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Cosmic acceleration and the theory of the microwave background

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C osm ic A cceleration and th e T heory of th e M icrowave Background
By
Benjam in M ark Gold
B.S. (Michigan S tate University) 1997
M.S. (University of California, Davis) 2001
DISSERTATION
Subm itted in partial satisfaction of the requirem ents for th e degree of
D octor of Philosophy
in
Physics
in the
O FFIC E OF GRADUATE STUDIES
of the
UNIVERSITY OF CALIFORNIA
DAVIS
Com m ittee in Charge
2005
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UMI Number: 3191126
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A bstract
Cosmic Acceleration and the Theory of the Microwave Background
by
Benjamin M ark Gold
Doctor of Philosophy in Physics
University of California, Davis
Professor Andreas Albrecht, Chair
This work represents an investigation of several details of the anisotropies of th e
cosmic microwave background.
The focus is on the response of the microwave
background to the conditions of the universe at very early and late times, during
both of which the expansion rate is thought to accelerate. During the radiation
era, any perturbations to th e otherwise smooth background will oscillate due to
the immense density and pressure at these early times. After th e universe cools
and m atter becomes dom inant, the photon perturbations travel freely to us while
the m atter perturbations continue to collapse under gravity to form galaxies and
stars. The initial conditions for these perturbations are set by an early period of
acceleration called inflation.
The increasing precision of cosmological datasets is opening up new opportuni­
ties to test predictions from cosmic inflation. In this work I study the impact of
high precision constraints on the prim ordial power spectrum and show how a new
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generation of observations can provide impressive new tests of the slow-roll infla­
tion paradigm , as well as produce significant discrim inating power among different
slow-roll models.
In particular, I consider next-generation measurements of the
Cosmic Microwave Background (CMB) tem perature anisotropies and (especially)
polarization. I emphasize relationships between the slope of the power spectrum
and its first derivative th a t are nearly universal among existing slow-roll inflation­
ary models, and show how these relationships can be tested on several scales with
new observations. Among other things, the results give additional motivation for
an all-out effort to measure CMB polarization.
W hile photons continue to travel almost freely during th e m atter era, changes
in the expansion rate due to acceleration at late tim es can subtly affect their dis­
tribution.
Such acceleration is posited to be due to the effects of an otherwise
unobserved dark energy. Also in this work I discuss several issues th a t arise when
trying to constrain the dark energy equation of state w = P / p using correlations
of the integrated Sachs-Wolfe effect w ith galaxy counts and lensing of th e cosmic
microwave background.
These techniques are com plem entary to others such as
galaxy shear surveys, and can use d a ta th a t will already be obtained from cur­
rently planned observations. In regimes where cosmic variance and shot noise are
the dom inant sources of error, constraints could be made on the mean equation of
state to w ithin 0.33 and its first derivative to w ithin 1.0. Perhaps more interesting
is th a t the determ ination of dark energy param eters by these types of experiments
depends strongly on the presence or absence of perturbations in the dark energy
fluid.
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C ontents
C ontents
iv
List o f Figures
vii
List o f Tables
ix
1 Introd u ction
1
2
4
M odels o f Inflation in th e Early U niverse
2.1
In tro d u c tio n ....................................................................................................
4
2.2
Definition of in fla tio n ....................................................................................
5
2.3 A scalar field in general r e l a t i v i t y .............................................................
6
2.4 Slow r o l l ...........................................................................................................
8
2.5 Power spectrum flu c tu a tio n s .......................................................................
9
2.6 Power spectrum am plitude and s h a p e ........................................................
12
2.7 Cosmic acceleration t o d a y ..........................................................................
13
3 T heory o f th e M icrowave Background
14
3.1 In tro d u c tio n .....................................................................................................
14
3.2 An overview of the m e th o d ..........................................................................
15
3.3 Choice of g a u g e ..............................................................................................
18
3.3.1
Synchronous g a u g e .........................................................................
18
3.3.2
Conformal Newtonian g a u g e .......................
20
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3.4
3.5
Evolution e q u a tio n s.......................................................................................
20
3.4.1
M e t r i c ..................................................................................................
20
3.4.2 General p a r t i c l e s ..............................................................................
21
3.4.3 P h o t o n s ...............................................................................................
24
3.4.4 N e u tr in o s ...........................................................................................
25
3.4.5
Cold dark m a t t e r .............................................................................
26
3.4.6 B a r y o n s ...............................................................................................
26
3.4.7
Dark e n e r g y .......................................................................................
27
Line of sight e f f e c ts .......................................................................................
28
3.5.1
Integrated Sachs-Wolfe e ffe c t.........................................................
29
...............................................................................................
29
3.5.2 Lensing
4
T esting Inflation w ith th e M icrowave Background
32
4.1
In tro d u c tio n ...................................................................................................
32
4.2
Scalar Field I n f la tio n ...................................................................................
34
4.2.1
Slow roll, ns, and n s .......................................................................
36
4.2.2
Model s p a c e ........................................................................................
38
...........................................
39
4.3.1
C M B ..................................................................................................
42
4.3.2
Lym an-ct...............................................................................................
44
4.3.3
Error contours for current and future d a t a ................................
45
Testing In flatio n .............................................................................................
50
4.4.1
R e s u lts .................................................................................................
50
4.4.2
Projected errors and c ro s s-c o rre la tio n s ......................................
55
4.4.3
The Fisher m atrix a p p ro x im a tio n ................................................
57
4.3
4.4
4.5
D etermining how well experim ents can do
C onclusion ................................................................................
5 Lim its o f Dark Energy from C ross-C orrelations
62
64
5.1
In tro d u c tio n ....................................................................................................
64
5.2
Background t h e o r y .......................................................................................
65
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5.3
5.4
5.5
5.2.1
The ISW effect and dark e n e r g y ...................................................
65
5.2.2
Lensing co rrelatio n s..........................................................................
66
5.2.3
Galaxy c o rre la tio n s ...........................................................................
67
Calculating the sensitivity to dark e n e rg y ................................................
69
5.3.1
Choosing a form for dark e n e rg y ...................................................
71
5.3.2
Galaxy survey characteristics
.......................................................
77
R e s u lts ..............................................................................................................
79
5.4.1
Dark energy p e r tu r b a tio n s .............................................................
81
C onclusion.......................................................................................................
83
Bibliography
85
A S elected Inflationary M odels
92
B
94
Full-sky expressions for angular power spectra
C U seful M athem atical D erivations
C .l
97
Param eter m arginalization for a m ultinorm al distribution in m atrix
f o r m .................................................................................................................
97
C .2 A lternate derivation of the evolution of the density of a general cos­
mological f lu id .................................................................................................
100
C.3 Connection between two forms of the Fisher m a t r i x ............................
103
C.4 Specific lensing noise e s ti m a to r s ...............................................................
105
D Code
108
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List of Figures
4.1
Values of 77.5 and n s 'a t CMB scales for several inflation models
4.2
Error in the prim ordial power spectrum from a WMAP-like experiment. 40
4.3
Error in the prim ordial power spectrum from an after-Planck-like
experim ent.........................................................................................................
41
4.4 68 % confidence regions in th e n s - n s ' plane for various experiments.
46
4.5 68 % confidence regions in th e ns~ns' plane for various experiments
w ith a different prior......................................................................................
47
4.6 Combining CMB and Lyman-cr experim ents............................................
48
4.7 Combined constraints in the n s ~ n s plane from Lyman-cr and CMB
e x p e r im e n ts ....................................................................................................
51
4.8 M arginalized (one-sigma) errors of n$ (solid line) and n s (dashed
line) as a function of pixel noise and beam w id th ..................................
56
4.9 Cross-correlation coefficients for all the param eter pairings of four
different e x p e r im e n ts ....................................................................................
58
4.10 Plots of the angular power spectrum as a function of cosmological
param eters.........................................................................................................
60
4.11 Plots of the angular power spectrum as a function of power spectrum
param eters.........................................................................................................
61
5.1 Noise levels for CMB lensing reconstruction
.........................................
68
5.2 Response of the tem perature power spectrum to dark energy bin
param eters.........................................................................................................
72
5.3 Response of the ISW -lensing power spectrum to dark energy bin
param eters.........................................................................................................
73
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. .
37
5.4 Response of the ISW -galaxy power spectrum to dark energy bin
param eters........................................................................................................
74
5.5 Three dark energy eigenvectors for an ideal ISW -lensing-galaxy count
correlation experim ent.....................................................
75
5.6 E rror in wo and wa as a function of to tal galaxy num ber.....................
76
5.7 E rror contours (68.3%) in the w0~wa plane..............................................
80
5.8 The effect of dark energy perturbations on error contours (68.3%) in
the w0-vja plane...............................................................................................
82
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List of Tables
4.1
Noise param eters used for sim ulating various experim ents.................
45
4.2
M arginalized errors in n s and n s for simulated experiments with
different priors on the Taylor expansion cutoff........................................
50
M arginalized errors in n s and n s ' for sim ulated experiments with
different priors on the consistency relation...............................................
52
Inflationary model p a ra m e te rs .................................................................
93
4.3
A .l
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1
C hapter 1
Introduction
The first direct observation of the cosmic microwave background was made by
chance by Penzias and W ilson [54] in 1964. By then the theory of the microwave
background had already begun to be worked out, most recently before then by
Dicke, Peebles, Roll and Wilkinson[16], which added to previous work by Alpher,
Herman, and Bethe. T he next m ajor step in observation came w ith th e launch of
the Cosmic Background Explorer (COBE) satellite in 1989, which m easured both
the frequency spectrum[48] and the anisotropy[5] (angular power spectrum ) of the
microwave background. The first was im portant in confirming th a t the microwave
background was indeed black-body as theorized nearly from the beginning, and the
latter measurement was the first to confirm th a t anisotropies beyond the dipole
were present. These anisotropies continue to be of great interest today because of
how much inform ation they contain about th e physics of th e early universe.
Meanwhile, the theory of the microwave background has continued to develop.
The general theory of linear perturbations in a homogeneous and anisotropic back-
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2
ground cosmology was first worked out by Liftshitz[41] in 1946. Later this work
was used to develop a theory of the anisotropies in the microwave background, first
for baryons and photons alone by Peebles & Yu [52] in 1970, w ith later additions
by many authors, continuing even today. This work represents an investigation of
several details of the anisotropies of the cosmic microwave background. In partic­
ular, the focus is on the response of the microwave background to the conditions
of the universe at very early and late times, during b o th of which the expansion
rate is thought to accelerate.
The acceleration at very early times sets up the
initial fluctuations th a t form the initial conditions for all the rest of the physics
of the microwave background. These initial conditions m ay contain information
about physics a t high energy scales, possibly as high as 1017 GeV. The acceleration
at late times affects the microwave background by changing the expansion rate of
the universe, which in tu rn alters th e rate of collapse of large-scale structure. This
structure can induce secondary effects in the microwave background, and measuring
them could give some clues as to w hat is causing this late acceleration.
The microwave background is now understood to have its origins in the early
universe.
During the radiation era, any perturbations to th e otherwise smooth
background will oscillate due to the immense density and pressure at these early
times. After the universe cools and m atter becomes dom inant, the photon p ertu r­
bations travel freely to us while the m atter perturbations continue to collapse under
gravity to form galaxies and stars. The initial conditions for these perturbations
are set by inflation, which is introduced in chapter 2. T he effect of these initial
conditions on the microwave background is investigated in chapter 4. The work
contained in th a t chapter first appeared in [24], which uses numerical techniques to
determine in detail how well future microwave background m easurem ents will be
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3
able to determ ine inflation param eters. This leads to some new ideas about how to
distinguish different inflationary models.
T he evolution during the radiation era of the photon perturbations from their
initial state to w hat we see today is governed by a complex set of differential equa­
tions discussed in chapter 3. W hile the photons continue to travel almost freely
during the m atter era, changes in the expansion rate can subtly affect their distri­
bution. These effects and how to observe them are discussed in chapter 5, which
contains work originally published in [23]. T h at work uncovers some fundamental
limits to how well correlations between the microwave background and other tracers
of structure can determ ine th e expansion history of th e universe. This m ethod is
one of many th a t can be used to determ ine the nature of dark energy, and chapter
5 includes some discussion of how feasible it will be.
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C hapter 2
M odels of Inflation in th e Early
U niverse
2.1
In tro d u ctio n
Inflation is a way of getting around the flatness and horizon problems in cos­
mology. The flatness problem is simply th a t a flat universe does not seem to be
stable: any deviation in energy density away from precisely th a t needed for zero
m ean curvature should cause the significance of curvature to monotonically increase
in tim e relative to m atter or radiation. Since we observe a nearly flat universe, the
origin of this flatness is a puzzle w ithout inflation. The horizon problem is th a t for
a conventional universe dom inated by m atter or radiation, the maximum distance
a causal signal may have traveled since the initial singularity increases faster th an
the expansion of the universe. This means th a t for any pair of comoving points
today, one can always find an early tim e before which those points were not in
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5
causal contact. In particular, the points of origin for the microwave background
on opposite sides of the sky should not have been in causal contact at th e tim e
th a t the photons were em itted, which means th e extremely sm ooth nature of the
background is also a mystery w ithout inflation.
Inflation as a solution to these problems and others was first proposed by
Guth[25] and soon after improved and refined by several other authors[ 2 , 42, 43,
26, 4], In this chapter I will present an overview of inflation and of the models
typically used to generate it, as well as a derivation of inflation’s most successful
prediction: a nearly scale-invariant spectrum of prim ordial density perturbations.
Much of this treatm ent is similar to th a t of Lyth [45] and can be found in detail
in Liddle & Lyth [40]; I have attem p ted to collect together the pieces essential for
finding the density perturbations.
2.2
D efin itio n o f in flation
T he general spacetime m etric which has the property of being b o th homogeneous
and isotropic is known as the Priedm ann-Lem aitre-Robertson-W alker m etric
ds 2 = dtl - a (t)
dr 2
1 + kr2
( 2 . 1)
where a is the tim e-dependent (but not space-dependent) scale factor, and k =
—1 , 0,1 describes the type of spatial curvature (closed, flat, or open, respectively).
Prom this m etric and Einstein’s equations come two differential expressions for how
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6
the scale factor evolves w ith time
2
H2
87tG
k
( 2 .2 )
(2.3)
where G is N ewton’s constant, and p and P are the to tal average energy density
and pressure of the universe. I have not explicitly included a cosmological constant
in the above expressions, but one can be incorporated by including a component of
the universe w ith energy density A / ( 8 itG) and pressure —A /(87tG).
