eters, Q
an d U. In tw o dim ensions, any S T F te n so r can be uniquely decom posed into a sym m etric
p a rt: A :af, — ( 1 /2 )gabA:cc an d an an tisy m m etric p art: B :acecb -F B :bc£ca w here A and B are
tw o scalar functions. T h is is rem iniscent of th e H elm oltz T heorem w hich contends th a t
a vector field (in 2 dim ensions) can be ex p an d ed into a p a r t which is th e gradient of a
scalar field a n d a p a rt which is th e cu rl o f a vector field; hence KKS use th e notation G
for “g ra d ie n t” an d C for “curl” . T h e correspondence is even stro n g er: if we have an m ap
of p o la rizatio n on th e celestial sphere we can easily d istin g u ish p o larizatio n p attern s with
a large “C ” co m p o n en t by locating divergence-free “sw irls” a n d “eddies” , from large “E ”
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regions which axe distinguished curl-free p attern s.
Follow ing KKS, we see th a t by in tegration by p a rts eq uations (2.34) transform into
integ rals over scalar spherical harm onics an d derivatives o f th e p o larization tensor:
“ S'") =
j
fi),
(2.40)
(2.41)
a (tm) = Y Qj
T h e second eq u atio n follows from e"6:c = 0. Since T an d Vab are real, all o f th e m ultipoles
m ust o b ey th e reality condition
(2.42)
S calar p e rtu rb a tio n s can produce only G -type po larizatio n and not C -ty p e polarization. O n th e o th e r h and, tensor or vector m etric p e rtu rb a tio n s will produce b o th types
[51]. S calar p e rtu rb a tio n s a re associated w ith curl-free m otion of th e photo n -b aryon fluid
prio r to decoupling an hence have no handedness so th ey can n o t produce any “curl” ,
w hereas v ector an d ten so r p ertu rb a tio n s do have a handedness. As p ointed o u t by KKS
an d Seljak, Z aldarriaga[52], an d Z aldarriaga [56], observation of prim ordial (i.e., not pro­
duced a t reionization o r by foregrounds) C -type p o larization (a nonzero
in th e CMB
would in d icate th e presence o f either vector or tensor (or b o th ) p e rtu rb a tio n s a t th e tim e
of last scatterin g . T h is would be stro n g evidence for th e existence of g rav itatio n al waves
(w hich pro d u ce ten so r p ertu rb a tio n s), an d provide th e first detection o f these objects.
G iven a foreground-free, low-noise m ap of th e p o larizatio n p a tte rn on th e sky we could
easily identify regions w ith non-zero “C” -ty p e polarizations, an d thus, we could literally
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confirm th e existence o f gravitational waves by eye!
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Chapter 3
An Introduction to Radiometry
A m icrowave ra d io m e te r is a class of radio receiver w hich can m easure the an ten n a tem ­
p e ra tu re o f d is ta n t sources.
R adio A stronom y a t m eter-w avelengths was pioneered by
K. Jan sk y in th e m id -1930's, b u t it was n o t u n til a fte r W orld W ar II th a t microwave
rad io m etry w as ap p lied to th e peaceful stu d ies of so u rces o f microwave emission. R o b ert
Dicke is c re d ite d w ith th e developm ent of th e first m o d e rn microwave radiom eter w hich
could m easure su b -K elv in differences in a n te n n a te m p e ra tu re w ith integration tim es o f a
few seconds [57]. T h e tech n iq u e known as “Dicke S w itch in g ” is a fundam ental o p eratin g
principle in all m o d e m d ay microwave rad io m eters, th o u g h its im plem entation takes on
num erous guises.
3.1
T o ta l P o w e r R a d io m ete r
T h e sim p lest ra d io m e te r is know n as the “T o tal Pow er R ad io m eter” . This device precedes
D icke’s m e asu re m en ts, a n d its shortcom ings m o tiv ated th e developm ent of th e technique
w hich b ears h is n am e.
T h e T o tal Power R ad io m eter consists of an antenna, a radio
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F i g u r e 3 .1 :
S c h e m a tic o f a S u p e r h e te r o d y n e T o ta l P o w er R a d io m e te r. T h e a n te n n a couples
ra d io fre q u e n c y (R F ) p o w er in to a h ig h -fre q u e n c y am p lifier, w hose o u tp u t is su b se q u e n tly dow n
c o n v e rte d in fre q u e n c y by a m ix e r a n d lo cal o sc illa to r. T h e in te rm e d ia te fre q u e n c y (IF ) b a n d is
th e n a m p lifie d a n d d e te c te d . T h e final s ta g e involves am p lific a tio n , a n d f iltra tio n (in te g ra tio n ) to
red u c e th e n o ise o n th e re c o rd e d D C sig n a l.
frequency (R F ) am plifier, a pow er d e te c to r ( “square-law ” d etecto r), a n d an integrating
elem ent. T h e square-law d e te c to r p ro d u ces a voltage whose DC level is p ro p o rtio n al to th e
power received by the radiom eter. R id in g on th is DC voltage is a noise-waveform, which
is sm oothed by th e in teg ratio n elem en t (usually a sim ple RC circu it).
In figure 3.1 a
slightly m ore com plicated receiver is show n which uses a dow n-conversion schem e (mixer)
to convert th e R F band to a lower, m o re easily processed band known as th e interm ediate
frequency (IF ) band. T his tech n iq u e, know n as heterodyning, is im plem ented in two of
P O L A R ’s channels: called “TPO” a n d “T P 1 ” , which d etect th e to ta l pow er in the two
orth o g o n al p o larizatio n s ta te s of th e in cid en t ra d ia tio n (which can b e p artially polarized
or unpolarized).
3.1.1
M in im u m D e te c ta b le S ig n a l
T h e m inim um d etectab le signal of a rad io m ete r is defined as th e te m p e ra tu re of a source
w hich produces a power level eq u al to th e pow er level produced by th e th erm al noise of
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the in stru m e n t itself in a given in teg ratio n tim e [20].
C onsider a rad io m eter w ith a system noise te m p eratu re Tays looking a t a load at
absolute zero.
We assum e th a t an te n n a accep ts only one sp atial electrom agnetic field
mode a n d one po larizatio n , the R F am plifier has a b an d w id th of A u p p , an d th e in teg rato r
has a b an d w id th of A v l f . T he power p ro d u ced by th e system is kbTay3A u n p , w here kb
is B o ltzm a n n ’s co n stan t. This produces a square-law voltage sp ectral d ensity (VSD) of
S FF oc kbTslJs. From C h a p te r 1 we note th a t th e RM S fluctuations in th e power produced
by th e sy stem are a 1 = J/^l/RF( S ^ F d u )2 = k^(A u R jr)(T Sy3)2. This noise signal is integrated
for a tim e r corresponding to A u ^ p oc i . From th e convolution theorem applied to linear
filters, as in C h a p te r 1, we find th a t th e to ta l system power RMS is:
a 2 = 2 (A vR f-A u iF )(kt,T sys)2
(3.1)
W hen looking a t a source of an ten n a te m p e ra tu re A T , th e square-law d e te c to r pro­
duces a D C signal voltage: V oc (fc&TAis). To get th e m inim um d etectab le signal in an
in teg ratio n tim e r . we find th e A T th a t produces th e sam e am ount of DC pow er as the
flu ctu atin g AC pow er produced by th e noise te m p e ra tu re of th e radiom eter. We find th a t:
AT =
/ J sys- .
(3.2)
y /A u R F T
3.1.2
L im ita tio n s o f th e Total P ow er R ad iom eter Technique
The b enefit of th e T o tal Power R adiom eter is its simplicity. U nfortunately, if th e gain of
the R F
am plifier varies, say w ith te m p e ra tu re o r voltage fluctuations in its pow er supply,
the u ltim a te sen sitiv ity to te m p eratu re v ariations of th e source willbe seriously degraded,
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an d e q u atio n 3.2 is no longer tru e. T e m p e ra tu re a n d voltage flu ctu atio n s will, in general,
be worse on long tim escales th a n o n sh o rt tim escales. This ty p e of flu c tu a tio n is known
as “ 1 //-n o is e ” . E q u atio n 3.2 in th e presence o f 1 / / noise becomes:
AT =
(3.3)
w here k a n d a are b o th positive c o n sta n ts [58]. So we see th a t th e effect of integrating
for longer p erio d s of tim e is offset b y th e p resence o f fluctuations o f th e o u tp u t of the
rad io m ete r on long tim escales. T h is w as D icke’s m otivation to sw itch th e a n te n n a between
two different loads a t a high frequency. In th is approach th e in stru m e n t does not have
enough tim e to change appreciably in such sh o rt tim escales. T h e fa ste r th e radiom eter
can be sw itched, the less im p o rta n t th e 1 / / flu ctu atio n s become. W h en o n e uses a single
receiver, a n d sw itches th e an ten n a betw een a celestial source an d a reference ta rg e t source
of know n te m p eratu re, A T is ac tu a lly a facto r o f two larger th a n in th e to ta l power
ap p ro ach . However, th e decrease in noise a t low -frequencies (n ear th e sw itch in g frequency)
m ore th a n co m pensates for this d ecrease in sensitivity.
3 .2
T h e C o rrela tio n R a d io m e te r T ech n iqu e
T h e co rrelatio n rad io m eter accom plishes th e Dicke sw itching referred to above by viewing
th e sam e source w ith two different receivers, a n d subsequently co rrelatin g th e ir o u tp u ts
a t R F frequencies using a m ultiplier. Physically, th e m ultiplier is usually b ased on a non­
linear device, such as a diode (see C h a p te r 4), w hich acts as a sw itch w hen provided w ith
a bias w aveform . T he bias power for th e d io d e com es from th e u n co rrelated R F power
in each arm , w hich being in th e R F b a n d , effectively (Dicke) sw itches a t th e se high radio
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HORN
ANTENNA
MULTIPLIERf----------- ’AMPLIFIER ----------- INTEGRATOR----------—
V««
R F a r ilF
SECTION
F ig u re 3.2: Schematic of a simple Correlation Polarimcter. Radio frequency (RF) fields arc split
into two linear polarization states by an orthomode transducer (OMT), and amplified. The field
amplitudes are multiplied, producing a DC voltage proportional to their product. The DC product
voltage is filtered and amplified before being integrated (low-pass filtered) prior to being recorded.
frequencies. T h e sy stem h as a v/2 noise advantage over th e single receiver Dicke sw itch
[20]. The only p en alty is th e com plication an d cost of th e second receiver. Consider th e
simplified co rrelatio n p o la rim e te r shown in 3.2.
R ad iatio n from th e sk y couples into a feed h o rn w hich p ro p ag ates b o th of the field’s
lin ear polarizations. T h e o rth o m o d e tran sd u cer (O M T ) sep arates th e tw o polarizations,
sending each one to its ow n receiver. T h e m u ltip lier form s th e p ro d u ct of th e two fields,
w hich are su b seq u en tly in te g ra te d . T his m u ltip licatio n a n d in teg ratio n is th e exact defi­
n itio n of th e cro ss-co rrelatio n of th e two p o larizatio n s ta te s , as defined in C h ap ter 1. As
we will show- in C h a p te r 4, for a single frequency, th e fields w hich e n ter th e m ultiplier are:
E y {t) = E Vo cos[i/f + 4>y {t)\ + n y {t)
E x (t) = E Xo cos[i/£ + <px (t)] + n-x(t),
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(3.4)
w here nx is th e noise waveform g en era ted by th e am plifier viewing p o la rizatio n sta te i.
T h ro u g h o u t we will assum e ( n i( t)n j( t)) = er2Sij. Following T h o m p so n [59], we will
express an electric field as:
E {0, u) =
E (9 , t) e ~ i2i:utd t
(3.5)
E (9 . u ) ^ vtdux
(3.6)
J —OC
E {0, t) =
J
— OC
w here E { 9 , t) is th e electric field produced a t tim e t w hen viewing a t a n angle 9 with
resp ect to th e axis o f th e horn. T h e h o rn ’s am p litu d e response fu n ctio n , G(0, u) (with
dim ension [length])1, is assum ed here to be axisym m etric. T he a n te n n a o u tp u t voltage for
polarization s ta te i € { r, y} is:
ViA(u) = r
J
Ei{0,Vi)G[O,Vi)d0.
(3.7)
— 7T
T he o u tp u t voltage, afte r b ein g am plified by th e H E M T /m ix e r/IF am plifier chain
(w ith to ta l rad io m eter voltage tra n sfe r function H ( u )), is V (iu ) = H ( u ) V iA (u).
T h e o u tp u t of th e co rrelato r is given by:
R(r) = lim
f
1 —*oo Z i J —oc
Vx ( t ) V ' ( t - r)d t.
(3.8)
'w hich is not equal to th e power response function, B(0, v), obtained, for exam ple, when m apping the
beam .
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R (t )
= l i m T_ o o 2 7 l Z > d t S ? 00dvx I ~ oad u y J ? 00d B J ^ 0 dff
x E x (9, ux )E y(6, vy )H x {9, ux )H y(0, uy )
(3.9)
* G X(.9', Vx)Gl{0, U y)^2av- te - i2KV^ t- TK
R em em bering th a t
et2wt^Ux v^ d t = 6(ux — uy ), we have:
« ( r ) = [ ° ° 7 W ) B ( v ) H x (v)H 'y(v)ei2^ d v .
J —OO
(3.10)
7(*/ ) = [Ex {vx )E-y {vy)}
(3.11)
where:
is th e source coherence function, and
B ( u ) = r d6' r Gx {9',u)G'y {9,i/)d9
J —7T
J —7T
(3.12)
is th e power response function of th e h orn, conventionally known as the beam pattern.
T h e p ro p erties
of th e source coherence function and th e beam p a tte rn com pletely
d eterm in e th e o u tp u t voltage.
3 .2 .1
M in im u m D e tec ta b le Signal
As in th e case of th e to ta l power rad io m ete r th e sensitivity depends on b o th the system
noise te m p e ra tu re and th e R F b an d w id th o f th e am plifiers. Since there are now tw o R F
am plifiers we take th e system te m p e ra tu re to be th e ir geom etric mean: T sya — \jT fy STsyS.
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For a n unpolarized source w hich fills th e an te n n a beam com pletely, each O M T polarization
p o rt sees only half o f th e to ta l in cid en t power.
Since th e correlation p o la rim e te r is essentially differencing th e pow er (a n te n n a tem ­
p e ra tu re ) o f each p o larizatio n s ta te , th e correlator o u tp u t can be sym bolized by V^n- <x
T y — T x . If we assum e th e noise o f th e tw o receivers are u n co rrelated (th o u g h approxi­
m ately equal) then by sim ple e rro r p ro p ag atio n we see th a t th e R M S noise o f th e correlator
o u tp u t is \/2 worse th a n th e to ta l pow er receiver. For th e Dicke rad io m ete r this result
also holds.
T h e m inim um d etectab le te m p e ra tu re difference in an in teg ratio n tim e r for th e cor­
rela tio n rad io m eter is:
(3.13)
We also note here, for co m p ariso n , th a t th e Dicke rad io m eter divides its integration
tim e equally between viewing th e signal an d the noise which is w hy it is y/2 tim es worse
th a n th e correlation radiom eter, a n d a full factor of 2 worse th a n th e total power radiometer[ 58].
58
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Chapter 4
POLAR: Experimental
Description
Before em b ark in g on th e technical d etails of th e P O L A R experim ent, it is a p p ro p ria te
here to give a b rief histo rical overview. T h e dev elo p m en t of th e correlation rad io m ete r
tech n iq u e preceeded th e discovery of th e CM B in 1965; see for exam ple [60].
In th e
early days of th e C O B E experim ent th e idea to a p p ly th e correlation rad io m eter to do
CM B w ork was proposed. A lthough n o t im p lem en ted on C O B E , developm ent correlation
rad io m eters a n d in terfero m eters as viable C M B receiver techniques continued for m any
decades, b o a stin g num erous experim ents in th e a re a s o f anisotropy research ([61], [62],
[63]) a n d th e sp e c tra l m easurem ents of [64]. T h e id ea to develop a correlation rad io m ete r
for in v estig atio n o f th e polarization of th e CM B c a n be tra c e d to Prof. D. W ilkinson of
P rin ceto n U niversity.
As a final in terestin g po stscrip t, we n ote th a t a (pseudo) correlation rad io m eter will
form th e h e a rt o f th e M A P [10] and P L A N C K [11] Low F requency In stru m e n t’s techniques
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to recover th e po larizatio n a n d an iso tro p y of th e CM B from space in th e early p a rt of the
next century.
4.1
P O L A R : E x p e r im e n ta l O v erv iew
P O L A R is th e first dedicated co rrelatio n p o larim eter designed for m easurem ents of the
CM B. It has th e w idest b an d w id th ( ~ 6 GHz) of any correlation receiver ever used for
CM B work. T h is ch ap ter sum m arizes th e m ajo r com ponents o f th e in stru m e n t as well as
all know n sy stem a tic effects. M easurem ent o f p o larizatio n of the C M B poses a wide variety
of ex p erim en tal challenges, m an y of w hich are fam iliar from the ex p erim en ts now m easur­
ing sp atial an iso tro p y in th e CM B. We describe below th e design o f P O L A R to illustrate
th e ex p erim en tal issues th a t m u st be addressed in any CM B p o larizatio n observation.
P O L A R m easures p o larization on 7° scales in th e K & band, which is th e sp ectral band
covering th e frequencies betw een 26 a n d 36 GHz. T his band is m u ltiplexed into three
sub-bands to allow for discrim in atio n ag ain st foreground sources an d to solve technical
problem s in th e developm ent o f a w id e-b an d w id th analog correlator.
T h e radiom eter
executes a d rift scan of th e zen ith w ith a FW H M = 7° beam p ro d u ced by a corrugated
feed horn an ten n a. In a prelim in ary engineering ru n , P O L A R has observed ~ 36 different
pixels for 5 days, a n d in a single night o f d a ta achieves a sensitivity level o f ~ 100/iK per
7° FW H M pixel.
P O L A R ’s design builds on techniques developed in previous searches for CM B po­
larization [13],
[65],
[15],
[66],
[17],
[18] and is driven by several factors: th e size
and an g u lar scale o f th e an ticip ated C M B signals, sp e c tra l removal o f foreground sources,
op tim ization o f th e observing schem e, a n d an ticip ated system atic effects.
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R adiation from th e sky couples into a co rru g ated circu lar h orn an ten n a (See Fig.
4.1). This a n te n n a h as ex trem ely low side lobes, n ear —80 dB a t 90° off axis, in bo th
polarizations, across a full waveguide b and. T h e a n te n n a o u tp u t couples to an o rth o ­
m ode tran sd u cer (O M T ), a waveguide device th a t decom poses th e incoming wave into
two orthogonal linear p o larizatio n com ponents.
T h e O M T defines th e x-y coordinate
system of th e a n te n n a . T h e en tire experim ental a p p a ra tu s ro ta te s ab o u t th e sym m etry
axis of the feed-horn in o rd er to m easure b o th linear p o larizatio n Stokes param eters.
The Q an d U Stokes p aram eters are defined in te rm s o f a co o rd in ate system fixed to
th e sky. T h ere are several approaches to m easuring Q an d U for a particu lar pixel on
th e sky. Lubin
S m o o t [15] em ploy a Dicke sw itch w hich a lte rn a te ly couples each of th e
polarization co m p o n en ts from th e O M T to a low-noise am plifier an d square-law d etecto r
[65], [15]. P hase-sensitive d etectio n at th e m o d u latio n frequency of th e sw itch yields th e
difference betw een th e se two com ponents, th e Q Stokes p aram eter, an d helps overcom e
1/ / noise from th e am plifier. A fter a 45° ro ta tio n a b o u t th e a n te n n a sym m etry axis th e
instrum ent m easures th e U Stokes p aram eter.
A second technique w hich has been used for p o la rim etry couples th e o u tp u t of an O M T
directly to two se p a ra te total-pow er radiom eters [67], [68]. T h e beam is switched on th e
sky to m easure th e s p a tia l anisotropy in two orth o g o n al polarizations. T his approach
m easures th e an iso tro p y in th e Q Stokes p aram eter o f th e incident radiation field, an d
currently provides th e m o st strin g en t upper lim its on th e p o larizatio n of the CM B.
An a ltern ate ap p ro ach , em ployed in PO L A R , is th e co rrelatio n radiom eter [60], [58].
In this in stru m en t th e tw o polarization com ponents are am plified in sep arate parallel
am plifier chains an d th e o u tp u t signals are correlated, resu ltin g in a signal proportional to
th e U Stokes p aram eter. T h is ty p e of in stru m en t effectively “chops” betw een th e two in p u t
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X
F ig u re 4.1:
nents.
Schematic of the POLAR ATa band Radiometer. See text for description of compo­
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R F signals a t a freq u en cy w hich is com parable to th a t o f th e R F signals them selves. A n
advantage of th is differencing m echanism is th a t it has n o m ag n etic o r moving p arts which
have tra d itio n ally co m p licated experim ents of th is ty p e. A fter a 45° ro tatio n th e co rrelato r
gives an o u tp u t p ro p o rtio n a l to th e Stokes Q p a ra m e te r. P O L A R ro tates continuously
ab o u t th e vertical a t 3 R P M . T h e ro tatio n m o d u lates th e o u tp u t sinusoidally betw een U
an d Q a t tw ice th e ro ta tio n frequency and allows th e rem oval o f a n in strum ental offset
an d o th er in stru m e n ta l effects th a t are not m o d u lated a t th is frequency.
4 .2
T h e P O L A R R a d io m ete r
T h e P O L A R rad io m e te r is com prised of 3 m ain sections:
• Cold receiver co m p o n en ts: optics, O M T , isolators, H E M T am plifiers.
• R o o m -tem p era tu re receiver com ponents: w arm R F am plifiers, heterodyne stage,
warm IF am plifiers, band-defining filters, d etecto rs.
• P o st-d etectio n co m p o n en ts: pre-am plifiers, low freq u en cy processing, and d a ta ac­
quisition.
4 .3
4 .3 .1
C o ld R e c e iv e r C o m p o n en ts
D ew ar
T h e com ponents co m p risin g th e “cold receiver sectio n ” are n a tu ra lly defined by those
contained in th e dew ar, w hich we will currently describe. T h e P O L A R dewar is a custom
fabricated dew ar c o n s tru c te d by Precision C ryogenic S y ste m s1 (see figure 4.2). T h e de-
'PCS: Indianapolis, In
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w ar is designed to house a C T I 350 cryocooler coldhead, and possesses no liquid cryogen
containers. T h e dew ar was designed to be q u ite flexible, accom m odating num erous pos­
sible upgrades. O ne possible up g rad e configuration would include th e a d d itio n of 2 more
co rru g ated feed horns in th e nom inal 20K (second stage) working volum e. W ith this in
m ind, th e cu rren t working cold-volum e is divided into th ree sections, w here th e
section
occupies o n e-th ird of th e shielded second stage. T h e first stage is used to cool a radiation
shield, which is nom inally m ain tain ed a t a te m p e ra tu re of ~ 80K.
F i g u r e 4 .2 : P O L A R D ew ar a n d
b a n d c o ld receiv er c o m p o n e n ts. T h e h o rn is lo c a te d off o f
th e s y m m e try a x is o f th e d e w a r in o rd e r to a llo w for fu tu re , h ig h e r-fre q u en cy receiv ers to perfo rm
sim u lta n e o u s o b se rv a tio n s.
4 .3 .2
V a cu u m S y ste m
For a large dew ar such as P O L A R ’s, a high vacuum is essential to achieve reasonable
cool-down tim es. T h e vacuum system is a n E dw ards T u rb o pum p system com posed of
a m echanical roughing pum p an d a tu rb o -m o lecu lar pum p. T he tu rb o -m o lecular pum p
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has an ex it ap ertu re, D , of 100 m m which is m aintained for th e m a jo rity o f the vacuum
length, L ~ 1750mm. We convert from ISO-100 to K F-50 (50 m m ) vacuum bellows hose
ju s t prio r to attach in g to th e d ew ar block-valve. As th e co n d u ctan ce o f a vacuum line
in th e molecular-flow regim e scales as D 3/ L , th e large d ia m e te r p u m p lines m ore th a n
com p en sate for th e relatively long length.
A fter leak-proofing th e dew ar, and cooling
dow n, it rem ains a t ~ 1 x 10-5 T o rr for m onths a t a tim e.
4 .3 .3
C T I C r y o c o o le r
Following pum p-dow n to ~ 1 x 10-4 T orr, th e pum p is d etach ed an d th e cryocooler’s
com pressor (C T I 8500 A ir C ooled) is activ ated . T he cryocooler is cap ab le o f ~ 50 W atts
of cooling power a t 80K (first stag e) an d ~ 5 W atts a t 20K (second stage).
In th e
field it is found th a t th e u ltim a te cold stag e tem p eratu res are c o rre la te d w ith the am bient
te m p e ra tu re of th e shelter in w hich P O L A R resides. Because ou r co m p resso r is air-cooled2,
as opposed to C T I's w ater-cooled version, th e com pressor’s com pression ra tio is a stro n g
function o f am bient te m p e ra tu re , w hich modifies its cooling efficiency. M aintaining th e
te m p e ra tu re stab ility o f th e co m p resso r is accom plished, to first o rd er, by a com m ercial
air-conditioner du rin g th e su m m er m o n th s which counters th e ~ 2k \ V h e a t o u tp u t from
th e com pressor. D uring th e w in ter, th e h eat o u tp u t by th e co m p resso r itself serves to
keep th e enclosed P O L A R sh elter a t a n early co n stan t te m p e ra tu re . T h e com pressor is
m echanically isolated from th e rad io m ete r by use of a sep arate ro ta tio n b earing coupled
loosely to th e m otor-driven m a in b earin g by copper braid.
T h e com pressor is fu rth er
isolated on its bearing by use o f ru b b e r p ad d in g on all su p p o rt s tru c tu re s.
2which is necessary due to th e fact th a t P O L A R rotates continuously
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T he cold ra d io m e te r com ponents are inside th e 80K stag e ra d ia tio n shield. O f course
it is essential to h av e v acuum -tight feedthroughs in to a n d o u t of th e dew ar. B o th w aveg­
uide o u tp u ts from th e H E M T s leave th e dewax th ro u g h v acuum -port W R -28 w aveguide
feedthroughs from A erow ave3. The feed th ro u g h s are m o u n ted on a single brass-disk flange,
which also serves as a feedthrough for th e H E M T bias w iring and th e Lakeshore4 # 1 0 T em ­
p eratu re D iode re a d o u t w iring. T he final m a jo r p o rt in th e dew ar is th e m a in vacuum
window th ro u g h w hich passes our
b an d sig n al. T h is p o rt is m ounted ~ 3” ra d ia lly off
the ro ta tio n axis o f th e c rv o sta t to allow for a d d itio n a l feed horns a t higher frequency as
m entioned above.
4 .4
O p tic s
P O L A R 's o p tical sy stem is sim plicity itself. T h e re axe no unw ieldy m irro rs, chopping,
flats, secondaries, te rtia rie s, etc. T h e m ain elem en t is a single co rru g ated feed horn. D ue
to the absence o f su p p lem en tal beam -form ing reflectors, spurious effects in tro d u c e d by
cross-polarization by o p tical elem ents is n ear th e m inim um possible level for a m illim eter
wave in stru m en t. O n ly o p tical polarim eters a re cap ab le o f achieving lower levels o f crosspolarization [69].
4 .4 .1
C o r r u g a t e d S c a la r F eed h o r n
T h e only tru e o p tic a l elem en t in P O L A R is its co rru g ated feed horn.
theoretical tre a tm e n t of these devices is discussed in
T h e canonical
[70]; this section m erely describes
3Aerowave C orp., M edford, MA
4Lakeshore C ryotronics, W esterville, OH
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th e relevant device p a ra m e te rs which are necessary and sufficient to c o n stru c t and model
the perform ance o f th e h orn.
P O L A R ’s horn was designed by D r. Josh G undersen, an d is b ased on th e procedure
outlined in [71], an d is n early identical to th e ATa-B and feed horn em ployed by th e
COBE
D M R ex p erim en t [72]. T h e goal of th e corrugated feed h orn is to p rovide a Iow-sidelobe,
low -cross-polarization G au ssian beam on th e sky. T he device sh o u ld e x h ib it a high degree
of sym m etry in its E a n d H planes and, if possible, produce a d iffraction-lim ited power re­
sponse w ith a narrow fu ll-w idth-at-half-m axim um (FW H M ). T h e final co n dition is sought
since, as show n in C h a p te r 2, th e an g u lar pow er spectrum of po larized an isotropy on the
sky is exp ected to p eak a t sm all an g u lar scales.
A lthough th e in p u t to th e o rth o m o d e tran sd u cer (O M T) is square-w aveguide, in which
we desire to sim u ltan eo u sly p ro p ag ate only th e TE °0 an d T E ^ m o d es5, th e horn has a
circular o u tp u t a n d is m o st n a tu ra lly tre a te d using the th eo ry of cy lin d rical waveguides. In
cylindrical co o rd in ates a cy lindrical waveguide will pro p ag ate a tran sv erse field m ode with
no azim uthal angle d ep en d en ce6 if certain conditions on th e w aveguide’s im pedance and
ad m ittan ce are satisfied. T h is m ode will have a pure co-polar ra d ia tio n field , i.e. on-axis,
its cross-polarization will vanish. T h e conditions required are th a t th e im pedance,
Z, and
adm ittan ce, Y , a t th e w aveguide wall are b o th identically zero. T h e im pedance condition
is ensured by v irtu e of th e fact th a t th e azim u th al com ponent of th e field is always locally
transverse to th e m etallic b o undary, i.e.