Inflation can be very broadly defined as any period during which the expansion
of the universe accelerates, i.e. an era during which a is positive. The requirement
for this is for the dom inant form of energy density to have an equation of state
w = P / p less th a n —1/3. Such an equation of state is sufficient to solve both the
flatness and horizon problems. More narrowly, inflation usually refers to a period
of accelerated expansion during the early universe, typically well before nucleosyn­
thesis but after the creation of undesirable relic particles. Also, inflation typically
refers to a period during which the dom inant energy density has an equation of
state w = —1 or very close to it.
2.3
A scalar field in gen eral r e la tiv ity
The action for a scalar field 4>w ith a potential V {</>) in a generic spacetime w ith
m etric
is
(2.4)
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7
where g refers to the determ inant of the metric. The energy density and pressure
of this field are
f = i r
w
+ vw ,
(2.5)
p = \ g “w9 ^ d r4, - V(<t>).
(2.6)
Prom this it is clear th a t if the field becomes dom inated by its potential energy
it will have the behavior of a cosmological constant, and if this energy density is
dom inant the universe will experience inflationary expansion. The Euler-Lagrange
equation of motion for the scalar field th en works out to
'< j > - ^ V 2<f>+ Z-i> + V'{<i>) = 0 .
cr
a
(2.7)
During an early period of inflation, two simplifying assum ptions are typically
made. First is to assume th a t the spatial geometry of th e universe is flat. Referring
back to equation 2 .2 , as long as the energy density dilutes w ith expansion more
slowly th a n 1/ a 2, it will dom inate th e behavior of th e universe and curvature will
rapidly become irrelevant. The second assum ption is th a t th e gradient term in
the equation of m otion vanishes, which will be true at later times even if it is
comparable to the other term s at earlier times. Since inflation is expected to last
for at least 60 e-foldings (representing an expansion of the scale factor by roughly
10 26), in typical models these two assum ptions are true to a high degree of accuracy
for all but the first few e-folds of inflation. Under these assum ptions along w ith the
assumption th a t the scalar field energy dominates, we can rew rite equation 2.2 and
the equation of m otion as
.\
a\
a)
2
,
1
3 Mj,
\4? + v w ] ,
4>+ 3 - ^ + V ,{(t)) = 0,
a
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( 2 .8 )
(2.9)
where I have now defined the Planck mass scale to be M p = (87xG)~1^2. These two
equations, together w ith a choice of potential, completely determ ine the (classical)
behavior of inflation.
2.4
Slow roll
“Slow roll” is the approxim ation where th e equation of m otion is taken to be
in the strongly dam ped regime w ith negligible <f> and small 0 relative to II, so
the equation of m otion simplifies to 3H<j) = —V'.
This also implies th a t V((f>)
dom inates the energy density, so in the slow roll approxim ation H 2 = P /3 Mp and
H is taken to be roughly constant. This leads to an exponentially increasing scale
factor a ~ exp(Ht), leading to a definition for the number of e-foldings A N = H A t
between two points of tim e during inflation.
There are several conventions for defining slow roll param eters. Here I will use
the convention of [40]:
MpV"
M tV 'V'"
( 2 . 10 )
T he slow roll approxim ation holds as long as these param eters are small.
The
motivation of this convention is so th a t e, 77, and £ will all be of similar order, which
makes collecting lowest order term s easier in complex expressions. Some authors
define slow roll param eters in term s of the Hubble param eter
these definitions
are equivalent to ones in term s of V (<f>) as long as the field is potential-dom inated,
because in th a t case H 2 ~ V. However, when the field has non-negligible kinetic
energy (such as very near the end of inflation) H 2 ~
+ V and the definitions
diverge.
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9
It will be useful later to express the derivatives of th e slow roll param eters with
respect to 4>in term s of other slow roll param eters
MP^
= r /V '2 i- ( 2 e ) I ,
M p ~T7 =
— r]V2e.
d<f>
\[2e
(2.11)
( 2 . 12)
The derivative w ith respect to (f>is related to scale factor during slow roll by
#
d in a
2.5
=
H
=
{2. 13)
V
Pow er sp ectru m flu ctu a tio n s
In the standard picture of inflation, small quantum fluctuations in the scalar
field have their length scale greatly increased by th e expansion of spacetime, so
much so th a t they rapidly become larger th a n the causal length scale and “freeze
out” . A fter inflation ends, the causal horizon can eventually catch up to these
length scales and the initial density fluctuations th en can grow into galaxies and
all the other structure th a t we see today.
If we separate the field into a homogeneous background piece and a perturbation
<p(x,t) = <j)(t) + 8<f>(x,t) and use th e equation of motion, then to linear order the
fluctuations themselves follow the equation (w ritten in Fourier space),
6(f) + 3-5<f> +
6(f>+ V"(4>)8<f) = 0.
a
\a J
(2.14)
Working through the details is made somewhat simpler by switching to conformal
time (defined by dr = d t / a ), and examining th e quantity u = a8(f). Also, we are
going to be interested in the fluctuations as generated on scales where th e potential
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10
is smooth, so the final term above will be dropped. After some m ath, the equation
of motion then can be rew ritten as
which has also made use of the fact th a t during inflationary expansion, r ~
(aH)-'.
Now th a t the equation of m otion has the form of a harmonic oscillator, quan­
tizing u(k,
t
)
proceeds in the norm al way; it is w ritten in term s of creation and
annihilation operators
u(k,
t
)
(2.16)
— w(k, r)a(k) + w*(k, r ) a ) ( —k),
where w ( k , r ) m ust satisfy th e same equation of m otion th a t u ( k , r ) does. This
leads to a solution for w(k, r), valid as long as the potential V (and hence H ) are
slowly varying,
w(k,
t
)
= —
.
(2-17)
(2.17)
Finally, w hat we’re after is the power spectrum of the fluctuations
e_
2
(I'M
2ir 2
2
27
+ 1 .
(2.18)
The am plitude of the fluctuations freezes out once their physical scale is much
larger than the horizon distance, and this am plitude is then w hat sets the initial
conditions for the universe after inflation. In this lim it k <C aH, so th e power
spectrum simply becomes ( H / 2 n ) 2.
The field perturbation 5<p is defined in a spatially flat slicing of spacetime. In the
comoving frame, the field pertu rb atio n vanishes by definition, b u t this frame has a
non-zero curvature perturbation. This curvature pertu rb atio n 77 is very generally
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11
related to the difference in tim e coordinate between the two frames St via the simple
relation 7Z = HSt. Using p S t = Sep, we can write the power spectrum P-ji of the
curvature fluctuations as
The field fluctuations originate during inflation, and as long as th e field is slowly
rolling during th a t tim e we can rew rite the power spectrum of curvature fluctuations
using the slow-roll param eters
H2
Pn = 8tr2M |e ’
(2'20)
where H and e are to evaluated a t th e tim e when the scale of interest exited the
horizon during inflation.
After inflation ends, the horizon grows faster th an physical scales and eventually
a curvature perturbation of a given wavelength will fall back into the horizon. W hen
this happens th e density pertu rb atio n of m atter or radiation at th a t scale will
depend on the curvature pertu rb atio n 1Z through its contribution to the effective
gravitational potential <f>. Density and the gravitational potential are related by
the Poisson equation
4 TrGSp = V 2$ ,
(2.21)
so in Fourier space 8 = 5p/p is given by
k\ 2_
4 = - U j
1
2 f k x2
*■
(222)
The gravitational potential itself is simply proportional to th e curvature p ertu r­
bation with a coefficient of order unity th a t depends on the details of the am ount
of various relativistic and non-relativistic species present.
Since each different
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12
wavenumber for the curvature perturbations falls back w ithin th e horizon at dif­
ferent times, this coefficient will be dependent on wavenumber. Thus the density
fluctuation power spectrum P$ of some species of particle after inflation is
p‘ =
ow(ii),T2{k)J k i ’
(2'23)
where T ( k ) is a transfer function normalized to unity on large scales th a t contains
the wavenumber dependence. The remaining constant of order unity is absorbed
into th e definition of the prim ordial power spectrum am plitude at horizon entry 5H
(sometimes w ritten as As) , defined by the expression
Pi =
2.6
(A)
T>(k)&l
(2.24)
Pow er sp ectru m am p litu d e and sh ap e
We now have the first im portant expression for connecting observations to th e­
ory, which is th a t the prim ordial power spectrum am plitude (observed through
density fluctuations at large scales) should be related to quantities related to the
potential energy density during inflation,
■>H
H
Mpy/e
(2.25)
Prom this several more shape param eters can be found. T he “tilt” or spectral
index of the prim ordial power spectrum of perturbations is defined to be
di n 5%
,
ns = 1 + i ^ k A given wavenumber k is
k/(aH)
( 2 ' 2 6 )
said to have left th e horizon during inflation
= 1, sod i n k = d in a.
W ith this in mind,
.
when
th e spectral index can be
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13
rew ritten
n s = 1 + —— In
a in a
1 —V 2 e — M p
(2.27)
= 1 + 277 —6 e.
The second derivative (first derivative of th e tilt) is
= —2£2 + 16ct7 - 24e2
2.7
(2.28)
C osm ic acceleration to d a y
Recent observations have led to the idea th a t th e expansion of th e universe may
have begun accelerating again sometime in th e recent (cosmologically speaking)
past, due to the influence of some hitherto unknown substance referred to as dark
energy. Once again, the idea th a t this effect might be due to a scalar field dom inated
by its potential energy is used to try and explain th e acceleration.
For recent
acceleration, this field is usually referred to as the quintessence field. There are
theories th a t unify the inflaton and quintessence fields, b u t in general this is difficult
to do since the energy scales involved are so far apart.
For quintessence, the quantum fluctuations would be unobservable and so the
development of section 2.5 is uninteresting. W hat is of interest and potentially
measurable is the equation of state for dark energy, and possibly even th e evolution
of large-scale fluctuations in its density.
These ideas are developed in the next
chapter.
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14
C hapter 3
T heory o f th e M icrowave
Background
3.1
In tro d u ctio n
The microwave background has become one of the m ajor sources for cosmo­
logical information in modern physics. In this chapter, I will provide background
m aterial for understanding th e theory of how the microwave background came to
look like it does today. Most of th e physics takes place during the radiation era,
when photons and baryons are strongly coupled together. The initial conditions
are a fairly simple set of param eters which can be obtained from an inflation model
(see chapter 2 for details), but because the microwave background’s deviations from
smoothness rem ain small even today most of the theory is completely linear w ith
respect to the initial conditions and can be discussed w ithout reference to them .
Large portions of this chapter essentially follow th e treatm en t of M a & Bert-
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15
schinger [47], with additional portions from Dodelson[17] and Hu[30]. However, I
have om itted m aterial not relevant to th e later chapters of this work, and have tried
to fill in some details and unify th e presentation to be consistent w ith the notation
used elsewhere in this work.
3.2
A n overview o f th e m eth o d
All techniques for solving for th e evolution of small perturbations in the uni­
verse sta rt w ith the Boltzm ann equation. The full phase space occupation function
f ( x , q , h , r ) for each species of particle is broken up into a background term fo(q)
and a first order perturbation f ^ ( x , q, h, r ) ,
(3.1)
where q is the m agnitude of the comoving particle m omentum , h is a unit vector
describing the direction, and r is conformal tim e1. The background distribution
fo(q) depends only on q because of th e assum ption th a t th e background is both
homogeneous and isotropic. It is also generally assumed th a t each particle species
started out at equilibrium in the early universe, and so the background distribution
has the form
(3.2)
where gs is the num ber of spin degrees of freedom, h is Planck’s constant, m is the
mass of the particle, k s is B oltzm ann’s constant, and T0 is th e tem perature of the
particles today (+ is for fermions, — for bosons).
1For th is chapter an d th is chapter only, for convenience overdots (such as in a) will represent
derivatives w ith respect to conform al tim e. T his is different from th e n o ta tio n of m ost of th e rest
of th is work.
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16
Because the photon perturbation is always small, we can define the tem perature
fluctuations of the photons A T /T w ith the expression
f { x , g, n, t ) = f 0 ^
>
(3 -3)
which means the photons have a therm al distribution w ith tem perature T + AT.
Combined w ith equation 3.1, some m anipulation reveals th a t as long as the tem ­
perature fluctuations are small
T- = - ( ^ f )
<3-4>
It will tu rn out later th a t the tem perature fluctuations are in fact independent of
g, so we can shift to Fourier space and expand n in Legendre polynomials to define
A e(k,r) as
f
AT
— (x,h,r)
.° ° ,
->
= / d3k e lk'x ]P (-i)* (2 f! + l ) A i (h,T)Pi (k ■
n).
(3.5)
e=o
Because the evolution of small perturbations is linear, different wavenumber modes
should evolve independently, and we can separate A e (k ,r) into a piece representing
the initial prim ordial perturbations 5a and a transfer function which depends only
on k = \k\,
A t (k, t ) = SH( k ) A e(k, r )
(3.6)
is the photon tem perature at th e origin (x = 0), today ( r =
W hat we observe
r 0), as a function only of the direction n. This function is th en usually expanded
in term s of the spherical harmonics
AT,
(n = ^ 2 ^ 2 aemYem(n),
( 3 . 7)
T
r=o m=—i
which, together with the previous expressions implies the following expression for
the spherical harmonic coefficients
aim
=
( -
0 ^471- [ cPkY?m (k)6H( k ) A e ( k , T = T0).
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( 3.8)
17
Finally, the results of observations are usually reported using the angular power
spectrum Ce of these spherical harmonics, which is defined by the expression
(O'imQ'l1m')
In theory, the angle brackets indicate an ensemble average over the probability
distribution of the prim ordial power spectrum (no theory gives exactly where the
density fluctuations occurred, only a probability distribution). In reality, however,
we can only observe one universe and the only averaging possible is over the 21 + 1
values of m for each multipole moment I.
Thus especially at low 7, there will
always be some uncertainty as to whether the m easured Ce is close to the “tru e”
average Ce for our universe. This uncertainty goes by th e nam e of cosmic variance.
Finally, combining the definition of the angular power spectrum w ith the relation of
the spherical harm onics coefficients to the photon distribution gives th e im portant
result
Ce = 4n
J
d3k P s ( k ) A j ( k , r = r 0),
(3.10)
where Ps(k) is th e power spectrum of prim ordial fluctuations.
T he basic procedure for calculating the Ce from a theoretical model, then, is to
obtain a set of coupled differential equations from the B oltzm ann equation for each
species of particle, and integrate the differential equations until today to solve for
A e ( k , t) . This, combined with a model for the prim ordial power spectrum (usually
given by inflation), gives the angular power spectrum CeHistorically, th e first numerical codes used the system of coupled differential
equations to solve for the photon distribution and hence A e(k, r ) , evolving all multipoles of interest forward in tim e until today. However, after recom bination the
universe becomes largely transparent to photons, and to lowest order they stream
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18
freely to us today. This means th a t the small angular scale (large £) fluctuations
we see today were really just larger scale (smaller £) fluctuations at the last scat­
tering surface th a t we see projected onto th e sky. Modern codes (starting with
C M B F A S T [59]) take advantage of this realization to evolve a much smaller set of
differential equations (because only lower multipoles are needed) and then project
the anisotropies onto the sky to find the angular power spectrum today.
3.3
C hoice o f gauge
To consider the evolution of perturbations during th e radiation era we need
to pick a coordinate system in which to m easure those perturbations. There are
two systems commonly used for tracking perturbations, each w ith advantages and
disadvantages. For this section and most of this chapter I will presume flatness of
the universe. Including curvature makes only minor changes to a few equations;
typically only equations for the m etric are directly affected, and I will note where
this occurs. Otherwise the most significant changes are in the expansions in spher­
ical harmonics; these functions simply change to their ultra-spherical counterparts,
which modifies the geometric factors in their recurrence relations.
3.3 .1
S y n ch ro n o u s g a u g e
The synchronous gauge is defined by requiring th a t all th e clocks of freely falling
observers agree (i.e. are synchronized, hence th e name). This is equivalent to requir­
ing th a t the tim e-tim e and time-space com ponents of the p ertu rb ed m etric rem ain
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19
identically zero. Hence the line pertu rb ed line element is
ds 2 = o 2(r) [—d r 2 + (<5y + /ip-(x, r ) ) dx ld x ^ ,
(3.11)
where r is conformal time, and the sym metric tensor h y ( x ,r ) is the perturbed
p art of the metric. This tensor has six degrees of freedom, and these are typically
divided up according to there spatial transform ation properties. Two degrees of
freedom transform as scalars, two as vectors, and two as tensors. The inclusion
of vector and tensor degrees of freedom is one of the strengths of the synchronous
gauge; i t ’s weakness is th a t the gauge conditions do not completely fix the gauge
degrees of freedom and so there will be some unphysical solutions for the density
perturbations. Almost all numerical codes work in th e synchronous gauge.
R ather th a n work in real space, i t ’s easier to use th e linearity of perturbations
to evolve each wavenumber k independently. The scalar p art of th e metric p ertu r­
bation can be Fourier transform ed
h-®caiar)(£, t ) =
J
d3k el*'x k i k j h ( k , r ) + ^
Qrj(k,r)
where h(k, r ) and r](k, r ) now represent th e two scalar degrees of freedom in Fourier
space. Vector perturbations decay for an expanding universe and are only sourced
by exotic forms of m atter (such as cosmic strings), so they will be ignored. Tensor
perturbations can propagate (these are essentially gravitational waves), bu t are
only sourced by prim ordial tensor fluctuations. Their contribution to the observed
microwave background tem perature anisotropy is mostly at low £, and even there
it may be impossible to separate from other effects, so I will not discuss them in
detail.
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20
3 .3 .2
C on form a l N e w t o n ia n g a u ge
T he conformal Newtonian gauge is m otivated by the Newtonian limit of general
relativity. The perturbed line element is
ds2 = a2( r ) [—(1 + 2 T ( x ,r ) ) d r 2 + (1 + 2 $ (T ,r ) ) d x ldxi\ ,
where the two fields
(3.13)
r ) and T (x, r ) are scalar potentials where in the New­
tonian limit 'h is simply the gravitational potential and $ = —T. Typically the
Fourier transform s of these fields are used directly in the evolution equations. The
advantages of this gauge are th a t the m etric remains diagonal and th a t the degrees
of freedom have a simple physical interpretation. The gauge degrees of freedom
are also fixed by this gauge so there are no unphysical solutions. The disadvan­
tage is th a t tensor degrees of freedom are not easily added to this formulation,
and there are some numerical issues which arise during com putation th a t make the
synchronous gauge somewhat easier to use.
3.4
E volu tion eq u ation s
3.4.1
M e tr ic
The background metric is still th a t of a Friedm ann-Lem aitre-Robertson-W alker
cosmology, so the evolution of the scale factor is given by the two equations (overdot
indicates a derivative with respect to conformal time)
a' 2
aJ
8-jrGa2
p,
3
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(3.14)
21
where the overbar indicates the background average of density p or pressure P.
Curvature can be included in th e first equation above as a component of the energy
density which scales as a -2 ; w ith this scaling the equation of state is P = —(1 /3 )p
and so curvature does not contribute to the second equation.
Plugging the perturbed m etric of equation 3.11 into E instein’s equation R ^ u —
\gnvR = T)W and carrying through to linear order gives a coupled system of equa­
tions for the evolution of h ( k , r ) and r](k,T)
k 2rj —
Z CL
= ^ k G oPST q ,
(3.16)
k 2f] = 4tt G aH V ST f,
(3.17)
h + 2 - h — 2k2r] = —87rG a 2hT/,
h + 677 + 2 ^ (h + 677) - 2k2rj = 24ttG o2 (kCkj - ^
3.4.2
(3.18)
[T] - ^ T fcfe), (3.19)
G e n e ra l p a rticles
The evolution of the particle distribution for each species follows the Boltzmann
equation
df
df
d f dx
d f dq
d f dn
. .
T r = T P T iT r + i T r + A T r = C^
^
where C represents any collisional term s th a t may be present. Recalling th a t we
split the distribution function into a background piece and a first-order term , we
wish to find the first-order part of th e Boltzm ann equation. T he first two term s are
easy, the fourth consists of two first-order pieces m ultiplied and so can be neglected.
The tricky one is the th ird term . The comoving m om entum q we have been using
does not include the metric perturbations and is thus not exactly equal to the
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22
conjugate m om entum to x. The true conjugate m om entum is
Pi =
V™3 -
(3 -2 1 )
Prom the requirem ent th a t the time derivative of the conjugate mom entum be equal
to zero we can find the first order expression for d q / d r ,
= -\qhijniU j.
clt
(3.22)
Z
Then the first order Boltzm ann equation (after dividing through by fo(q)) is
1 i
d l n f 0(q) _ l n [f l
2 q lj7llUj
dq
~ f0 ^ ’
<9/(1) ,
dr + e f
(0 oqN
(
^
or, after a Fourier transform ation
^
or
+ i «(jU )/<» +
d in g
e
= 1 C[/].
f0
(3.24)
M assless particles
For massless or extremely non-relativistic particles, either th e simplification
q —>e or q —►0 can be made. In either case, it is then simple to integrate out the
g-dependence of the Boltzm ann equation to obtain a new variable
g
S
S t d e M t m U n , T)
(3.25)
J g dq g /0(g)
Neglecting the collision term , the B oltzm ann equation in this variable is
dF
+ ik fiF = 4 rj — |( f i + 6fi)/i2
or
(massless),
(3.26)
(3.27)
where f i = k - n , and the factor of 4 comes from th e integral of d l n / 0/d ln g .
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23
Even further, the directional dependence can be decomposed into spherical har­
monics. A further assum ption th a t the m om entum distribution depends only on
the angle between k and h (i.e. th a t there is no externally specified preferred an-—+
gle for the particle distribution) means th a t F( k, h, r ) can be w ritten as a sum of
Legendre polynomials
OO
F(ic, h, r ) = 2£ + 1)Fe(k, r ) P e{k ■h).
e=o
(3.28)
The Fe(k,r) have been defined in such a way th a t they are simply the harm on­
ics of the (Fourier transform of) the fractional particle density, thus the density
perturbation S = 8p/p is simply equal to F0. The fluid velocity and shear term s
simply correspond to the dipole and quadrapole pieces of F as well. Using the
recursion relation for the Legendre polynomials, their orthogonality, and the fact
th a t P^{p) = |(3^r2 — 1), the Boltzm ann equation can th en be further reduced to a
hierarchy of multipole equations
dF
2
= —kFi — - h
f
= ^ ( F o - 2F2),
f)F
(massless),
(3.29)
k
a 7 = 2 7 T T ^ - ‘ - < < + 1>F« ! -
(<>2)
for massless particles. Ideally, massive particles are am enable to a similar treatm ent.
In practice the two im portant non-relativistic fluids are cold dark m atter, which
can be dealt w ith even more simply, and baryons, which couple to photons and
have significant corrections due to collision term s and finite (even if small) sound
speed.
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24
3 .4 .3
P h oton s
The starting point for the photon equations is the hierarchy of equations already
derived for ideal massless particles. The photons only differ through th e addition
of scattering term s proportional to aneaT (derived in [7]), where n e is the electron
density and aT the Thom son scattering cross-section. The scattering depends on the
photon polarization; in order to take this account properly we can introduce another
variable G(k, h, r ) which is the difference of th e two polarization components. The
original F ( k , h , r ) simply represents the sum over polarization states'.
The two
multipole hierarchies of equations are then
2
—— = —kF\ — - h
or
3
(photons),
(3.30)
? I ± = h F o - 2F2) + a n ea r (F1baryon - Fx),
or
o
f)P
k
4
1
—— = —(2Fi — 3F3) + — (h + 6fj) + — aneOT(Go + G 2 — 9 F2),
or
5
15
10
^o r
= IdI T+T T1 [ ^ - 1 -
~ ane ^ F e,
(£ > 2)
for the sum of polarizations, and
8G
1
—— = —kG\ + - a n eOT(F2 — Go + Gq)
or
2
(photons),
— — = —(Go — 2 G 2) —aneoTGi,
or
3
<9G
k
1
—— = —(2Gi — 3G3) + — arz.e<Jr(Go + F 2 — 9 G 2),
or
5
10
= 2Z T 1 {lGl- x ~ {£ + 1)G' +l] " an*UTG^
^ > 2)
for the difference.
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(3.31)
25
3 .4 .4
N e u tr in o s
Massless neutrinos follow exactly th e equations derived earlier for generic mass­
less particles. Massive neutrinos species, however, cause a bit of a problem since
interesting mass ranges (roughly .1-10 eV) correspond to a regime where the neu­
trinos are transitioning from relativistic to non-relativistic during th e radiation and
recombination era. Thus neither the massless nor extremely massive approxima­
tions are appropriate.
The traditional solution is to take the original distribution function of all vari­
ables f ^ ( k , q, n, t ) and define a new set of variables dq by expanding in the Leg­
endre polynomials
00
f {1\ k , q, n, t ) = ^ 2 ( - i ) e(2£ + 1
q, r ) P t (k ■n)
(3.32)
t=o
This results in a hierarchy of differential equations th a t depend not only on wavenumber (and tim e), but on particle m om entum as well
dTo
= —fc -T i +
e
6
dr
dr
= ^-(T
O6
o-
am q
(massive neutrinos),
(3.33)
2 T 2),
d^2
dr
cMe
d r = 2 ? y r f ( » ' - ■ - (« + 1)'f<+l1'
(0
2)
To relate back to things like the density or velocity pertu rb atio n one needs to
solve these equations and then integrate th e appropriate multipole moment over
q. Computationally, this can become cumbersome because not only does one need
to grid over wavenumber k bu t also over particle m om entum q, using enough grid
points th a t the numeric integration over q needed at the end can be done accurately.
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26
3 .4 .5
C o ld dark m a tte r
In the synchronous gauge, coordinates are defined by freely falling observers.
Since cold dark m atter m atter is by definition a pressureless fluid of particles th a t
interact only gravitationally, th e dark m atter particle coordinates can be used as the
definition of the coordinates in the synchronous gauge. Thus in the synchronous
gauge, the cold dark m atter particle velocities are effectively absorbed into the
coordinates, and the fluid can be described by a single equation
flT?
1
^
= -± h
(cdm),
(3.34)
where the factor of 1/2 is related to the equation of state.
3 .4 .6
Baryons
Baryons are massive particles, but their velocities do not necessarily match
those of the cold dark m atter and so may be non-zero in th e synchronous gauge.
They also interact strongly w ith photons before recom bination and thus a collision
term is present. It is a reasonable approxim ation, however, to neglect the higher
moments (t > 1) of the m omentum distribution. The two equations describing the
baryon distribution are
= -kF i - -h
or
(baryons),
(3.35)
= _ “ F 1 + c2A;Fo + ^ a n et7T(F 1 - F 1photoIl) )
a
3
where R is the background photon-to-baryon energy density ratio and
is the
square of the baryon
sound speed. For an isolated non-relativistic fluid the
speed would effectively
be zero, bu t interaction w ith th e photons results in an
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sound
27
im portant correction. The final term of the differential equation for F\ comes from
Thom son scattering; it is the m irror of th e corresponding term in the equation for
^photon aiKj ^ ie coefficieilt is the consequence of m om entum conservation.
3 .4 .7
D a rk e n er g y
If the observed cosmological acceleration is due to a cosmological fluid, th en it
too can be described with a distribution function and may have perturbations. For
a general fluid w ith a constant equation of state w which relates the background
density and pressure P = wp, an alternate approach to deriving the distribution
equations (see appendix C.2 for details) gives
(3'36)
- 3aa { % - W) S - { 1 + w ) ( e + > i ) '
Q = - - ( 1 - 3w)9 - - ^ — e + k 2 (
ay
’
1+ w
\ l + w5p
- a] ,
J
where 5 = Fo = 8p/p, 0 = T \ / ( l + w) and is th e divergence of the fluid velocity,
and a = F2/2, sometimes called the shear p ertu rb atio n 2.
The above pair of equations is not closed, and a different derivation is needed
to provide differential equations for a and higher moments of the distribution. For
dark energy, however, it is sufficient to take th e lim it k —►0 as dark energy models
m eant to approxim ate the behavior of a cosmological constant tend not to fluctuate
strongly on small scales.
The above equations also illustrate some of th e problems faced when considering
dark energy perturbations. For an isentropic fluid, 5P/5p is the square of the sound
C o n v e n tio n s v ary on th e definition of th e shear p ertu rb atio n .
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28
speed cs, which in tu rn is equal to
o
Cs =
dP
dp
—
dw
w + p—
dp
(3.37)
As long as re is a fixed constant, c?s = |u>| and the above differential equations
simplify to (for large scales)
-(1 + w) \ 0 +
h
(3.38)
6 = — (1 — 3w)9.
CL
There would already seem to be causality problems for models w ith w < —1, how­
ever if w is slowly varying (as it does in many dark energy models) the sound speed
as defined above formally diverges as w approaches —1, simply because for w = —1
the density p becomes constant and thus d w / d p diverges. This poses difficulty b o th
in model building and numerical calculation of dark energy perturbations.
3.5
Line o f sigh t effects
As stated in section 3.2, the most common technique to solve for the microwave
background anisotropies today is to evolve the entire coupled system of equations
described in the previous section up until the after the photons decouple, and
then basically project the result onto th e sky we see today.
After decoupling,
photons travel almost completely freely until we observe them , experiencing only the
cosmological redshift as the universe expands. However, there are two corrections
to this picture due to secondary effects during this tim e th a t will be im portant in
later chapters.