Etrans E<t>\r=Tl■ M axw ell’s eq uations near the
=
surface of a m etal s tip u la te th a t th e ta n g en tial com ponent of th e field vanishes a t the
5This is th e so-called “h y brid-m ode” condition
6T his is a “b alanced h y b rid m ode”
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bo undary of a co n d u cto r a t r = r i , so Z oc E $ = 0. T h u s, th e use of a m etal has paid
off. B u t w h at a b o u t th e effect o f th e corrugations? S urely if th e walls of th e guide are
not sm ooth, it is im possible to sim ultaneously have th e im pedance vanish a t all locations
along th e guide. However, as long as the corrugations are sm aller th an
th e field does
not sense th e presence o f th e corrugations, and the surface is th en ju st as conductive as a
sm ooth m etal surface.
However, th e co rru g atio n s have their role: they null o u t any residual ad m ittan ce a t
th e wall of th e guide, a n d effectively present a high resistan ce to currents which a tte m p t
to flow along th e walls o f th e guide. T he corrugations are £ in height, so th a t a field is
180° o u t of phase w ith itself afte r clim bing up an d th en dow n th e corrugation. T his phase
cancelation is only s tric tly tru e for one frequency of course, b u t in practice the frequency
dependence can be offset by to leratin g a slight m ode im balance.
T hus we have an electric field of the following form:
E x (r) = A J 0( K cr) exp
E y (r) = 0
where r is th e rad ial d istan ce in ou r cylindrical co o rd in ate system , A is th e am p litu d e
coefficient, an d K cr \ = 2.405 is th e root of th e zero-order Bessel function J o (K cr).
The com plete m o d al-d istrib u tio n of th e E x field is given by expanding th e Bessel
function in eq u atio n 4.1 in G auss-L aguerre modes, viz:
J 0( K cr) = £ A pL°p
p=o
2 r2
exp ( —r 2/ w 2)
VJ
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(4.1)
w here L®(x) is a zero-order L aguerre polynom ial a n d w = w ( z ) is th e locus of points
w here th e electric field in th e horn is —tim es sm ailer th a n th e on-axis field. W ylde [73]
d em o n strates th a t in o rd er to m axim ize th e pow er (cx E 2) in th e fu n d am en tal m ode, the
function w (z ) should have a m inim um a t a p o in t located w Q = 0.6435a in w ard from the
h o rn ’s ap e rtu re ; here a is th e radius o f th e horn a p e rtu re . T h e m inim um o f w { z ) is know n
as "the beam w aist” , an d is th e G au ssian O ptics g en eralization o f th e g eo m etric optics
"focus” .
T able 4.1, rep rin ted from [74], in tu rn rep rin ted from [70] gives th e calcu la ted power
in each m ode for th e first 11 m odes o f th e expansion in eq u atio n 4.1. We h ave used this
m odel to p red ict th e far-field beam p a tte r n o f th e P O L A R K A band h orn o u t to ~ 20°.
T he agreem ent is q u ite im pressive for such a sim ple m odel, o b v iating th e need for a pricey,
com m ercial finite-elem ent field-analysis code.
T able 4.1: N orm alized Pow er Coefficients for G auss-L aguerre M odes
M ode
Pow er coefficient
0
1
2
3
4
5
6
7
8
9
10
0.9792
4.90 x 10~u
1.45 x 10“ 'J
1.86 x 10 "4
3.81 x 10-4
1.16 x 10-8
3.97 x 10~4
1.50 x 1 0 "8
1.59 x 10"4
2.33 x 10 "4
1.12 x 10"4
If th e horn h ad an infinitely w ide a p e rtu re , its an g u lar response7 w ould be a deltafunction. W e would have a true “p en cil-b eam ” - one w ith no off-axis response. U nfor­
7(^r [£ ( r ) ] ) 2, w here T is th e Fourier tran sfo rm o f th e field in th e ap ertu re, E(r)\ z= l -
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tun ately , we could n ot acc o m m o d a te such an idealization a n d o u r h o rn ’s ap ertu re field
p a tte rn is necessarily tr u n c a te d sh arp ly a t th e physical rad iu s o f th e horn, a. This sharp
tra n sitio n introduces d iffractiv e “ringing” in th e an g u lar pow er response of the horn, a
co n dition know n as a “sid e-lo b e” . T h e simple G au ss-L ag u erre m odel breaks down a t lowpow er levels, which for us tra n s la te s to the far off-axis sidelobes o f th e horn located a t
~ 40° a n d beyond. In th e ab sen ce o f a reliable m odel for th e far off-axis behavior of our
feed, we have m easured th e b e a m response for a variety o f frequencies, for b o th polariza­
tions. as well as th e cro ss-p o larizatio n response. T h e resu lts o f th e sim ple G auss-Laguerre
m odel are sum m arized in figure 4.3.
2 9 G H z E —p l a n e No L en s , C o m p a r i s o n w ith T h e o r y
-20
M e a s u re d
-3 0
6)
Tneory
cn
c
o
a.
in
O
a:
-50
A
-6 0
0
-20
20
40
Angie (D egrees)
Figure 4.3:
Shown here arc the results of the Gauss-Laguerre model described in the text (solid
lin e), compared with the E-plane beam map at 29 GHz (triangles).
T h e final, b u t by n o-m eans least im p o rtan t p a rt, of th e feed h o rn is th e mode converter
which is a sep arate electro fo rm ed elem ent placed a t th e th r o a t o f th e horn. The mode
converter com bines th e T E ° n an d T M ° n circular w aveguide m odes to crea te the H E °n
co rru g ated waveguide m ode.
T h e m ode converter’s c o rru g atio n s are o f varying height,
unlike th o se in th e flare se c tio n o f th e horn. T h e p rim ary p u rp o se of th e variable height
70
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corrugations is to define th e b an d p ass of th e horn, w ith o u t sacrificing th e sidelobe or
cross-polarization levels achieved in th e flare section o f horn [70], [71], [75], [76].
Up u n til th is p o in t we have been rath e r am plicentric, i.e. we have largely ignored
phase considerations. T h e electrical p ath length from points in th e ap ertu re plane to th e
horn th ro at is a fu n ctio n o f r, im plying th a t th e p h ase o f a wavefront which is in-phase
upon im pinging on th e a p e rtu re will be progressively o u t of phase as a function o f rad ia l
distance off th e h o rn axis. T h is phase error leads to a decoherance o f the electrom agnetic
field. T he phase e rro r, A , is usually norm alized to a p artic u la r w avelength, generally
th e nom inal b a n d ’s c e n te r w avelength, A0. From elem en tary geom etric considerations we
have:
A ^
/»0
_ c o s 0o) _ « ta n ( ^ )
Aq
£,
(4.2)
where Zs/an£ is th e le n g th from th e h o rn ’s apex to its a p e rtu re radius, a, and 0o is th e horn
flare’s semi-angle. S m all-A horns produce diffraction-lim ited pixels on th e sky, which are
necessarily frequency-dependent.
Large-A horns p ro d u ce nearly frequency-independent
beam s because th e frequency inform ation is sm eared o u t across th e band. For 0.2 > A
and A > 1.2, th e p h ase-cen ter is frequency in d ep en d en t, though, in practice, difficult to
realize mechanically. F or w ide-band horns, th e m inim um w id th o f th e b eam ’s £ co n to u r
(th e beam w aist) is lo c ated in th e th ro a t of th e horn an d is given by: wq =
For th e
P O L A R horn, A ~ 0.6, p lacing th e phase center closer to th e a p e rtu re th a n to th e apex,
resulting in a frequency d ep en d e n t beam size as sum m arized 8in ta b le 4.4.1.
8T he a u th o r wishes to recognize th e Herculean efforts of C hris O ’Dell a n d his assistant K ip H y att
for producing o u tsta n d in g m easurem ents of the Km band b eam param eters (and figure 4.3), as well as
N athan S tebor who assisted th e a u th o r in earlier pioneering, though less-sophisticated, beam p a tte rn
71
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31
G H z H —P l a n e
i
10
-2
10
10
-6
-1 0 0
-5 0
0
50
100
Angle [cleg]
F i g u r e 4 .4 :
T h e P O L A R A a-b a n d 31 G H z (m id d le frequency b a n d ) H -P la n e B eam P a tte r n is
sh o w n . T h e E -P la n e b eam m a p is sim ila r to w ith in ~ 1% o u t to ~ —20dB =>• 30°.
T able 4.2: P O L A R A a-B and M easured and M odeled FW H M B eam W idths.
P lane t/ [GHz] Qfwhm ± 0.1°
26
7.9°
E
7.6°
29
E
7.1°
36
E
7.5°
29
H
36
6.9°
H
72
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4.4.2
O p tica l C ross P olarization
As m entioned above, th e co rru g ated scalar feed h o rn b o a sts a low-level of cross-polarization.
T he cylindrical s y m m e try o f th e feed m ight lead one to ex p ec t th a t th e cross-polarization
of the h orn should b e id entically zero. However, even for a p erfectly designed, com pletely
sym m etric feed, th is is n o t tru e. T h e reason for th is is g ro u n d ed in th e geometry' of th e
horn and source w hen tre a te d as a scatterin g problem .
Define th e (x,y) p lan e to be th e a p ertu re plane, a n d 0 to be th e angle m easured
clockwise from th e y-axis in th e ap ertu re plane to th e p o la rizatio n axis of th e field. We
first consider a field p o larized along + y ,a n d th e x — z p la n e to be th e scatterin g plane. Let
+ k be th e w ave-vector, a n d 0 ( 0 , 0) be th e angle betw een —x an d + k. 0 ( 0 , 0) can have
both p o lar angle, 9, an d azim u th al angle, 0 dependence. T rea tin g th e wail of th e horn
as a perfectly sm o o th co n d u cto r,9 we find th a t for th is scenario, th e reflected field has
the sam e p o la rizatio n a fte r scatterin g , independent o f Q ( 9 ,0 ) . T hus, for an ideal horn,
the cross- p o la rizatio n induced by scatterin g in a p lan e co n tain in g th e polarization axis
is identically zero since th e re has been no p o larizatio n conversion. T his is also m anifestly
true for sc a tte rin g in a plane p erp en d icu lar to th e p o la rizatio n axis.
Now consider a n incident field, again w ith a p o la riz a tio n in th e + y direction, b u t
now im pinging on a n elem ent of surface area a t an angle 0 = 45°. Using th e fact th a t
the electric field inside th e co n d u cto r vanishes we o b ta in co n d itio n s on th e reflected field’s
parallel and p erp e n d ic u la r com ponents to th e surface, E ± , vanishes an d we find th a t there
m e asu rem en ts.
90 f co u rse th is is o n ly s tr ic tly tr u e fo r an u n c o rru g a te d feed , b u t a s we hav e show n, in th e flare
section o f th e h o rn th e c o rru g a tio n s a re A /4 in d e p th , ca u sin g th e field to b e 180° o u t o f p h ase a fte r
trav e rsin g o n e c o rru g a tio n . T h is c o n d itio n is id en tical to th a t o f s c a tte r in g from a p erfect c o n d u c to r, so
o ur a p p ro x im a tio n h ere is rea so n ab le .
73
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is p o larizatio n conversion. For such a field we find th a t th e incident field’s y-com ponent
will be con v erted to a n x-co m p o n en t upon reflection, th u s p roducing a p o larized signal in
th e cro ss-p o larizatio n d irectio n , x . T h e m ag n itu d e o f th e induced cross-polarization will
vary as sin 2 d>, an d will b e peaked a t <p = 45°, 135°, 225°, 315°.
-ioE-
-20
5
c
^ -3 0 ^
t
4)
£
P
§F
S - 4 0 F04
t
-5 0
-1 0 0
-5 0
0
50
100
Angle ( D e g )
F igure 4.5: 29 GHz Cross-Polarization Beam Map. Here the solid line is the co-polar E-planc
(field parallel to the +6 direction) power response pattern, the dotted line is the cross-polarization
response as measured along the azimuthal direction, 4>, and the dashed line is the cross-polarization
measured along 0 + <p. As mentioned in the text, both cross-polarization responses should vanish
at 0 = 0.
4 .4 .3
O rth o m o d e T ransducer: O M T
Following th e th ro a t in th e o p tical p a th , there is an electroform ed a d ia b a tic 10 transition
from th e th r o a t’s circu lar o u tp u t w aveguide to th e sq u are-in p u t waveguide o f th e OMT.
T his device was d esigned by ap p licatio n of th e P y le C o n d ition, which seeks to m atch the
10See ([77] for useful d efin itio n o f a d ia b a tic ity as a p p lie d to w av eg u id e tra n sitio n s
74
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cutoff w avelengths o f th e T E ^ 0 an d th e T E ° u m odes [78]. T h e tra n sitio n was m anufac­
tu red by C u sto m M icrowave Inc., of Longm ont, CO.
The O M T 11 is a waveguide device used to sep arate th e tw o linear p o larization states
in th e incident field. P O L A R ’s O M T was fabricated by A tlan tic M icrow ave12, and is a
th ree-p o rt device w ith a sq u are in p u t p o rt, and two rectan g u lar o u tp u t p o rts containing
th e orthogonal p o larizatio n signals.
A lthough n o t s tric tly representative o f P O L A R ’s O M T , th e p a p e r by C h attopadhyay
[79] presents a n equivalent circuit m odel of a functionally equivalent device. T h e O M T ’s
entrance p o rt is K a b an d sq u are guide which su p p o rts b o th T E qX & T E ° 0 m odes sim ulta­
neously. Inside th e sq u are guide is a th in sep tu m which behaves as a 3-dB pow er divider
for the T E ° 0 m ode, an d reflects th e T E qX m ode tow ards a sh o rt-slo t coupling p o rt tran s­
verse to th e in cid en t field. T he slot is often reduced in size by th e ad d itio n of an inductive
iris to im prove coupling to th e T E qX m ode, an d reduce th e coupling to th e T E ^ 0 mode.
To com pensate for th e added reactance o f th e iris, a canceling (capacitive) reactance is
added in p arallel [79]. See tab le 4.3 for a su m m ary o f P O L A R ’s O M T properties.
Table 4.3: P ro p erties of P O L A R ’s O M T: A tlan tic M icrowave M odel 2800.
P ro p e rty
Value
N otes
Isolation
V SW R
C ross-polarization
-35 dB
< 1.2
-30 dB
Specified an d M easured
Specified an d M easured
E stim ated
" V a rio u s ly re fe rre d to in th e co m m u n ic atio n s lite r a tu r e as: p o la riz a tio n d ip lex ers, d u a l-m o d e tra n s­
du cers, o rth o -m o d e tees, a n d o rth o m o d e ju n c tio n s. C o m m u n ic a tio n s a p p lic a tio n s use th e se devices to
b ro a d c a st a n d s e p a r a te d is tin c t ch an n els w hich s h a re id e n tic al b a n d p asses. S A T C O M T V ap p licatio n s,
b ro a d c a st o d d T V ch a n n els in o n e p o la riz a tio n , a n d even T V c h a n n e ls in th e o rth o g o n a l p o la rizatio n .
12 B oston, M A
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N o n -B e a m -F o rm in g O ptics: V acuum W in d o w a n d G rou n d S creen s
T he vacuum w indow is given th e task o f allowing ra d ia tio n to land on o u r detectors, while
keeping th e in n a rd s of th e radiom eter cool an d pressure-free. To this end, a sophisticated
m ulti-elem ent ap p ro ach was adopted. T h e w indow was designed by C hris O ’Dell, and
based on th e extensive investigations of P ete r T im b ie’s M S A M II/T op H at work. T h e
m ain features o f th e w indow are a 3 mil v acuum -tight polypropylene vacuum barrier and
n o n-vacuum -tight G ore-Tex window s u p p o rt w hich b ears th e several h u n d red pounds of
force on th e window .
A layer of V olara13 (expanded polyethylene) serves to seal in a
d ry-nitrogen layer betw een th e polypropylene layer a n d an y precipitable w ater vapor in
the atm o sp h ere w hich would otherwise condense a n d freeze on th e vacuum window. T h e
window is exceedingly leak-free, allowing pressures o f < 10“ ° T orr to be m aintained for
m onths a t a tim e.
T h e final p seudo-optical elements of th e P O L A R in stru m en t are its tw o concentric
ground screens, see figure 4.6. The use of two g ro u n d screens is not unusual in the field,
although P O L A R ’s screens are optim ized to reject polarized spillover, ra th e r th a n totalpower spillover.
T h e th eo ry o f ground screen o peration is q u ite sim ple: one a tte m p ts to steer th e
power received in th e sidelobes of th e op tical sy stem to a well-known, co n stan t, and
preferably lo w -tem p eratu re source, ra th e r th a n allow ing th e m to land on th e e a rth ’s 300K
surface. It is essen tial th a t th e sidelobe response n o t be m o d u lated by th e in stru m en t’s
m o d ulation schem e o r else synchronous signals will be produced which, in principle, are
indistinguishable from th e feeble cosmic signals we are a tte m p tin g to m easure.
13V oltek C o rp .
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L
Dewmr
Roof Level
x:
T
i
Electronics/
Data
Acquisition
I
"T----------- T
F i g u r e 4 .6 : P O L A R ’s G ro u n d sc re e n s. T w o s e ts o f g ro u n d sc re e n s a re u se d to re d u c e th e polarized
.spillover from th e e a r th , a s w ell a s p o la riz e d em issio n from th e sh ie ld s th e m se lv e s. T h e o u te r shield
is fixed to th e s tr u c tu r e in w h ich P O L A R resid es, a n d is co m p o se d o f a lig h tw e ig h t ste e l skeleton
co v ered b y 0 .0 5 ” a lu m in u m s h e e ts. T h e in n e r g ro u n d sc re e n is covered w ith flat E c co so rb panels
(to re d u c e th e ir p o la riz e d e m issio n ), a n d c o -ro ta te w ith th e P O L A R r a d io m e te r . A lso show n is
th e m o to r-d riv e n , fib e rg la ss c la m sh e ll-d o m e w hich c a n b e re m o te ly o p e r a te d v ia th e W orld W ide
W eb in th e e v e n t o f in c le m e n t w e a th e r.
77
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P O L A R ’s g ro u n d screen ap p ro ach is d u al-p u rp o sed .
F irs t a n in n er conical ground
screen ro ta te s w ith th e in stru m en t an d is co ated w ith Eccosorb. T h e Eccosorb panels ab­
sorb. ra th e r th a n reflect th e sidelobes to th e sky. T h is ab so rp tiv e ap p ro ac h is uncom m on
in CM B an iso tro p y ex p erim en ts as it increases th e to ta l pow er loading on th e detectors,
which is p a rtic u la rly troublesom e in th e case o f u ltra-sen sitiv e bo lo m etric d etecto rs. How­
ever, we e stim a te th a t th e a n te n n a te m p e ra tu re o f th e in n er shield to b e < IK . P olarization
g en erated by em ission from th e bare m etal surface o f th e uncovered shield is believed to
be m uch m ore troublesom e th a n th e slight increase in sy stem te m p e ra tu re . A dditionally
P O L A R ’s in n er g ro u n d screen co-rotates w ith th e receiver, w hich en su res th a t if there
is any resid u al p o larized power produced by th e inner screen, it will p ro d u ce a constant
polarized offset, r a th e r th a n a less-tractable, ro ta tio n -m o d u la te d offset.
T h e second level of shielding is o f th e m ore conventional reflective-scoop design, e.g.
W ollack [17]. T h e sco o p 14 is m o u n ted to th e side o f th e P O L A R observatory, a n d is made
of alum inum p anels 8 ’ w ide an d 6’ high. To e stim a te th e level o f sidelobe suppression
induced by th is shield we have em ployed S om m erfeld’s diffraction calcu latio n for points
deep in th e shadow region of a knife-edge s c a tte re r [55]. We e stim a te th e suppression
to be ~ —40 dB , w hich in com bination w ith a sim ilar (m easured) figure from th e inner
ground screen, a n d th e low-sidelobe response o f o u r feed horn, gives a to ta l estim ated
sidelobe supp ressio n o f over -100 dB . We note th a t th e diffraction calcu latio ns employed
th ro u g h o u t th e design process were based on scala r diffraction th e o ry ra th e r th a t th e more
com plete vector theory, which is m ost ap p ro p ria te for a p o larizatio n ex p erim en t.
14d esig n ed a n d b u ilt b y N a th a n S te b o r a n d K ip H y a tt
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4 .5
H E M T A m p lifiers
T h e developm ent an d im p lem en tatio n o f w ide-band low-noise am plifiers has revolutionized
th e field o f coherent CM B radiom etry. T here exist no com m ercial devices which can m atch
th e perform ance of th e am plifiers produced by th e N ational R adio A stronom y O bservatory
(NR_AO), under th e d irectio n of M .W . Pospieszalski and W .J. L akatosh. C u rren t devices,
capable of up to 20 G Hz b an d w id th s w ith noise te m p eratu res lower th a n 50 K, are available
for all waveguide b an d s up to W -B and (75 - 110 G H z).
T h e P O L A R A"a-b an d H E M T am plifiers were co n stru cted in th e Fall of 1994 and are
no longer considered “sta te -o f-th e -a rt” . O u r am plifiers are based a ro u n d H F E T transistors
produced by Hughes Electronics.
SOURCE
GATE
DRAIN
n' GaAs
n AlGaAs LAYER
------------------ _ -----AJGaAj
SPACER
UNDOPED GaAs
SEMI-INSULATING GaAs SUBSTRATE
F igure 4.7:
HEMT Structure
A generic stru c tu re is show n in figure 4.7. T h e H F E T itself h as electron 2D EG sheet
densities o f ~ 1012c m - 2 , a n d a m obility of 104cm 2/V s , thus ju stify in g th e nam e High
E lectro n Mobility T ran sisto r w hen com pared w ith th e m obility o f a “garden-variety” lowfrequency, coolable, low noise F E T , which has a m obility of 5 x 103cm 2/V s and noise
79
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te m p eratu res o f ~ 100 — 200K. P O L A R ’s N R A O am plifiers15 are com posed of four indi­
vidual tra n sisto r stages, w ith th e lowest noise device preceeding th e following three. E ach
stage provides roughly 7-8 dB of gain, resu ltin g in an overall gain of ~ 30 dB. C u rren t am ­
plifiers utilize In P based devices for th e first sta g e (w hich have lower noise-tem peratures
th a n GaAs devices) a t th e expense of slig h tly increased 1 / / noise. However, th e lowfrequency sp e c tra l pro p erties of these am plifiers are largely irrelevant (at least in theory)
for radiom eters such as PO L A R which em ploy th e co rrelatio n technique.
NRAO su p p lied th e am plifiers as well as su p p o rt electronics which provide regulated
voltages an d c u rren ts to each of the four in d iv id u al tran sisto rs. Supplied w ith th e devices
are d a ta sh eets w hich specify th e N R A O -optim ized values for th e g ate bias-current an d
th e drain-source voltage. A fter tu ning these p aram eters, th e device’s transconductance,
gm =
is com pletely determ ined. A t a fixed Vd3, th e noise te m p eratu re depends on th e
d rain current weakly, b u t w ith a w ell-determ ined m inim um . U nfortunately, this m inim um
n o ise-tem p eratu re cu rren t results in a ra th e r low tran sco n d u ctan ce, and thus low-gain.
So, an o p tim izatio n is carried o u t in a tw o-dim ensional param eter-space for each of th e
four tran sisto rs in each of th e two P O L A R H E M T S . W ith ap p ro p ria te p aram eter tu n in g
th e devices can be used a t ro o m -tem p eratu re which is q u ite convenient for p ro to ty p in g a
radiom eter w hile still in th e user-friendly confines o f th e lab.
P O L A R ’s tw o am plifiers had noise te m p e ra tu re s o f ~ 65 K w hen m easured a t N RA O
5 years ago.
C u rren tly we m easure th e noise te m p e ra tu re o f th e entire system , which
includes c o n trib u tio n s from num erous lossy an d em issive com ponents which preceed an d
I5S erial N u m b ers: A 29 a n d A30
80
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follow th e H E M T s16, to be: ~ 75K (See C h a p te r 5). W e can assum e th a t th e H E M T s
do m inate th is value, th o u g h th e aforem entioned n o n -id ealities o f th e system c o n trib u te
~ 15 K. A fu tu re g o al o f th e P O L A R K & ban d receiver is to o b ta in upgraded In P devices,
which should d ecrease o u r sy stem tem p eratu re by a facto r o f a t least two.
4 .6
R o o m T e m p e r a tu r e R a d io m e te r B ox: R T R B
A fter am plification, th e signals leave the dew ar th ro u g h cu sto m -m ad e coin-silver waveg­
uides which ro u te th e signals along a com plicated, b e n d in g 3D p a th from th e H E M T s in
parallel to 6” stain less steel waveguides which provide a th e rm a l break from th e 300K de­
w ar walls to th e 20K H E M T s. T h e stainless guides a re b o lted to a vacuum -tight K A b an d
w aveguide feed th ro u g h m an u factu red by Aerowave. O u tsid e th e dew ar, stra ig h t sections
of R hodium p la te d , b razed -co p p er waveguides are used to co m p en sate for th e p a th -len g th
differences betw een th e tw o polarizations incurred by th e 3D bends. As m entioned in
C h ap ter 4. it is e ssen tial th a t th e two signals trav erse id en tical electrical p a th lengths so
th a t th e electric fields will be in-phase a t th e co rrelato rs. Finally, th e waveguides en ter th e
R TR B , where th e sig n als are converted from w aveguide to coax to m atch th e in p u ts of th e
M IT E Q [JS426004000-30-8P] R'a-band w arm H E M T am plifiers. T hese devices are based
on H EM T technology ju s t as th e NRAO devices are, how ever th e ir noise te m p eratu res
are significantly h igher: T ^ ITE<^ ~ 250K. To com p en sate, th ese devices o u tp erfo rm th e ir
m ore refined cousins in th e following all-im p o rtan t qualities: gain, delivery tim e, and n o t
surprisingly, price!
Following th is seco n d -stag e of am plification, th e signals are dow n-converted in fre­
16see C h a p te r 5 for a d isc u ssio n o n sp u rio u s lo ad in g in tro d u c e d b y th e se co m p o n en ts.
81
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q u en cy by th e su p erh ete ro d y n e co m p o n en ts, as delineated in th e following subsection.
4 .6 .1
S u p e r h e te r o d y n e C o m p o n en ts
W e dow n-convert in frequency because th e u ltim a te m ultip licatio n o f th e signals is m ost
easily perform ed a t th e lowest to lerab le frequency which still preserves th e nom inal 10
G H z b an d w id th .
D ow n-conversion m a in tain s th e ban d w id th , an d allows us to fu rth er
am plify th e signals before th e y are d etecte d .
S u p erh etero d y n e techniques are com m on in alm ost all m o d e m com m ercial com m uni­
catio n s ap p licatio n s, th o u g h in
recent y ears th e y have been d e-th ro n e d from th e ir position
as th e CM B co m m u n ity ’s “receiver-of-choice” w here they reigned in th e 60’s a n d 70’s. We
now' p resen t a b rief review of the technique.
Following [69], we F ourier expand th e in p u t signal as:
VI
A = ^2
cos(2rcvit + 0 t ),
(4-3)
V\
an d th e local o scillato r’s pure-h arm o n ic signal as:
B = 6cOs(27TJ/mt +
<£m )-
(4.4)
T h e low signal level o u tp u t of th e m ix er is approxim ately: ( A + B ) 2 = A 2 -h B 2 -I- 2 A B ,
w hich is:
V2
= ^2
V2
a »GJ COs(27TI/t-£ -I- 4>i) COS(27TI/j< + 0J-) -(- b2 COS2(27TI/m£ -I- 4>m) +
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in
26 ^ 2 Oi cos(2Tri/it + 0i)cos(27ri/mi + (j>m ).
(4.5)
Ui = I / l
U sing som e sim ple trigonom etric relatio n s, we can conclude th a t th e first te rm on th e
HRS o f eq u atio n 4.5 contains term s:
• betw een 2u\ an d 2i/2
• betw een \i>\ — zv2| an d \i/2 ~ i'llT erm 2 co n tain s th e frequencies 2um a n d 0 (i.e., D C), and th e th ird te rm contains
frequencies:
• betw een i/\ + i/m an d v>2 + um .
• betw een \u\ — i/m \ a n d \u2 — um \.
T h e IF p o rt of th e M IT E Q [TB0440] trip le-b alan ced m ixer is transform er coupled and
passes only frequencies betw een 2 - 1 2 G H z.