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29
3 .5 .1
In te g r a te d S ach s-W o lfe effect
Even if the photons have decoupled from other m atter, they still feel the effects
of gravity. Once we have the distribution function F for photons after decoupling
we could project it onto the sky to find the tem perature anisotropies today, except
for the presence of the h term in th e evolution equations. Because of this, the full
expression for the anisotropies today will contain an integral over h, or in the more
familiar conformal Newtonian gauge, a term th a t looks like
/■today
/-today
F dr = [other terms] + /
/
J decoupling
(T — $ )e lfc^ T-Ttoday)d r
(3.39)
Jdecoupling
After decoupling, $ = - $ to a high degree of accuracy and so this effect amounts
to an integral over —2$.
3 .5 .2
L en sin g
The integrated Sachs-Wolfe effect directly changes the tem perature of each pho­
ton as it travels through changing gravitational potentials. G ravity can also deflect
the photon paths themselves, leading to a shift in the observed anisotropy power
spectrum. This effect is most easily analyzed using w hat is called th e projected
potential <f)(h), which is defined in such a way th a t the deflection angle of a photon
observed in direction n is equal to V<p{n). W ith this definition, the projected poten­
tial in term s of the full three-dim ensional gravitational potential (in the Newtonian
gauge) is
/•today
(j>(n) = - 2
d r ^ ( r ) $ ( f ,r ) ,
(3.40)
J decoupling
where the integral over r is equivalent to an integral over distance for th e photon
traveling in direction n, and
is a geometrical factor th a t expresses the lensing
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30
efficiency. The factor of two is the same factor of two th a t always arises when com­
paring photon deflections in general relativity to th a t of the Newtonian prediction.
T he relationship between the lensed and unlensed tem perature perturbations is
therefore
A y le n s e d
A T
— y — (ft) = - jr (ft+ W )
(3.41)
AT
AT
= — (ft) + V ^(ft) ■V — (ft)
+ i v . « f t ) V ^ ( n ) V iV ' ^ ( f t ) + ■■. ,
including up to second order term s. The full analysis using spherical harmonics
involves some difficult m athem atics, but it is possible to capture the physics by
using a Fourier transform instead. This is equivalent to treatin g th e sky as a plane
(the fiat-sky approxim ation), and is valid as long as the angles of interest are small.
The transform is defined by
f(i) =
J
dhf(n)e-'l \
(3.42)
so the lensed tem perature perturbations are
- A £ ■k - A
1 ff
2 JJ
where
= l\ +
« , A T (- }
(2tt)4 T
j
^
<3-43)
^^
— I.
The power spectra are defined by
( 2
7
(2tt)2<5(£--
i)CjT = ( h f
,
.
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(3.44)
31
Plugging in the expression for the lensing tem perature anisotropies to find the
lensed tem perature Ci is made easier by sticking to second order or lower terms, in
which case there are only four combinations of the term s from equation 3.43. Also,
the term th a t has the form
AT1*
f H2f
AT1
J
\
4) 4 • ( « - « ) ,
(3.45)
will be equal to zero, since the delta functions in the definition of the spectra force
£x = £ = £' for this expression. This leaves three term s, which after accounting for
all the delta functions leads to
lensed C j T = C j T +
f£ x
£x ■ ( £ - £ ( )
( 2t t ) 2
2
c r ^<ir«i
c t!
(3.46)
n2r TT f d£2 p4_n 4><i>
-2
The final term simply acts to remove power added by th e second term , as a conse­
quence of th e familiar notion th a t lensing should conserve photons over the whole
sky. Thus given a relatively sm ooth lens power spectrum , lensing effectively smears
out features in the tem perature angular power spectrum .
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32
C hapter 4
T esting Inflation w ith the
M icrowave Background
4.1
In tro d u ctio n
Over the last several years, extraordinary observational support has emerged
for the idea th a t key features of our universe were formed by a period of cosmic in­
flation. During inflation, the universe enters a period of “superlum inal expansion”
which im prints certain features on the universe. The physical degree of freedom
responsible for inflation, generically called th e “inflaton” , has yet to find a com­
fortable home in fundam ental theory, and there are many competing ideas for how
fundam ental aspects of inflation could play out. None th e less, at the phenomeno­
logical level a standard picture of inflation has emerged. For details, please refer to
chapter 2 of this work.
From the observational point of view, th e standard picture is defined by a set
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33
of observable characteristics th a t are the same across virtually all proposed models
for th e inflaton. The most well known predictions from the standard picture of in­
flation are th a t the universe has critical density (to w ithin roughly one part in 105),
th a t the prim ordial perturbations are coherent, (leading, for example, to acoustic
peaks in the microwave background power spectrum ), and th a t the power spec­
tru m of prim ordial perturbations is nearly scale invariant, w ith th e tilt param eter
n s constrained to be close to unity. A unique spectrum of coherent gravitational
waves is also predicted, which could eventually come w ithin range of direct gravi­
tational wave detectors, and which could also be observed indirectly via signals in
the microwave background polarization.
B ut inflation makes many more predictions th an these. Specifically, a given
model for the inflaton will predict a detailed shape for the prim ordial power spec­
tru m th a t goes way beyond w hat can be described simply by a single tilt param eter.
The detailed shape of the power spectrum is a reflection of the particular evolu­
tion of the inflaton during inflation, som ething th a t is precisely specified in a given
model. In this chapter we show how th e next generation of experim ents could bring
studies of the power spectrum shape to a whole new level. These studies present
two kinds of opportunities: One opportunity is to make additional tests of the
standard picture of inflation. To this end, we focus on particular power spectrum
features th a t are known to exist across essentially all inflation models. The search
for these features could either confirm or falsify the stan d ard picture of inflation.
The second opportunity is to go beyond broad tests of the standard picture.
The next generation of experiments which we consider here will provide im portant
additional information. This inform ation could actually distinguish among different
specific inflaton models, assuming th e stan d ard model is not falsified, or it could
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34
provide very useful constraints on the alternatives if th e standard model is ruled
out.
Our approach here is similar to and inspired by a large body of earlier work on
this subject [13, 36, 39, 33, 12, 18, 34, 27, 28]. Our emphasis here is identifying
w hat useful inform ation about the prim ordial power spectrum and inflation might
be revealed by a new generation of experiments. For this work we assume th a t
the prim ordial perturbations are adiabatic.
As emphasized in [9], relaxing this
assum ption would result in more degeneracies and would lead to somewhat weaker
constraints on param eters.
The organization of this chapter is as follows: Section 4.2 gives background
inform ation about slow roll inflation. Section 4.2.1 introduces the aspects of slow
roll inflation we intend to test. Section 4.3 discusses the CMB and Lym an-a d a ta
(existing and simulated) we use to test inflation. Section 4.4 gives our m ain results
and 4.5 gives our conclusions. A ppendix A gives details of the inflation models we
use for our plots. This work was first published in [24].
4.2
Scalar F ield In flation
In the standard picture, inflation occurs when the potential energy density V (</>)
of a scalar field (j) (the inflaton) dom inates th e stress-energy [25, 2, 42, 43]. This
scalar field may be a tru e scalar field or an effective field obtained from some more
complicated theory. The period of potential dom ination is usually closely connected
to very slow evolution of the inflaton field, the so-called “slow roll” behavior, and
it is this slow evolution th a t produces a nearly scale invariant spectrum of p e rtu r­
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35
bations. In the slow roll inflationary scenario, however, 0, and therefore V(<j>), are
not completely constant during inflation, and this leads to deviations from total
scale invariance.
The dynamic behavior of the potential is determ ined by the equation of motion
for a scalar field in an expanding universe
0 + 3 H(f> + V'{4>) = 0.
(4.1)
The gradient of the field is ignored, as even if present it will be quickly damped
by the inflationary expansion to th e degree th a t it is irrelevant for th e classical
evolution of the background spacetime, which is w hat we determ ine from Eqn. 4.1.
The field is considered to be in a slow roll regime if 0 is negligible. The Hubble
constant H = a /a is related to the to ta l energy density of th e universe which if
dom inated by th e scalar field is
(4.2)
where M P = 1/ x/SttG has been set to unity. It is custom ary to define slow roll
param eters such as
(4.3)
although several other conventions also exist in the literature. Assuming th a t these
param eters (and the higher derivatives of V ) are small leads to expressions for the
primordial am plitude of density p ertu rb atio n s1 [3]
(4.4)
1T he am plitude squared A $ 2{k) is sim ply k 3Ps(k) from c h ap te r 3
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36
the spectral index of these perturbations
d l n k A s 2(k)
ns{k)=
dink
= 1 + 2’> - f e >
<4 -5>
and the derivative of the spectral index
n s '{k) =
flln l
= —2£2 + 1677c - 24e2.
(4.6)
The right-hand side of each equation above is to be evaluated during inflation at
the tim e when the scale of interest k exits th e Hubble radius.
4 .2 .1
S low roll, ns, an d n s
A general feature of slow roll is th a t ns' is higher order in the slow roll param ­
eters th an ns- Thus if we assume higher order term s become increasingly small,
then barring a conspiracy of cancellation between term s Iris'| ~ (ns — l ) 2 or less.
As pointed out in [19], while this assum ption is commonly m ade it is an addition
to the common assum ption th a t the slow roll param eters are small, at least when
formally considering “the space of all possible inflation models” . We emphasize
here th a t in practice the slow roll hierarchy between n s and (ns)' is indeed realized
in the vast m ajority of published models, so detection of a large ns', while not
completely ruling out slow roll, would force a rethinking of the standard picture of
inflation. This relation can be generalized to higher derivatives, resulting in a kind
of consistency relation
dnns
|n + l
< It s - 1
d in kn
(4.7)
which could be taken as defining a kind of ‘norm al’ class of inflationary models.
For inflationary scenarios involving m ultiple fields (often called hybrid models)
this condition is relaxed. W ith multiple fields the extra freedom introduced makes
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37
0.08
n
1
r
j
L
i- . r l.
“i
r
~\
r
J
L
0.06
0.04
0.02
0.002
n
“i
r
r
J
L
n
r
j e r
0.001
□
........
0
0
J
l
0.02
I
L
□
0.04
ln s -
J
I
0.06
L
0.08
!|
Figure 4.1: The values of n s and n s ' at CMB scales are plotted for several models.
The lower graph is a zoomed view of th e b ottom of the upper graph. In both, th e
upper line is ns' = n s — 1 and th e lower line is ns' = (ns — l ) 2. The thick line
on the lower graph shows the evolution of n s and n s w ith scale for two orders of
m agnitude in each direction.
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38
it easier to remain on the edge of violating the slow roll conditions over many efoldings. Thus the tendency for models to lie under the curve |n s'| = (ns — l ) 2
is not as strong for hybrid models. 'Of course, theories which generate prim ordial
density perturbations from something other th an inflation also have no need to
obey the above consistency relation.
To discuss observational constraints, we take the point of view of [63] th a t
the prim ordial power spectrum is an unknown function, which may be sampled by
experiments at one or more scales. Statem ents about th e slope of this function (and
higher derivatives) then can only be tested by effectively sampling the function to
high accuracy at several nearby scales. C urrent analysis tends to use all th e d ata to
provide only lim ited inform ation about th e power spectrum. We wish to emphasize
th a t higher quality d a ta over a range of scales will allow us extract significantly
more inform ation about the prim ordial power spectrum , inform ation th a t can have
a great impact on tests of the inflationary picture.
4.2 .2
M o d e l sp ace
One of the simplest models to evaluate is a pure exponential. For V = V0 exp(A0)
the spectral index n s — 1 —A2 for all scales, and thus ns' and all higher derivatives
are zero. Thus for this model m easurem ents of n s simply m ap into constraints on A,
w ithout presenting an opportunity to falsify the general model. B ut a measurem ent
of ns' which excludes zero can rule this type of inflaton potential altogether.
This model is special in th a t the potential is constructed to form th e simplest
possible power law spectrum of perturbations. Most inflationary models have more
complicated forms, but many proposed models approxim ate th e exponential behav­
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39
ior on the scales which cosmological measurements probe.
A previous survey of models of inflation and their spectral index can be found
in [46]. In figure 4.1 several different models taken from a sampling of the literature
have been plotted. (Specific inform ation about each model plotted is given in the
Appendix.) The types of models range from high-order polynomial to mass-term
to brane-world inspired scenarios [40, 20, 26, 60, 43]. Despite the difference in
the form of each m odel’s potential, almost all of these live on or below the line
\ns '\ = (ns - l)2.
C ertain models can exhibit more exotic behavior, such as the running-mass
model described in [14] (an example this type of model is m arked by the star on
Figure 1), or the interesting type of potentials given by Stew art and Lyth[19]. These
can give |n s'| ~ \ns — 1| over a range of scales, resulting in a markedly different
prim ordial power spectrum . These models form an im portant ‘altern ate’ class of
models which will be easy for future d a ta to confirm or rule out.
4.3
D eterm in in g how w ell ex p erim en ts can do
To find the possible im pact of CMB and Lyman-a: experim ents, we model the
primordial power spectrum w ith a Taylor series expansion of the spectral index
around a particular scale
/ 7 \ - l + n s + n s ' x + \ n s " x 2+...
A s 2(k) = P ( p )
.
(4.8)
where x = In(k/k*), k* is th e pivot point, and P is an overall normalization. We
then use Fisher m atrix techniques to jointly estim ate param eters for each experi­
ment.
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40
101
1 1 1 11111--------- 1— 1 1 1 11111
1
1 1 11 m i i
1— r
I I 1IIII
I I I I I I14-
A
r*
10 °
id "
T
t
itI
10"1
s
w
<
10-2
10-3
10-4
10-4
I
I
I
10- 3
10"2
I
I
I
1
I I I 1 I
1
10- 1
1
I I I
I
I I I I I
I
I
10 °
1
I
I I I I
101
k ( h M p c -1)
Figure 4.2: One-sigma error in the (binned) prim ordial power spectrum from
a WMAP-like microwave anisotropy tem perature and polarization experim ent
(squares w ith solid error bars), and from a Lyman-o: experim ent as in [15] (crosses
with dashed error bars).
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41
10°
io
02
-1
CO
W
02
C
<
*0
10 °
k ( h M p c -1)
Figure 4.3: One-sigma error in the (binned) prim ordial power spectrum from
an after-Planck tem perature and polarization experim ent (squares), and from a
Lyman-o: experiment w ith to tal uncertainties one hundred tim es smaller th a n cur­
rent experiments (triangles).
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42
Param eterizing the power spectrum a particular way has its own set of advan­
tages and disadvantages. One nice feature of the step-wise form of [64] is th a t it
is easy for each bin am plitude to be a nearly statistically independent param eter
in th e likelihood analysis. The coefficients of a Taylor series expansion generally
have larger covariances. The disadvantage of th e step-wise form, however, is th a t
th e quantities of interest to us (such as th e spectral index) are not simply related
to th e shape param eters.
4 .3 .1
CMB
We first consider CMB experim ents which measure b o th th e tem perature and
polarization anisotropies. For scalar perturbations there are three power spectra
( T E C)
described by C\ ' ' , where (T, E, C) indicate the tem perature, ZTmode polariza­
tion, and cross-correlation power spectra. These Q all have similar dependence on
the prim ordial power spectrum , and are found by
Cl = (4x)2
J
" A s 2( k ) |A ,(k,
t
=
To) | 2
,
(4.9)
where A ?(k,T = r 0)are transfer functions for th e CMB, and A s 2(k) is the squared
am plitude of the prim ordial power spectrum . The functions A e ( k , r = To) depend
on cosmological param eters, and can be conveniently calculated using th e CMBFAST
code [59].
Error in a cosmological param eter Sj can be estim ated as y j (F -1 )^, where F is
the Fisher m atrix
*■«=
£
I,J=(T,E,C)
E g ( 0 - ' ) 4
i
3
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(4.10)
43
The covariance m atrix C has elements which can be approxim ated by [36, 65]
CtT
=
( U + l ) U y (Cj + N l ) 2 '
(4U)
c “
-
W
(4 1 2 )
Ccc
=
^
[ c f + N f f ’
(2£ + l)/.kj.
x [ ( C f )2 + [Cj + Nj) (Cf + N f ) ] ,
(4.13)
for the diagonal elements, and th e off-diagonal elements are
^
In equations
4 .1 1
C tc
=
c-
=
through
4 .1 6 ,
(4 1 4 )
( 2 1 + 1 ) U y C‘ { C l + N ' ) '
(415)
( 4 ' 1 6 )
f sky is th e fractional sky coverage of the experiment,
and we have defined a noise term
n
where
<J(t ,e )
( T, e )
2
til
(O.4250fwhmq 2
— 0(T,JS)Pfwhme
i
(
a
1?\
is the noise per pixel in th e tem perature and polarization measurements
and #fwhm is the w idth of the beam. For experiments like W M A P which obtain
tem perature and polarization d a ta by adding and differencing two polarization
states, the noise per pixel for each is related by o r 2 = cte2/2.
The derivatives dC t/ d s i are evaluated via finite difference using a numerical
code derived from CMBFAST and DASh [37]. We consider only flat models, using as
param eters
2the acoustic angular scale £&,£lmh2, exp(—2 r), the
prim ordial
power spectrum norm alization P, and the first seven coefficients in th e expansion
2 For an analytic expression for £a , see [29], These cosmological p aram e te rs correspond closely
th e th e A , B, A4, and Z p aram eters of [38].
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44
of n$. We use 0.05h M pc-1 as the pivot. The Fisher m atrix calculation for the
errors in the param eters also assumes a “tru e” model around which the derivatives
are taken; for this we use a ACDM model with fh,h2 = 0.0222, Q,mh2 = 0.136,
Qa = 0.73, r = 0.1, and a flat prim ordial power spectrum .
4.3.2
Lyman-ct
For the Lyman-a: data, the error bars th a t were reported in [15] for the linear
m atter power spectrum are used. The prim ordial power spectrum is related to the
linear m atter power spectrum by
PLu(k) = P0k A s 2( k ) T 2(k),
(4.18)
where T 2(k ) is the transfer function and contains the dependence upon cosmological
param eters, and Pq is a normalizing constant. T hen th e error bars for the power
spectrum param eters are calculated via standard error propagation techniques using
the previous equation and the analytic form for the transfer function [51]
=
[q)
ln (l + 2.34 q)
2.34q
x [l + aq + (bq)2 + (c q f + (dq)4]
,
(4.19)
where q= k / ilti2 exp(—2fh) and a , b, c, d are fit param eters which are irrelevant to
the error analysis. Weuse the results of the CMB param eter estim ation as inputs
for determ ining the errors in h, fl, and Q(,.
The large system atic norm alization error reported in [15] is a problem for esti­
m ating prim ordial power spectrum am plitudes, bu t does not affect local estim ates
of the slope or higher derivatives, so we do not include it.
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45
Table 4.1: Noise param eters used for simulating various experiments.
Experim ent
W MAP-like
PLANCK-like
SPT
CMB-pol
4 .3 .3
aT
20 fxK
10 /rK
12 fjK
3 /iK
6>fwhm
18’
6’
0.9’
3’
/ sky
0.7
0.7
0.1
0.7
l max
1500
3000
4000
3000
Error co n to u rs for cu rren t and fu tu r e d a ta
An illustration of CMB and Lyman-o: constraints on th e prim ordial power spec­
tru m is shown in figures 4.2 and 4.3. If we fit a function to all the d ata points,
and assume th a t function to be linear th en of course th e slope will be tightly con­
strained. If we allow the function to have a more complicated shape, the slope
at any point becomes less well constrained. Figure 4.2 roughly represents current
experim ental limits. The main point of this chapter is th a t future d ata can become
good enough to loosen the assumptions on the shape and still produce very tight
constraints. Figure 4.3 gives an illustration, by showing constraints on the binned
power spectrum form some future experim ents accurate enough to clearly distin­
guish a model w ith n s — 0.05 from a model w ith rts = (0.05)2, even using d ata
spanning only one order of m agnitude in wavenumber.
The ultim ate limiting factor in how precise all these measurem ents can be is due
to cosmic variance. For the CMB, the fractional error from this effect is A C t/C t ~
1/y/I , which m eans even for large i each individual Ci can only ever be known to
w ithin a few percent. Thus the error in figure 4.3 is m ostly cosmic variance limited,
and the only way to further reduce the error is to assume some smoothness for th e
prim ordial power spectrum and bin the d ata. Figures 4.2 and 4.3 use a step-wise
param etrization to constrain th e prim ordial power spectrum in bins in In k, which
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46
0.05
P L A N C K -lik e
■SPT
■CMB-pol
0
- 0 .0 5
0.05
0
n0 -
0.05
n fidu cial
Figure 4.4: 68% confidence regions in the n s ~ n s plane for various experiments of
the sort described in table 4.1. The SPT constraints are shown dashed, PLANCKlike dotted, and CMB-pol constraints solid. We use prior 1-6 as described in section
4.4. The thin lines are constraints w ithout polarization information.
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47
0.05
■PLANCK-like
■SPT
’f id u cial
■CMB-pol
0
- 0 .0 5
0.05
0
ns -
0.05
n£ducial
Figure 4.5: 68% confidence regions in th e n s~ ns plane for the same experim ents
as in figure 4.4, only now using prior lib as described in section 4.4.
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48
0.08
CMB
■p rojected
0.06
^S SSW M A P ext
0.04
0.02
—
f
0
0
0.02
- - i~r i
0.04
ns -
0.06
0.08
1
Figure 4.6: Solid contours show constraints in the ns ~n s' plane from the CMB
d a ta of the “CMB-pol” experim ent of table 4.1 (using prior 1-6) for scales near
k = 0.0565. Several different fiducial models are shown, two representative of the
types shown in figure 4.1 and one large-ns' model of th e type described in [19]. The
dotted contours are the result of assuming th e consistency relation of equation 4.7
and extrapolating from CMB scales to Lym an-a scales (k = 2.39). If the relation
is valid, constraints from th e two types of d a ta will be correlated in this plane as
illustrated by the pairs of adjacent solid and dashed contours. All contours are
drawn enclosing regions of 68% confidence level.
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49
is meaningful as long as the power spectrum is a sm ooth function of wavenumber
relative to the bin width. For the best CMB d a ta points, which here occur near
scales corresponding to t ~ 1000, th e bin w idth is roughly equivalent to binning a
few hundred Ci together.
From figure 4.3 we can see th a t the CMB accurately probes the primordial
power spectrum at somewhat larger scales th an those where th e Lym an-a d ata is
most accurate. This means the two experiments provide constraints on n s and ns
at different scales, allowing us to further test our models by looking at how they
predict these quantities should change w ith wavenumber. We will explore this idea
further in section 4.4.
To then make error contours in th e n s - n s ' plane, we use the covariance m atrix
from the param eter analysis to marginalize over other param eters and determine
the covariance m atrix for just n s and n s ' ■ Of th e four hypothetical CMB experi­
ments listed in table 4.1, we show how well the last three place constraints in the
n s - n s plane (for two different priors) in figures 4.4 and 4.5. We examine the fourth
experiment (the best) in detail (using th e weakest prior) in figure 4.6. These results
are discussed at length in the following section. For hypothetical Lym an-a experi­
ments, we do not understand the physics connecting th e prim ordial power spectrum
to measurements as well as for the CMB. We therefore “sim ulate” improvements in
experiments as an overall reduction in statistical uncertainty due to larger samples,
and an improved knowledge of the transfer function due to decreased errors in h, Q,
and f4. For Lym an-a experiments to provide constraints com petitive w ith those
expected from Planck will require datasets roughly 100 tim es larger th an current
ones, and more im portantly, an understanding of system atics (or at least those th a t
affect estim ates of n s and ns') down to the percent level. Given th a t these sys-
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50
Table 4.2: M arginalized errors (xlOOO) in n s and ns' for sim ulated experiments
w ith different priors on truncating the Taylor expansion of the spectral index. Power
spectrum param eters are amplitude, tilt (ns), and for column I-ra, the first m
derivatives of the tilt.
Experim ent
WMAP-like
w polarization
PLANCK-like
w polarization
SPT
w polarization
CMB-pol
w polarization
prior 1-6
Sns , Sns '
(xlOOO)
prior 1-3
5ns , 5ns '
(xlOOO)
prior 1-1
5nS) 5ns '
(xlOOO)
309,
203,
20.3,
16.0,
24.6,
15.2,
9.43,
5.42,
258,
195,
16.6,
12.4,
21.5,
10.7,
8.28,
4.08,
69.1,
67.8,
11.6,
6.90,
16.3,
6.37,
6.17,
2.32,
274
260
13.9
10.7
18.6
13.8
6.71
4.37
152
132
10.7
7.48
14.6
8.67
5.67
2.42
33.9
20.6
9.79
3.94
14.5
4.78
5.60
1.75
tem atics represent a lack of understanding of the (scale-dependent) light-to-mass
and baryon-to-dark m atter ratios, such an improvement may not appear soon. We
hope to study the error for future Lym an-a experim ents in more realistic detail in
future work.
4.4
T estin g Inflation
4 .4 .1
R e su lts
For completeness, we discuss a range of possible priors, each of which represents
a different points of view on w hat one wants to take as an assum ption and w hat
one is trying to test. In tables 4.2 and 4.3 we report the constraints th a t various
CMB experiments can place on ns and ns'. Our weakest prior, which we refer to as
1-6, is simply to use the four cosmological param eters (I a , &bh2, Om/i2, r) and eight
power spectrum param eters (P and the first seven coefficients in the expansion of
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51
0.08
■ L y m a n -a
•p r o je c te d
0.06
■ com b in ed
0.04
0.02
0
0
0.02
0.04
ns -
0.06
0.08
1
Figure 4.7: C onstraints in the ns ~ns ' plane from the Lyman-ct d ata of figure 4.3
for scales near k = 2.39 (thin large ovals). The projected ovals from CMB scales
(dotted ovals, from Fig. 4.6) are then combined with the Lym an-a d a ta to form
improved constraints (thick small ovals). Again, 68% confidence level regions are
plotted.
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52
Table 4.3: Marginalized errors (xlOOO) in n s and n s ' for sim ulated experiments
w ith different priors on the higher derivatives of the spectral index. Prior 1-6 from
Table 4.2 above is for reference. Priors Ha and lib use all six derivatives in the
joint param eter estim ation, bu t Ila imposes the constraint th a t all higher (ns" and
above) derivatives are w ithin 0.01 of scale invariant, and lib imposes the consistency
relation (Eqn. 4.7) on the higher derivatives.
Experim ent
WMAP-like
w polarization
PLANCK-like
w polarization
SPT
w polarization
CMB-pol
w polarization
prior 1-6
5ns , Sns '
(xlOOO)
prior Ila
5ns , 5ns '
(xlOOO)
prior lib
Sns , Sns '
(xlOOO)
309,
203,
20.3,
16.0,
24.6,
15.2,
9.43,
5.42,
97.8,
93.1,
12.3,
7.50,
16.6,
7.36,
6.81,
3.38,
70.6,
69.4,
11.6,
7.26,
16.5,
7.00,
6.36,
3.13,
274
260
13.9
10.7
18.6
13.8
6.71
4.37
38.9
34.1
10.1
5.14
14.5
5.19
5.72
2.26
34.6
22.4
9.79
4.60
14.5
4.80
5.61
1.82
ns). Having so many param eters for the power spectrum allows the shape to vary
quite a bit, and loosens constraints on each term of the expansion. We include so
many param eters not so much because constraints on all of them will be interesting
(some, in fact, will probably always be unm easurable), bu t to show the effect various
assumptions about them will have on th e constraints on n s and ns'. If the reader
dislikes these param eters, our prior 1-1 is equivalent to not including them at all.
To get b etter constraints requires either using a more restrictive prior or im­
proving the experiment. In table 4.2 we change th e prior by using fewer power
spectrum param eters. Prior 1-3 uses only th e first four term s of the expansion of
ns (i.e. up to third order), and prior 1-1 uses only the first two terms, such th a t
the only power spectrum param eters for prior 1-1 are am plitude, ns, and ns'. In
table 4.3 we change the prior by placing a priori constraints on th e higher derivative
term s of the expansion, rather th a n dropping them completely. Prior Ila supposes
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53
th a t all the higher derivative term s (ns" and higher) are “small” , less th an 0.01.
Prior lib also imposes the constraint th a t the higher derivatives be “small” , but
supposes they fall off in the form of th e consistency relation of equation 4.7.
Prior 1-6 represents the weakest assumptions, and prior 1-1 the tightest as­
sumptions. (Prior 1-1 basically adds ju st one new param eter, ns', to the canonical
set.) Prior lib represents an assum ption of the standard inflationary picture for
the higher derivatives, b u t places no prior constraint in th e n s — ns' plane so those
param eters can be used to test th e standard picture. (Of course, if the the standard
slow roll picture fails, prior lib may no longer be of interest)
In the tim e since th e preprint of th e paper describing this work first appeared,
the W M AP collaboration announced their results [6]. O ur predicted error for ns
and ns' of 0.069 and 0.034 matches up quite well to their reported errors of 0.060 and
0.038, (using our prior 1-1, which m ost closely m atches the “W M A Pext” analysis
of [62],)3 W hile the W M A P results are not statistically very significant for our
purposes, we plot the error contour in figure 4.6 for comparison.