T h e final spectrum of th e “in term ed iate
frequency-’ (IF ) - i.e. dow n converted, p o r t’s o u tp u t is:
OiCOs[2Tt(Ui - Um)t +4>i -
6
(4.6)
V , = U m + l/a
w hich is a (scaled) replica of th e in p u t, R F , w ith an identical ban d w idth, though
now a t lower frequencies: IF . T he IF signal is subsequently amplified in th e IF band
to provide th e a p p ro p ria te bias power level in to th e m ultiplier. Each m ultiplier requires
~ 6dB m , or 4 m W , of bias power to function as a bilinear m ultiplier. We m ust am plify th e
signal s u b sta n tia lly to m eet th is req u irem en t as th e th ree m ultipliers are preceeded by a
trip lex er w hich a tte n u a te s each signal by a facto r o f th ree in power. We use two stages of
IF am plification, again provided by M IT E Q devices w ith 2-12 GHz band passes. T h e gain
83
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of each device falls steeply above a frequency of fzdB — 12 G H z, a tte n u a tin g any residual
o u t-of-band frequencies w hich m ight otherw ise be p ro p ag atin g in th e coax t r a n s m is s io n
lines.
4 .6 .2
T h e C o r r e la to r
A fter m ixing an d IF am plification, th e signals are frequency m u ltiplexed by R eactel triplex­
ers. T h e function o f th e trip lex ers is two-fold. F irst th ey serve to provide us w ith three
(ideally) independent bands from which we m ay investigate th e sp e c tra l behavior of our
signals. Secondly, these devices allow us to “flatten ” th e gain o f th e system across the
wide R F -b an d w id th th a t we achieve w ith th e H EM Ts. T h is ensures th a t the effective
ban d w id th will not be reduced from its nom inal specifications. Following th e triplexers,
the signals from th e two p o larizatio n sta te s are fed into th ree s e p a ra te w ide-band m ultipli­
ers (one for each su b-band). T h e m ultipliers themselves are M IT E Q [DBP1I2HA] double
balanced m ixers w ith R F b an d passes from 1-12 GHz an d a n IF b an d p ass from 0-500
MHz. T h e IF o u tp u t p o rt is n o t tran sfo rm er coupled, an d can th u s pro p ag ate the DC
signal pro p o rtio n al to th e co rrelatio n betw een signals in th e x a n d y polarization states.
P erform an ce o f an Id eal M u ltip lier
T here are several ways to im plem ent a co rrelato r for use in a co rrelatio n radiom eter [10],
[61], [64], [62], [80]. P O L A R utilizes a co rrelato r based on S ch o ttk y -d io d e m ixer tech­
nology. In th is m anifestation, th e ideal co rrelato r is realized by a double balanced mixer,
a phase m odu latin g elem ent, an d lock-in am plification. A t th e h e a rt of th e correlator is
the w ide-band double-balanced analog m ultiplier. This ty p e o f m u ltip lier isolates b o th
R F in p u t p o rts from on e-an o th er, while th e IF p o rt is coupled to b o th . T he prim ary
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a______
-a
■
■
Figure 4.8:
Electrical Schematic of M ultiplier
difference betw een a m ultiplier an d a conventional m ixer is th a t th e IF b an d w idth is m ade
in ten tio n ally narrow to suppress frequency com ponents g re a te r th a n ~ 100 MHz. As the
R F p o rts have bandw idths from 1-12 G H z th e narrow IF b a n d p rev en ts higher order term s
th a n essentially D C from p ro p ag atin g in th e o u tp u t. In m a n y circles, a narrow R F band
m ultip lier is know n as a ‘phase d e te c to r’, for reasons which will becom e ap p are n t shortly.
T h e tran sfo rm e r coupled in p u ts co h eren tly ad d the signals a t b o th R F po rts, trad itio n ­
ally labeled by ‘L ’, for LO (i.e. Local O scillato r), and ‘R ’, for R F . D ue to th e sym m etrical
arran g em en t o f transform ers an d d io d es, in practice, th e re is no electrical distinction be­
tw een th e tw o R F ports. A schem atic o f th e m ultiplier is show n in figure 4.8.
T h e m u ltip ly in g elem ent itself is a bridge arran g em en t o f S ch o ttk y diodes. T he Schottk y diode is a m etal-sem iconductor ju n c tio n device w ith a n I-V ch ara cteristic of:
I(V) I0(et&
=
-
1)
85
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w here I 0 is th e reverse s a tu ra tio n c u rren t, kb is B o ltzm an n ’s co n stan t, e is th e charge
o f th e electro n , an d
77
~
1
is th e “id eality facto r” param eterizin g non-idealities o f th e
ju n c tio n . Following M aas [81], we m odel th e diode as a c u rren t source, I ( V ) in parallel w ith
a ju n c tio n cap acitance, to g e th e r in series w ith a resistor, R s . T h e ju n ctio n cap acitan ce is:
C (V ) =
°°
U - &<t>)b, *
w here C 0 is th e ju n c tio n cap acitan ce w ith zero voltage difference, an d 4>bi is th e ju n c ­
tio n ’s b u ilt-in voltage. C ( V ) is im plicitly defined by:
c m =fis
w here Qd is th e depletion-region charge. T h e c a p a c ito r’s o u tp u t cu rren t is:
m
=
c{v{t)) x ^ £ 1
.
We will also need th e ju n c tio n ’s conductance:
Now we can express th e d io d e’s o u tp u t voltage as
= I R s = Z~l V = Y V R S, w here
Z is th e equivalent circu it im pedance, an d Y is th e corresponding ad m ittan ce. We have
th a t:
Z — {g + iu>C) * + Rs,
so :
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7 (0 = _______ W ) _______
(g(t) + iujC {t))~ x + R s
4 .6 .3
M u lt ip lic a t io n
E x p an d in g th e ideal I-V curve for sm all ju n c tio n voltages, we find th a t :
V o u t(t) oc c o n s t a n t + a V ( t ) -+- (3V2(t)
show ing th e non-linear dependence o f V out on V . Here, th e am plitude term s in boldface
are com plex qu an tities, so we m ust p ay a tte n tio n to phases. For th e m ixer, V = V l o + V r f
so th a t Vout oc |V LO|2 + V ^ 0 V r f + V lo V J j j. + | V r f | 2. T h e q u ad ratic term s result in a
DC co m p o n en t of m agnitude | V l o | 2 + |V r j - |2 , while th e bilinear term s produce the de­
sired “m ixed” term s. In th e case o f th e correlato r, th e DC com ponent will also contain a
co n trib u tio n from th e cross-correlated com ponents o f th e signals a t th e R F and LO ports.
We assum e th a t Vout(t) •will be in teg rated , for a tim e period 2T, which kills off the term
lin ear in V ( t) .
For th e bilinear term s we have:
(V R F (t)V L o (t - t )> =
lim - i - [ T V R F (t)V £.0 ( t - r ) d t .
T — o c Z 1 J —T
(4.7)
W e now choose an explicit rep resen tatio n for th e signals:
V
V
r f (w ,
lo^
t) = V RFe ,<wt+* " r>
, t ) = V LOe i^ +* * ° ),
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(4.8)
(4.9)
(4.10)
so eq u atio n 4.7 becomes:
Voutir) = ^ ( V R F ( t) V J (0 ( t - r)>
= l i m r ^ o o / J r VRFe i<-utf++*F)VLOe - i W
=
lim r-o o
=
(4.11)
- T)+ + L °)dt'
Sir VRFVLoei^-'/)t'+,/Te^o-,i‘^ dt'
( ’V r f
Vl
o
)6{u -
v ) e i& L O -* R F + V T )
A t zero lag ( r = 0), th e o u tp u t is (V r f Vl o )6 (u>—v ) e l^ LO~'t>RF\ showing th a t no o u tp u t
results w hen th e R F an d LO p o rts are a t different frequencies. W hen the frequencies a re
m atched, th e o u tp u t varies sinusoidally as cos{<Pl o ~ 4>r f )~. ju stify in g th e nam e “phase
detecto r” as m en tio n ed above.
T he above analysis has only trea ted a single frequency com ponent. In practice b o th
the LO an d R F signals are com posed of a finite b an d of frequencies:
rl/o + A tU F
V lo/r f W = /
J Vo
„
.
V l o / r f C^ t ) e 7r‘ di/.
(4.12)
S u b stitu tin g th e above into eq. 4.11, we find:
R l r {0) = /
(V l o (i/)V Ji f ( i/) ) c o s ( ^ lo ( ^ )
- 4>ne{v))dv-
J VQ
88
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(4.13)
P e r fo r m a n c e o f th e N o n -Id e a l C o r r e la tio n R a d io m eter
T h e sim plest non-ideal b eh av io r o f th e co rrelatio n radiom eter is th e effect o f electrical
p a th length m ism atch betw een th e in p u t arm s. From eq. 4.11 th e c o rre la to r’s D C o u tp u t
is p ro p o rtio n al to cos{4>lo ~4>r f )- If th e p hases them selves are frequency d ep en d en t, as in
th e case o f a p a th length m ism atch, th e n th e D C o u tp u t of th e c o rre la to r will be reduced
by th e cosine term . The p a th len g th difference A L introduces a d ispersive phase shift via:
4>{uj. A L ) = 2 ttA L / X gvide = 2tt A-~ a . F rom eq. 4.13, we have:
R l r (0 ) = r ° +^ l'RF VLO{ ^ t ) V RF( u ,t) e 2^
JVo
t coS( 2 7 r ^ ^ - ) d u .
C
(4.14)
T h e c o n trib u tio n of each sp e c tra l co m p o n en t is th u s w eighted by th e cosine of its
phase. Equivalently, th e b an d p ass of th e co rrelato r is m o dulated by th e cosine term . It is
therefore im p erativ e to acc u rately m a tc h th e p a th lengths in th e sy stem . In p ractice this
is accom plished by injecting a com p letely polarized signal into th e O M T in p u t which is
sw ept in frequency across th e R F b a n d . B y m easuring th e m o d u latio n of th e sp ectru m
of th e co rrelato r by the cosine envelope, we can d eterm in e the eq uivalent electrical p ath
len g th im balance. From these m easu rem en ts one can also d eterm ine th e b an d p ass of the
co rrelatio n rad io m eter once it has b een phase-optim ized. T he electrical p a th difference
m easu rem en ts agree qu ite well w ith m easu rem en ts of th e physical w aveguide p a th differ­
ence. To balance th e p a th len g th s one sim ply adds th e ap p ro p riate le n g th of wave guide
to th e s h o rte r arm of th e receiver.
T h e rem ain in g co n trib u tio n s to th e n o n-ideality of th e co rrelatio n rad io m ete r result
p rim arily from gain and p h ase a sy m m e try betw een arm s, across th e b an d passes. T he
effects can be caused by m ism atch ed b an d s, te m p e ra tu re dependence, a n d p h ase instability
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Table 4.4: Tolerances o n F requency Response V ariations for a 2.5% R eduction in SN R
T y p e o f V ariation
P erm issab le Level
A m p litu d e Slope
3.5 d B across b an d
Sinusoidal R ipple
2.9 dB p eak -p eak
In te r b an d C en tro id Offset
5% o f A i/rf
P h a se V ariatio n across band
12.8°
of th e am plifiers a n d /o r th e co rrelato r. In p ractice it is im possible to elim inate all such
effects, an d so we provide a n e stim ate o f th e tolerable level o f a few of these effects such th a t
th e y would co n trib u te to a 2.5% d eg rad atio n of th e signal-to-noise ra tio of th e correlation
receiver following T h o m p so n e t al. [59]:
The final n on-ideality o f th e correlation rad io m eter w hich m u st be confronted is its
te m p eratu re dependence. T h ere are two relevant effects caused by te m p eratu re fluctua­
tions: gain in sta b ility a n d p h ase instability. Even for an ideal m u ltiplier th ere is a DC
com ponent of th e o u tp u t w hich is pro p o rtio n al to th e sum o f th e to ta l R F power in each
arm . Let us exam ine th e effect of a change in th e to ta l pow er o u tp u t (m odulated, for exam ­
ple, by changing atm o sp h eric a n te n n a tem p eratu re, T ) on th e o u tp u t from th e correlator.
VVe will assum e th a t th e rad io m ete r is viewing an unpolarized source which produces fields
E y ( T ) and E X( T) , w hich are them selves functions of th e am b ien t te m p eratu re. We will
furth er assum e th a t th e feed horn is sym m etric w .r.t. th e E an d H plane response and all
ban d passes are id entical betw een th e two arm s. T h e o u tp u t from th e correlator is given
by:
Vout{T) oc ( |E X( T ) |2 + E x ( T ) E y (T ) + E y ( T ) E x (T ) + |E y ( T ) |2).
The two bi-linear te rm s in E y , E x vanish due to th e u n co rrelated assum ption. T h e two
q u ad ratic te rm s survive an d we assum e th a t th e audio frequency pow er sp ectra of th e two
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fields are given by:
w here th e frequency dependent term is caused by fluctuations in th e pow er response
due to te m p e ra tu re variations. Since this
o u tp u t, it too will have a
1/
1/
/ sp ectru m now ap p ears in o u r m ultiplier
/ spectrum ; ap p are n tly defeating th e purpose o f utilizing the
correlation technique in th e first place! T h ere are tw o m ethods to remove th e contribution
from th e to ta l pow er channels.
One is a hardw are based approach, an d th e o th er is
based on softw are rem oval of th e to tal power co n trib u tio n . In th e final analysis, PO L A R
inco rp o rates techniques from b oth approaches to achieve a high-level of suppression of
residual c o n tam in a tio n from the effects of th e e a r th ’s atm osphere.
S oftw are B a se d S o lu tio n to C orrelator D rifts
T he softw are-based solution does not utilize th e phase m o d u lato r. Instead, th e DC o u t­
p u t from th e co rrelato r is m easured, including co rrelated an d uncorrelated contributions.
A fter b in n in g th e co rrelato r signal and b o th to ta l pow er signals, fits are perform ed to the
co rrelato r o u tp u t a n d th e best-fit reconstructions of th e to ta l pow er signals are regressed
ou t of th e co rrelato r signals. This technique removes m ost, b u t n o t all, of th e to ta l power,
1/ / c o n tam in a ted , co n tributions to th e co rrelato r o u tp u t. T h e prim ary reason lim itation
is th a t in p ractice th e sam pling of the d etecto r o u tp u ts is n ot sufficiently fast to accurately
sam ple th e to ta l pow er channels which have uknee ~ 4 Hz. We would need to sam ple the
d etecto rs m uch faster th a n 2Vknee ~ 10 Hz to avoid aliasing. However, ou r audio b an d ­
w idth is 0 — 5 Hz, so we would certainly lose high-frequency inform ation from th e to ta l
pow er channels sim ply by ou r choice of in teg ratio n and d a ta acquisition m ethods. To
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address th is p ro b lem we bin th e raw to tal power o u tp u t a t
1
s a m p le /.05 seconds to
1
s a m p le /.5 seconds , w hich effectively introduces a low-pass filter in to o u r d a ta stream a t
~ 2 Hz. T h u s we lose in fo rm atio n a t timescales sh o rter th a n ap p ro x im ate ly twice the bin
size, which c e rtain ly excludes a large portion of th e H E M T low -frequency power spectrum .
However, th e softw are so lu tio n offers a great deal o f flexibility in th e analysis, an d has
been applied w ith success in th e anisotropy detections by F em enia et. al [82], Additionally,
we avoid th e offsets w hich are associated w ith ad d in g activ e com ponents, such as the
m o dulator, w ith its a tte n d a n t differential loss. We have sim u late d th e perform ance of the
DC offset rem oval tech n iq ue in various scenarios. F irst, we g en era te random , noise like,
d a ta rep resen tin g signals for TPO an d T P 1 . T h en we c a n sim u late a correlator signal
w ith an d w ith o u t a co m p o n en t w hich possesses a variable am o u n t o f correlation w ith the
to tal pow er channels. W e can th e n observe th e power s p e c tr a o f th e correlator before
and afte r regressing o u t th e co rrelated com ponent. In all cases th e technique reduces the
RMS flu ctu atio n s from th e sim u lated correlator o u tp u t. N o te th a t since this sim ulated
d a ta is w hite noise it is n o t a n acc u rate rep resen tatio n o f th e a c tu a l signals in th e real
radiom eter. T h erefo re we also need to test our technique o n real d a ta ta k en while viewing
the sky o r a te m p e ra tu re stab ilized black-body calib rato r. T h e la tte r m easurem ent only
contains flu ctu atio n s from th e radiom eter, while th e form er co n tain s contributions both
from th e rad io m ete r an d th e atm osphere.
H ardw are B a se d S o lu tio n to C orrelator D rifts: P h a s e M o d u la tio n
T he h ard w are-b ased so lu tio n places a phase m o d u lato r in one arm o f th e local oscillator
stage. T h e p h ase is square-w ave chopped from 0 to tt a t 1024 Hz. T h is has the effect of
m o d u latin g only th e c o rrelated com ponent of th e R F fields before th e m ixers a t a n AC
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frequency w hich is m uch higher th a n th e
1/
/ knee o f th e to ta l power channels, d o m in ated
by th e H E M T s them selves. T he signal o u t o f th e co rrelato r is then d em o d u lated a t the
chop frequency of th e phase m o d u lato r using s ta n d a rd lock-in techniques.
P O L A R ’s im p lem en tatio n of th e phase m o d u latio n technique employs:
• an an alo g m u ltip lier (M ITEQ M odel D B P0112H A 2)
• an electro n ic 0° — 180° phase shifter (Pacific M illim eter P roducts)
• a sw itch-referenced synchronous d e m o d u la to r d etecto r and in teg rato r (A nalog De­
vices A D 630)
T h e im p lem en tatio n o f the technique req u ires n o t only m ultiplication b u t also phase
sw itching a n d phase-sensitive d etectio n (lock-in). In doing so, we elim inate th e A 2 -t- B 2
term s in e q u a tio n 4.5 w hich correspond to th e su m of th e to ta l power in b o th p olarizations.
These signals a p p e a r a t DC, an d do n o t su rv iv e th e lock-in process. T h e o u tp u t of th e
lock-in d e te c to rs is p ro p o rtio n al to only th e co rrelated com ponent com m on in each arm of
the p o la rim eter. In p ractice, the phase o f th e square-w ave chop signal sw itch is slightly out
of phase w ith th e signal ap p earin g a t th e lock-in signal in p u t. This undesired phase shift
is caused by p ro p ag atio n delays in th e sy stem , a n d any p arasitic reactance (in d u ctan ce or
capacitance) in th e m ultipliers them selves, o r in th e subsequent stages of au d io -b an d signal
conditioning. W e AC couple our co rrelato r signals before am plification in th e second gain
stage o f th e pre-am p s. W e do n o t low-pass filter th e co rrelato r signals in th e audio pre­
am p; if we d id th e co rrelato r signal would a p p e a r as if its high-frequency co m p onents were
rolled-off (th e high-frequency “co m ers” of th e ideal square-w ave m odulated signal would
be slightly rounded-off, equivalently in tro d u c in g a phase shift prior to lock-in d etectio n ).
T h e p a ra s itic reactan ces m entioned above effectively convert a p o rtio n o f th e sig93
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n a l’s in-phase com ponent to a n 90° out-of-phase co m ponent (w hich we call “quadratureph ase” ). If we used only o n e lock-in, referenced to th e no m in al “in-phase” of the phase
sw itch, we would lose in fo rm atio n on th e correlated com p o n en t o f th e signal. To recover
th is inform ation, we utilize two lock-in detectors p er c o rrelato r, one in-phase w ith the
chop, an d th e o th e r q u a d ratu re-p h ase com ponent. See figure 4.9.
O nce th e phase sh ifter is supplied w ith its square-w ave c u rre n t chop signal the actual
choice o f lock-in technique is e ith er hardw are based or softw are based.
The com mon
pro b lem is to find th e o p tim u m reference phase for th e p h ase sensitive detector, which is
im plem ented in hardw are o r in softw are. A poor reference w aveform will reduce the signalto-noise (SNR) of th e sy stem considerably (see th e following subsection for a discussion
of th e lock-in am plifier’s S N R ). T h e lock-in signal, being a com plex quantity, has an
asso ciated m odulus an d phase. T h e real and im aginary co m p o n en ts m u st be completely
determ in ed to recover th e un d erly in g signal.
B o th approaches have ad vantages and disadvantages.
T h e softw are based solution
acquires th e signal d irectly from th e pre-am ps, and th en p erfo rm s a best-fit to the phase
of th e (known) phase-shifter drive signal.
T his technique has th e advantage th a t the
reference frequency is easily m odified as system atics w arran t. Its m ain disadvantages are:
• th e re m ust eith er be a significant am ount of correlated signal such th a t the correlated
co m p o n en t’s m o d u latio n is easily resolved from th e noise; or, in th e case th a t the
real-tim e correlated sig n al’s phase is buried in th e noise, th e phase solution obtained
from a source w ith a large correlation betw een arm s m u st be tru s te d to be constant
betw een calibrations (w hich m ay be days a p a rt).
• for a high frequency p h ase sw itch, th e d etecto r o u tp u t m u st be sam pled faster th a n a t
94
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least tw ice th e phase-shift frequency. T h e phase sw itch frequency is usually a factor
of ~ 10 higher th a n the
1/
/ knee of th e first-stage am plifiers, which in P O L A R ’s
case would require a sam pling frequency o f 2 x 10 x ~ 5 H z~ 100Hz. For th e roughly
10
channels o f d a ta and housekeeping th is resu lts in an effective d a ta ra te of ~
K b /S or roughly
1
100
GB per day which is ra th e r im practical to reduce, unless th e
softw are is perform ed in real tim e. However, w ith
8
d a ta channels sam pled a t 1
K Hz, this to o is im practical.
L ock-In D e te c to r s
To im plem ent th e “hardw are-based solution” , a cu sto m lock-in detector was co n stru cted .
The d e te c to r is centered on the AD630 Synchronous M o d u lato r/D em o d u lato r chip. Sig­
nals leave th e p re-am p card and en ter a se p arate R F tig h t box containing six sep arate
lock-in circuits. T h e num ber “six” corresponds to phase sensitive detection of th ree cor­
relators. each w ith tw o reference phases, “in -p h ase” an d “qu ad ratu re-p h ase” . A fter m ul­
tip licatio n of th e p re-am p o u tp u t signal by th e two reference waveforms, the resulting
product is low-pass filtered at 5 Hz, which also serves as o u r anti-aliasing filter for our 20
Hz DAQ sam pling system .
For diag n o stic purposes, we have an ad d itio n al variable-phase reference signal which
can be used in place of th e fixed phase reference signal into th e AD630. T his gives us
the flexibility to “tu n e ” th e phase of th e reference signal such th a t the q u a d ra tu re phase
com ponent of th e co rrelato r o u tp u t can be nulled.
For observations, we reference th e
lock-ins to e ith e r th e fixed in-phase or q u ad ratu re-p h ase waveforms. O ur phase-sw itch
driver circu it produces b o th the in-phase an d q u a d ra tu re phase reference waveforms. We
are able to change th e phase of th e reference waveforms w ith respect to th e physical chop
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w aveform by discrete step s o f 0.18°. T h is technique allows u s to m axim ize th e signal-tonoise ra tio of the co rrelato r ch an n els as well as to remove a n y signal com ponent from the
q u a d ra tu re phase d etecto rs pro v id in g us w ith extrem ely pow erful m o n ito rs o f the noise of
th e co rrelato r channels.
We have m easured th e com plex tran sfer function of th e p re-am p s to determ ine both
th e ir voltage gain and relative p h ase sh ift. T h e la tte r is esp ecially im p o rta n t in th e case of
o u r phase-sensitive d etectio n schem e o u tlin ed above. Any u se o f signal-conditioning ele­
m ents, such as filtration, of th e sig n al o u t of th e pre-am ps w ould c o n trib u te to a nontrivial
p h ase sh ift between th e c o rrelato r o u tp u t an d th e signal w aveform in p u t to the AD630’s.
Such a p hase shift would be d isa stro u s if we were only m e asu rin g th e real com ponent of
th e lock-in signal. O u r du al-referen ce p hase approach solves th is p roblem of course.
T h e o u tp u t of th e lock-ins w ith te rm in a te d inputs are m e asu re d to determ ine their
offsets.
T h is is done in ad d itio n to th e corresponding m e asu re m en t for th e pre-am ps.
F u rth erm o re , the tran sfer fu n ctio n o f th e pre-am ps m ust be fully characterized in order
to d eterm in e d the effective in te g ra tio n tim e for each channel.
4 .6 .4
E le c tr o n ic s B o x a n d H o u s e k e e p in g
T h e rm a l regulation o f th e R T R B is essential to th e s ta b ility of th e in stru m en t over
long p erio d s of time. We have identified a num ber of co m p o n en ts which are extrem ely
tem peratu re-sen sitiv e. T h e m ost sen sitiv e com ponents are th e non -lin ear devices such as
th e m ix ers/m u ltip liers, an d especially th e G u n n O scillator. T o reg u late th e te m p eratu re
we have con stru cted a th e rm al co n tro l circu it which em ploys feed b ack from a sensor inside
th e R T R B . T his control circu it is c en tered on a com m ercial m icro p ro cesso r-b ased P ID con­
tro l (O m ega), and can reg u late u p to 300W of power applied d ire c tly to M IN C O H eaterfoil
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pads. T h e pow er an d control of ou r ap p ro ac h allows us to regulate th e te m p e ra tu re o f all
elem ents in th e R T R B to b e tte r th a n 100 m K /tftrs per-day.
W e also m o n ito r several o th e r housekeeping signals, including: 4 Lakeshore # 10 te m ­
p e ra tu re sensor diodes inside th e cry o stat (o n th e H EM T S, cold plate, an d feed horn) an d
th e dew ar pressure. We em ploy a m u lti-stag e pow er reg u latio n approach, using precision
voltage reg u lato rs an d reference th ro u g h o u t th e R TR B ; all signal circu itry (H E M T bias
card s, p o st-d etectio n electronics, etc.) are d o u b le regulated a n d EM I shielded.
4 .6 .5
P o s t - D e t e c t i o n E le c tr o n ic s : P D E
T h e p re-am plifier is th e final com ponent o f th e signal chain for the to ta l pow er d e te c to rs,
and th e p en u ltim a te co m ponent for th e co rrelato rs as th e y are detected via th e lock-in
circu its describ ed above. To minimize th e su scep tib ility to electrom agnetic interference
(E M I), th e signals are am plified and filtered before leaving th e rad io m eter box. A single
V ector card co n tain s 5 identically c o n stru c te d circuits, a single one of which is displayed
in figure 4.9. T h e card is m o u n ted in close p ro x im ity to th e detecto rs an d shares th e sam e
th e rm ally reg u lated environm ent.
T h e first stag e o f th e pre-am plifier consists o f th e gain stag e set by a low-noise A nalog
Devices O P -27 P recision O p-A m p. T h e g ain is ad ju stab le by selection o f th e feedback
resistor. Following th e gain set stage is a 4-pole, 5Hz Frequency Devices A nti aliasing
filter. T h e b an d p ass o f th e anti-aliasing filter also serves to set our fundam ental in teg ratio n
tim e, r . In o rd e r to m easure th e effective in te g ra tio n tim e we calculate th e pow er sp e c tra l
d en sity (PSD ) o f th e o u tp u t o f th e pre-am plifier circu it w ith a 50fi te rm in ated in p u t. T h e
effective in teg ratio n tim e is th e tim e lag betw een which sam ples from th e d e te c to r can
be considered in d ep en d en t. D enote th e vo ltag e tran sfer function of th e p re -a m p s/a n ti-
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F i g u r e 4 .9 :
P o s t-D e te c tio n P re -a m p lifie r c irc u its for th e to ta l p o w er c h a n n e ls (to p ) an d th e
c o rre la to r c h a n n e ls (b o tto m ) . T h e o v e rall g a in o f th e p re-a m p s c a n b e sw itc h e d b etw een low -gain
( d u rin g c a lib ra tio n ) o r h ig h -g a in ( d u rin g o b se rv a tio n s). T h e g a in is c o n tro lle d by a T T L signal
su p p lie d b y th e d a t a a c q u is itio n s y ste m .
aliasing filters as = H ( u ) . T h e pow er sp ectru m o f H(u>) is S h (u ). T h e n we have that:
1
7
t p s d
1
s g~ 7m /
S
h
{0 ) 7 - o o
S h {uj)(Lj = - —
9
-5 f/(0 )
2
/
S H(u)dw = 2A i/ = 7o
t
We have th a t r = 2 t p s d - To convert Noise Equivalent V oltage (N EV ) in [ V /\ /H z \ to
[V-v/s] we have: N E V =
where S is th e PSD w ith units [V 2 s]. So if we are m easuring
th e P S D an d converting to N EV we have N E V = sj ^ psD >ju stify in g th e h ith erto ad hoc
practice of dividing P S D ’s by sp urious factors o f \/2.
4 .6 .6
D A Q H a r d w a r e a n d S o ftw a r e
T h e d a ta acquisition system is com posed o f th re e main sections:
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• N ational In stru m en ts D I0 -M I0 -1 6 , 16 B it D aq p ad A nalog-to-D igital C onverter
M odule
• C om paq P en tiu m II notebook com puter
• N ational In stru m en ts Labview software
T h e m ain hardw are co n strain t is th a t all co m p o n en ts m ust ro tate w ith th e polarim eter,
elim in atin g the use of a sta n d a rd desktop co m p u ter for acquisition an d sto rag e purposes.