(Note th a t if
the central value does not change much as the d a ta improve th e implications for
inflation will be very interesting.)
We have sim ulated CMB experim ents w ith and w ithout polarization measure­
ments. Since the prim ordial power spectrum affects th e CMB tem perature and
polarization in exactly the same way, naively polarization simply adds a second
way of measuring the same thing and should only reduce uncertainty by a factor
of V2. However, allowing joint estim ation of other cosmological param eters in ad­
dition to those describing the shape of the prim ordial power spectrum introduces
3O ur characterization of th e W M A P noise and beam size is som ew hat m ore pessim istic th a n
th eir rep o rted num bers, b u t we suspect th is is com pensated by fu rth er experim ental details we
do not include.
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54
confusion and near-degeneracies. The value of measuring polarization is not so
much th a t it directly puts limits on the power spectrum , b u t in reducing the confu­
sion w ith other param eters. O ur analysis shows the increasing value of polarization
as CMB experiments improve.
In figures 4.6 and 4.7 we see the error contours in th e n s - n s ' plane for a hy­
pothetical super-Planck CMB tem perature and polarization experiment and for a
“super” Ly-a; survey. We see th a t such d a ta would provide very significant con­
straints in this space. In particular, these experim ents are good enough to clearly
distinguish points on the line n s — n s — 1 from the line n s' = (ns — l ) 2 for all
but very small values of ns, and thus would offer significant tests of th e standard
inflationary picture.
Combining d a ta from bo th experiments will provide additional constraints and
tests. Each experim ent provides constraints in the n s - n s ' plane, b u t on somewhat
different scales. These am ount to providing constraints on th e inflaton potential
V(4>) near a particular wavenumber k. There are several possible options for com­
bining d ata from several experiments. One approach is to use a single param etrization for the prim ordial power spectrum and then to jointly estim ate all param eters
using the full dataset.
If both experiments were at th e same scale, this would
am ount to simply overlapping their individual error contours. To perform joint
estim ation for experiments at different scales, we would want to find param eters
th a t are “good” across different scales, b u t this conflicts w ith our aim to test how
good (i.e. constant) a param eter n s really is across a large range of scales. Also,
a more immediate concern, is th a t different experim ents often have a (sometimes
poorly characterized) system atic error in th eir relative norm alization which causes
problems for a joint param eter analysis. For a recent discussion of these issues, see
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55
[21].
For simplicity, then, we choose a second option, which is to look at the two
experim ents separately and then use th e consistency equation (Eqn. 4.7) to produce
the inequality
Ts(^Lya) — Ts(^CMb)
/ /,
N 1\2
■. ,
— 7—j------- < (n s (khya) - l ) z
In k iya
— In kcu B
as an additional inflationary test.
/, on\
(4.20)
By looking at only derivatives of the power
spectrum we avoid the relative norm alization problems. Visually, this am ounts to
projecting the CMB contours for n s and ns' up to Lyman-a: scales (or vice-versa)
and checking to see if the contours overlap, as shown in figure 4.7. This third
test is not redundant because we use different pivot points {k\jya and /ccmb) for
the different datasets.
A potential w ith large higher derivatives could pass the
ns' < (ns — l ) 2 test at a particular scale and yet fail it when d a ta from different
scales is used.
Finally, we would like to rem ind th e reader th a t for figures 4.6 and 4.7, th e real
inform ation is in the size of the error contours rath er th a n their actual placement. In
the absence of real data, we show contours from sim ulated d a ta for a small sample of
models all of which show correlations between th e two experim ents consistent w ith
the standard inflationary picture. U ltim ately n atu re will tell us if such correlations
are really there.
4 .4 .2
P r o je c te d errors a n d c r o ss-co r rela tio n s
To further show the value of b e tte r experim ents, we have investigated how the
projected errors in n s and ns' should change as a result of improving both resolution
and noise levels. Figure 4.8 shows the marginalized errors (with prior 1-6) w ith and
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56
0.03
0.02
1111»*LLi
0.01
0
4
2
£
6
10
8
ctt ( f i K )
0 .0 3
0.02
0.01
0
1
2
3
4
5
6
0 fwhm ( a r c m i n u t e s )
Figure 4.8: M arginalized (one-sigma) errors of n s (solid line) and ns' (dashed line)
as a function of pixel noise and beam w idth. Values w ith(w ithout) polarization are
shown as thick(thin) lines. For the lower plot, beam w idth has been fixed at 6',
and for the upper noise has been fixed at 10/rK.
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57
w ithout polarization, as functions of beam w idth and pixel noise.
A source of worry is possible cross-correlation between our power spectrum pa­
ram eters and other cosmological param eters. Figure 4.9 shows th e cross-correlation
coefficient for the experiments of table 4.1 as defined by
r
- =
_______ ( F
' I h _______
(A91f
A “good” experiment will be able to distinguish param eters and ten d to have low
correlations between them , however a very noisy experim ent will also have low cor­
relations between param eters (presuming th e noise is uncorrelated) so this figure
m ust be interpreted w ith some care. Nevertheless, some interesting features can be
pointed out. Confusion between tilt and optical depth is well-known, and polariza­
tion helps greatly a t reducing such confusion. We did find th a t there is generally a
correlation between I a and our power spectrum param eters (of which n s is of the
most interest). The correlation seems to arise because power spectrum param eters
can combine to mimic a slight horizontal shift of a peak. M easuring th e locations of
multiple peaks makes this conspiracy of power spectrum param eters more difficult,
however, and for the best experim ents th e correlation disappears. For the near
future, an accurate determ ination of th e angular scale of the sound horizon at last
scattering will be im portant for placing constraints on inflationary param eters.
4 .4 .3
T h e F ish er m a tr ix a p p r o x im a tio n
One possible source of error in our calculations stems from the approxim ations
th a t go into the Fisher m atrix analysis technique. For low i, th e covariance m atrix
C is dom inated by cosmic variance, and equation 4.10 can be rew ritten in term s
of a new variable Zt = lnC^. As pointed out by Bond et. al. [8], these Zt are
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58
A§ n s n s ’ n j ^ n ^ n ^ n j p n j , 8) u\>um ** T
# • • • • ■ • o «0
A§ n gn g’n lp J n ^ n ^ n ^ n ^ ^ b “ m 1A T
0
•#•••0000
• 0 0 • O o o •
• 0 0 • O O o o
...............................................O
0o of o• O° o•O• o• O00• o°
0 * 1 8 OO O o • O
« O 80 • • • •
• O O • 0 0 0 - • '
■• • 0 • • • •
O O • 0 0 0 •
• o o • 0 0 0 ■
• Oo • 0 0 0
° ° • 0 0 0
. oo• # 0 0 •
•
0 O0 °
• 0 • O• • • • 0O0O
•
OO....................O0O0
O • • • *• O 0O *
0 o o •
•0O 0 °
• 0 • • • • 8 # O 0 O
■0000
Oo °
WMAP-like
0•
...............o
o • • ■ 0 • O l •
0 »
o0 • O
• • 0 • •
O •0 •o • •
O
O
•
•
o O • P8la*n •c k -lik' 0e * 0
^
O
O
o 0 . O
• • 0
• •
• O
• • °
• O • 0 8OO o • • o
•
•
0 - 0 - o
• • 8O • 0 0
o • • O o 0 0
• • 8O O 0 0 0
8 8 • o o • 0 0
o
•
o
••
•o
• °
• •
O 0 O
•00
•
SPT
° o o
O °00*
•
•
• •
. O
•
8 • • o
8
•
•
O
O
O 80
•
0O •
•
O^
8 8O•
•0-0-00
•
^
o
0
O •
o 0O
• • O 0
.................
CMB-pol
Figure 4.9: Cross-correlation coefficients for all th e param eter pairings of four
different experiments. Filled circles represent positive correlations; open circles
indicate negative correlations. The radius of each circle is proportional to the
cross-correlation coefficient r^. For reference, note th a t
= 1 for all diagonal
elements.
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59
b etter variables th a n the C\ for Fisher analysis in th e sense th a t they are Gaussian
distributed. For large enough I, the difference between using Zi and Ct is minor,
however.
Ju st having variables th a t are Gaussian distributed is not enough; the variables
should respond linearly to the param eters. Linearity in response to the cosmological
param eters is a well-studied problem and has been discussed extensively in the
literature (see [38] for a recent discussion). W hat remains is to check our power
spectrum param eters. We have done this by examining Ct as a function of the
power spectrum param eters.
Figures 4.10 and 4.11 show Ce as a function of param eters for several choices
of I. The vertical scale is arbitrary; th e thing to look for in these figures is th a t
ideally dCe/dSi should be constant, so th e slope of each line should be as well.
For the ranges we use for each param eter, deviations from linearity are small for
most param eters. The least linear param eter is the angular scale £a , which is not
surprising since varying this param eter prim arily shifts the power spectrum (which
is oscillating) left or right. Even so, for smaller changes in
the slope becomes
close to linear, and the worst deviations are at high m ultipole moments where the
signal-to-noise ratio drops. Only the region w ithin the central five points was used
for the Fisher m atrix construction.
For the power spectrum param eters we are m ost interested in (ns and ns'),
deviations from linearity are less th a n one percent for param eter values w ithin all
the errorbars th a t we report in tables 4.2 and 4.3. For th e higher order param eters,
the worst deviations from linearity occur only for small £ (~ 10), for which we
found the linear approxim ation to be valid w ithin one percent for param eter values
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60
0.02
0.022
0.024
0.13
0.135
0.14
nm
mh z
300
305
1.
310
0.78
0.8
0.82
0.84
0.86
e x p (-2 -r)
Figure 4.10: Plots of the angular power spectrum as a function of cosmological
param eters for several choices of t. The vertical axis of each plot is Ci in ar­
bitrary units; the different lines in each plot are for (from dark to light colors)
I = 2 0 ,2 2 0 ,4 2 0 ,.... The horizontal axis for each plot is a different cosmological
param eter: baryon density (upper left), m a tte r density (upper right), angular scale
(lower left), and optical depth (lower right).
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61
96
-
98
0 . 0 2 - 0.01
100 102 104
0
n.
iiIiii ih
0.01 0.02
-
0.96 0.98
1
0 . 0 2 - 0.01
0
1.02 1.04
0.01
0.02
na
Figure 4.11: Plots of the angular power spectrum as a function of power specrum
param eters for several choices of i. The vertical axis of each plot is Ce in ar­
bitrary units; the different lines in each plot are for (from dark to light colors)
t = 20, 220,420,___ The horizontal axis for each plot is a different power spectrum
param eter: am plitude (upper left), tilt (ns) (upper right), first derivative of the
tilt (lower left), and second derivative of th e tilt (lower right).
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62
less th a n around 0.02. In our calculations, we varied the higher order param eters
by am ounts several times smaller th an this, and while the resulting errorbars were
sometimes larger th a n this, at these small i the error from cosmic variance is also
large, so the effect on the overall param eter analysis is small. We checked this
by redoing the param eter analysis w ithout including low t values at all, and the
qualitative results of our work do not change. For higher t (~ 200 and above), the
linear approxim ation remains good up to param eter values of order unity.
4.5
C on clusion
We have shown th a t the next generation of cosmological experiments should
determine the shape of the prim ordial power spectrum sufficiently to allow new
tests of the the standard picture of inflation. If the stan d ard picture is upheld,
a new level of differentiation among different inflaton potentials will be possible.
We have investigated the potential im pact of new d a ta on b o th the CMB and the
Lym an-a forest.
The Lym an-a d a ta offers a promising route to testing slow-roll models both
on its own, and in conjunction w ith CMB data. C urrently published d ata does
not get too far w ith this enterprise, bu t next-generation observation could have
considerable im pact4.
For the CMB data, if we are interested in general constraints w ithout placing
restrictive priors on th e prim ordial power spectrum , Planck-like CMB experiments
4As this work was com pleted we learned th a t th e Sloan D igital Sky Survey is preparing to re­
lease a new L y m an -a d a ta se t. W hile n o t as large as our survey which is “n ex t generation”L y m an -a
d a tase t sim ulated in th is chap ter, it m ay have considerable im portance to th e issues raised in this
chap ter [58]
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63
do not quite have the precision to pu t the strong limits on ns' th a t we desire. It
is not until relatively high t th a t the uncertainty from cosmological variance is low
enough for our requirements, and it is precisely at these high I th a t the Planck ex­
perim ent noise rapidly becomes dom inant so a further improvement beyond Planck
is needed. Our hypothetical “CMB-pol” experiment should start placing interest­
ing constraints in the n g- ng plane. Also im portant is the measurement of the
(E-mode) polarization channel, which is vital to reducing degeneracies th a t make
the tests more challenging.
For constraints on the power spectrum itself (as opposed to other cosmological
param eters), inform ation from low multipole moments (£ < 500) contributes very
little due to cosmic variance. However, coverage of a reasonable fraction of the sky is
needed to retain high resolution in £, and simply to beat down statistical noise. The
proposed South Pole Telescope (SPT) may do well in this regard. Higher multipole
moments are useful up until Silk dam ping reduces th e overall CMB signal. As
CMB experim ents improve, polarization will become more im portant as the key
to breaking degeneracies between th e effects of the power spectrum shape, which
affects tem perature and polarization identically, and other cosmological param eters,
which generally do not.
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64
C hapter 5
Lim its o f Dark Energy from
C ross-C orrelations
5.1
In trod u ction
U nderstanding cosmic acceleration is one of the largest problems facing physics
today. So far the most direct m easurem ents of acceleration have come from distanceredshift measurem ents [55, 56, 35]. This acceleration is thought to be due to the
effect of dark energy, a new form of energy density th a t dominates the current
universe. However, a universe w ith dark energy exhibits a different evolution for
density perturbations, and this has observable consequences.
In this chapter, I will discuss how well certain m easurem ents of growth can
constrain the expansion history of the universe, and thus dark energy. In particular
I will discuss the lensing of the cosmic microwave background (CMB) by intervening
structure combined w ith measurements of galaxy num ber density correlations from
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65
surveys. These types of measurements are attractive since they can be obtained
essentially “for free” from experiments already planned.
I will also discuss the
sensitivity of these measurements to the presence of perturbations in the dark
energy fluid, which can have an im portant effect on the results [10]. In this chapter
“lensing” refers specifically to lensing of the CMB by foreground structure, rather
th an th e extremely productive research area of studying the shape distortion of
background galaxies due to lensing from foreground galaxies (the theory of which
is covered extensively in [32, 61]). This work was first published in [23].