T h e D aqpad is a co m p act 16-Bit, 16 channel (single-ended) analog-to-digital converter
(A D C ). It sam ples all
8
d a ta channels as well as
8
housekeeping channels a t a sam pling
ra te of 20 Hz, which over sam ples the o u tp u ts from th e detectors by a factor of 2 since
th e N yquist frequency for o u r anti-aliasing filters is 10 Hz. T he D aqpad interfaces to
th e notebook co m p u ter via a lm parallel p o rt interface cable. T he Labview softw are is
custo m w ritten to sam ple all channels a t 20Hz. B y sam p lin g and sto rin g all o f th e d a ta in
close physical proxim ity to th e detectors, we m inim ize co rru p tio n due to R F I which m ight
otherw ise occur if th e A D C to o k place off of th e ro ta tio n platform .
T h e d a ta files are indexed by calendar d ate, w ith several hundred files stored p er day.
A fter one day of acquisition, th e d a ta files are tran sfered from th e ro ta tin g notebook
co m p u ter to a desktop co m p u ter via a local a re a netw ork E th ern et connection. T h e coax
E th e rn e t connection leaves th e ro tatin g electronics box th ro u g h 2 channels of a 10 channel
shielded slip-ring (5 th D im ension). No R F I is noticed du rin g d a ta transfer.
4 .7
R o ta tio n M o u n t and D riv e S y s te m
Since P O L A R ’s recovery o f th e Stokes p aram eters is based upon th e ir m od u lation under
ro tatio n s, we have co n stru cted a 30” d iam eter b earin g platform and AC m o to r system to
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ro ta te th e cry o stat a t 2 R P M ( ~ 30m H z see figure 4.10).
Aafmuth Motor
ao*
F i g u r e 4 .1 0 : R o ta tio n m o u n t, sh o w in g m o to r, b earin g , a n d e n c o d e r p o s itio n . T h e w h eels allow
for fine p o sitio n in g o f th e in s tr u m e n t o n its p la tfo rm , a n d a re rem o v ed o n c e th e in s tru m e n t has
b e e n alig n ed .
T h e AC m o to r is sm o o th e r th a n a s te p p e r m otor approach(w hich we tried originally),
an d is ideal for continuous ro ta tio n such as ours. T he dew ar b e a rin g rides on a 0.100”
stainless-steel ball-bearings in a lu b ricate d channel, which co n tain s a b o u t 400 balls. The
m o to r pulley has a 1000 b it/r o ta tio n T T L com patible relative angle en co d er w hich reads
o u t th e ro tatio n angle. A c u sto m m ade on e-b it absolute encoder is trig g ered once p er rev­
olu tio n a t a specific an g u lar p o sitio n w hich serves to set th e zero an g le for th e polarization
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recovery.
4.8
I n s tr u m e n t B a n d p a sses
We present here a m e asu re m en t o f the b an d p ass of th e in stru m e n t. For this m easurem ent,
an HP 83751A S ynthesized Sw eeper is used to p ro d u ce a sw ept signal from 13 to 18 G H z,
which is su b seq u en tly d o u b led in frequency by an activ e do u b ler (producing 26 - 36 G H z)
an d fed into a pow er s p itte r. T h e o u tp u ts from th e pow er sp litte r are, of course, 100%
correlated, a n d th e se signals are fed into th e w aveguide in p u t p o rts of th e R TR B . T his
allows us to m easu re th e b an d p ass of all w arm R F co m p o n en ts. T he correlated signals
are a tte n u a te d by 60 d B to provide a pow er level sim ilar to th a t obtained w hen view ing
th e sky.
To illu s tra te th e b an d p ass m easurem ent an d phase chop m eth o d we refer th e read er to
figure 4.11 which show s a 600 M Hz section o f th e b a n d p ass of co rrelato r J3 before lock-in
detection. T h e p h ase sw itch is chopped a t 1 KHz, cau sin g sign reversal of th e co rrelato r
DC o u tp u t. T h is sig n al is subsequently fed in to th e lock-in d etecto rs, and th e DC level
o u t is recorded to m easu re th e response as th e in p u t signal is sw ept in frequency. All
three co rrelato r ch an n els are m easured this way, an d th e resu ltin g bandpasses are show n
in figure 4.12.
Table 4.5: R ad io m eter C entroic s, B andw idths,-----------------------------an d O b serv in g Sensitivities
(T^nt — 15K)
7--C h an n el uc[GHz] A u [G H z]
TP0
TP1
J3
J2
J1
7H
31.9
30.8
28.0
31.5
35.0
8.0
1.2
2.7
2.9
40.0
10.1
8.5
7.8
4.5
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J 3 C o r r e l a t o r Output
2 7 .0 0
2 7 .1 0
2 7 .2 0
2 7 .3 0
Frequency [GHz]
2 7 .4 0
2 7 .5 0
2 7 .6 0
F i g u r e 4 .1 1 :
A 600 M H z se c tio n (o u t o f 1.2 G H z) o f c o rre la to r J 3 ’s b a n d is show n. T h e
m o d u la tio n is c a u s e d b y a sq u a re -w a v e p h a se c h o p o f th e lo cal-o scillato r sig n a l. T h e low -frequency
o s c illa tio n is th e re s u lt o f g a in a n d p h a s e v a ria tio n s a c ro ss th e b a n d .
Co rre la to r Bandpasses
0.8
<15
■o
u
• f 0 .6
E
<
J3
0.2
J2
-
0.0
28
30
32
34
F re q u e n cy [G H z]
F i g u r e 4 .1 2 : A ll th r e e c o rre la to r b a n d p a s s e s a re sh o w n . T h e re is sig n ific an t re sp o n se to c o rre la ted
sig n a ls o v e r th e full /Ga b a n d .
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T able 4.6: P O L A R K & B and R ad io m eter C o m p o n en ts
M anufacturer
M odel
C ircular-S quare T ran sitio n
C ustom Microwave
—
OM T
A tlantic Microwave
OM 2800
H EM Ts
NRAO
A29 & A30
W arm R F A m ps
M IT E Q
J S426004000-30-8P
Mixers
M IT E Q
TB0440LW1
G unn O scillator
M illim e te r W ave O s c illa to r C o .
—
W arm IF A m ps
M IT E Q
AFS6-00101200-40-10P-6
Triplexers
R eactel
—
C orrelators
M IT E Q
D B P112H A
Total Power D etecto rs
H ew lett Packard
H P 8474C
Lock-In Am plifiers
Analog Devices
AD630
Kg, band P h ase S w itch
P a c ific M illim e te r P r o d u c ts
—
Dewar
P re c isio n C ry o g e n ic S y s te m s
—
Cryocooler
C T I Cryogenics
8 5 0 0 C o m p re s s o r, 350 C o ld H e ad
Device
T able 4.7: PO L A R 0 jserving P a ra m e te rs
P a r a m e te r
P O L A R 1999
D e c lin a tio n o f D rift Scan
B e a m w id th
F ra c tio n a l S k y coverage
R o ta tio n R a te
P o in t S o u rce S en sitiv ity
P o s t d e te c tio n B an d w id th
S a m p lin g F req u en cy
43.03°
7°
255° x T F W H M a 5%
0.03H z
0 .7 n K J y - i
5H z
20 Hz
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Chapter 5
Calibration
5 .1
C a lib r a tio n desiderata
A n ac c u ra te calib ratio n is essential to th e success o f the P O L A R ex p erim en t. T h e ideal
c alib ratio n source will allow us to d eterm in e th e following q u an tities o f interest:
• v o ltag e-to -a n ten n a te m p e ra tu re conversion coefficient for each ch an n el
• sy stem noise te m p e ra tu re for each channel
• m inim um d etecta b le polarized signal in one second o f in teg ratio n , th e Noise Equiv­
alent T em p e ra tu re (N E T ), for all d etecto rs
• offsets a n d long-term stab ility of o u r in stru m en t
An ideal source would be a p o larized astro p h y sical point source w ith enough power to
be seen in “real-tim e” . T h is w ould allow real-tim e beam -m aps as well as calibration. We
can estim a te th e necessary power such th a t a 5- a detectio n is m ad e in th e fundam ental
0 .2
s in teg ratio n p erio d o f th e receiver back-end - which would be b rig h t enough to see
clearly in real-tim e.
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T h e a n te n n a te m p e ra tu re seen by P O L A R ’s to ta l pow er d etecto rs when viewing a
source o f flux d en sity S ( v ) [Jy] is given by:
w here kb is B o ltz m a n n ’s co n stan t an d
is th e solid angle o f th e m ain beam . For
P O L A R . Q b = 0.047sr, w hich implies an a n te n n a te m p e ra tu re of 0.7fiK for a 1 J y source.
N ote th a t e q u atio n 5.1 holds for a single m ode d e te c to r for a single polarization, as indi­
cated by th e presence o f th e factor “2” in th e d en o m in ato r. For a 5 —a detection, w ith
a receiver w hose N E T ~ 3 m K / V H z , a source o f a n te n n a te m p eratu re T { y ) ~ 33mK
is required. T h is is eq uivalent to a 335 J y source a t 31 GHz. For com parison, Cas-A ,
the b rig h test know n rad io source, has a flux density of 206 J y a t 31 GHz. We note th a t
this d eriv atio n assum es we are m easuring a change in a n te n n a te m p e ra tu re by th e source,
not a ro ta tio n m o d u la te d p o larization signal. Since th e p o larizatio n of Cas-A is less th a n
10% a t 31 G H z, for th e p o larization signal to be d e te c te d in real tim e would require a
signal te n tim es larger! C learly, we can n o t ex p ect to use a n astro p h y sical point source for
calib ratio n o f P O L A R . W e n o te here, for possible fu tu re relevancy, th a t M adison, W I is
alm ost ideally s itu a te d for a zen ith scan w ith a sm aller beam size, for Cyg-A, th e second
m ost pow erful rad io source, lies a t a declination of 6 = 41°, placing it only 2° from th e
zenith. T h e p o la rizatio n p ro p erties of Cyg-A have been m easured a t arcsecond scales a t 15
GHz to be ~ 10%, d ecreasing due to F arad ay ro ta tio n to 2% a t 5 G H z [83]. A sub-degree
p o larizatio n ex p erim en t a t 31 G Hz could d e te c t th is o b je c t a t th e several—a level.
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5 .2
T w iste d -C o ld L oad (T C L ) C alib ration s
For o u r in itial lab o rato ry calib ratio n s we have developed an internal calib ratio n m ethod
w hich p roduces a correlated electric field a t th e in p u t to th e orthom ode tra n sd u c e r (OM T).
T h e in ten sity of th is field can be varied above 30 K, and produces a polarized signal a t
th e in p u t to th e O M T which calib rates th e en tire radiom eter from th e feedhom on. We
call th is c a lib ra to r the '‘T w isted C o ld L oad” , o r TCL, an d it allows us to determ ine
th e ra d io m e te r’s noise tem p eratu re, noise equivalent tem p eratu re (N E T ), a n d calibration
coefficients.
T h e T C L is co n stru cted from a section o f circular copper waveguide, w ith castable
Eccosorb epoxy CR-114 coated walls.
T h e Eccosorb coats th e walls in such a way as
to m axim ize th e num ber of reflectio n s/ab so rp tio n s for a wave incident on th e calibrator.
To accom plish th is an “inverse-spike” geom etry was adopted. T he m easured re tu rn loss
across th e K A b an d b etter th a n -30 dB . A section of circular stainless-steel waveguide is
atta c h e d betw een th e load and th e O M T to act as a th erm al break, a n d a copper strap
connects th e load to the 20K cold sta g e (see figure 5.1).
T h e load is polarized by coupling its circu lar o u tp u t flange to a W R -28 circularto -re c ta n g u la r tran sitio n , which acts as a polarizing filter.
Following is a co-aligned
rectan g u lar-to -circu lar tran sitio n w hich th e n couples to th e circu lar-to -sq u are transition
w hich feeds th e O M T . By ro ta tin g th e El-plane of the back-to-back tran sitio n s 45° with
resp ect to th e sq u are ap ertu re o f th e O M T we are injecting in a 100% co rrelated field into
each p o rt o f th e O M T. T his allows us to have a tem p eratu re controlled th erm al source,
which m im ics th e effect a
100%
polarized source would produce in our co rrelato r channels.
T h e ro ta tio n by 45° m otivates th e n am e o f th e device: th e T w isted C old L oad (TCL).
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Cu STRAP TO
COLD PLATE
A A A A ______________ HEATER
Cu
~
MOLDED
ECCOSORB
COPPER
CYLINDRICAL
W AVEGUIDE
STAINLESS
STEEL
CYLINDRICAL
W AVEGUIDE
FROM ABOVE
THERMOMETER
THERMAL
------------- BREAK
------------------------------
\
/
CIRCULAR TO
WR-28
—------------ RECTANGULAR
WR-28 TO
CIRCULAR
CIRCULAR TO
SQUARE
FROM ABOVE
OMT
Figure 5.1: The Twisted Cold Load Calibrator (TCL). T he antenna temperature of the load is
varied by heating an Eccosorb loaded section of cylindrical waveguide. The back-to-back circularto-rcctangular waveguide transitions polarize the load along the E-plane of the rectangular waveg­
uide. This plane is rotated along the main-diagonal of the OM T to produce correlated fields in
the E and H plane ou tp u t ports of the OMT.
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T h is ro ta tio n affects th e pow er ( = a n te n n a tem p eratu re) seen by b o th th e to ta l power
d etecto rs an d th e correlators. F or th e to ta l pow er detectors, th e use o f th e rectangular
section o f w aveguide converts th e d u al-p o lariz atio n circular waveguide in to one which only
p ro p ag ates a single-m ode. B ecause th e load itself is unpolarized, an d we acc ep t only one
p o larizatio n , we have cut th e pow er in to th e O M T from 2kbTioad&i'RF to ki,Tioad&VRFT h e effect of th e 45° tw ist is th a t th e a m p litu d e of th e field amplitude delivered to each
to ta l pow er d e te c to r is reduced by cos 45° = - 7- .
T hus a change in load te m p eratu re
&Tioad p roduces a change in th e to ta l pow er d etected te m p eratu re of A7^gg* .
For th e co rrelato rs, th e red u ctio n in pow er is th e sam e factor of two. T h e correlators
m u ltip ly th e field am p litu d e in each o f th e polarizations. E ach p o larizatio n receives -jof th e field p ro d u ced after th e m ode conversion.
M ultiplying these tw o factors in th e
co rrelato rs p ro d u ces th e sam e facto r of tw o red u ctio n as for th e to ta l pow er detectors.
5.2.1
S y ste m N o ise T em p eratu re
T h ere are several m ethods to co m p u te th e noise te m p eratu re of th e ra d io m ete r using the
T C L or w ith several different te m p e ra tu re ex tern al loads. W hile th e form er m ethod pro­
duces lower an d m ore stable te m p e ra tu re s, th e la tte r is much faster to im plem ent, and
also in co rp o rates th e effect of th e feed h o rn . W e have tested th e T C L m e th o d vs. the
am b ien t te m p e ra tu re ex tern al load m e th o d to en su re consistency. F u rth e r te sts using the
T C L are p lan n ed in order to o b ta in h igher precision results a t te m p e ra tu re s lower th a n
th e lowest te m p e ra tu re external load available: 77K. However, th e noise te m p e ra tu re cal­
ib ratio n resu lts q u o te d in th is th esis are o b ta in e d from the am b ien t te m p e ra tu re external
load m eth o d .
108
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M ethod 1: Y-Factor M easurem ents
For th e to ta l power channels we use a com m on technique to o b ta in a first-order estim ate
of th e noise te m p eratu re o f each a rm o f th e radiom eter. G iven two load tem p eratu res, 7 //
an d T c , an d th e corresponding D C voltages produced in th e to ta l pow er detectors, Vff
an d Vq , th e y-factor is defined to be: y —
Solving for th e receiver noise
te m p e ra tu re , T tv , we find:
„V c T h - V h T c
T n = 13 VH - V c
'
w here 0 = ^ for th e T C L a n d ( 3 = 1 for th e am bient te m p e ra tu re calibrations.
M ethod 2: Bandwidth Technique
K now ing th e b an d w id th A u of th e system , th e voltage flu ctu atio n s A K m j in a n integra­
tion tim e r , an d the calib ratio n coefficient, g in [V/K] allows us to estim ate th e noise
te m p e ra tu re of th e receiver via th e rad io m ete r equation:
T sys = Kg
w here
k
=
1
1A
VRMs ' y A i/t - (3Tioad,
for th e to ta l pow er channels, and
k
=
y /2
(5.2)
for th e co rrelato r channels, and
(3 = ^ for th e T C L and 0 = 1 for th e am b ien t tem p eratu re calibrations.
M ethod 3: Linear Intercept Technique
For th e tw o to ta l power channels an d th e th ree correlators we can use th e RMS noise
on th e D C d etecto r voltages to d eterm in e th e noise te m p e ra tu re . T his m eth o d has th e
ad v an tag e th a t it does n o t req u ire th e knowledge of th e ra d io m e te r’s b an d w id th . T he
109
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noise m odel for th e to ta l pow er detecto rs is:
VA ut
= OcT,load +
7
i
(5.3)
(5.4)
while for th e co rrelato rs we have:
A T rm s =
v^2 ^ ('T s Y S o T s Y S i+ P T io a d )
(5.5)
=
(5.6)
iT lo a d
+
6.
A gain, (3 — ^ for th e T C L an d 0 — 1 for th e am bient te m p e ra tu re calibrations.
We see th a t for b o th th e to ta l power channels and th e co rrelators ATVms is a linear
function of th e load te m p e ra tu re .
T he x-intercept o f th ese lines will be equal to the
negative of th e system noise tem p eratu re. We note again th a t these results apply to the
am b ien t load m eth o d , n o t th e T C L m ethod.
For th e T C L m eth o d it is necessary to
inco rp o rate th e effect of th e single-m ode waveguide which reduces th e load tem perature
seen by th e d etecto rs by a factor o f two.
N o is e Temperature Contribution from Lossy Com ponents
T h e system noise te m p e ra tu re is dom inated by the noise te m p e ra tu re of th e HEM T am ­
plifiers. However, th e co n trib u tio n of th e following ro o m -tem p eratu re am plifiers, as well
as loss in com ponents preceeding th e H EM TS cannot be neglected. T h e dom inant lossy
110
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elem ents preceeding th e H E M T s are th e tw o P A M T E C H cryogenic iso lato rs and the dew ar's vacuum window. T h e iso lato rs’ e x a c t physical te m p e ra tu re is unknow n, b u t we may
estim ate th e m to be a t ~ 40K, which is th e physical te m p e ra tu re o f th e horn, so this is a
w orst-case estim ate. T h eir in sertio n loss is: L = l / e lso = O .ldR = 0.03, w here etso is the
tran sm issio n o f the isolator. T h e loss o f th e vacuum window is conservatively estim ated
at
1 %.
T h e noise co n trib u tio n from am plifiers in series w ith th e H E M T s are reduced by
th e gain of th e H EM TS [20], w hich is w hy we p u t o u r best am plifiers first in th e signal
chain. T h e room te m p eratu re M IT E Q am plifiers have noise figures o f F = 2.5d B = 1.8
which tra n sla te s to a noise te m p e ra tu re o f ( F — 1)290FT = 232K.
T h e to ta l estim ated sy stem noise te m p e ra tu re including all o f th e se ad d itio n al factors
is:
F„
=
T h e m t + 7;
T m i t e q + ( --------- i ) F 30 + ( -------------- 1) T window
C rH EM T
Ktis o
'
=
232
6 5 K + -316
^ K + 0 03 x 40A' + 0 01 x 30° F
~
70 K ,
V^ w i n d o w
'
w here th e gain an d noise te m p e ra tu re of th e H EM TS have b een ta k en from their
specifications after co n stru ctio n a t N R A O in 1994. T h is n u m b er agrees q u ite well w ith
th e m easured values of Tn discussed below.
R e su lts o f N o is e T em p era tu re M ea su r e m en ts
Table 5.2.1 displays th e noise te m p e ra tu re of th e system o b ta in ed from th e correlator
channels q u ad ratu re-p h ase co m ponent using th e linear intercept m e th o d o u tlined above.
Ill
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T able 5.1: System Noise T e m p e ra tu re O b tain ed From C o rre la to r C hannels Using Linear
In terc ep t M ethod
______________________________________
C h an n el Tjy from Linear In te rc e p t
J1
64.7 K
J2
79.0 K
77.4
K
J3
A p lo t of th e y-factor m easu rem en ts for th e th re e co rrelato r channels is shown in
figure 5.2. This figure allows us to o b ta in th e system noise te m p e ra tu re derived from each
c o rre la to r (x-intercept) as well as a n in d ependent m easu rem en t of each co rrelato r’s N E T
(y -in tercep t). Some com pression can be discerned for th e highest am bient te m p eratu re
load used (300 K). H igher precision m easurem ents using th e T C L will be perform ed to
o b ta in th e lowest possible u n co rrelated load te m p eratu res ( ~ 30K ). T h e values o f A T rjv/s
for co rrelato rs J 1 a n d J2 agree q u ite well w ith th e m easu red values from the PSD d a ta
discussed a t the end o f th is C h a p te r. C o rrelato r J3 shows th e highest level o f com pression
an d not-coincidently, o b ta in s th e h ighest N E T - a value inconsistent w ith th e N E T derived
from its PSD .
5 .3
W ir e G rid C alib rator: W G C
W hile th e T C L calib ratio n s describ ed above are th e m o st a c c u ra te m eth o d to calibrate th e
detecto rs, th e TC L does n o t ac c u ra te ly represent the tru e co n fig u ratio n of th e radiom eter
as used d u rin g observations.
T h e absence of the feed h orn an d vacuum window is a
strik in g shortcom ing o f th e T C L technique. A dditionally, th e T C L calib ration m ethod is
tim e-consum ing as it requires com p lete disassem bly o f th e cry o stat to install. Therefore,
we req u ire a calibration m e th o d w hich is easily im plem ented, an d rep resen tative of th e
ob serv atio n s we are a tte m p tin g to perform . T h e conventional ap p ro ach to these issues for
112
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Correlator AT,™ v s . L oad T e m p e r a t u r e
60
J3
50
>
E
J2
30
i
E<3
20
50
100
150
200
Load T e m p e r a t u r e [K]
25 0
F i g u r e 5 .2 : Y -fa c to r m e a s u re m e n ts o f all th re e c o rre la to rs a re show n. T h e d a s h -d o t, d a s h , a n d
d o tte d lin es a r c th e b e s t lin e a r fits to th e c o rre s p o n d in g c o rre la to r d a ta . T h e r e s u lts o b ta in e d
from th e se m e a s u r e m e n t for J1 a n d J 2 a re c o n s is te n t w ith th e c o rre sp o n d in g values o b ta in e d usin g
th e o th e r tw o m e th o d s d iscu ssed in th is C h a p te r (P S D a n d b a n d w id th m e th o d s). A ll c o rre la to r
c h a n n e ls a p p e a r to su ffe r so m e c o m p re ssio n . C o rre la to r J 3 d isplays th e g re a te s t c o m p re ssio n ,
a n d th e N E T d e riv e d from th is m e a su re m e n t is n o t c o n s is te n t w ith th a t o b ta in e d fro m th e P S D
m e a s u re m e n t d is c u s s e d a t th e en d o f th is C h a p te r.
113
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a polarim eter is to em ploy a wire g rid calib rato r (W G C ) [65], [84], [85]. T h e W G C is a
passive source which can provide electrom agnetic fields which are co rrelated in each arm
of th e receiver, and is placed o u tsid e th e cry o stat for rapid im plem entation.
C o ld
Load
or
Sky
<n
> Warn
/
o
Grid _
\
Load
>
1
D ry
•M il
Figure 5.3:
.
N itro g e n
Side view of wire grid calibrator (WGC) in place during calibration.
T h e grid functions by tra n s m ittin g th erm al rad ia tio n from a black b o d y source in one
polarization, an d reflecting th erm al rad iatio n from a second blackbody source (at a dif114
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ferent te m p e ra tu re ) in to th e o rth o g o n al polarization, see figure 5.4.
For th e P O L A R
calib rato r, th e cold load is lo cated above th e grid a n d p ro d u ces fields E \ an d H \, and
th e w arm load p ro d u ces fields
£2
an d H i- T h e grid’s w ires reflect fields polarized along
th e ir axes an d tra n s m it fields o rth o g o n al to th eir axes. T h e resu lt is th a t, ideally, H \ is
tra n s m itte d an d
£2
is reflected into th e feed-horn producing, as we will show, a
100%
polarized diffuse so u rce w ith an a n te n n a te m p eratu re eq u al to th e te m p e ra tu re difference
betw een th e two loads.
ni '
IN CID EN T F IE L D
FOR.
REFLEC TIO N
FROM H O T L O A D
IN C ID EN T F IE L D
FO R
T R A N SM ISSIO N
FRO M CO LD
LO A D O R SK Y
F ig u re 5.4:
Geometry of the Wire Grid Calibrator
O u r w ire-grid c a lib ra to r 1 was fab ricated by deposition o f c o p p er o n to a 50 m il m ylar
su b stra te . T h e w ires them selves are 0.008” wide w ith 0.008” spaces. For su p p o rt the grid
is sandw iched betw een Dow C o rn in g “p in k ” Styrofoam sheets (em issivity o f ~ 1%), and
th e sandw ich is m o u n ted a t 45° to th e a p e rtu re plane. See figure 5.3 for th e orientation
of th e grid d u rin g calib ratio n . T h e grid h as an in teg rated b earin g sy stem w hich allows it
'D e sig n e d by C h ris O ’D ell.
115
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to ro ta te d irectly over th e v acu u m window. T h is allows us to keep th e dew ar s ta tio n a ry
an d sim ply ro ta te th e grid to c a lib ra te P O L A R .
As show n in C h ap ter 3, th e p ro p erties of th e source function,
7
(u), defined in C h ap ter
4 d e te rm in e th e o u tp u t v oltage recorded by th e correlation channels. T h e function
7
( 1/)
d ep en d s on th e coherence o f th e electric fields produced by th e th e rm a l ra d ia to rs. How­
ever, we only know th e a n te n n a temperature o f th e h o t and cold loads, n o t th e electric
fields p ro d u ced in th e x a n d y d irections. F o rtunately, as we will see, o nly th e a n te n n a
te m p e ra tu re s are needed in th e end.
T h e resu ltin g field seen by th e feed-horn is th e superposition o f th e tra n s m itte d field
H i, an d th e reflected field E 2 , In te rm s o f th e (x', y1) basis of th e feed-horn an d O M T and
th e (x, y) basis of th e rest fram e of th e W G C , th e electric field p ro d u ced by th e W G C as
th e p o la rim eter is ro ta te d by a n angle a is given by:
= E x (t) cos a + E y (t) sin a
E y> = —£ x ( t ) s i n a -(- E y (t) cos a.
(5.7)
T h e o u tp u t of th e c o rrelato r from th e coherence function given by 3.11 is:
K u t oc ( E x ' ( u ) E ^ ( u ) )
= ((E x {t) cos a + E y {t) s i n a ) ( —E x (t) s i n a + E y (t) s in a ) ) ,
w ith th e load fields E x , E y given by:
116
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(5.8)
Ey(t) = E yo cos \yt + <j>y{t)\
Ex(t) = E Xo cos[i/t + <t>x{t)\.
(5.9)
W e o b ta in :
Vm t oc ( —E x (t) cos a E x (t) sin ct + Ey(t) sin a E y ( t ) sin a )
oc (E yE * — E XE *) sin a cos a
<x s in 2 a [ ( E y E ;) - (EXE'X)]
oc Q sin 2a
=
w here
7
7
(Ty — T x ) sin 2 o
converts a n te n n a te m p e ra tu re (m easured by th e rad io m eter) to intensity (th e
u n its of th e Stokes p aram eter, Q ). N ote th a t a t q = 0°, 90°, 180°, 270° the correlators
have zero o u tp u t as th e fields are com pletely aligned along only one p o rt o f the O M T and
th u s do n o t p ro d u ce co rrelated fields betw een th e two arm s.
Ideally, th e grid would reflect T^ot from th e side in 100% horizonal polarization and
tra n s m it Tcon from th e to p in
Th.ot) sin 2 a .
100%
vertical polarization, resu ltin g in Vout oc 7 (^')(Tcow —
In p ractice, due to loss an d reflection, th e g rid is n o t perfect and instead
we observe th e following an ten n a te m p e ra tu re s at th e feed-horn in th e two orthogonal
p o larizatio n s [84]:
117
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T able 5.2: E stim ated P ro p erties of G rid a n d Loads Used For C alib ratio n
P ro p e rty
Value
0.995
m
0 .0 2
n
0.95
t±
290 K
Tbg
A T i = T w o * — T3ky
256 K
196 K
A T 2 = T 300 K — Tt7K
Thot'
Tcoid'
=
=
7~|| [(1
*_l [ ( 1
—r i)Thot
■+* nTbg]
+
—n)Tcoid + nTbg) +
^||)[(1 — Tl)Tcoid + riTbg]
(5.10)
— * x )[(l - rfiThot + nTbg],
(5.11)
(1 —
(1
where: ry is th e g rid ’s reflectivity to rad iatio n polarized p arallel to th e wires, t± is the
grid ’s tran sm issio n for rad ia tio n polarized p erp en d icu lar to th e wires, ry is th e reflectivity of
the load, an d Tbg is th e effective background te m p e ra tu re of th e ra d ia tio n field surrounding
the calib rato r. In th e above equations, we have neglected th e effects o f th e em issivities
and dielectric co n stan ts o f th e m ylar and Styrofoam .