5.2
B ackground th eo r y
5 .2 .1
T h e IS W effect a n d d ark en er g y
As photons travel from the last scattering surface (LSS) to us, they fall into and
climb out of potential wells th a t lie along their path. If the gravitational potential
$ does not change w ith time, th en the accompanying blueshifts and redshifts will
cancel each other out, leaving no net effect. However, if the potential does change
over tim e there may be some overall change in each p h o to n ’s wavelength, and hence
the observed tem perature. This is the integrated Sachs-Wolfe (ISW) effect, w ith
the change in tem perature A T ISW observed in a direction n expressed simply as
(5.1)
where the gravitational potential is w ritten as a function of position x and lookback
distance D (used as a proxy for conformal tim e).
For a flat, m atter-dom inated universe, th e potential rem ains constant over tim e
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66
even though the density perturbations themselves do not. This means th a t any
ISW effect originates either from early times, when the density of radiation was
still significant enough to affect the expansion rate, or from late tim es when dark
energy became dom inant. Thus if th e late ISW signal can be separated out from
others, it provides a clean measurement of dark energy.
• Normally the ISW effect itself is buried by the prim ary tem perature anisotropies
from the LSS. However, the prim ary anisotropies are set at the last scattering
surface, at a completely different epoch and at different length scales th a n those of
the structure growth responsible for the ISW effect. The prim ary anisotropy should
thereby be uncorrelated w ith the ISW effect and other m easurem ents of growth.
Thus cross-correlating other m easurem ents w ith CMB tem perature m aps can be
a useful tool for bringing out inform ation about growth, as has been discussed
recently by [1, 53] and dem onstrated by [57].
5.2 .2
L en sin g co r re la tio n s
The first measurement I will consider correlating w ith th e tem perature map is
a measurement of gravitational lensing of the CMB. The microwave background
is gravitationally lensed by m atter th a t lies between us and the LSS. The m ap of
photon deflection angles over the sky can be w ritten as the gradient of a scalar field
cj) called the projected potential, which depends on the 3D gravitational potential
$ as
(5.2)
where Ds is the distance to th e last scattering surface.
Just as w ith CMB tem perature maps, th e m ap of the projected potential can be
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67
decomposed into spherical harmonics. Then a two-point function of the modes can
be put together to construct an angular power spectrum. The expression for the
power spectrum can be w ritten as a line-of-sight integral over th e 3D gravitational
potential
r
/ _
n\ 2
Cf * ~ f ddDDdPtk.
DD<&2( kD) ,( T>
DS )° p m
(5,3)
k-tZ#
W hile this equation is only exact in the flat-sky (large £) limit, th e im portant feature
is th a t it captures the physics of how the angular power spectrum depends on a
line-of-sight integral of the gravitational potential m ultiplied by a kind of window
function and the primordial (i.e.
the growth function has been separated out)
power spectrum P${k). The exact expressions and further details can be found
in appendix B; they are used for the com putations in this chapter for multipole
moments where the difference is im portant.
However, for th e rem ainder of this
section I will w rite only the Lim ber-approxim ated integrals for clarity. The cross
correlation between the tem perature and lensing is due to th e presence of the ISW
effect in the tem perature, and thus its power spectrum has th e simple flat-sky form
Cj*~
5 .2 .3
J d D D 9 (k ,D )i{ k ,D )^ P -P m )
(5.4)
k=e2*
G a la x y co rrela tio n s
The second type of observation I will consider is counting galaxies projected
on the sky. On large scales, fluctuations in the num ber density of galaxies should
track fluctuations in the gravitational potential (possibly w ith some bias). Prom
the map of num ber density over the sky, we can obtain th e auto-correlation power
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68
100 nK
c\2
t—
H
+
< 1 0 nK
< 1 0 nK E - m o d e
10
100
1000
I
Figure 5.1: Noise levels for CMB lensing reconstruction. The dark curve is th e
lensing power spectrum for a typical cosmology. The different do tted lines are noise
levels for CMB tem perature experim ents w ith a 4' beam and 0.1 /iff noise per pixel
(dotted line) or 0.01 / i K noise per pixel or less (dashed line). The lower dashed line
is for an E-mode CMB polarization measurement.
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69
spectrum
Cf ~
J
dDD<S>2( k , D ) n ] { D ) P l ( k ) \ k_ t!g ,
(5.5)
which depends on the potential as viewed by some window function n g(D) describ­
ing the distribution of galaxies th a t are actually observed. The galaxy spectrum
should be correlated b oth w ith the ISW p a rt of the tem perature spectrum
C j 9 ~ / d D D $ ( k , D ) $ { k , D ) n g(D)P%{k)
k=eQ
(5.6)
and w ith th e lensing potential power spectrum
C f-
(5.7)
dD D3 >\k, D ) n g( D ) ^ ^ - P f k )
k=et£
Again, in the calculations described later these Lim ber-approxim ated integrals are
used only for high multipole moments.
The exact expressions are used for low
multipole m oments (£ < 100).
5.3
C alcu latin g th e s e n sitiv ity to dark energy
The power spectra themselves are com puted numerically using the techniques
described in the previous section and a version of the CMBFAST code [59] which I
modified to ou tp u t lensing spectra and other information.
The basis for all the analysis is the Fisher inform ation m atrix. Given a fiducial
model, the Fisher m atrix describes how sensitive the model is to changes in its
param eters. First, all the power spectra are p u t together into a covariance m atrix
/ c j T + NT
C/ =
cT*
c 7g
c f
c f + Nf
C f9
c f
cf9
\
C f + JVf )
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(5.8)
70
Experim ental noise can be modeled as a contribution to the diagonal term s of
this m atrix. In our case I take the noise contribution to the prim ary tem perature
anisotropy N j to be negligible. However, even a small am ount of noise is significant
when reconstructing lensing information, and I take this into account using the
derivations for lensing error as a function of tem perature noise found in [49] and
shown in Fig. 5.1. I take the prim ary contribution to the noise for galaxy surveys
N f to be shot noise (bias for each survey is treated as a free model param eter).
Some other possible noise sources are discussed in section 5.3.2.
W hen I consider combining three galaxy surveys at different redshifts, each
galaxy survey gets its own row and column, resulting in a 5 x 5 covariance m atrix.
The Fisher m atrix is then constructed as a sum over multipole moments,
(5.9)
where the s* are labels for the actual model param eters.
T he model param eters actually varied are the dark energy param eters discussed
in th e next section, plus an angular scale param eter £a , baryon density Qbh2, m atter
density Qmh2, prim ordial power am plitude A s and tilt ns, optical depth r , and a
bias param eter bi for each galaxy survey. All models were constrained to be flat.
I investigated two fiducial models; b o th had param eters equivalent to a m odern
concordance cosmology (fid.e. = 0.75,
ns = 0.9).
= 0.04, flcdm = 0.21, h — 0.65, r = 0.1,
The difference between the two was in the dark energy evolution;
one model had a “pure A” equation of state of w = —1, whereas the other used
a quintessence-type model w ith w = —0.9 today running smoothly to w = —1
at high redshift. The latter has a somewhat enhanced sensitivity to dark energy
param eters due to the evolution of dark energy at low redshifts, and is used for
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71
reporting the final constraints on dark energy param eters in section 5.4. Using the
“pure A” model instead degrades the dark energy param eter constraints by about
20% overall.
5.3 .1
C h o o sin g a form for dark en erg y
Different experim ents have differing sensitivities to dark energy. W ith no knowl­
edge of the fundam ental physics behind cosmic acceleration, there is little reason to
favor one functional form for the equation of state w(z) over another. In th a t light,
principal mode analysis is useful for revealing w hat the experim ents can actually
constrain. In such an analysis one chooses some basis set of eigenfunctions (cut off
to a finite set size), and then expresses the fundam ental modes for each experiment
as combinations of basis functions. In our case the basis functions will simply be a
set of 25 boxcar functions covering the redshift range from z = 0 to z = 5.
Figures 5.2-5.4 show how the various power spectra respond to changing the
equation of state w(z) w ithin a bin. The top panel of each figure shows th e fiducial
Ce, and the different lines on th e lower panel are dC i/d w i where i labels the bin,
and each line is for a different i. Lighter colored lines are for higher redshift bins,
and the bin centers have a spacing of A z = 0.25. From th e figures it is possible
to see when dCijdWi ~ dCe/dwj, which means th a t those two bin param eters are
impossible to distinguish using th a t power spectrum alone.
A few example eigenvectors are shown in Fig. 5.5.
T he solid line connects
the am plitude for each of th e 25 w(z) bin param eters.
In general, dark energy
information is mixed w ith other cosmological param eters so th a t the eigenmodes are
complicated combinations of several param eters. However, w ith the Fisher m atrix
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72
O
£o
<
1x10
10
100
m ultipole
1000
Figure 5.2: Response of the tem perature power spectrum to variations in the dark
energy equation of state. The top panel shows th e fiducial C f T , and the b ottom
panel shows d C j T/ dwi where Wi labels the value of w(z) in each redshift bin. The
bins have a width of A z = 0.25 and bins from z = 0 t oz = 2.5 are shown, with
lighter (more red) color indicating higher redshift. In this figure it can be seen th a t
other th an at low i, the effects of different dark energy param eters on the power
spectrum are nearly degenerate.
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73
7--- 1--- 1—TT
T----1—I I I I I |
■e-
E- —
<
10
100
multipole
Figure 5.3: Response of the ISW -lensing cross-power spectrum to variations in
the dark energy equation of state. The top panel shows the fiducial C j ^ , and th e
bottom panel shows d C j ^ / d w i where Wi labels the value of w(z) in each redshift
bin. The bins have a w idth of A z = 0.25 and bins from z = 0 to z = 2.5 are shown,
w ith lighter (more red) color indicating higher redshift. For this figure the effects of
different dark energy param eters are much less degenerate th an for the tem perature
power spectrum.
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74
o
T----1—I I I I I
TTTF
£?_
o
<1
10
100
multipole
Figure 5.4: Response of the ISW -galaxy power spectrum to variations in the dark
energy equation of state. T he to p panel shows the fiducial C j 9, and th e bottom
panel shows d C j 9/d w i where Wi labels th e value of w(z) in each redshift bin. The
bins have a w idth of A z — 0.25 and bins from z = 0 t oz — 2.5 are shown, with
lighter (more red) color indicating higher redshift.
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75
i
i
i
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i
i i i
i
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1
r e d s h if t
Figure 5.5: Three dark energy eigenvectors for an ideal ISW -lensing-galaxy count
correlation experiment. The left p art of each plot shows th e 25 w(z) param eters,
and on the right are the 9 other param eters (in order): angular scale I a , baryon
density
m atter density Vtmh 2, optical depth r, prim ordial power spectrum tilt
ns, primordial power am plitude A s , and the bias param eters for each galaxy survey
bi,b2,bz. The error in each eigenmode for a cosmic-variance lim ited experim ent is
shown for each mode.
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76
(5w
1
(5w
0.2
1 0 10
1 0 11
1 0 12
n u m b e r of g a la x ie s
Figure 5.6: Error in wq and wa as a function of to ta l galaxy num ber with lensing
noise fixed at the “< 10 nK E-mode” level from Fig. 5.1. The solid lines w ith
squares are for a survey with three redshift bins, dashed lines w ith triangles for a
survey with one high redshift bin, and d otted lines w ith stars for one w ith a single
low redshift bin.
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77
in hand not only can one perform principal mode analysis, one can also marginalize
over some param eters and switch to more familiar dark energy param eters with
simple m atrix operations. For example, it is simple to switch from the 25 w(z)
param eters to the uwan param etrization [11, 44], which takes the form
%
w(z) = w0 + Wa — — .
(5.10)
1 “r Z
The full eigenmode analysis shows th a t even for ideal experiments, no more th an
two or three dark energy param eters will be well-constrained, so for the rest of this
chapter the “wa” param eterization will be assumed.
5 .3 .2
G a la x y su rv ey c h a ra c te ristic s
My intent is to approxim ate the behavior of galaxy surveys which do not include
spectra, b u t are able to obtain approxim ate photom etric redshifts through color
information. I therefore consider the situation where one has a large num ber of
galaxies th a t can be assigned to one of three redshift bins, centered around z = 0.5,
z = 1.0, and z = 1.5. Each of these bins is approxim ated as a rounded boxcar
function similar to the technique of Hu and Scranton [31]. This approxim ation
is best for survey slices th a t are not severely m agnitude or volume limited. For
those cases the overall effect is to smear out the edges of the bins, weakening the
advantage of having several redshift bins.
Galaxy surveys contain numerous system atics.
Stellar contam ination should
not be a m ajor problem since for faint surveys the m ajority of observed objects
are galaxies. M odern m ethods can reduce stellar contam ination to a few percent or
sm aller[50], and stars tend to sm ooth out the galaxy clustering rath er th a n strongly
bias it in a particular direction, so th e overall effect on th e angular correlation power
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78
spectrum (which is already nearly smooth) should be weak.
More worrisome is th a t the analysis presumes knowledge of the redshift distri­
bution of galaxies in the sample, which of course could be in error. We can estim ate
how small the overall redshift calibration error needs to be by converting the red­
shift error into an error in the angular scale of galaxy correlations, and examining
the change in the correlation power spectrum compared to th e shot noise. Because
of the large angular scales we’re interested in, it tu rn s out th a t even for galaxy
surveys with 1012 galaxies (corresponding to roughly 104 per square arcm inute),
shot noise at these scales is still large enough th a t an overall redshift calibration
error of up to a percent is tolerable.
Another worry is th a t the distribution of dust, which can b o th obscure galaxies
and have its own long-wavelength emission, likely has its own anisotropies which
may dom inate the small galaxy num ber count anisotropies we are interested in
measuring.
For cross-correlating w ith the tem perature spectrum , however, the
angular scales of interest are quite large (~ 10°) and one can be hopeful th a t dust
anisotropies on such large scales will be relatively local and it will be possible to
measure and correct for them , as in [57]. For the rem ainder of this chapter I will
assume any systematics in the galaxy correlations are below th e shot noise level
(which itself is quite lim iting), bu t new experiments may well reveal im portant new
information about w hat system atics to consider.
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79
5.4
R esu lts
In the absence of noise, over 40% of the improvement in dark energy constraints
gained by adding CMB lensing d a ta to the tem perature inform ation comes from the
CT
f cross-correlation channel. Even though in the ideal case th e noise in the CMB
and lensing potential anisotropies is uncorrelated, th e inversion of the covariance
m atrix in equation 5.9 means th a t noise in the auto-correlation spectra will find
its way into the cross-correlation channel. This then means th a t the noise in th e
reconstructed C f ^ spectrum will be th e limiting factor as to how well CMB lensing
can constrain dark energy.