We have tw o pairs of te m p e ra tu re differences w ith which to calib rate P O L A R . Using
a 300 K load (in reflection) an d th e sky (in transm ission) we o b ta in a polarized an ten n a
te m p eratu re of 256 K. U sing a 300 K load (in reflection) an d a 77 K Liquid N itrogen load
(in transm ission) we o b tain an an te n n a te m p e ra tu re o f 196 K. H ere we have assum ed the
properties of th e grid as listed in tab le 5.2.
A plot o f a calib ratio n ru n is shown in figure 5.5. T h e sinusoidal oscillations are the
result of ro ta tio n o f th e grid over th e feedhom . T h e first set o f oscillations corresponds to
th e 300 K L oad an d th e sky an d th e second set corresponds to 300 K L oad and th e 77 K
Liquid N itrogen Load.
118
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Total P o w e r D e t e c t o r Ca li br at io n Run
-2
«
"o
>
15K
300K -
—3
77K
-4
300K -
77K
15K
-5 £
2000
0
4000
6000
10000
8000
samples
C o r r e l a t o r J 2 In —P h a s e C a l i b r a ti o n Run
<n
o
>
-5
-10
2000
0
4000
6000
10000
8000
samples
C o r r e l a t o r J 2 Q u a d r a t u r e —P h a s e C a l i b r a t i o n Run
0 .0 4 0
0 .0 3 0
0 .0 2 0
o
0.010
>
-
8:898
15K
J T —
2000
W
4000
6000
8000
#
:
10000
samples
F i g u r e 5 .5 : V o lta g e s o u t o f c o r re la to r J 2 a n d a to ta l pow er d e te c to r d u r in g c a lib ra tio n w ith the
W ire G rid C a lib r a to r (WGC). T h e m id d le figure show s th e v o lta g e o u t o f J 2 ’s in -p h a se lock-in
d e te c to r , th e b o tto m fig u re sh o w s th e c o rre sp o n d in g v o ltag e o u t o f J 2 ’s q u a d r a tu r e p h a s e lock-in
d e te c to r . T h e v a rio u s te m p e r a tu r e lo a d s a r e in d ic a te d a t th e tim e th e y a r e a p p lie d . T h e first set
o f o sc illa tio n s c o rre s p o n d s to a p o la riz e d te m p e r a tu re o b ta in e d by u sin g a 3 00 K lo a d a n d th e sky,
w h ich p ro d u c e s 256 K sig n a l. T h e se c o n d s e t o f o scillatio n s c o rre sp o n d s to a p o la riz e d te m p e ra tu r e
o b ta in e d by u s in g a 300 K lo ad a n d a L iq u id N itro g en L oad p ro d u c in g a 196 K sig n a l. N o te the
su p p re s s io n o f th e c o rre la te d c o m p o n e n t in th e q u a d ra tu re p h a se d e te c to r , a n d a lso n o te t h a t the
no ise e n v e lo p e o f t h a t d e te c to r is a fu n c tio n o f th e load te m p e r a tu r e m aiking it a n effectiv e noise
m o n ito r.
119
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5 .3 .1
G ain M atrices
T h e g rid furnishes us w ith a well-defined th e rm a l load w hich is 100% polarized, which is
ex ac tly w h at is needed to calib rate P O L A R ’s voltage o u tp u t in term s o f a n te n n a tem ­
p e ra tu re . Following [84] we m odel th e o u tp u t o f th e p o la rim eter versus ro ta tio n angle as
a lin ear co m b in atio n o f th e Stokes p a ram eters a t th e feed horn. T h e o u tp u t voltage is
m odeled as a v ecto r v:
(
V =
Vy \
vx
\ VQ /
=
( 9yy
9yx
9yQ N
9xy
9xx
9 xQ
\ 9Qy
9Qx
9Q q )
( Oy \
( TFy )
T Fx
\
tfq
+
)
Ox
+ Tl
(5.12)
\Oq )
or
v = gTf + o + n
(5.13)
w here g d en o tes th e gain m atrix , ( T f ) is th e v ecto r o f ac tu a l a n te n n a te m p e ra tu re s
p ro d u ced a t th e feed by th e grid, and o an d n , respectively, represent offset an d noise
co n trib u tio n s to v. Ideally, g would have on ly o n -d iag o n al elem ents. T h e off-diagonal
elem ents of g correspond to various non-idealities o f th e in stru m en t which will resu lt in
offsets in o u r m easurem ents. We will ela b o ra te on th e se te rm s in th e following.
In p ractice, th e grid is placed d irectly over th e feed h o m a p e rtu re form ed by th e vacuum
w indow an d ro ta te d while th e p o larim eter is held fixed. T h e resu ltin g vector o f voltages is
recorded an d a least-squares fit is m ade to th e d a ta using th e m odel o f eq u atio n 5.12. T h e
gain m a trix p aram eters, including th e off-diagonal cro ss-talk elem ents, and th e offsets are
recovered for each calib ratio n ru n .
120
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5.3.2
C a lib ra tio n U sin g th e W G C
W ith th e a n te n n a te m p e ra tu re s of th e loads given by eq u atio n s 5.11, T h e voltages out of
the two to tal-p o w er channels an d th e co rrelato r channel are then:
( vy ( a ) \
v=
vx {a)
\
v q {q
)J
^
9yyTc
"f" 9yxT3 4-
9yQ (That'
9xyTc 4“ 9xxT3 -F 9xQ (Th0t>
) s i n 2o: 4-
Oy
\
Tcold.') sin 2or 4- ox
4- h
(5.14)
\ 9 vq T c + 9 xq T s 4- gQQ{Thot' — Tcoid') sin 2 a + oq J
where
Tc = Th0t>cos 2 a 4- X ^ ' sin 2 q
and
T3 = That' sin 2 q 4- T ^ d ' cos 2 a
To recover g . we first in teg rate long enough th a t th e noise term , h is negligible, and
then m easure th e offsets, o. T h en we can invert 5.14 to o b ta in g. Since we have two pairs
of te m p eratu re differences we can m easure th e calib ratio n co n stan ts as a function of this
difference an d check for linearity. O u r two loads p ro d u ce effective po larization antenna
te m p eratu res o f 256 K a n d 196 K and it is verified th a t th e calib ratio n co n stan ts are equal
to b e tte r th a n 10% over th is range. T h e linearity o f th e co rrelato rs over th is wide range in
te m p eratu res, a n d a t such high tem p eratu res, suggests th a t th e rad io m eter is linear over
6
dB , which is q u ite im pressive for an in stru m en t w ith ~ 100 dB of R F gain and which
uses diode-based m u ltip liers to im plem ent th e co rrelatio n rad io m eter technique.
121
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5 .3 .3
R e s u lt s o f W G C : G a in M a tr ic e s a n d S y s t e m a t i c E ffe c ts
We recover th ree se p arate gain m atrices; one p er co rrelato r:
In th e following we show th e gain m a trix G in [K /V ]; G ij = l / g i j . T h e off-diagonal
elem ents are ind icated by th e ir relative size co m p ared w ith th e corresponding on-diagonal
term s, i.e.. for a given co rrelato r G xq = / G q q , w here / is given as a percent. For all
th ree co rrelators, J l , J2 , J3 we obtain:
f - 9 6 .4
3.8%
6 .1 %
8.3%
59.3
-6 .7 %
6 .1%
-6 .7 %
55.8
Gji =
\
f - 9 6 .4
3.8%
8.3%
59.3
-9 .2 %
%
-9 .2 %
29.3
/ -9 6 .4
3.8%
8.3%
59.3
8 .2 %
-9 .3 %
G.J 2 =
i,
G j3 =
^
8 .6
8 .6
\
(5.15)
% \
(5.16)
8.6% \
-9 .3 %
(5.17)
55.5 )
T he entries of th e gain m a trix g tell us a g re a t deal a b o u t th e perform ance o f our
in stru m en t. T h e on-diagonal elem ents o f G , ( G u , Gyy, G q q ) d o m in ate th e m atrix; they
are th e term s which m easure th e system calib ratio n in [K /V ],
However, to th e experim entalist, th e off-diagonal elem ents a re n early equal in im­
p o rtan c e for th ey encode inform ation regarding th e sy stem ’s im balance, cross-talk, and
im perfect isolation betw een polarization states. A ll of these effects resu lt in system atic
differences betw een th e ac tu a l polarization, an d th a t w hich is m easu red by PO LA R . T hese
effects are know n as “offsets” , and m ay be in co rrectly in te rp re te d as celestial signals if
122
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th e y are n o t rem oved o r axe u n stab le in tim e.
T h ere are tw o in d ep en d en t effects responsible for th e off-diagonal elem ents, corre­
sp o n d in g to th e two different off-diagonal elem ents gxy = gyx a n d gxQ = 9yQ- Let us
exam in e each of th ese effects in detail. F irs t we will analyze th e effects o f gxy ^ 0, so we
set g yQ = n y = oy = 0. T h e n th e first non-ideality, gxy (or equivalently, gyx), im plies th a t
at a =
0
. w hen we should only see Thot', we actu ally observe v y = gyyT^ot>+ 9yxTcoid'-
So th e n o n-idealitv gxy has introduced a n offset in th e to ta l pow er o u tp u ts . We can now
iden tify th e term s which co n trib u te to th e to ta l offset in th e to ta l pow er channels. F irst,
we m ake som e p relim in ary definitions.
W e define th e isolation o f th e O M T to be th e
fractio n o f tra n s m itte d pow er into one rectan g u lar (p o larizatio n ) p o rt w hich appears a t
th e o th e r (o rthogonal) p o rt. T he m ain co n trib u tio n to th e to ta l pow er offset is from cross­
p o la rizatio n effects in th e horn or th e O M T
2
T hese term s c o n trib u te to th e gxy term s
in th e g ain -m atrix eq u atio n s used for calib ratio n . A sim ilar calcu la tio n shows th a t th e
co n trib u tio n s to th e co rrelato r off-diagonal elem ents, e.g., gxQ, are p rim arily a ttrib u te d
to gain differences in th e h o rn ’s E an d H plane pow er response, as well as gain im balances
in th e tw o H E M T am plifiers.
To ad d ress th ese problem s we have co n stru cted a feed horn w ith a high-level of sym ­
metry- betw een E an d H planes, along w ith a low level o f cro ss-p o larizatio n of ~ —30dj9,
see C h a p te r 4.
m easu red ( |5 |J ^
T h e O M T also has a h igh degree of E /H p o la rizatio n p o rt isolation;
|2
~ —30d B ) . T h o u g h its cross-polarization has n o t been m easured, it
2A lthough rarely refereed to as such, cross polarization of th e O M T is a well-defined concept; distinct
from th e m ore fam iliar 6gure-of-m erit: isolation. Conceptually, we can consider th e a c tu al O M T to be
an in-series com bination o f a feed-horn (w ith non-vanishing cross-polarization) a n d a n ideal O M T (i.e .
one w ith no cross-polarization). T hen, th e cross-polarization of th e actual O M T is th a t o f th e series-feed.
T h is definition a sserts th a t no practical m easurem ent of a h o m -O M T com bination can decom pose th e
cross-po larizatio n into com ponents definitively associated w ith e ith e r sub-com ponent.
123
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is believed th a t it is less th a n -30dB. G ain im balances are m inim ized by installing fixed
coaxial a tte n u a tio n in th e IF b and, ju s t p rio r to co rrelatio n , to b e tte r th an 1d B .
From 5.15, 5.16 , 5.17 , we see th a t th e off-diagonal elem ents are uniform ly less th a n
10% of th e ir on-diagonal co u n te rp a rts. T h is in d icates th a t th e re are correlations betw een
th e co rrelato rs an d th e to ta l power d etecto rs. T h is co rrelatio n causes th e correlators to
ex h ib it all of th e un d esirab le tr a its o f th e to ta l pow er d etecto rs: decreased sensitivity and
1 / / behavior as d e m o n stra te d in section s:psd. T hese effects are com ing in a t a low level,
however. F u rth e r investig atio n is needed: m u ltip le calib ratio n s over longer periods of tim e
will provide us w ith e s tim a te s o f th e errors in th e gain m atrices as well as indicating th e
long term sta b ility o f th e calib ratio n s them selves.
5 .4
N o is e A n a ly s e s an d N o ise E q u iv a len t T em p eratu res: N E T s
O nce we know th e calib ratio n betw een voltage an d te m p e ra tu re , by m easuring th e voltage
RM S we in tu r n o b ta in th e te m p e ra tu re RM S. T h e noise eq uivalent tem p eratu re (N ET)
o f th e d e te c to r is th e in stru m e n t noise as m easured in a b an d w id th of V l H z , which can
be converted to th e noise in a one-second in teg ratio n , A T ra/ s , by dividing by \/2 . We
have th a t th e noise in a n a rb itra ry in teg ratio n tim e, r , is:
A T RAfS = N E T / V f .
(5.18)
A gain, th e m odel of th e noise o f th e to ta l pow er d etecto rs differs from th a t of th e
correlato rs. From eq u atio n 5.3 for th e T o tal Pow er d etecto rs:
N E T = T s r s + Tioad^
VAv
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^
19^
while for the correlators, from equation 5.3, we have that:
N E T = y/2-
\J ( T t PO + Tload )((T T P l + Tload)
y /K u
(5.20)
T h e m o st naive technique to o b ta in th e N E T s is sim ply to calcu late th e RMS of the
tim e stre a m in a one-second segm ent a n d convert from voltage to te m p e ra tu re . This ap ­
proach. however, only applies w hen th e signal is a wide-sense s ta tio n a ry G aussian process,
as defined in C h a p te r
1.
O f course, th e tim e stream is not pure G au ssian w hite noise;
th e sp e c tru m will exhibit an excess o f low frequency power ( “ 1 / f noise ” ). In practice,
low -frequency com ponents in th e tim e-stream will d o m inate th e RM S if th ey are not re­
moved. T h is, o f course, is th e m o tiv atio n for th e m odu latio n of th e sig n al by rotation of
th e in stru m e n t. A b e tte r approach is to co m p u te th e power sp ectru m o f th e d ata, and use
it to e stim a te th e RM S. To do this we m u st first m easure th e tran sfer function of the pre­
am plifiers. G(u ) . T h is is accom plished by te rm in atin g th e in p u t to th e pre-am plifier and
recording its PSD . T he equivalent in teg ratio n tim e of th e pre-am plifier circuit is related
to its au d io b an d w id th by:
2&Vaudio
G audio (0)
2 /q G audioi^dlS
(5.21)
T his r is th e equivalent in teg ratio n tim e of th e pre-am plifier for each channel to be
used in th e rad io m eter equation.
G iven a m easurem ent of a ch an n el’s P S D , S ( u ) [V 2 /H z], an d pre-am plifier transfer
function, G ( v ) , we can estim ate th e variance of th e tim e stream as:
a
2
Sjy)G{y)dv
!™ G{v)dv
125
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(5.22)
In practice, a given c h a n n e l’s PSD is only “w h ite” (i.e. flat) above a certain frequency,
^kneei and so th e lower lim it in th e n u m erato r’s in teg ral is replaced by th is value. T h e
am p litu d e o f th e PSD in th e “flat” region provides us w ith an e stim a te of th e tim e s tre a m ’s
variance w ith o u t
1/
/ noise. M aking use of th e W ien er-K h in tch in T heorem from C h a p te r
4, we can estim ate th e a u to c o rre la tio n function (A C F) o f th e tim e-stream from th e inverse
Fourier tran sfo rm of its P S D , an d evaluate it a t zero-lag. O f course this e stim ate is n o t
ind ep en d en t of th e PSD e stim a te since th e a u to co rrelatio n function, R ( r ) , of th e t ime
stream , y ( t ), is given by:
1 r 1'*
R ( r ) = lim — /
y ( t' ) y (t ' + r ) d t'
(5.23)
T —oo i J - T / 2
w hich, w hen r =
0
i
, becom es:
r
^
T J-T/2
^
i
o
>
-
a
r y
Jo G ( v ) d u
.
(5.24,
A lthough these two m e th o d s are not m a th em a tic ally in d ep en d en t, it is inform ative
to ev alu ate b o th in p ractice. E ach has its ow n region o f applicability, an d each can be
co m p u ted independently a n d th e n com pared to th e F o u rier tran sfo rm estim ate given by
the o th er. T h e ACF a t zero-lag of th e raw d a ta suffers from co n tam in atio n due to the
low -frequency drifts a p p a re n t in th e PSD, an d it is th erefo re m ore convienient to calculate
th e PS D an d then invert it to get th e ACF. O n th e o th e r h an d , correlations induced in th e
tim e s tre a m by in stru m e n ta l effects (e.g., th e low-pass filter in teg ratio n stage) are m ost
clearly visible in the A C F.
For b o th th e A CF a n d P S D estim ations o f th e noise it is im p ractical to tran sfo rm large
sections of d a ta a t a single tim e. A dditionally, this m e th o d w ould result in low-precision
126
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estim ates o f th e desired functions in ex ch an g e for high-resolution (in e ith e r frequency or
te m p o ral bins). In stead , we choose to b re a k th e d a ta sets into several sm aller sections
an d su b seq u en tly average th e transform s. T h is ap p ro ach also yields a n e s tim a te o f the
d isp ersio n o f th e A C F o r PSD a b o u t its m e an value.
N o is e A fte r L ock -In D e te c tio n
As discussed earlier, afte r m ultiplication th e signals are detected via a lock-in circuit
cen tered on th e A nalog Devices 630 M o d u lato r/D e m o d u la to r. T h e A D630 m u ltip lies the
signal o u t o f th e m ultipliers by a reference w aveform , which in ou r case is a square-w ave
g en era ted by a ded icated crystal o scillator.
T h e oscillator g enerates th e phase-sw itch
waveform as well as th e in-phase an d q u a d ra tu re reference phases for th e AD 630s.
We n o te th a t o u r lock-in d etectio n a fte r m u ltip licatio n is essentially an im p lem en ta­
tio n o f th e Dicke radiom eter, sw itched a t au d io frequencies i/switch = 1 K H z. Follow ing
m u ltip licatio n , th e p ro d u ct of th e m u ltip lier o u tp u t a n d th e reference w aveform is inte­
g ra te d for a n a m o u n t of tim e 3 ^ —tll.Y—— • O u r
after th e lock-in is effectively the
difference betw een th e m ultiplier signal a t tim es se p arated by, roughly, th e p e rio d of the
phase sw itch.
T stV i t c h
|ti — tj | >
v i p ~ 0.1 nsec. So, since T ^ t c h 3> 0.1 nsec th e PSD o f th e lock-in d e tecto rs
1/A
— 1 msec. T h e m u ltip lier o u tp u t a t tim es
£,
an d
tj,
is u n c o rrelated if
will be P S D o f th e noise waveform p ro d u ced by o u r d etecto rs when view ing a n unpolarized
source will be
2
tim es larger th a n th e co rresp o n d in g value w ithout th e lock-in d etecto rs.
T h e signal-to-noise ra tio will th u s be red u ced by a facto r o f y/2, which is in a d d itio n to the
facto r of y/2 a lre ad y included in th e ra d io m e te r eq u atio n for th e co rrelatio n rad io m eter.
R elativ e to a n ideal to ta l power rad io m eter w ith a n N E T of a [mK H z~5], o u r co rrelato r
channels have po st-d etectio n N E T ’s o f 2cr. F or com parison w ith pred ictio n s, as in table
127
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5.3, we alw ays quote noise e stim a te s after lock-in detection.
We n ote th a t th e b ehavior o f th e noise should be in d ep en d en t o f th e phase of th e refer­
ence w aveform supplied to th e lock-in. To illu strate this we co n sid er figure 5.7 where the
in-phase a n d q u a d ra tu re phase low -frequency power s p e c tra o f co rrelato r J1 are shown.
We have developed a p hase-sw itch driver circu it which also p ro d u ces th e in-phase and
q u a d ra tu re phase reference w aveform s. For added flexibility we are able to change the
phase o f th e reference w aveform s w ith respect to the physical chop waveform by discrete
step s of 0.18°. The q u a d ra tu re p h ase detecto rs provide us w ith ex tre m ely powerful moni­
tors of th e noise of th e co rrelato r channels.
5 .5
R e s u lts o f N o is e A n a ly se s
We begin by referring th e read er to figure 5.6 which shows th e pow er s p e c tra of all three
in-phase co rrelato r channels a n d b o th to ta l pow er detectors. It is clear from th e spectra
th a t th e correlators are far m ore sensitive th a n th e to ta l pow er d etecto rs. T he dram atic
1/
/ rise in th e to tal pow er d e tecto rs is ab sen t in th e co rrelato r ch an n els, an d allows us to
slowly m o d u late th e in stru m en t o u tp u t by ro ta tio n of th e ra d io m e te r a t 30 mHz, rath e r
th a n a t several Hz.
N ext, we exam ine th e b ehavior o f th e co rrelato r power s p e c tra a t low frequencies. In
figure
5 .7
we display th e pow er s p e c tra a t frequencies com parable to th e ro ta tio n frequency
of th e in stru m en t. B o th th e in-phase an d q u ad ratu re-p h ase co m p o n en ts of th e lock-in
d etecto rs are shown, an d th e agreem ent a t frequencies g reater th a n th e ro tatio n frequency
allows us to use these channels as noise m onitors. T he presence o f signals a t th e rotation
frequency a n d its harm onics is ev id en t in th e in-phase d etecto rs, su g g estin g th a t th ere are
128
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Low F r e q u e n c y P o w e r S p e c t r a l D ensity of All Data C h a n n e l s
0.100
o
a>
0.010
0.00 1
2
4
6
F r e q u e n c y [Hz]
8
10
F i g u r e 5 .6 : T h e p o w er s p e c tr a o f all five P O L A R d a t a ch a n n els w hile view ing th e s k y (TAnt =
15K ) a rc sh o w n . T h e c o rre la to r s p e c tr a a re p lo tte d as solid lines a n d th e to ta l p o w er a re d o tte d .
S ev eral fe a tu re s a re e v id e n t fro m th e se p lo ts. T h e 1 / / b e h a v io r o f th e to ta l p o w er d e te c to rs , a n d
th e low -p ass filte rin g o f th e a n ti-a lia s in g filters on all c h a n n e ls ab o v e 5 Hz a re e v id e n t. T h e C T I
c o ld h e a d e x p a n s io n /c o m p re s s io n cycle is a t 1.2 Hz, a n d th is fe a tu re a n d its h a rm o n ic s a re clearly
v isib le in th e to ta l p o w er c h a n n e ls. F rom h igh-noise (to p ) to low -noise (b o tto m ) th e c h an n e ls a re
T PO , T P 1 , J 3 , J 2 , J l .
129
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C o r r e l a t o r J1 Low F r e q u e n c y P o w e r S p e c t r a l D ensity
0 .0 1 0
o
a>
">
i
i—
<3
0.001
0 .0 2
0 .0 4
F r e q u e n c y [Hz]
0 .0 6
F i g u r e 5 .7 : T h e Io w -frcq u en cv p o w e r s p e c tr a o f th e low est noise c o r r e la to r ( J l ) , in -p h a se (solid)
a n d q u a d r a tu r c - p h a s e lock-in d e te c to r s (d a s h e d ), w hile v iew ing th e sk y (Ty\nt = 15K ). T h e q u a d ra ­
tu r e p h a se d e te c to rs b eh a v e a s e ffec tiv e no ise m o n ito rs. T h e r o ta tio n fre q u e n c y is 0.03 Hz, a n d
so m e sy n c h ro n o u s sig n a l m o d u la tio n is e v id e n t in th e in -p h a se d e te c to r a t th is fre q u en c y a n d its
h a rm o n ic s. A b se n t in th e q u a d r a tu r e - p h a s e ch a n n els is th e 1 / / n o ise rise a t freq u en cies below
0.005 Hz.
correlated signals in th e raw tim e-stream s which are being m o d u la te d by th e ro tatio n of
th e in stru m en t. T h e sta b ility o f th ese offsets over a single ro ta tio n o f th e instrum ent is
crucial to th e recovery of th e Stokes param eters.
O verall, th e perform ance o f P O L A R is q u ite satisfactory.
T able 5.3 com pares the
m easured N E T values an d w ith predictions based upo n th a t c o rre la to r’s system noise
and bandw idth. T h e lo n g er-term perform ance o f P O L A R is fu rth e r elab o rated upon in
C h a p te r 10.
From table 5.3 several p h en o m en a are observed. F irst we see th a t th e noise estim ates
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T ab le 5.3: N E T E stim a te s from P S D an d RM S C o m p ared W ith P red icted N E T
C hannel
N E T fro m P S D [ m K i/je c ]
NETT fro m R M S [m K -ysec]
3.6
6.5
7.1
4.4
7.5
13.5
Jl
J2
J3
P r e d ic te d fro m T,v A: A i/ [m K v 'a e c j
2.3
2.9
4.3
based on th e RM S are sy stem atically higher th a n those o b ta in e d from the PSD estim ates.
T his is to be exp ected : th e RM S suffers m ore-severely from 1 / / contam ination, w hich is
difficult to rem ove from th e tim e-strea m directly. T h e P S D estim ates are ta k e n as th e
m ost a c c u ra te estim a te s of th e N E T because th e
1/
/ b eh av io r can be easily d istinguished
an d s u b tra c te d .
N ext we see th a t th e co rrelato rs differ su b sta n tia lly from one-another. T h is is p rim arily
a ttr ib u te d to th e p h ase o p tim iz atio n discussed earlier. T h e variable phase-shifter, shown
in figure 4.1, has been o p tim ized for channel J l , w hich has th e effect of increasing i t ’s
c alib ratio n in [V /K ] relativ e to th e o th e r two co rrelato rs. A t p resen t it is n o t possible to
ind iv id u ally tu n e th e phase of each frequency b an d p rio r to correlation. T his featu re will
be im plem en ted p rio r to P O L A R ’s observing ru n th is w inter. We expect to th e n o b ta in
th e sam e sen sitiv ity for all th re e co rrelato rs allow ing us to o b ta in a co-added se n sitiv ity
for all co rrelato rs o f ~ 3 m K v /s e c /\/3 ~ 1.7m K y/sec.
131
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Chapter 6
Systematic Effects
B ecause th e an ticip ated p o larizatio n signal is a factor o f ~ 10 tim es sm aller th a n th e
te m p e ra tu re anisotropy cu rren tly being detected, a th o ro u g h u n d erstan d in g of system atic
erro rs is crucial. P o larizatio n experim ents have several ad v an tag es, however, th a t prom ise
to m ake th is effort possible. F irst, as shown in C h ap ter 7, th e atm o sp h ere is known to be
polarized only a t a very low level, far below th e expected level o f CM B polarization. A ddi­
tionally, P O L A R m easures th e polarization of each pixel in a m an n er which is (nearly) in­
d ep en d e n t o f neighboring pixels. It does not require com parison o f pixels through different
airm asses, a n d a t different tim es. In anisotropy observations, beam switching often adds
noise an d ad d itio n al chop -d ep en d en t signals. P otentially, atm o sp h eric effects will have a
sm aller co n trib u tio n to th is ty p e o f experim ent th a n to g ro u n d -b ased CMB anisotropy
ex p erim en ts an d will allow longer observation tim es th a n have been possible in th e p ast.
L o n g -term observations are key to u n d erstanding an d rem oving system atic effects [8 6 ];
[87]. M any spurious in stru m e n ta l effects have been isolated from astrophysical effects by
lon g -term in teg ratio n te sts w ith th e horn an ten n a replaced by a cold term ination.
T h e m o st tro u b lin g a sp ect o f these effects is th a t th e y m ay n o t be stable in tim e. For
132
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th e co rrelato r channels, th e most pernicious co n trib u tio n arises from gain flu ctuations in
th e H EM T am plifiers and in the conversion efficiency an d phase stab ility o f th e heterodyne
stage. T h e conversion efficiency of th e m ixers is d ep en d en t on th e G unn O scilator power
which fluctuates ju s t like an amplifier, th u s intro d u cin g gain fluctuation, w hich can be
m isin terp reted as signals if some com ponent o f th e variation is com m ensurate w ith th e
m od u latio n frequency of th e instrum ent. T h e phase stab ility of th e oscilator is equally
troublesom e [59], since phase fluctuations betw een th e two arm s of th e rad io m eter reduce
its effective ban d w id th . T here are several sta n d a rd m ethods to improve th e sta b ility of the
h eterodyne stag e o f th e receiver, including phase m odu latio n at frequencies o f ~
1
KHz,
and phase-locked loops to stabilize th e LO. T h e la tte r is quite com m on in conventional
radiotelescopes, th o u g h it is not incorporated in th e P O L A R K &band system . T h e form er
technique is perform ed, however, as described in C h ap ter 4.
In Table
6
we list som e im p o rtan t sy stem atic effects encountered in previous polariza­
tion m easurem ents and sum m arize th e solution ad o p ted by POLAR.
Effect
M echanical S train
M agnetic C oupling
M icrophonics
EM I and R F I
T h erm al V ariations
Sidelobe P ickup
Table 6.1: E xpected S ystem atic Effects
C o ntrol M ethod
O rigin
In stru m en t R o tatio n
Z en ith Scan
Minimal
Ferrite
Components
(Isolators
Only)
R otatio n in B earth
Isolation
M echanical V ibration
S h ie ld /F ilte r
Local Sources
D iu rn al/E n v iro n m en t
T em p C ontrol
Low Sidelobe Antenna and Ground Screens
S u n /M o o n /E a rth
G r a v ita tio n a l effects
A problem w ith any rad io m eter th a t m u st move in th e e a rth ’s gravitational field is positiond ep en d en t stress a n d stra in on waveguide jo in ts, etc. In th e experim ents o f L ubin and
S m oot, for exam ple, observations off-zenith produced polarized offsets w hich were 1 to 10
133
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tim es la rg e r th a n th e resulting u p p er-lim it. P O L A R a tte m p ts to m inim ize these problem s
by s ta rin g a t the zenith, so th a t to first order no g rav ita tio n a l to rq u es on waveguide
c o m p o n en ts are present. T he ro ta tio n speed is slow, ~ 2 rp m , a n d c o n sta n t so accelerations
on s to p p in g an d sta rtin g ro ta tio n are non-existent.
M a g n e tic F ield effects
A p a rtic u la r concern is th e coupling o f th e E a r th ’s m agnetic field to th e radiom eter. T h e
CO B E D M R had ferrite Dicke sw itches which produced a sp u rio u s signal a t th e ~ 0.1 mK
level (K o g u t e t al. 1996b). P O L A R ’s o nly ferrite com ponents are its isolaters. However,
o th e r co m p o n en ts such as am plifiers, etc., m ay have a low-level m ag n etic field dependence.
M o d u latio n o f these effects can be m inim ized by m ain tain in g a c o n sta n t o rientation of
ro ta tio n axis w ith respect to th e E a r th ’s field. T h e m a g n itu d e o f th is effect has been
e stim a te d by generating a ~ 10 G au ss DC field, an d was u n d e te c ta b le for a one hour
in teg ratio n . F u tu re AC field te sts are planned.
M ic r o p h o n ic s
T h e effects o f vibrations th a t o ccu r d u rin g ro tatio n are reduced by use o f H EM TS rath e r
th a n high im pedance devices such as bolom eters, and by th e fact th a t we take d a ta only
w hile th e in stru m en t is ro ta tin g a t a c o n sta n t rate, using a sm o o th -d riv in g A C m otor. We
utilize num ero u s v ibration isolation techniques, as described in C h a p te r 4, to decouple the
p rim a ry source of m echanical v ib ratio n s: th e cryocooler. W aveguide ju n c tio n s can be a
p a rtic u la r concern as vib ratio n a t th e interface between w aveguide flanges can cause m od­
u la tio n o f th e ju n ctio n im pedance, w hich will be present in th e R F signals. To minimize
th is effect we have m inim ized th e n u m b er o f joints by c o n stru ctin g cu sto m m anufactured
134
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waveguide sections, and stiffened all waveguides and support structures.
EM I and R F I
T hese effects can be controlled by Faraday shielding th e in stru m e n t and by filtering electri­
cal lines in to an d o u t o f th e dew ar. A dditionally, we em p lo y tw o-stages o f pow er reg u latio n
to all co m p o n en ts, b o th inside the dew ar as well as in th e w arm -IF section. R F sources
th a t occur in th e rad io m ete r R F band or IF b an d are becom ing increasingly troublesom e.
O f p a rtic u la r concern in th e future will be com m u n icatio n s satellites o p eratin g in th e b a n d s
of in terest. N o R F I from terestrial sources (such as a irp o rt RA DA R) has been d e te c te d
in th e d a ta , a ttr ib u ta b le prim arily to the relative iso latio n o f th e Pine Bluff O bservatory.
G iven th e sm all sky coverage an d low point-source se n sitiv ity of the P O L A R 7° e x p e r­
im ent. it is unlikely th a t there is contam ination by a n y know n satellite com m unication
system .
T e m p e r a tu r e D e p e n d e n t E ffects
T em p e ra tu re variatio n s in th e radiom eter can be m itig a te d by active te m p e ra tu re co n tro l
an d by sh ield in g th e in stru m en t from the Sun. T h e la tte r fu n ctio n is n atu ra lly perform ed
by th e g ro u n d shields so t h a t th e an tenna an d receiver a re com pletely shielded. T he form er
fuction is p rovided by a tw o-stage tem p eratu re co n tro l sy stem . T he first, coarse level of
control is p rovided by a 300 W a tt PID controled h e a tin g elem en t on th e m ain electronics
box. A finer level o f co n tro l is provided by a second P ID con tro led , therm o-electric cooler
(P eltier C ycle) which provides 10 W atts o f h e a tin g /c o o lin g power.
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S id elo b e P ick u p an d Sp illover
T h e p o larim eter m u st be able to reject o r d iscrim inate ag ain st em ission from th e Sun,
M oon, an d E a rth , which ap p ear only in th e sidelobes of th e beam . N one o f these sources
are expected to be significantly polarized, b u t asy m m etry in th e a n te n n a response to the
two linear polarizatio n s will create sp u rio u s signals.
T h is effect is responsible for the
false-detection claim ed by Nanos[13], w ho la te r a ttrib u te d his d etectio n to pick-up from
a elevator tow er in close proxim ity to his observing site. P O L A R requires th a t th e total
power from these sources lie below
1
fj.K, n ecessitatin g 75 an d 63 dB sidelobe rejection for
th e Sun an d M oon, respectively. A ssum ing 30 dB rejection from th e g ro u n d screen, this
level of rejection can be achieved w ith th e co rru g ated h orn an te n n a if d a ta are rejected
w hen these sources lie closer to th e zen ith th a n 50° and 30° respectively. B inning of the
d a ta in S un-centered or M oon-centered co o rd in ates will allow us to uncover correlations
betw een th e p o sition of these ob jects a n d th e response of th e p o larim eter. Spillover from
th e ground an d surro u n d in g stru ctu re s is m inim ized by inco p o ratin g tw-o-levels of ground
screening. T h e first ground screen ro ta te s w ith th e polarim eter an d provides ~ 30 dB
of suppression. T his ground screen is covered w ith eccosorb foam sh eets which absorb
atm ospheric rad ia tio n , rath e r th a n p olarizing it v ia reflection off o f a m e tal surface. T he
o u te r gro u n d screen is fixed w ith respect to th e p o larim eter an d provides an additional
30 dB suppression. T his screen reflects th e sidelobe response which diffracts over th e first
screen, to th e 10K sky rath e r th a n th e 300K ground. T h e com bination o f two levels of
gro u n d screening, along w ith th e co rru g ated feedhorn’s intrinsically low sidelobe level,
reduces th e co n trib u tio n from off-axis ra d ia tio n by ~ 100 dB.
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Chapter 7
Large Angular Scale Foregrounds
in the
K
a
band
A fu n d am en tal question for any a tte m p t to m easure th e p o larizatio n of the CM B is
w h eth er th e cosmological signal can be distinguished from polarized foreground sources.
W hile astrophysical (non-cosm ological) sources of polarized ra d ia tio n are of interest for
o th e r fields, th e m easurem ent of CM B polarization is o u r m ain objective, so these sources
are spurious effects.
T hese foreground sources all have sp ectra th a t are d istinct from
th a t of th e CM B, an d in principle can be distinguished from it by m ulti-frequency m ea­
surem en ts. T his technique has been employed for observations o f CM B anisotropy [8 8 ].
However, polarized foreground sp e c tra have not been stu d ied as extensively. To estim ate
th e in ten sity and sp e c tra of these foreground sources, we rely on th eo retical predictions and
ex tra p o latio n s from m easurem ents a t different frequencies of th e a n te n n a tem p eratures of
these foregrounds. Here we sum m arize th e properties o f atm ospheric and astrophysical
(th o u g h non-cosmological) foreground sources.
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Microwave P o la r iz a tio n S ig n a l S o u r c e s
1 0 0 0 .0 0
1 0 0 .0 0
a.
c
ua>
a
u
a
E
o
E-
a
re m .
10%
o ta l
1 0 .0 0
D u st 100%
1.00
c
c
v
75%
c
CMBR
o la r iz e d 10'(
<
0.10
0.01
100
10
1000
F r e q u e n c y in GHz
F ig u re 7.1:
Polarized Foreground Spectra a t Millimeter Wavelengths. Spectra of expected
polarized radiation sources at high galactic latitudes, are shown for a 7° beam. A 3 fiK. polarized
CMB signal is shown, corresponding to 10% of the 10- 5 CMB anisotropy. At frequencies lower
than 90 GHz the polarization signal is dominated by galactic synchrotron emission (up to 75%
polarized, as shown). Galactic bremsstrahlung radiation is not polarized in direct emission, but can
be up to 10% polarized (as shown) after Thomson scattering. Galactic dust is shown conservatively
with 1 0 0 % polarization.
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7 .1
S y n ch ro tro n E m issio n
Diffuse g alactic sy nchrotron ra d ia tio n arises from ionized regions o f o u r galax y th a t posses
m ag n etic fields. C onsider th e re la tiv istic m otion o f a charged p a rtic le of charge q and mass
777.
We have th a t:
^-(■ymv) = - v x B
etc
c
(7.1)
- ^ ( 7 T7ic2) = q v E = 0.
(7.2)
T h e second eq u atio n im plies t h a t
7
= co n stan t, so we have: nri'y^ft = | v x B . E xpanding
this eq u atio n into term s parallel a n d p erp en d icu lar to th e m a g n etic field, we find th a t
U|| ^ c o n s ta n t, and
frequency u B =
= |v j_ x B which th e equation o f unifo rm circu lar m otion w ith
Since th e a n g u la r acceleration is a± = u>b v ± , we find th a t th e to tal
ra d ia te d pow er is:
2 q AB 2 2 2
„
“
3c?m2^ V±
'
For an isotropic d istrib u tio n o f p itch angles a , we find P =
w here cr? is the
T hom son cross-section an d U b is th e energy d en sity of th e m a g n etic field: U b = B 2/ 8 tt.
Following [89], we can o b ta in a q u a lita tiv e description o f th e s p e c tru m of synchrotron
rad ia tio n . T h e individual electro n s will p ro d u ce a brief pulse o f ra d ia tio n once p er revo­
lution. T hese pulses have a b ro ad sp ectru m , w hose square is p ro p o rtio n a l to th e power
sp e c tru m o f th e rad iatio n . If we assu m e th a t th e charges have a n energy sp ectru m of:
N ( E ) d E = C ( a ) E ~ pd E , w here th e p article d istrib u tio n index, p, is re la te d to th e spectral
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index, s, by: s =
^
we o b ta in th e following spectru m :
\fZqz C B s i n a
27T77ic2(p 4-1)
/p
19W
^4
12'
1 V
mcuJ N ~ ( p —* ) / 2
1 2 ' '3<7-Bsina^
?
M
an d polarization:
n = * ± I.
P+ 3
T h e a n ten n a te m p e ra tu re o f sy n ch ro tro n em ission obeys a power law:
^ s y n c h ro tro n ( ^ )
^ i
w here (3 is referred to as th e synchrotron spectral index. T he p olarization level II of
sy n ch ro tro n rad iatio n is rela ted to th e sp ectra l index [90]:
M + 3
3/3 + 1
F arad ay ro tatio n an d non-uniform m agnetic fields will reduce th e level of polarization
given by this equation. T h e rad ia tio n is linearly polarized between app ro x im ately 10%
and 75%, depending on galactic coordinates. Below 80 G H z th e polarized synchrotron
em ission dom inates all sources, including th e C M B if it is polarized a t th e 1 x 10 - 6 level,
as show n in Figure 7.1. In figure 7.2 we estim ate syn ch ro tro n emission by e x tra p o la tin g
th e Brouw & S p o elstra [91] m easu rem en t a t 1411 M Hz to m illim eter w avelengths w ith
th e m odified power-law sp e c tru m used to fit th e C O BE'D M R. d a ta [92]. For o ur m odeling
purposes we choose /3 = —2.9 [93] , [92], an d n = 75% [91].
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O.OOOOOK
0 . 0 0 0 2 4 1926K
RA
F igure 7.2: Polarized Synchrotron Emission at 1411 MHz ([91]) extrapolated to 31 GHz using
a power law spectrum. From right to left the range of RA is Oh < R A < 24h, and from top to
bottom the range of Declination is 20° < 6 < 60°. POLAR’s observing strip appears in the middle
of the figure.
7.2
P o la r iz a tio n P r o d u c e d b y In terstella r D u s t
T h ere are tw o m echanism s by w hich in terstellar d u st m ay produce a p o larized signal. The
m ore fam iliar therm al, vib ratio n al, em ission is trea ted first. A fterw ards, we sum m arize
recent developm ents in th e m odeling o f polarizatio n produced by ro ta tin g d u s t grains.
E x tin ctio n of unpolarized rad ia tio n from dielectric cylinders will p ro d u ce polarization
because th e cylinder’s scatterin g efficiency (or extinction cross-section) is different along
the E an d H planes of th e rad iatio n . T h e reason for this rests in th e an iso tro p ic n a tu re of
the g ra in ’s electric an d m agnetic m om ents. For cylinders w ith radii m uch sm aller th a n the
w avelength of the scatterin g (as in th e case o f sim ple dipole a n ten n a), cu rren ts will flow
along th e cylinder axis. D epending on th e real an d im aginary co m p o n en ts o f th e g rain ’s
com plex index of refraction, m .
Follow ing Spitzer [94], we define x =
w here a is th e cylinder rad iu s. W e find th a t
the polarizing effectiveness of an infinite cylinder w ith a refractive index o f m = 1.331,
r =
1
var‘es from r — 1 (*-e., 100% polarization) a t i =
1,
t o r ~ 0.4 a t x = 5.
‘ For exam ple, for ice m = 1.33
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7 .2.1
T herm al D u st E m ission
T h e po larizatio n level o f in te rste lla r d u st is n ot well know n. D epending on th e shape an d
th e alignm ent o f d u st p a rtic le s, em ission from d u st particles m ay be highly polarized w ith
th e po larizatio n I ~ 0.07 , u su ally aligned parallel to th e g alactic plane [94].
A t low g alactic la titu d e s th e rm al em ission from d u st p article s dom inates th e n ear in­
frared sp ectru m . U sing th e d u s t sp ectru m m easured by th e C O B E FIRAS [95] norm alized
to th e IRAS 100 m icron m a p , we find th a t high galactic la titu d e d u st em ission is negligible
below 80 GHz, even w hen assum ed to be 100% polarized. W e use the two te m p e ra tu re
d u st model [95] :
T^ “
K 2P it ( 9 0 0 ^ ib ) 2 !B‘'(2°'4K) + 6-7A(4.77K)].
(7.3)
A t high galactic la titu d e s Tdust ~ 10pK a t 200 G Hz ([93]; [92]).
7 .2.2
D iffuse E m issio n from R o ta tin g In terstellar D u st G rains
R ecen tly there has been a n effort to explain th e “anom alous” correlation betw een infrared
d u st em ission a t 100^m a n d m easurem ents m ade a t 30 - 50 G Hz. T he em ission does
n o t ap p ea r to be co n sisten t w ith H a em ission [96] (to be discussed in th e following sec­
tio n ). Instead, an ex p la n a tio n in term s o f spinning in terstellar d u s t has been proposed by
D rain e and L azarian [97]. T h e em ission arises from d u st g rain s com posed o f 100-1000 p a r­
ticles which are “sp u n -u p ” by incident starlig h t due to th e ir non-vanishing electric dipole
m om ents. T h e precise s p e c tru m o f e m itte d rad ia tio n d ep en d s sensitively on th e m odels
used for the com ponents o f th e In terste lla r M edium (ISM ) an d for th e grains them selves.
G enerically, however, th e ro ta tio n a l em ission is expected to d o m in ate th a t from th e rm al
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d u st em ission below 70 G H z [97].
T h e m ain q u estio n for th e P O L A R ex p erim en t is th a t th e em ission m ay be polarized.
T h e d u st grains a re also ex p ec ted to have a non-vanishing m agnetic d ip o le m om ent, which
will ten d to alig n w ith th e m ag n etic field in th e ISM . L azarian an d D rain e [98] find th a t
th e p o larizatio n p ro d u c e d by such grains is expected to be in th e ran g e 0.1-10%. This
deleterious effect is ex p ec ted to be reduced slightly, however, as th e alignm ent of th e
m agnetic field is u n c o rre la te d over large regions o f th e ISM. We have n o t considered this
foreground in an y o f o u r analyses, th o u g h in th e fu tu re we m ay in v estig ate its effect on our
d a ta set. As u su al, each foreground co n tam in a n t necessitates a w ider sp e c tra l coverage in
o rd er to s u b tra c t.
7 .3
B r e m s str a h lu n g E m issio n
C o n tinuum em ission from electro n s enco u n terin g p rotons which does n o t resu lt in recom bi­
n atio n is known as free-free em ission. T h e rad iatio n produced is th e fam iliar b rem sstrahlung
em ission found w herever th e re are copious q u an tities of free electrons an d protons. O ne
ty p e of ionized region are th e H II regions, usually identified in o p tic a l wavelengths by
H Q light.
H II regions are ty p ically “S trom gren Spheres” - spheres o f ionized plasm a
surro u n d in g en erg etic sta rs.
B rem m strah lu n g sc a tte rin g of unpolarized rad ia tio n does n o t p ro d u ce polarized rad ia­
tio n [89]. However, b re m sstra h lu n g em ission will be polarized via T h o m so n scatterin g by
th e electrons in th e H II region itself. T h e rescattered rad iatio n will b e polarized tangentially to th e edges o f th e cloud, a t a m axim um level o f approxim ately
th ick cloud.
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10%
for an optically
As a toy model, we will co n sid er an incident plane wave em itted from th e cloud center
to a sm all scatterin g region a t th e rad iu s o f th e cloud. From C h a p te r 2 we ex p ect th a t the
p o larizatio n will be generated by th e plane wave, so all we need to know to estim ate the
p o larizatio n produced by this sc a tte rin g is th e em ission of th e th e rm al brem sstrahlung.
Follow ing Spitzer [94], we e stim a te th e free-free em ission factor for electro n -p ro to n scat­
te rin g as :
j u = 5.44 x
c m ' V ' s r - ^ - 1,
(7.4)
Ti
w'here n , is th e pro to n density, n e is th e electron density, an d g / f is th e G au n t factor,
w hich is a slowly varying function o f frequency, w hich in the radio region is well-modeled
by:
T ’3 /2
9 f f = 9 .7 7 (1 + 0 .1 3 0 lo g —— ).
(7.5)
To o b ta in the to ta l rad ia ted energy, e, we in teg rate equation 7.4 over A~du and obtain:
e = 1.4 x
1 0 _ 2 7 n en ,T ’2
(p y /)e rg c m - 3 s - 1s r _ 1 H z- 1 ,
(7 6 )
w here th e m ean G au n t factor, ( g/ f ) , has a w eak dependence on te m p e ra tu re , varying
betw een
1 .1
and 1.4 over reasonable H II te m p e ra tu re ranges.
YVe now need th e radius of th e S tro m g ren Sphere surrounding th e s ta r (w hich produces
th e free-electrons). T his radius is a function of th e s ta r ’s spectral ty p e. F rom Spitzer we
have ta b le 7.3.
D ue to th e ro tatio n al sy m m etry a b o u t th e line of sight of th e spheres we are considering
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T able 7.1: P ro p erties o f H II Regions
S p ectral T y p e Te f f [ ° K]
R /R s u n
47,000
05
13.8
09
34,500
7.9
BO
22,600
6 .2
we ex p ect th a t th e intrinsic p o larizatio n p roduced by these clouds will be zero. However
sc a tte rin g o f th e C M B q u ad ru p o le (w hich is estim ated to be th a t m easu red by th e C O B E
D M R ) by th e free-electrons in th e H II region will p roduce po larizatio n w ith a m agnitude
of
Q ~
\J -^ Q r m s ~ l — 1 0 f i K
(7.7)
T h e p o larizatio n produced by th is ty p e of scatterin g will peak a t an g u lar scales com ­
parab le to th e angle su b ten d ed by th e H II regions, an d th u s for P O L A R ’s 7° beam they
are negligible. T h e locations an d em issivities o f galactic H II regions a re n o t well known,
b u t B e n n e tt e t al. ([92]) m odel th e b rem sstrah lu n g em ission in th e galax y by s u b tra c t­
ing a sy n ch ro tro n m odel from m icrowave sky m aps. In any case th e p o larizatio n in the
re scattered b rem sstrah lu n g em ission will be a t least an o rd er of m ag n itu d e sm aller th a n
the p olarized sy n ch ro tro n signal a t frequencies g reater th a n 10 GHz. W e q u o te th e result
of B en n e tt et al. [92] th a t
Tbremsstrahlung OC IS
,
(7-8)
w ith to ta l in ten sity ~ 40/xK a t 30 G H z ([93]; [92]).
D u st g rain s are know n to be p resen t in H II regions. Since we know from th e previous
section th a t in terstellar d u st polarizes starlig h t, we also ex p ect th a t th e d u st associated
w ith H II regions m ay produce p o larizatio n in add itio n th e po larizatio n we have considered
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—0 .4 9 0 1 6 9 m K
1.9861 4m K
F re e T ree E m issio n in 4 0 P e g . S trip
a
01
o
RA
F ig u re 7.3: COBE Estimated Free-Free Antenna Temperature Map Centered on Declination
S = 43°. From right to left the range of RA is Oh < R A < 24h, and from top to bottom the range
of Declination is 20° < 6 < 60°. POLAR’s observing strip appears in the middle of the figure.
by T h o m so n scatterin g . W e have n o t e stim ated th is effect, however.
7 .4
E x tr a g a la c tic P o in t S o u rces
T h e d o m in an t ra d ia tio n m echanism for ex tra g alac tic radio sources is sy n ch ro tro n em ission
[99]. T h ese sources have a n et p o larizatio n o f < 20%. C alculations m ad e by Franceschini e t
al. [100] of th e te m p e ra tu re flu ctu atio n s in m easurem ents o f an iso tro p y o f th e CM B arising
from unresolved, ran d o m ly d is trib u te d sources show th a t th ey c o n trib u te negligibly a t 30
G H z to a 7° an iso tro p y ex p erim en t. If th e o rien tatio n s of th e p o la rizatio n vectors o f these
sources axe u n co rrelated over 7° regions, we would also ex p ect a negligible co n trib u tio n to
th e signal observed P O L A R .
7 .5
A tm o s p h e r ic C o n ta m in a tio n
A lth o u g h s tric tly n o t a foreground, em ission from th e atm o sp h ere provides an equally
form id ab le o b stacle to th e d etectio n o f p o larizatio n of th e C M B . T h e a n te n n a te m p e ra tu re
of th e e a r th ’s atm o sp h ere betw een 10 a n d 60 G H z is d o m in ated by a n em ission feature
a t ~ 22 GHz caused by atm o sp h eric w ater vapor, and a series o f em ission lines a t ~ 60
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GHz due to m olecular oxygen. W e now investigate w h eth er o r n o t th is em ission is linearly
polarized.
7.5.1
P olarized E m issio n from th e E a rth ’s A tm osp h ere in th e K & B an d
In th e absence o f e x te rn a l fields, n eith er of these atm ospheric com ponents is known to
em it polarized ra d ia tio n in th e frequency range o f in terest. However, Zeem an sp littin g of
the energy levels o f atm o sp h eric molecules by th e m agnetic field o f th e e a rth can produce
polarized em ission.
T h e valence band of w ater is com pletely full, an d th us, does not
exhibit Zeem an sp littin g . H owever, th e O 2 m olecule has a non-zero m agnetic m om ent due
to its two u n p aired valence electrons which in teract w ith th e E a r th ’s m agnetic field. We
here discuss polarized em ission from m esospheric oxygen, and show th a t it is negligible in
com parison w ith th e e x p e c te d polarized in ten sity o f th e CM B.
T he Zeem an effect b reak s th e energy degeneracy of th e tw o unp aired valence elec­
trons of m olecular oxygen. T h e to ta l angular m om entum q u an tu m num ber of th e oxygen
molecule is j =
1,
w hich im plies th a t th e oxygen m olecule’s ro ta tio n a l sp ectra l lines are
Zeeman sp lit into 2 j -(-1 = 3 d istin ct lines. D ipole rad iatio n selection rules for tra n si­
tions betw een these levels p e rm it tran sitio n s as long as th e change in m agnetic q u an tu m
num ber, m , is: A m = 0, ± 1 . T ransitions w ith A m = + 1 , for exam ple, correspond to
the ab so rp tio n of a rig h t circu larly polarized p h o to n o r th e em ission of a left circularly
polarized ph o to n . T h e a b so rp tio n an d em ission p ro p erties depend, therefore, on b o th the
frequency an d p o la rizatio n o f th e rad iatio n . T h e frequency o f each Zeem an sp lit level is
[ 101 ] ,
[ 102 ]:
uz = i/q + 2.803 x 10
3 i? 77(A m )[GH z]
147
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(7.9)
w here vq is th e u n p e rtu rb e d frequency,
77
is a sh ift facto r w ith |r?| <
1,
and B is the
m ag n itu d e o f th e e a r th ’s m agnetic field, typically 0.5 G auss th ro u g h o u t th e mesosphere.
T h e largest possible frequency shifts occur for
77
= ± 1 , which im ply th a t th e center
frequencies for th e polarized em ission com ponents will be confined to w ithin 1.4 MHz of
th e u n sp lit c en ter frequency. In principle, em ission a t these split frequencies could be up to
100% circu larly polarized. Away from th e center frequencies, th e to ta l intensity of em itted
[103]. For a sm all shift in frequency, A uq,
rad ia tio n decays w ith frequency as:
away from th e cen ter frequency, th e first order fractional change in em issivity can be shown
to be:
^
(7.10)
I
V — I/O
For a single Zeem an sp lit com ponent,
A l _ 2A vz ,A m
I
1/ — v z
where A vz.Am = ^ z — v 0 = 2.803 x
1 0 _ 3 Ht 7 (A 77i)[GHz],
from eq u atio n 7.9. To obtain
th e to ta l co n trib u tio n to th e emission o f b o th p o larizatio n com ponents we m ust sum over
left-handed a n d rig h t-h an d ed contributions:
A /tot _
2 A v z ,A m
However, we have for th e shift factor in eq u atio n 7.9: T}(Am = + 1 ) = —r](Am = —1)
so th e n et effect on th e em issivity is exactly canceled o u t by th e two circularly polarized
com ponents [103]. A ny second o rd er co n tributions to th e em ission scale as
148
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~
{y
^ z )2 '
which im plies a co n trib u tio n o f < 10- 8 K for 26 < u < 46 G H z, i.e. th e frequency
band which P O L A R will probe. For these frequencies of o b serv atio n th e re is also a small
F araday ro ta tio n o f th e plane o f po larizatio n of th e C M B . R o sen cran z &: S taelin [104]
show th a t th e ro ta tio n o f th e plane o f polarization will be less th a n ~
10-2
degrees for
these frequencies. T herefore, b o th th e polarized em ission a n d F arad ay ro ta tio n of the
atm osphere a re negligible effects in th e range of frequencies w hich P O L A R probes.
7.6
S u m m a r y o f A str o p h y sic a l F o reg ro u n d s
O f all th e relev an t foreground sources, only diffuse g alactic sy n ch ro tro n rad iatio n and
ro ta tin g d u s t a re ex p ected to a p p ea r a t a level com parable to th e an ticip a ted polarized
CMB signals.
We have sim ulated th e perform ance o f P O L A R a tte m p tin g to measure
polarization in th e presence o f foregrounds based upo n th e g en eral least squares m ethod
developed by [105]. To co rro b o rate th e results of th e an aly tic e rro r calculation, we have
also perform ed a m ore explicit foreground removal sim u latio n sim ilar to th a t of [8 8 ]. We
find th a t th e effect of th e foregrounds is to increase effective p er-p ix el noise on th e recovered
Stokes p a ra m e te rs by a facto r o f ~ 3 [106]. If P O L A R can achieve a per-pixel noise lower
th a n
1
fj.K it w ill be capable of discrim in atin g a ~
1
—3/zK CM B p o larizatio n signal from
polarized g alactic sy n ch ro tro n rad iatio n .
149
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Chapter 8
Obser vat ions
8 .1
S ite
P O L A R 's observations are co n d u cted from a cu sto m -b u ilt o b serv ato ry located a t the
U niversity of W isconsin’s P in e B luff O bservatory, (P B O ). P B O is lo cated a t Longitude
-f89°45/. L a titu d e -t-43°01', ap p ro x im ate ly 10 m iles west o f th e cam p u s an d downtown
M adison . Its bucolic location places it in a relatively R F q u ie t region.
T h e observ­
ing p latfo rm itself is leftover from th e U W A stronom y d e p a rtm e n t’s W isconsin H -A lpha
M ap p er (W H A M ).
A m otorized dom e encloses th e rad io m eter and ro ta tin g g ro u n d screen, keeping precip­
itatio n o u t, an d m ain tain in g a m o d e ra te ly th erm ally stab ilized enclosure. T h e dom e itself
can be o p e ra te d m anually, o r rem o tely via a W W W page in case o f inclem ent w eather
developing w hile th e ex p erim en talist is elsewhere. T h e p la tfo rm has a high-voltage power
supply for o p eratio n of th e C T I 8500 com pressor, which requires 220V a t roughly 10A.
Also ru n n in g to an d from th e p a d is a n E th e rn e t hub a n d cables w hich provide an in­
tra n e t for d a ta to be tran sferred from th e ro ta tin g co m p u ter a tta c h e d to th e radiom eter
150
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
to a d esk to p w ork statio n located ~
8 .2
100’
aw ay in a sep arate building.
A tm o sp h e r ic effects
A lth o u g h th e atm o sp h ere is n o t ex p ec ted to produce appreciable linearly polarized ra­
d ia tio n , it produces a non-negligible c o n trib u tio n to th e system te m p e ra tu re o f th e ra­
d io m eter. A dditionally, significant flu ctu atio n s of atm ospheric loading increase th e lowfrequency noise sp ectru m of th e receiver. W e sum m arize th e co n trib u tio n to th e an ten n a
te m p e ra tu re seen by th e rad io m eter in th e K & band by co m p u tin g th e pow er sp ectru m of
th e atm o sp h ere using a com m ercial code, A T 1. To co m p u te th e a n te n n a te m p e ra tu re AT
requires as in p u t th e desired level o f p recip itab le w ater vapor (P W V ). W ith th is specified,
A T can co m p u te th e an ten n a te m p e ra tu re vs. frequency using a s ta n d a rd m odel of the
e a r th 's atm o sp h ere. Figure 8.1 shows th e atm o sp h eric an te n n a te m p e ra tu re vs. frequency
for various levels o f P W V 2.
P O L A R convolves the sp ectru m o f th e atm o sp h ere w ith th e (power) its tran sfer func­
tion . From th is we can com pute a d irec t relatio n sh ip betw een P W V an d Ta*m ; see figure
8. 2.
Since th e d o m in an t co n trib u tio n to th e atm ospheric te m p e ra tu re com es from th e 22
G H z H 2 O line ra th e r th an on th e O 2 line a t ~ 60GHz, th e dependence on P W V is
q u ite noticeable. D a ta on P W V is provided by th e FAA which uses atm ospheric w eather
balloons (radiosondes) launched a t reg u lar intervals, from several m ajo r US airp o rts. In
figure 8.3 we show a plot of P V W vs. d a te for 1998 from th e n earest so unding balloon
'w ritte n by Erich G rossm ann, A irhead Softw are, Boulder, Co.
2T h e a u th o r wishes to than k C hris O ’Dell for th e prep aratio n o f th e AT figures
151
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
30
€7
3
O
«7
Q.
E
c
oc
c
c
<
V
•C
Q.
sn
O
E
<
26
28
32
34
36
freq u e n cy [CHz]
F ig u r e 8 .1 :
S p e c tru m o f A tm o sp h e ric A n te n n a T e m p e r a tu re in th e
b a n d vs. P re c ip ita b le
W a te r V a p o r (P W V ) . P ro m th e to p dow n, th e fo u r levels o f P W V arc: 3 0m m , 20 m m , 10m m ,
Omm.
15
10
5
0
0
20
10
Precipitoble Woter V apor [m m ]
30
F ig u r e 8 .2 :
I n te g r a te d A tm o sp h e ric A n te n n a T e m p e ra tu re in th e K * b a n d vs.
W a te r V a p o r (P W V )
152
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P re c ip ita b le
site in G reen Bay, W L C learly, observation in th e w in te r is preferred as w ater vapor in
th e atm o sp h ere “freezes o u t” , resu ltin g in a low P W V a n d equivalently, a low atm ospheric
c o n trib u tio n to o u r a n te n n a te m p eratu re.
G r e e n B a y . Wl 9 8
«0
30
E
E
O.
g 20
a
Jo n
Feb
Wcr
Apr
Mcy
Jun
Jui
Ween:
F ig u r e 8 .3 :
Bay, VVI.
Aug
S ep
Oct
Nov
Dec
'7 .0 3 9 7 m m
R a d io s o n d e -m e a s u re d P re c ip ita b le W a te r V a p o r (P W V ) vs. d a y o f 1998 from G reen
As far as th e s ta b ility of th e atm o sp h ere is concerned, we are guided only by theory
as we have n ot yet com piled records o f th e power sp e c tru m o f atm o sp h eric fluctuations a t
PB O . It is well know n th a t th e atm o sp h ere obeys a K olom ogorev sp ectru m [107], [108].
We can com pare th e pow er sp ectru m of d a ta taken on d a y s w hich a p p ea r to be stab le vs.
un stab le. T h e effect o f atm o sp h eric fluctuations can also b e e stim ated by com paring d a ta
taken w ith th e in tern al cold load, as in C h a p te r 5, w ith re a l d a ta tak en while observing.
8 .3
O b se r v a tio n S tr a te g y a n d S k y C o v er a g e
O ver a single night, P O L A R sweeps o u t a 7° x 360° x cos 43° = 1844°2F W H M sw ath of
th e sky. T h e 36 7° F W H M pixels com prise 5% of th e sky. T h e d a ta is binned in to Stokes
153
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Q an d U vs. R A , a n d m u ltip le n ig h ts of d a ta are coadded. T h e scan passes th ro u g h th e
galaxy tw ice per d a y a t R A ~ 19h a n d again a t R A ~ 6h.
C o n stra in ts on, o r d etectio n of, th e po larizatio n o f th e C M B an d its asso ciated power
sp ectru m d ep en d g reatly o n th e am o u n t of sky coverage of th e observation as well as
th e sen sitiv ity o f th e rad io m eter.
S ensitivity considerations are com m on to all CM B
observations: tim e lim itatio n s re s tric t signal in teg ratio n , and co n strain th e a m o u n t of
sky coverage.
We m u st reach a com prom ise betw een th e in tegration tim e required to
achieve th e desired signal-to-noise ra tio while also sam p lin g a representative d istrib u tio n
of celestial regions. We now discuss o u r observing stra te g y in th e context of th e achievable
level o f sen sitiv ity o f P O L A R .
8 .3 .1
S e n s it iv i t y t o t h e P o w e r S p e c t r u m
P O L A R 's sen sitiv ity to polarized C M B fluctuations is q u an tified by its w indow function,
W(. T h e observed tw o-point co rrelatio n function is re la te d to th e power sp e c tru m and
window fu n ctio n as follows:
( Q i n ^ Q i h i ) + U ( n i ) U ( n 2 )) = j - £ ( 2 * + 1) x Q n W / x Pe(cosO),
(8.1)
1=0
where, for exam ple, Q ( n ) is th e Stokes p a ra m e te r m easured for a pixel lo cated in the
d irection n . C j 1 is th e pow er sp e c tru m describing th e degree of polarization on angular
scales ch ara cterized by m ultip o le £, W f is th e w indow function of this observing scheme,
and cos( 0 ) = fix • n 2 is th e s e p a ra tio n betw een pixels in th is observing scheme.
T he an aly sis o f P O L A R differs from th a t o f m ost an iso tro p y experim ents in several
respects. T h e p rim ary difference is t h a t th e observations a re total-pow er in n a tu re , ra th e r
154
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th a n differential.
T he window functions for this experim ent will reflect th e fact th a t
th ere is no "chopping” of th e beam in sky position inherent in th e observation. Single
pixels will be formed by binning th e acquired d a ta , and differencing betw een pixels can
be perform ed during analysis o f th e d a ta ; n o t during acquisition. T h is approach avoids
sy stem atic effects which can arise from m echanical chopping m echanism s.
D a ta from
P O L A R will be analyzed using a v ariety of synthesized window functions, each sensitive
to a different angular scale. In th is resp ect th e analysis will be sim ilar to th a t of the
S askatoon Big P late observations [18,
6 8 ].
W indow functions for observations w ith less th a n full-sky coverage are specified by
th ree functions: th e beam profile function, th e beam position fu nction, and the weight­
ing or ‘lock-in’ function [109]. T h e beam profile function, G(8 ,9i , o b ), where a b is the
b eam w id th , quantifies th e d irectional response o f th e anten n a, w hich is roughly G aussian
as seen from th e beam -m aps of C h a p te r 4. G ( 6 ,8 x, o b ) effectively sam ples all angular
scales larg er th a n , approxim ately, th e an g u lar size o f th e b eam .
T h e angular coordi­
n ates of th e center of th e beam are specified by th e beam p o sitio n function, 0*. T he
lock-in function, w f , is th e w eighting o f each of th e N binned pixels indexed by f, for
th e scan stra te g y denoted by a.
W e have G(O,0i,&B) =
exp
where
o B = F W H M / 2 > / 2 ) n 2 = 0.052.
Following (W hite & Srednicki), th e w indow functions axe:
W f 0 = j d k x f dX 2 H a ( x i ) H lS(x 2 )P /( x i - * i )
w here P( are th e Legendre polynom ials, an d H a ( x =
(8.2)
w ?G (6 ,9 i, o b ) quantifies the
response o f th e an ten n a (for a differencing stra te g y indexed by a ) , w hen pointed in the
155
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d irectio n of x. For a g aussian beam : H a (Z) = Y h w? 2-n<jT exP
^ •
T h e u ltim a te sen sitiv ity o f th e in stru m en t to th e power sp ectru m , Ce is given by [110]:
A Ce
r
2
1 i N
~cT" l(2 ? T T y l
.
( A Q r m s &f w h m ) 2 _ e W \
c,
i1 +
e
1
* J ^
,0
(8'3)
where: cr*, = 0A256 f w h M i A Q r a i s is th e per-pixel Stokes p a ra m e te r sensitivity, and
th e fractio n o f th e sky covered, f aky , for P O L A R is ~ 5%, as m entioned above.
At
large an g u lar scales, £ is sm all, an d since th ere are only 2£ 4- 1 a^ m to e stim ate Ce, the
co n trib u tio n
to
the variance in th e
recovered Ce is large ( “C osm ic V ariance” ).T his is
th e u ltim a te lim it to th e erro r in th e power sp ectra as th e per-pixel in stru m en t noise is
reduced - th ere is no gain m ade by in teg ratin g for longer tim es. However, since P O L A R
has a large in stru m en t noise, we m u st in teg rate for long periods o f tim e such th a t the
second te rm in equation 8.3 is reduced to th e level of 1fj.K. T h e long term stab ility of
P O L A R is crucial to achieving th e levels of in stru m en t noise per-pixel required to make
detectio n s of non-zero low-£ values of Ce-
156
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Chapter 9
Data Reduction and Analysis
9.1
D a ta A n a ly sis M e th o d o lo g y
T h e raw d a ta from th e p o larim eter consists o f eight d a ta channels ( 2 to ta l power diode
d etecto rs, 3 correlators x two lock-in phase references, in-phase a n d q u a d ra tu re phase,
th e ab so lu te one-bit en co d er (A O E ) a n d encoder signals, an d num erous housekeeping sig­
nals (te m p e ra tu re s o f b o th H E M T s, th e h orn tem p eratu re, th e cold stag e tem p eratu re,
th e d ew ar’s pressure, an d
2
te m p e ra tu re sensors in th e ro o m -tem p eratu re receiver box,
an d R T R B h eater P ID m o n ito r ch an n el). T h e raw d a ta from th e to ta l pow er detectors
are sam p led a t 20 Hz afte r leaving th e 5 Hz low-pass anti-aliasing filter. For th e corre­
lato rs th e raw signal is am plified th e n sen t to a m o d u la to r/d e m o d u la to r w hich is driven
synchronously by e ith e r th e in-phase o r o u t-p h ase reference w aveform , a n d th e n low-pass
filters a t 5 Hz. D ata is collected for 7.5 m inutes and th e n sto red to a n A S C II file.
T h e signal processing p ro ced u re is com posed of th e following steps:
• B in n in g of d a ta by ro ta tio n angle an d corresponding initial estim a tio n of noise a t
th is bin size.
157
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• E stim ates/rem o v a l o f atm o sp h eric co n tam in a tio n in th e co rrelato rs using th e to ta l
pow er d etecto rs.
• F its to recover Stokes Q an d U for each ro tatio n .
• B inning tim estream d a t a in to R .A . pixels on th e sky. an d estim atio n of noise for a
single nig h t of d a ta .
• C o -ad d in g of RA pixels for all nights o f d a ta
9 .2
B in n in g o f D a ta
T h e raw d a ta contains an average of 600 sam ples p e r ro ta tio n ta k en a t a sam pling ra te
of 20 Hz.
T h e first step in th e analysis is to bin th e d a ta from th e n atu ra l sam pling
an g u lar binsize,
=
0 .6 °
to larger bins in o rd er to have enough d a ta points to have an
acc u rate noise estim ate for th e su b seq u en t m inim um -;^ 2 -fits. C learly, it would be nice to
have m any sam ples p er b in to e stim ate th e noise on th e tim escale o f th e bin. However,
w ith a c o n sta n t ro tatio n r a te o f u ~ 2R P M , “sm earing” o r co rrelatio n between bins would
be ap p reciab le as we increase th e binsize. W ith th e fu n d am en tal in tegration tim e of 0.2
sec defined by th e an ti-aliasin g filters, th e re are only
20
/ \/E ~ 9 independent sam ples
per second. F igure 9.1 show s th e first level o f binning: in to ro ta tio n angle only, for ten
ro tatio n s o f th e p o larim eter ( ~ 5 m inutes of d a ta ). T h e d a ta p lo tte d here are from a
single high-galactic la titu d e pixel d u rin g excellent observing conditions.
T h e n ex t ste p is to co -ad d all such ro ta tio n s (w ith in a given tim e period, thus defining
ou r pixel o n th e sky) to form a “sin g le-ro tatio n ” (for th a t pixel). T h is d a ta is th e n fit for
th e Stokes p aram eters, as show n in figure 9.2.
158
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C o r r e l a t o r J1: 1 0 R ota ti ons B in n e d Da ta v s . Rotation Angle
0 .0 0 4
0 .0 0 2
"5
Q_
3
o
0 .0 0 0
o
©
u.
L.
<-> -
0 .0 0 2
- 0 .0 0 4
0
100
200
300
Angle [ d e g r e e s ]
F i g u r e 9 .1 : S how n h e re is th e o u tp u t o f c o rre la to r J1 for te n r o ta tio n s o f th e p o la rim e tc r. T h e
d a t a a re b in n e d in to a n g le , a n d all ro ta tio n s a rc o v e rp lo tte d . N o o th e r a v e ra g in g o f th e d a ta has
b e e n p e rfo rm e d .
159
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C o r re la to r J1: 10 R ot at i on s Bi nned Data vs. Ro tation Angle
0 .0 0 3 0
0 .0 0 2 0
0.0010
3
Q_
"3
o
0 .0 0 0 0
L_
o
a
©
u.
(5 -0 .0 0 1 0
-
2 3 .4 ± 4 0 .5 ,u K
—3 5 .0 ± 56.5uK
2 0 .0 i5 7 .3 y u K
1 7 9 .5 / 1 4 4
0 .0 0 2 0
-0 .0 0 3 0
0
100
2 00
Angle [ d e g r e e s ]
300
F ig u r e 9 .2 : Show n h e re is th e o u tp u t o f c o rre la to r J1 for te n ro ta tio n s o f t h e p o la rim e te r, w ith
all r o ta tio n s av erag ed a n d fitte d to o b ta in th e S to k es p a r a m e te rs a s sh o w n . T h e d a t a a re first
b in n e d in to an g le for e a c h ro ta tio n , a n d th e n all r o ta tio n s a re co -ad d cd to fo rm a sin g le -ro ta tio n .
T h e so lid line is th e fit to th e m o d e l 5 ( 0 ) = / + Q cos 20 4 U sin 28. N o r o ta tio n s y n c h ro n o u s e.g.,
cos 8 o r sin 8 c o m p o n e n ts h a v e b ee n rem oved.
160
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9 .2.1
In ter -b in C orrelation s cau sed by rotation o f th e p o la rim eter
Let us m odel th e o u tp u t of th e p o la rim eter viewing a polarized so u rce as:
v(t) = P ( t ) s m 2 i s t + n(t)
(9.1)
w here P = \ J Q 2 4- U*2 . We defer for th e m om ent th e “b eam -sm earin g ” an d its effect
on sky pixel-pixel co rrelation . A lso here P ( t ) = c o n s ta n t
= P a, an d n (f) is G aussian
d is trib u te d noise w ith variance cr2 .
W ith v = 0.03Hz, we sam ple v(t) 600 tim es p er ro ta tio n an d average th e resu lts into
N bins: u,- = ^ I t ' - a / 2 v (t)dt, w ith A = 30 / N (binsize in seconds). F rom C h a p te r
convolution th eo rem asserts th a t if y(t) =
1,
the
h{t — t?)x(t!)dt?, th e n :
y{t) =
(9.2)
J —OC
w here H { uj) & X(u>) are th e F ourier tran sfo rm s of h(t) <k x ( t) , respectively.
p resen t case, h(t) = ^ for t
e wA/a sitwA / 2
6
In the
{ —A /2 , A /2 } , an d is zero elsew here. T herefore, H ( uj) =
tsjow we need th e sp ectru m o f v(t) = P c s in 2 i/t -I- n(t):
V(") = f-o o v W e ^ d t
=
[Pa sin 2ut -I- n(*)]e*‘rtd i
= Pa[ i ( 6 (2 v - u ) - 6(2u + u,)) + N ( t) ]
(9.3)
(9.4)
(9.5)
O u r u ltim a te goal is th e pow er sp ectru m , Sy (u), of th e b in n ed d a ta , Sy (u>) = \Y \2 =
\H\2\V\2 = S h(u )Sv (u).
161
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This is easily seen to be:
s»(">=i [T/t/2l2 [?M2"- ">+{(2"+")!+T1 '
(9'6)
To d eterm in e th e effect o f th e binning on o u r sig n al an d noise, we exam ine the a u to ­
correlation fu n ctio n (A C F ), R y ( t ) , of y(t) by Fourier tran sfo rm in g S y(uj) — S n (u>), (w here
S n (u>) is th e PSD o f th e noise):
= /“o
2 [ ? t«P " - “>)] + «(2" + “Oil
= [£^
A ]2cos2ut
0 -7 )
(9.8)
T he A C F o f th e b in n ed noise is :
L A w /2
I
'99>
2
We now define q(t) such th a t:
/
oo
f& l2
q(t)dt =
dt
(9.10)
J - A/ 2
-oo
Then:
i / “«
=
i
e<“ ‘du>
162
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(9.11)
and then:
H
J —oo
q(t)q(t')dt = ^
[ S1^ ^ / 2 ] 2e ^ ^ .
J —OO L G*Vi
J
(g>13)
so we need to com pute th e A C F of q(t), w ith:
Sc q(t<) = 1, f e { - f , f },
<,(() = 1, t S
and:
/
J — OO
q(t)q(t')dt = A q(t').
(9.14)
S u b stitu tin g into eq. 9.9. we have:
R«(t) =
V e< “W
(9.15)
= <’'2 [ 1 - | s | ] ■ l‘ I S A ,
=
0
otherw ise.
T h e b est way to quantify th e effect o f th e binning is to look a t th e signal-to-noise ratio
of th e b in n ed d a ta , an d com pare it to th e SN R of th e unbinned d a ta w hich is ju st
p2
T he SN R is :
S N R = yjol/ol.
F irst, we have for th e signal:
163
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(9.16)
for v = 0.03Hz an d A = [4 ^ "
for a m ultiple of ^ =>• a 2 =
0
= 0.2 sec/b in , a* = 0 . 0 4 . N ote th a t if we integrated
. as expected since we w ould be averaging our signal to zero.
T h e variance of th e noise is:
o r°° f sin
- 1 Jo [ Acj/ 2
2
2
2
4duj.
T h e S N R 2 th en becom es:
p 2 [ s in i/A 1 ‘
0
L ^ J
ff"
(9.17)
*•>
We see th a t A ~ 2 im plies th a t SiVi? ~
1.
T h is bin size would give us only a b o u t 10
poin ts p e r ro ta tio n to recover Q an d U, so to be conservative, we tak e A = 0.2s which
gives 144 points p er ro ta tio n .
164
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9 .2 .2
Is O u r B in n in g S t r a t e g y O p t im a l?
We can now co m p are th e perform ance o f o u r bin n in g technique w ith th a t of o u r fore
bearers L ubin a n d Sm oot [15], h ereafter LS, by asking th e following: given a ro ta tio n tim e
of T Spin , w h at is th e optim al d istrib u tio n o f N sam ples o f th e o u tp u t of th e p o larim eter?
For a p o larized source, th e o u tp u t of th e rad io m ete r is given by: v(t) = Qcos2rp(t) +
U sin 2ip(t). LS m easured th e o u tp u t a t
8
locations in angle sep arated by 45°. To recover
th e Stokes p a ra m e te rs LS effectively p erfo rm a lin ear least-squares fit to a m odel, w hich
is p aram eterized by a model o f th e ir ex p ected o u tp u t. T h eir observables a re essentially
two eight-dim ensioned vectors, one for Q a n d one for U. To p ro ject o u t th e Q (U ) co n ten t
of th e eight-dim ensional d a ta vector w hich th e y observe th e y m ultiply by X i ( X 2 ):
0
1
-1
0
0
i
-1
(9.18)
11
£
1_
*i = -S
V2
1
0
0
1
-1
0
V o /
\-i)
To perform th e linear least-squares fit, LS form th e design m a fr ix [ lll] : A i j =
X (t )
y
w ith i € {1,2} a n d j e { 1 ,...8 } . T h is resu lts in th e following covariance m a trix , C a =
[q ,j]- 1 , w here a tJ = 5 ll= i
W .
A ssum ing equal noise p e r in teg ratio n a t each
position, <Tk = o’/y T a p tn / 8 , we have:
165
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showing that the estim ates for Q and U are uncorrelated.
For P O L A R binned to 144 p o in ts p e r ro tatio n , th e co rresponding design m atrix is
d eterm in e d bv:
f cos 2u t\ \
X 1=
/ s in 2 z/ti N
,
\ COS
2l / t 1 4 4
*2
;
=
\ sin
/
21^144
(9.20)
J
We find:
( cos 2ist\ \
A
13
Xi{tj)
cr(tj)
( s in 2 i/<i \
1
(9.21)
(Tk
\ COS 2 ut 144
\ sin 2 i/t 144 /
Now we co nstruct a = A t A , a n d th e associated covariance m a trix C y = [O y ]-1:
1 4 4 ^ -2
/ 5(144 - E i l 4i cos Auti)
C ™ LAR = x (
lspin
V
5 E i i ! sin Auti
5
E ! = l sin 4 v t t
\
).
(9.22)
1 (1 4 4 + E i = l cos 4 ^ 7
A t first glance th is covariance m a trix ap p ears to have off-diagonal elem ents - i.e., we
seem to have correlated o u r erro rs o n Q an d U. However, by sy m m e try 1, all of th e sum s in
e q u atio n 9.22 are equal to zero, leaving us w ith th e resulting d iag o n al covariance m atrix:
J '
sia2ni/IV =
0
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(923>
Now we can co m p are eq u atio n s 9.19 an d 9.23. We assum e th a t th e en tire ro tatio n tim e
is used for in teg ratio n s for each m ethod. T h e LS m ethod will com bine
w ith variance
8
x a 2/Tspin to achieve a final variance of
2
8
m easurem ents,
x a 2. Similarly, P O L A R will
com bine 144 sam ples w ith variance 144 x <r2 /T spm to o b tain final variance o f 2 x a 2, which
is equivalent to th e m inim um detectab le te m p e ra tu re for th e LS m ethod. T hus, for both
LS and P O L A R th e s ta n d a rd deviations o f Q o r U will be a factor o f y/2 tim es larger th a n
would be o b tain ed for an in ten sity m easu rem en t given the sam e length of tim e.
In practice, n o t all of th e tim e for a given ro ta tio n can be used for in tegrations in
th e LS m ethod. T im e is lost m oving betw een an g u lar positions, and th e p o larim eter is
no t capable o f sto p p in g in stan tan eo u sly so it m u st be slowed to a stop. T herefore, given
a fixed tim e p e r rev o lu tio n , set perh ap s by th e characteristic tim escale o f fluctuations of
atm ospheric em ission, th e m eth o d used for P O L A R is preferable.
Finally, a lth o u g h th e errors o n Q and U a re uncorrelated, the noise in th e tim e stream
is correlated betw een ad jacen t bins for tw o reasons:
• the an ti aliasing filter has non-vanishing correlation on tim escales less th a n
0 .2
sec.
• th e o u tp u t from th e co rrelato r is co rrelated w ith the o u tp u t from th e to ta l power
channels due to th e presence of th e co rrelated atm ospheric com ponent present in
each to ta l pow er channel.
We will ex p an d u p o n th e abrogation of th e second o f these phenom ena in th e following
section.
167
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9 .3
C o rrela tio n B e tw e e n S k y P ix e ls
A deficiency of th e d rift scan m e th o d em ployed by P O L A R is t h a t ad jacen t pixels share
significant correlated signal. W e now present a form alism to ad d ress th is concern, and
estim ate its effect on o u r resu lts.
Let us model th e o u tp u t o f th e p olarim eter when
viewing in the d irectio n 0 as:
P i t ) = P (0 ) sin 4>{Clt) + n(Clt)
(9-24)
w here n = y / Q 2 + U 2 an d <t>is th e RA , Cl is th e ro ta tio n ra te o f th e e a rth . The zenith
in M adison, WT corresponds to a d eclination of 8 = 43°. In 24 h o u rs o f right ascension
th e re are 360° cos 43° ~ 255° a lo n g o u r scan. W ith th e a n g u lar ro ta tio n ra te of the earth:
uj ~ 1.2 x 10~ 5 Hz, we co-add 7.5 m inutes o f d a ta to form a single b in o f a n g u lar size 1°.32.
O u r beam size is ~ 7° w hich im plies th a t adjacent bins will sh are significantly correlated
signal,
u p to a lag of ~ 5 bins. F rom th e individual bins weth erefo re co-add several bins
to form a single larger b in w ith (hopefully) larger signal an d lower noise. T h e behavior of
th e noise during this b in n in g is o u r first indication of th e p erfo rm an ce o f th e instrum ent,
as th e signal co n trib u tio n is clearly su bdom inant.
F irst we m odel th e o u tp u t signal as th e convolution o f th e tr u e sky signal, 11(0), w ith
th e norm alized a n te n n a pow er response function2, B(6):
P ( 0 ) = f ° ° n (0 )5 (0 7 —0 0
2assum ed to be azim uthally sy m m etric
168
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(9.25)
From the convolution theorem we have:
P (0 ) =
f°°
(9.26)
J —OO
w here II(fc) a n d B ( k ) are th e Fourier tra n sfo rm s o f 11(0) an d 5 ( 0 ) .
We would like to know how much co rrelatio n o u r binned pixels have betw een them .
T he b inning in an g le effectively introduces a low -pass filter, H ( k ) . T h e sp e c tru m o f th e
binned pixels w ill b e given by: P ( k ) = H ( k ) U ( k ) . T h e binned pixels are th e n P (0 ) =
s f - o O P { k ) e 2*'k6dk . H ( k ) is given by:
ff(Jfc) = i
/ " '+a
= 1
A J ox- a
A
^
1
27rfc
2
2
^
2ttfcA
(9.27)
where 2A = 2 0 0 b in s/2 5 5 ° x 5 ~ 1°.32.
T h e m ost d ire c t m e th o d is to com pute th e au to c o rre la tio n function o f th e m easured
sky d a ta
P i{9)
a n d find th e lag in degrees w here it falls to zero. T h e au to co rrelatio n
function o f P ,, (P t (0 )P i(0 ')), assum ing s ta tis tic a l iso tro p y o f P,(0) is only d ep en d e n t on
the an g u lar separation 6 — 0 ', an d is given by:
P ,(0 )P ,(0 - 0')de
(P t (O)Pt-(0 ')) =
(9.28)
= /f^ n (A :)n (fc)B (fc)P (A :)e 27riA:<?dA:
(9.29)
= f^oo C k W k e 2nikedk,
(9.30)
where th e w avenum ber k ~ £.
An a lte rn a tiv e m e th o d to check for no n -v an ish in g correlation betw een sky pixels is to
169
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exam ine th e pow er s p e c tra of th e b in n ed d a ta . We can th in k of th e averaging procedure
used d u rin g co n stru c tio n of th e sky bins as a low-pass filter, an d th e n convolve th e filter
response fu n ctio n w ith th e d a ta o f th e 7.5 m in u te bins. T his is b est done in th e frequency
dom ain w here th e convolution is th e inverse F ourier transform of th e p ro d u c t of th e equiv­
alent tra n sfe r fu n ctio n {i.e. sp ectru m ) o f th e filter and th e a n g u la r/tim e sp ectru m of the
7.5 m in u te d a ta . T h is Fourier analysis w ill also allow us to identify p erio d icities in th e raw
7.5 m in u te bins w hich, since th e astro n o m ical signal is negligible, can only be attrib u te d
to sy stem atic effects in our d a ta . B y looking a t the m easured pow er sp ectru m we can
determ in e th e ran g e o f angles over w hich o u r binning strateg y in tro d u ces correlations,
ju st as in th e case o f an RC low-pass filter w here frequencies less th a n th e reciprocal of
the tim e-co n sta n t can be considered in d e p en d en t.
<pv(o)p,(0')> = n o
pernio - we
= I - o o U { k ) U { k ) B { k ) B { k ) e 2^ k0dk
(9.31)
= J ^ C tW .e ^d k .
H ere we have in troduced th e n o ta tio n C* = n 2(/;) and Wk = B 2(k). O f course C/t
is ou r m ain q u arry , so it is n ot possible to su b stitu te its value into eq. 9.31 a priori.
However, over a sm all range of a n g u la r scales (in P O L A R ’s case, at large an g u lar scales),
we take Ck = c o n s t a n t = CnT h e m easu red pow er sp ectru m in te rm s o f th e true power sp e c tru m is given by:
=
c n Sml X
^ W k-
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(9 32)
T he m easured w indow function, W k , is given by:
Wk
=
|fl(A:) 2 =
1
If
1J —OO
B(0) e2nik0dd\2
e - { e * l2 6 l) e 2 * ik 6 d 9 2
Ir
‘ J — OO
cx e - %^ kHo,
(9.33)
showing th e co rrelatio n induced by “beam sm earing” is largest at sm all scales since
the power sp e c tru m is largest a t large angular scales. T h is window leads to:
C r as = C n ^
| ^
e - fc2 0 o/ 2 .
(9 .3 4 )
Notice th a t it is n o t possible to choose a binning w hich com pletely removes pixel-pixel
correlations. O n bins of 3°.5 which correspond to th e tim escale of th e individual d a ta files,
we see th a t th e co rrelatio n is ~ 50%, as in tu itio n would p redict.
171
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Chapter 10
Results
T his thesis has d escribed a w ork-in-progres: th e K a b an d in c arn atio n o f P O L A R . PO L A R
is cu rren tly acq u irin g d a ta an d th e perform ance o f th e in stru m en t can now be assessed.
P relim in ary in d icatio n s suggest th a t th e in stru m e n t is functioning q u ite well, though only
by con d u ctin g long-term observations will we be able to d eterm in e low-level system atic
effects “lurking” in th e d a ta set.
In th is C h a p te r we p resen t resu lts of an in itial d a ta ru n o f P O L A R observing th e sky
durin g th e late su m m er-m o n th s o f 1999. Finally, we present calcu latio n s o f th e expected
signal level for m odels w ith early reionizatoin a n d o u tlin e an analysis form alism which
will be im plem en ted in an effort to cross-correlate th e fu tu re P O L A R d a ta s e t w ith the
existing C O B E D M R d a ta set. T h is la tte r analysis is exp ected to give us a significant
signal-to-noise ad v an tag e over th e calculation o f th e p o larizatio n au to -c o rrelatio n alone.
172
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10.1
E x p e c te d L on g-T erm P er fo rm a n ce o f P O L A R
As m en tio n ed in C h a p te r 4, th e sensitivity o f th e p o la rim eter is p rim arily determ ined by
its cooled H E M T am plifiers w hich have noise te m p e ra tu re s o f ~ 65 K; far g reater th a n the
sta te-o f-th e a rt devices w hich now achieve noise te m p e ra tu re s o f ~ 10K in th e K & band
[112] ,
[113]. T h is noise te m p e ra tu re is com parable to th e a n te n n a te m p eratu re o f the
atm o sp h ere a t a g o o d ob serv in g site. As show n in C h a p te r 5, P O L A R ’s A"a-band H EM Ts
are m uch noisier, resu ltin g in a co n trib u tio n to th e sy stem noise te m p eratu re o f nearly
five tim es th e atm o sp h eric level in th e K & b and.
However, even w ith th ese devices we believe P O L A R will be able to produce cosmologically significant u p p e r lim its on th e po larizatio n o f th e C M B for a universe w ith no
reio n izatio n , o r possib ly a d e te c tio n of po larizatio n for a universe w ith early reionization.
Long in teg ratio n p erio d s will be required to reach a sen sitiv ity level ~
1
—10/xK per-pixel,
which w ould allow us to p ro d u ce these results. We recall th a t th e RM S noise in a m ea­
su rem en t of e ith e r Q o r U (in an ten n a te m p eratu re) is given by th e radiom eter equation
[20], which for th e Q Stokes p a ra m e te r is:
a /-»
K ( T rec ■+* T a im + T c M b )
& Q r m s = ------------ /A
/=----------- ,
v/A i/T / 2
ei n i \
(1 0 .1 )
w here Trec an d T atm are th e receiver and atm ospheric noise te m p eratu res, respectively.
r is th e to ta l tim e sp e n t observing the CM B; th e tim e s p e n t e ith e r Q or U is r / 2 . A u
is th e rad io frequency (R F ) b an d w id th an d as we have show n in C h a p te r 3, k =
> /2
for
a co rrelatio n rad io m ete r. F o r th e K a polarim eter, Trec cz 75K a n d T atm — 12K. If all
th ree co rrelato rs ca n be com bined Ai/ is ~
6
G Hz, re su ltin g in a sen sitiv ity to Q or U of
N E T = A Qrms — 2 m K s 1/ 2. For th e to ta l polarized in ten sity we have: I poi = \JQ 2 4- U^.
173
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The error in /poi is A /p d = >/2A Qrms = AtKs1/ 2, before foreground subtraction.
10.2
L o n g -T erm In teg ra tio n T e sts
We note th a t e q u atio n 10.1 indicates th a t log[ATftAfs(i")] vs. r should be linear w ith
slope equal to —5 . F igure 10.1 shows a p lo t o f log[AC?flA./s(T)] an d log[At//LMs(T)] vs. r
formed by com b in in g correlator channels J 1 an d J2 for a single night o f d a ta . T h e d a ta
points are co m p ared w ith th e theoretical ex p e c ta tio n values. Judging from th e continuing
tren d tow ards d ecreasing noise in P O L A R ’s co rrelato r channels we fully e x p ec t th a t the
long-term s ta b ility o f th e instrum ent will allow us to se t th e most restric tiv e lim its on the
polarization o f th e CM B.
Single Night Stokes Q P a r a m e t e r AQ,,*
vs. T im e
1 0 . 0
E
s
<
Cr
100
1000
10000
Integration Time [sec]
Single Night Stokes U P a r a m e t e r All**
vs. Tim e
1 0 . 0
E
2
<
100
1000
10000
Integration Time [sec]
Figure 10.1: Long term integration behavior of the noise in the Stokes Q and U parameters
for one night ~ 10 hours of data. The solid line indicated the expected behavior of the noise vs.
integration time A Q r m s or U r m s °c 1 /v/t-
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10.3
E stim a te d P o la r iz e d Signal L evel a n d U n cer ta in ty
10.3.1
E stim a ted T otal S ign al Level in M odels w ith E arly R eionization
To sim u late the perform ance o f P O L A R , we have used th e s ta te -o f-th e -a rt Boltzmann
code C M BFA ST to g enerate all relevant pow er-spectra. Since th e pow er sp ectra are not
im m ediately useful we have invoked th e stan d ard procedure o f m ap-generation; see e.g.
[114]. P olarization m aps are tra d itio n a lly plotted as “sticks” w ith am p litu d e (Q 2 -rU 2) 1^2
and o rien tatio n angle (1 /2 ) t a n _ 1 ( f //Q ) , following from th e definitions o f th e Stokes pa­
ram eters. For all figures w hich d ep ict th e polarization we p lot th e p o larizatio n am plitude,
not Q o r U directly. T h e m a p s 1 are q u ite helpfull for use in sim u latio n s o f th e instrum ent’s
perform ance in th e presence o f foregrounds and o th e r ex p erim en tal non-idealities.
F irst we show th e te m p e ra tu re anisotropy sky realization for a “s ta n d a rd CDM ” model
w ith no reionization in figure
RM S F l u c t u a t i o n
■
1 0 .2
.
1 5 . 9 7 4 8 uK
- 4 9 . 6 3 0 0 uK________________________
< 6 .6 0 0 0 uK
T e m p e r a t u r e Map: No R e i o n i z a t i o n
o*
To>
U
(J
O
F i g u r e 1 0 .2 : S im u la te d te m p e r a tu r e a n iso tro p y m ap m a d e u sin g C M B F A S T . A “s ta n d a rd C D M ”
m o d el w ith n o re io n iz a tio n p ro d u c e s th e u n d e rly in g pow er s p e c tru m w h ich is used to g enerate a
re a liz a tio n o f th e sk y w hich is s u b s e q u e n tly convolved w ith th e b e a m p a t t e r n o f P O L A R to c re ate
th is figure. F rom rig h t to left th e ra n g e o f RA is Oh < R A < 24h, a n d from to p to b o tto m th e
ra n g e o f D e c lin a tio n is 20° < 6 < 60°. P O L A R ’s o b serv in g s tr ip a p p e a r s in th e m iddle of th e
figure.
N ext we show th e te m p e ra tu re anisotropy sky realization for a “s ta n d a rd CDM ” model
w ith reionization a t z = 50 in figure 10.3.
1P re p a re d by N a te S te b o r
175
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RMS F lu c tu a tio n ■
1 6 .6 0 7 7 uK
- 5 1 . 0 * 0 0 uK
4 7 . 9 9 0 0 uK
T e m p e r a t u r e Map: 1 0 0 % R e i o n i z a t i o n a t Z = 5 0
R
RA [ * ]
Figure 10.3: Simulated tem perature anisotropy map made using CMBFAST. A “standard CDM”
model with reionization a t z = 50 produces the underlying power spectrum which is used to
generate a realization of the sky which is subsequently convolved with the beam pattern of POLAR
to create this figure. From right to left the range of RA is O/i < R A < 24h, and from top to bottom
the range of Declination is 20° < 6 < 60°. POLAR’s observing strip appears in the middle of the
figure.
From figures 10.2 a n d 10.3 we see th a t th e effect o f early reionization on th e tem ­
p e ra tu re an iso tro p y a t large an g u lar scales is u n d etectab le.
T h is is to be expected as
th e prim ary effect o f reio n izatio n is to suppress th e te m p e ra tu re aniso tro p y a t sub-degree
an g u lar scales [56]. Now we show th e polarization sky realizatio n for a “s ta n d a rd C D M ”
m odel w ith no reio n izatio n in figure 10.4.
RMS F lu c tu a tio n =
0 . 0 0 5 3 8 2 4 5 uK
0 . 0 2 9 4 3 0 0 uK
0 . 0 0 0 5 9 5 5 0 0 uK
Polarization
Ma p : No R e i o n i z a t i o n
RA [h]
Figure 10.4: Simulated polarization map made using CMBFAST. A “standard CDM” model no
reionization produces the underlying power spectrum which is used to generate a realization of the
sky which is subsequently convolved with the beam pattern of POLAR to create this figure. From
right to left the range of RA is Oh < R A < 24h, and from top to bottom the range of Declination
is 20° < <5 < 60°. POLAR’s observing strip appears in the middle of the figure.
F inally we show th e d ra m a tic enhancem ent of th e p o la rizatio n for a m odel w ith early
reionization a t a red sh ift o f z = 50 in 10.5.
176
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RMS F l u c t u a t i o n
0 . 0 6 9 1 1 4 4 uK
0 . 0 0 7 3 9 6 0 0 uK
P o l a r i z a t i o n Ma p :
0 . 3 7 5 5 0 0 uK
100% Reionization a t Z = 50
RA [h]
F ig u re 10.5: Simulated polarization map made using CMBFAST. A “standard CDM” model
with reionization at z = 50 produces the underlying power spectrum which is used to generate a
realization of the sky and subsequently convolved with the beam pattern of POLAR to create this
figure. From right to left the range of RA is Oh < R A < 24h, and from top to bottom the range
of Declination is 20° < S < 60°. POLAR’s observing strip appears in the middle of the figure.
C o m p arin g figures 10.4 and 10.5, we see t h a t th e p o larizatio n of th e C M B is enhanced
by a facto r of ten , to levels which are w ith in reach o f P O L A R assum ing a sen sitiv ity of
~ IfiK p er-pixel for all 36 pixels. T h is figure reinforces th e conclusions o f C h a p te r 2: th e
level of p o la rizatio n is extrem ely sensitive to th e ionization h isto ry o f th e universe, b o th
before a n d a fte r recom bination. We ex p ect, th e n , th a t th e observed p o la rizatio n signal
will d ep en d critica lly on th e o p tical d e p th , r , for p h o to n s back to th e last sc a tte rin g
surface. A p relim in ary estim ate of th e effect o f reio n izatio n can be o b ta in e d by com p u tin g
th e ex p ec ted RM S polarization and asso ciated ex p erim en tal u n certain ty for m odels of
a reionized universe. Figure 10.6 reinforces o u r claim s th a t th e effect o f reionization is
m ost p ro n o u n ced in th e polarization, n o t th e C M B . T h is effect allows th e degeneracy
betw een p a ra m e te rs such as H 0 and r to b e broken , w hich will u ltim a te ly allow [56] for
higher-precison estim ates of b o th [56].
R ecall in C h a p te r 2 we presenteed figure 2.5 w hich displayed C P , for th e pow er spec­
tru m co m p u ted u sin g CM BFA ST for various to ta lly reionized (ionized fractio n x = 1)
scenarios, p aram eterized by th e redshift o f reio n izatio n z Tl. In F igure 10.7 we plot th e
ex p ected RM S po larizatio n vs. zri, for 0 <
< 105, along w ith th e s ta tis tic a l l a un-
177
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40
>~
O
■5
Q_
u
- 20
< -4 0
-6 0
0
100
200
R.A. [deg]
300
0
100
200
R.A. [deg]
300
2.5
c
o
o
N
_o
o 0.5
Q.
0.0
Figure 10.6: Simulated observations of CMB anisotropy and polarization from Madison, WI.
Power spectra are generated using CMBFAST and convolved with the POLAR beam size and
observing strategy to create realizations of the sky as seen from our observing location. Correlation
between anisotropy and polarization is taken into account. The solid lines correspond to total
reionization at z = 50, and the dotted lines correspond to no reionization. Note that the effect of
reionization is nearly unnoticeable for the anisotropy, while for the polarization its effect is quite
dramatic.
178
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2.0
1.5
S’
J i.
c
o
*—>
a
•E
a
o
1 -0
cn
S
a:
0 .5
0.0
0
20
40
60
80
100
120
z«
F ig u r e 1 0 .7 : S im u la te d P e r-P ix c l RAIS p o la riz a tio n vs. re d sh ift o f re io n iz a tio n . A lso show n is
P O L A R ’s e x p e c te d 1<t p e r-p ix e l e rro r b a rs for ~ o n e -y e a r o f o b se rv a tio n s. N o c o rre la tio n s have
been ta k e n in to a c c o u n t. C o m b in in g all 36 pixels w o u ld re s u lt in a re d u c tio n in th e e rro rs by a
fa c to r o f ~ 6.
certainties we ex p ec t based on ou r N E T an d o b serv atio n tim e. T h e u n derlying power
spectrum is a generic CD M model w ith D =
= 0.05, h = 0.65, A = 0. and pure
scalar p ertu rb a tio n s. T h e inclusion of a tensor co m p o n en t should enhance th e large an­
gular scale p o la rizatio n [46] , [115], so th is figure u n d erestim ates th e RM S polarization
predicted by som e cosm ological models. T his figure suggests th a t P O L A R could begin to
detect p o larizatio n o f th e CM B a t th e lcr level if th e universe becam e com pletely reionized
at a redshift zrj > 45.
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1 0 .4
T e m p e r a tu r e -P o la r iz a tio n C ro ss-C o rre la tio n an d C O B E
CM B p o la riz a tio n can be decom posed in to tw o com ponents: one o f w hich is spatially
co rrelated w ith th e te m p e ra tu re anisotropy, a n d an o th er, larger co m p o n en t which is un­
co rrelated . Ng &c Ng [116] an d C ritten d en , C oulson, & T urok [115], d e m o n strate th a t,
given a high-resolution CM B te m p e ra tu re m a p , it would be possible to identify celestial
regions w hich a re sta tistic a lly m ore likely to posses higher levels o f th e co rrelated polariza­
tion co m p o n en t. As shown in [117], th e u n c o rre la te d p o larizatio n co m ponent dom inates
th e c o rre la te d co m ponent by a factor of a t le a st three.
F or d etecto r-n o ise lim ited po larizatio n ex p erim en ts, it can be advantageous to search
for p o la rizatio n -an iso tro p y (Q T ) correlation in a d d itio n to p o larizatio n -p o larizatio n (QQ)
cro ss-correlation. If th e noise in th e te m p e ra tu re an iso tro p y m ap is negligible in com par­
ison w ith th e noise o f th e p o larizatio n m easu rem en t, <r, th e e rro r in (Q T ) will be linear
in a w hile th e variance in th e po larizatio n cross-correlation function grow s as a 2. In this
lim it it becom es ad vantageous to search for cross-correlation.
1 0 .4 .1
C o r r e la tio n s B e t w e e n T e m p e r a t u r e a n d S to k e s P a r a m e te r s
Due to th e assu m p tio n th a t th e universe is s ta tistic a lly isotropic we ex p ect th a t twopoint co rrelatio n functions o f th e Stokes p a ra m e te rs will only dep en d on th e angular
se p a ra tio n betw een th e tw o p oints. T h is a ssu m p tio n is im plicit, for exam ple, w hen we
ex p an d th e te m p e ra tu re au to co rrelatio n fu n ctio n in to Legendre P olynom ials ra th e r th a t
spherical harm onics. U nfortunately, due th e exp licit reference to a p a rtic u la r coordinate
system for th e definition o f o u r exp erim en tally m easured Stokes P aram e ters, i.e ., Q =
T n s —T e w -, sim p ly co rrelatin g Q an d U in a p a rtic u la r co o rd in ate system gives correlation
functions w hich dep en d on th e positions o f th e p o in ts being co rrelated as well th e angular
180
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s e p a ra tio n [118]. T h is was th e s itu a tio n p rio r to K am ionkow ski, Kosowsky, a n d S tebbins
(K K S) an d Seljak an d Z aldarriaga [54] , [51].
As m en tio n ed in C h a p te r 2, two p rescrip tio n s to specify co o rd in ate in d e p en d en t p aram ­
eters ex ist, e ith e r th e expansion into ra n k - 2 sy m m e tric ten so rs on th e celestial 2 -sphere, or
in to sp in-w eighted spherical harm onics. To o b ta in th e real-space co rrelatio n functions of
Stokes p a ra m e te rs we define Q an d U w ith resp ect to axes w hich are parallel an d p erp en ­
d icu lar to th e g re a t arc (or geodesic) co n n ec tin g th e tw o p o in ts being co rrelated . So Q r is
th e difference in intensities in two lin ear-p o la rizatio n s ta te s parallel and p erp en d icu lar to
th e g re a t arc co n n ectin g th e two p o in ts, a n d Ur is th e difference in two lin ear-polarization
s ta te s w hich lie 45° aw ay from th e p a ra lle l a n d perpendicular[56]. T his is rem iniscent of
th e m ore fam iliar notion of parallel tr a n s p o r t in differential geom etry, except now we are
c o m p u tin g th e derivative of a sp in o r o n a curved m anifold ra th e r th a n a (m ore-sim ple)
vector.
Since we have th ree observables T , Q , U we ex p ect 3! different tw o-point correlation
fun ctio n s to be relevant if th e CM B c a n b e tre a te d as a g aussian random field. W e denote
th e six co rrelatio n functions as:
( T T ) , (Ur Ur ), (Q rQ r) , ( Q r T ) , ( Q r U r ) , a n d ( Ut T ). D ue to th e ex p ected sy m m e try of
th e universe u n d e r p a rity tran sfo rm a tio n s we will find th a t Q r is invariant u n d er reflection
along th e g re a t arc connecting th e tw o p o in ts being co rrelated , Ur will change sign under
th is sy m m e try tran sfo rm atio n . T , b ein g a scala r fu nction, will obviously be invariant.
So we ex p ect th a t th e ensem ble averages (Q r Ur ) an d ( UrT ) will be zero. A h in t in this
d irec tio n was provided in C h ap ter 2 w h ere we found th a t th e re were only four nonzero sets
o f m o m en ts C p , C p , C p , and C p G. C o rrespondingly, four nonzero co rrelatio n functions
provide a n eq u iv alen t statistical d escrip tio n .
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Following K K S, we now exam ine th e a u to an d cross-correlation functions. F irst, for
reference, we s ta r t w ith th e te m p e ra tu re au to -co rrelatio n function: (T T ):
g TW = ( ^
- - - ^ 2 ))
•
\ -to
to / ni fi2=co66
( 1 0 .2 )
Since, as m entioned above, th e co rrelatio n function depends only on th e angular sepa­
ratio n of th e tw o points, w ith o u t loss o f generality, we m ay choose one p o in t to be a t the
n o rth pole. ( 0 , 0 ), and th e o th e r to be on th e <j>=
0
longitude a t a d istan ce
0
from the
n o rth pole, (0 ,0 ). Next we ex p an d T ( n ) in term s of spherical h arm onics and note th a t
Y {lrn)(0 ,0 ) = < ! W ( 2 Z+ 1 )/(4 tt). So
c r m _ ^ ( 0 . 0 ) r ^ 0))
=
£
( « & a J m')v-(;m)( 0 , 0 ) Y
- ( ,0)
Iml'm'
=
C l b l l'S m m ' J
E
S m o Y f j i 'm ') (0 ,0 )
Iml'm1
=
E
l
i
^ r ^ C /T P /(cos0).
(10.3)
47T
For th e (Q Q ) correlation function we have th a t:
c « W = (5 d 6 i2 5 d M \
\
0
^0
,
(io.4)
/ fii-n2 = co sff
w here th e Stokes p aram eter Q r is defined as th e difference in brig h tn ess between axes
parallel an d p erpendicular to th e g re a t arc connecting n i and &2 - A gain we are free to
choose one p o in t to be a t th e n o rth p ole an d an o th er a d istance
0
aw ay along the
0
=
0
longitude. T his choice has th e ad d ed ad v an tag e th a t th e g reat arc con n ectin g these two
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points is along th e 9 direction, so we can use th e Q defined in th e (9, <t>) coordinate system .
As a consequence o f th e “hairy -b all” theorem of topology ([119]), in all choices of spherical
coordinate system s th e re will b e tw o points (poles) w here th e azim u th al coordinate is illdefined. In th e p a rtic u la r spherical co ordinate sy stem we are considering, th e definition
of Q a t th e n o rth pole is am biguous. As long as we agree n o t to discuss th e polarization
at th e pole we can always consider a point on th e 0 = 0 longitude which is infinitesim ally
close to the n o rth pole: in o th e r words, Q (0 ,0 ) really m eans lim#—o Q ( 9 .0). For th e (UU)
correlation function, th e derivation is sim ilar, giving
C u {9) = - £ H L t i ^ 2[CiCG j 2)(cos0) + C f G ^ c o s * ) ] .
(10.5)
For th e (T Q ) cross-correlation function,
=
/ r (»i) Qr(n2) \
Tq
\
=
£
1
1 0 .4 .2
Tq
/ n i- n 2 = c o s 0
—7 — N i C j G P?{cos 9).
47r
(10.6)
T h e C O B E D M R I n s tr u m e n t
T h e C O B E D M R ex p erim en t perform ed its p rim ary ta sk exceptionally well: it m ade
th e first unam biguous d etection of th e aniso tro p y of th e CM B [120].
T h e success of
th e in stru m en t is largely a ttrib u ta b le to th e ro b u st co n stru ctio n of th e in strum ent, th e
stab le observing lo cation, an d observing stra te g y an d analysis which were subsequently
carried
o ut. In a d d itio n to its ab ility to d e te c t th e anisotropy o f th e CM B, C O B E also
had th e ability to d e te c t th e p o larizatio n of th e C M B . D ue to th e com plicated observation
strateg y im plem ented in an effort to glean m axim um inform ation regarding th e anisotropy,
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p o la rizatio n resu lts from C O B E are n ot p a rtic u la rly restrictiv e [121].
T h e g re a te st usefulness o f th e C O B E in s tru m e n t tow ards a d etectio n o f p o larizatio n
of th e C M B m ay be y et to come. As we will show , it is possible for an in stru m e n t such as
P O L A R to d e te c t th e cross-correlation o f p o la riz a tio n w ith the te m p e ra tu re an iso tro p y
w ithout d etectin g th e polarization a u to -c o rrelatio n itself. In a loose sense, since th e te m ­
p e ra tu re an iso tro p y of th e CMB observed to d a y o n all scales is th e b ears th e im p rin t of
th e q u ad ru p o le aniso tro p y a t decoupling, an d th e p o larizatio n o f th e C M B is co rrelated
w ith th e th is q u ad ru p o le anisotropy as well, we ex p ect th a t po larizatio n and te m p e ra ­
tu re an iso tro p y will be correlated. In fact, cro ss-co rrelatio n m ay provide th e only link
betw een p revious detectio n s (of anisotropy), a n d th e proposed m easurem ents discussed in
th is thesis.
1 0 .4 .3
M o d e l a n d D a ta In p u t
To place in terestin g lim its on the p o la riz a tio n -te m p e ra tu re cross-correlation function we
require tw o in p u ts in ad d itio n to th e P O L A R d a ta s e t: a w ell-sam pled te m p e ra tu re an isotropy
m ap . a n d a th e o re tic al model of th e pow er s p e c tru m . T h e power sp ectru m will depend
up o n several cosm ological param eters, one of w hich r - th e optical d e p th due to reioniza­
tion , h as a d ra m a tic effect on the large an g u lar scale polarizatio n , an d n o t th e te m p e ra tu re
anisotropy. T h u s, a significant detection o f p o la riz a tio n -te m p e ra tu re cross-correlation m ay
a ctu ally prove to be th e initial observable d e te c te d by P O L A R . To p rep are for th is an aly ­
sis, in figure 10.8 we show th e C O B E te m p e ra tu re an iso tro p y m ap centered on P O L A R ’s
observ in g fields.
G iven th is m ap , we now require a m odel for th e te m p eratu re-p o lariz atio n cross-correlation
function. In ad d itio n to th e anisotropy an d p o la riz a tio n sp ectra , C M B FA ST also com putes
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RMS f l u c t u a t i o n in s t r i p =
0 . 1168l6m K
—0 .4 0 1 191 mK
0 .4 7 8 3 8 3 m K
rn R F
T ^ m n arn tn rA
A m ent
n Ntrin
a
<
u
a
F ig u re 10.8: COBE Tem perature Anisotropy Centered on POLAR’s Observing Fields. From
right to left the range of RA is Oh < R A < 24h, and from top to bottom the range of Declination
is 20° < h < 60°. POLAR’s observing strip appears in the middle of the figure.
RMS" F lu c tu a tio n =
2 .2 6 5 1 1 u k '2
- 6 . 9 3 6 3 4 uK-2
6 . 4 1 3 0 0 uK-2
T e m p e r a t u r e —P o l a r i z a t i o n C r o s s C o r r e l a t i o n Ma p
RA [h]
F ig u re 10.9: Simulated polarization-tem perature cross-correlation map made using CMBFAST.
Total reionization at z = 50 is assumed. From right to left the range of FLA is Oh < R A < 24h, and
from top to bottom the range of Declination is 20° < <5 < 60°. POLAR’s observing strip appears
in the middle of the figure.
(Q T ), w hich can be used to g e n e ra te a sky-realization. Figure 10.9 show s a sim ulation of
th e p o la riz a tio n -te m p e ra tu re cross-correlation ex p ected for a m odel w ith com plete reion­
ization a t z = 50.
T h is com pletes th e list o f q u a n titie s needed to p ro b e th e ex istan ce o f a cosmologically
significant d e tectio n o f p o la riz a tio n -te m p e ra tu re cross-correlation. O nce a com plete season
of d a ta from P O L A R is o b ta in e d we will begin th e process of pro b in g th e cross-correlation
using a s ta n d a rd likelihood an aly sis such as th a t p resen ted in [1 2 2 ].
185
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Chapter 11
Conclusions
We have d em o n strated th a t th e detection o f th e po larizatio n o f th e CM B is difficult b u t
technologically feasible. A d etectio n would p erm it th e discrim ination betw een heretofore
degenerate theo retical predictions. P o larizatio n o f th e C M B has a unique sig n atu re in
bo th real an d Fourier space, as well as d istin ct sp ectra l characteristics. A d etectio n o f po ­
larization, in co n ju n ctio n w ith th e cu rren t d etectio n s of CM B an isotropy could be th e best
available probe o f th e io n izatio n history of th e pre-galactic m edium . T his epoch of cosmic
evolution is of g reat in terest, a n d su p p lem en tal inform ation from p o larization detection
could greatly advance o u r knowledge of th e form ation o f s tru c tu re in th e early universe.
T h e current g eneration o f an iso tro p y m easurem ents are sufficiently refined th a t th e fun­
d am en tal p aram eters o f classical cosmology are beginning to be determ ined. D etection
of polarization of th e C M B also prom ises num erous dividends th ro u g h o u t cosmology, and
one readily observes th a t th e s ta tu s o f p o larizatio n observations to d ay is rem iniscent of
the s ta tu s of an iso tro p y m easurem ents a decade ago.
“B etter is th e end o f a th in g th a n th e beg in n in g thereof.” - Ecclesiastes
186
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