In Fig. 5.1 I have calculated the noise in th e reconstructed C f lensing power
spectrum for several different types of experiments. At small enough tem perature
noise levels (a bit above 10 n K per pixel for 4' pixels) all of the noise in the lensing
C f is in fact coming from cosmic variance in th e tem perature C j T . Note th a t even
at this limit, the noise in the lensing spectrum is still roughly th e same order of
m agnitude as the signal. P a rt of th e low signal-to-noise ratio appears to be related
to the choice of a red tilt (ns < 1) in the prim ordial spectrum of the fiducial
model. This reduces CMB power at th e small scales from which th e lensing signal
is reconstructed, leading to more relative noise in the lensing power spectrum . The
to tal noise in the lensing power spectrum can also be reduced further by combining
several polarization modes as per [49], which results in an improvement by a factor
of a few, so it may not be necessary to drive th e detector tem perature all th e way
down to 10n K to get the desired precision. In any case cosmic variance is the
ultim ate limit th a t I will consider here, and we shall see th a t it is a fairly restrictive
one, at least as far as w(z) constraints are concerned.
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80
2
1
0
1
2
- 0 .5
0
0 .5
<5w0
Figure 5.7: Error contours (68.3%) in th e w0- w a plane. From innerm ost to out­
ermost they represent (a) a “perfect” CMB lensing-galaxy count cross correlation
measurement where cosmic variance is the only lim itation, (b) a more realistic mea­
surement w ith galaxy density of ~ 102 arcm in-2 and the “< 10 nK E-mode” lensing
noise of Fig. 5.1, (c) an ISW-lensing only experiment (no galaxy counts).
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81
How about cross-correlating galaxies w ith th e ISW effect in order to augment
our inform ation about dark energy? This can provide interesting results, especially
w ith th e possibility of adding even very lim ited redshift inform ation into the mix. I
show the results in Fig. 5.6. Galaxy information, especially th a t coming from high
redshift galaxies, can tighten the dark energy constraints by more th an a factor of
two.
Finally, the constraint contours in the WQ-wa plane are shown in Fig. 5.7. These
are constraints made on w{z) simultaneously w ith constraints on all the other cos­
mological param eters described above. In th e absence of any other experiments,
CMB lensing and galaxy counts combined can realistically constrain (contour b)
wo to a precision of about ±0.33 when all other param eters are marginalized over.
5 .4 .1
D a rk en er g y p e r tu r b a tio n s
T he above results were obtained assuming th e absence of perturbations in the
dark energy fluid. A tru e cosmological constant has w = —1 and is perfectly sm ooth
w ith no perturbations. However, many quintessence models are built with some sort
of scalar field which in general can have its own density fluctuations. These density
fluctuations become im portant at late tim es when the structure formation respon­
sible for the ISW effect is occurring, and thus significantly affect the above results.
This effect is shown in Fig. 5.8. Note th a t including perturbations mostly increases
the uncertainties along the degeneracy direction, especially for experiments lim ited
only by cosmic variance, w ith the result th a t limits on both dark energy param eters
are degraded by nearly a factor of three.
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82
2
1
0
1
-2
- 0 .5
0
6
0 .5
w0
Figure 5.8: The effect of dark energy perturbations on error contours (68.3%) in the
w 0- w a plane. Contours (a) and (b) are as before in Fig. 5.7. Contours (c) and (d)
show the effect of including dark energy perturbations for experiments considered
in (a) and (b), respectively.
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83
5.5
C onclu sion
Cross-correlating the ISW effect w ith CMB lensing and galaxy counts can in
principle place limits on dark energy; ideally (if cosmic variance were the only
lim itation) one could measure w0 to ±0.093 and wa to ±0.32, though in practice
realistic limits are probably worse by about a factor of three. These measurements
depend on dark energy’s effect on the growth of structure, and thus have a depen­
dence on the equation of state quite different from distance-redshift measurements.
Observation of such growth effects would bolster the case for dark energy as the
source of cosmic acceleration. Also, these measurements are strongly sensitive to
perturbations in the dark energy fluid (which do not affect distance observations)
and thus may ultim ately be more useful as a measurement of perturbations th an
as a precise determ ination of the equation of state.
There is certainly room for improvement for these kinds of m easurements. The
am ount of lensing being measured is small, so cosmic variance and the finite reso­
lution of experiments become im portant. In this work I have used the CMB tem ­
perature m ap to reconstruct the lensing spectrum ; this m ethod essentially makes
use of four-point correlations in th e original tem perature map. The three-point
correlation function (known as th e bispectrum ) also contains contributions from
lensing which are known to be significant [22] and potentially less noisy, although
how well dark energy inform ation can be extracted from realistic d a ta is still under
investigation.
The scales on which galaxy densities correlate w ith the ISW effect are very large
(degree scale or larger), so the deviations from a sm ooth background are small and
even for large numbers of galaxies shot noise is significant. One way of overcoming
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84
this problem is simply to use more information from galaxies th a n merely their
number density. D edicated telescopes th a t can find galaxy shapes and redshifts are
predicted to put good constraints on dark energy [61]. In th e future a wide array
of complementary observations will be available to determ ine the nature of cosmic
acceleration, each w ith their own sensitivity and lim itations.
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85
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92
A ppendix A
Selected Inflationary M odels
The potentials used in the models shown in figure 4.1, from left to right are (in
units where M P = 1)
M 4 [1 + A In <f>
(A.l)
M 4 [ l - e - > / 2] 2
(A.2)
M 4 [1 - e“ *]
(A.3)
M 4 [1 + A<f)2]
(A.4)
M 4 cos2 -^,
M4
(A.5)
M 4 [1 - A(j)12]
(A.6)
M 4 [1 - Acj)2]
(A.7)
M 4 [1 - A 04]
(A.8)
+ ^4? (A + 0.6)
(A.9)
other param eters were chosen as in table A .l to
produce a dp/p of roughly 2 x 10 5. For m ost models above, the ending point for
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93
Table A .l: Inflationary model param eters
9/ 4
2 x 10~4
1CT3
10“ 6
1(T16
OCO
IO -
A
5 x lO-1
—
—
10~3
100
5 x 10-'
10~2
X
M
1(T3
ltr3
1(T3
1(T4
Mi—
1
Model #
1
2
3
4
5
6
7
8
9
0.139
inflation can be found by numerically evaluating where th e slow-roll param eters
become equal to unity. A numerical code was w ritten to facilitate this, and to then
numerically integrate the equation of motion from a suitable startin g point to deter­
mine the detailed shape of the prim ordial power spectrum of density perturbations.
This code is made publicly available in its entirety in appendix D.
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94
A ppen d ix B
Full-sky expressions for angular
power spectra
In chapter 5, th a t flat-sky approxim ations for power spectra were used for clarity.
However, the full expressions as given by Okamoto and Hu[30] are necessary for
accurate numerical calculation. Recall th a t th e projected potential is defined as
« n ) = - 2 |< i r ) ^ ^ * ( x ( n ) ,D ) .
(B .l)
As w ith any function on the sky, this can be broken up into spherical harmonics
«n) = ^ ^ y 7 " (n ),
(B.2)
and averaging over the ensemble of coefficients defines the power spectrum
(■(ftmfa'm1) =
(B.3)
Since the projected potential depends only on the full gravitational potential, the
power spectra will depend on the power spectrum of th e gravitational potential
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95
A$(fc, z). This dependence will come w ith a window function closely related to the
lensing efficiency
= 47T
l2
dk
A|(fc,z)
k
(B.4)
where the j( are spherical Bessel functions.
For regimes where the growth of
the potential rem ains linear, the power spectrum A $ ( k , z ) can be replaced by
F 2(D) A%(k,0) where F(z) is the growth rate for density perturbations defined
by 5{z) =
The projected potential power spectrum is then
dk
Cf* = 4tt I y A | ( M )
dD2F(D)
Ds - D
j e ( k D / H 0)
DD,
(B.5)
T he galaxy-galaxy correlation spectrum is similar to the projected potential
power spectrum except th a t instead of the window function expressing th e lens­
ing efficiency, the window function for galaxies depends on th e number density of
galaxies n g(D) as a function of distance or redshift for the particular survey one
wishes to describe. Otherwise th e equation is th e same
C f = 4tr
dk
k
d D 2 F ( D ) n g(D )je( k D / H 0)
0)
(B.6)
Remembering th a t the 1SW portion of the tem perature power spectrum is given by
A T ISW(h) = - 2
I dD <3>(x(n), D),
(B.7)
the cross-correlation between tem perature and lensing is given by a similar expres­
sion
CJ*
=
4tr j y A | ( M )
J
dD2F{D)
f d D 2 F ( D ) j e( k D / H 0)
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(Bd
96
T he other two cross-correlation spectra follow in direct analogy
C j9 =
x
47t
J
^A|(&,0)
d D 2 F ( D ) n g( D) j e ( kD / H 0)
(B.9)
d D 2 F ( D ) j e ( k D / H 0)
for tem perature-galaxy correlations, and
d D 2 F ( D ) n g{ D) j e{ k D / H 0
x
d D 2 F ( D ) D * n £ ) j t { k D / H 0)
for galaxy-lensing correlations.
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(B.10)
97
A ppendix C
U seful M athem atical D erivations
C .l
P aram eter m argin alization for a m u ltin or­
m al d istrib u tio n in m a trix form
Often we have to deal w ith probability distributions th a t are Gaussian in N
dimensions, b u t where there is much correlation between dimensions. Such a prob­
ability distribution can be w ritten com pactly as a function of an TV-dimensional
vector of param eters x = { xi , X2, ■■■, x jy} using m atrix notation
(C.l)
where No is a norm alization (which we’ll be unconcerned w ith), and F is the inverse
of the N x N symmetric m atrix of covariances C.
Its is frequently desirable to integrate th e probability distribution over one or
more of the param eters to obtain a marginalized probability distribution of a smaller
set of param eters. For example, marginalizing over one of th e param eters Xi to
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98
obtain the distribution as a function of x which is an N — 1 dimensional vector
/
OO
dxiPi*).
(C.2)
-OO
If the m atrix F were diagonal, the integral would be easy to perform algebraically,
however if N is large and there are many non-zero covariances between x, and
the other param eters then the algebraic expression can become quite cumbersome.
Fortunately, there is a convenient technique using the m atrix notation which I offer
here w ithout rigorous proof.
Taking advantage of the fact th a t the resultant probability distribution will still
be a m ultivariate Gaussian, we can rewrite th e previous expression as
iY0 exp ^ - i x r F x ^ =
J
dxi N 0 exp ^ ~ ^ x TF x j ,
(C.3)
where F is an N —1 x N —1 m atrix. The im portant inform ation about the probability
distribution P (x) is contained in this m atrix, and the question is to determine how
F can be obtained from the N x N m atrix F . Define
to be th e m atrix minor
of F , i.e. the N —l x N —1 m atrix formed by removing the z-th row and j -th column
of F. Also define
to be the N — 1 dimensional vector formed by taking the z-th
column of F and removing the entry in the z-th row. Then th e m atrix F describing
the probability distribution obtained by marginalizing over th e param eter aq is
F =
- - L v (i)v (i)T,
*a
(C.4)
where Fa is the z, z-th entry of th e m atrix F.
For a simple concrete example, imagine a two-dimensional probability distribu­
tion P(x, y ) = N q exp[—(ax2+ 2bx y+cy2)/2}, thus th e m atrix F for this distribution
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99
is given by
F =
(C.5)
\ bc/
M arginalizing over the param eter y results in th e integral
p ( X) = r
dyNo exp
1
(iax2 + 2bxy + cy2)
(C.6)
J —c
which is easy enough to do analytically
2?r
P ( x ) = N 0\ — exp
—| a
2 V
b2
| x~
c
(C.7)
We can check th a t the m atrix technique gives the same answer. In this case M-22-1
is simply the l x l m atrix with entry a, and
a one-dimensional vector w ith
entry b. T hen the ( l x l ) m atrix F can be found through the m atrix technique to
be
F = a
c
b x b,
(C.8)
which m atches the algebraic expression found earlier. This m atrix technique has
been checked to m atch the algebraic result for up to 5 x 5 m atrices using com puter
algebra, b u t a rigorous proof remains beyond th e author at this time.
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100
C .2
A ltern a te d erivation o f th e ev o lu tio n o f th e
d en sity o f a gen eral cosm ological fluid
Consider a sm ooth fluid w ith background density p and pressure P , with small
perturbations. One way of defining this fluid is through its stress-energy tensor
Tg=-(i> + Sp),
(C.9)
Tf = ( p + P ) v u
Tj = (P + dPjS] + a),
where dp and 5P represent density and pressure perturbations, Uj represents veloc­
ity perturbations, and u* represents shear perturbations. We are working in the
synchronous gauge and in conformal tim e, where the to tal m etric tensor can be
w ritten as
g^vd x ^d x 1' = a2 [—d r 2 + (5^ + hij)dxldx^1
\,
(C.10)
where the tensor hij represents the m etric perturbations.
Conservation of energy and m om entum is represented in general relativity by
the equations
+ r va0T af3 +
where the Christoffel symbol
= 0,
(C.ll)
9a0- v) ,
(C .1 2 )
is defined as
=
2 ^ (W q ;/3 +
9ug- a ~
and the semicolon subscript indicates th e derivative x-a = dax.
Looking at the first-order p art of th e u = 0 component of th e conservation
equation will give the equation for the evolution of density perturbations. To obtain
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101
this, it is useful to explicitly break T IW,
and g,iv into background and first-order
pieces. The metric is easy to break up by inspection
Qyuv — 9[iv T Sg^v = a [diag( 1 ,1 ,1 ,1 ) -f- h ^ ] ,
(C.13)
but we will also need to make use of the inverse metric, which to first order in
is
^ [d ia g (-l, 1,1,1) - h i j ] ,
a*
where the lowered indices on
The stress-energy tensor
(C.14)
are intentional.
is also easy to break up by inspection. The unper­
tu rb ed tensor T ‘“ is simply diag(—p, P, P, P), and we will only need the following
perturbed components
5T0° = - 5 p ,
(C.15)
ST f = (p + P ) v u
STi = 8P,
w ith no implicit summing over i in the last expression. A slight complication is
th a t we need to find the stress-energy tensor to first order w ith both indices raised.
We can make use of the m etric to write
T n» =
= + 5 T »gW + f £ 5 g av.
(C.16)
The first term is the background tensor T ^ u = diag(p, P, P, P ) / a 2, and th e second
two term s form the first order p ertu rb atio n S T ^ w ith the components we are
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102
interested in as
(C.17)
a2
a
again w ith no implicit summation.
W ith the perturbed metric in hand we can also com pute th e Christoffel symbols
to first order and w rite them as a background term plus a perturbed piece
=
^a0 + STU ' After some tedious com putation, th e only non-zero unperturbed pieces
of the Christoffel symbols are
-pO
00
pO
ii
_
(Jj
pi
Oi
(C.18)
There are also only two sets of nonzero perturbed com ponents which we will need
1•
(C.19)
(C.20)
Finally we are prepared to find th e first order piece of th e energy conservation
equation
+
sr°a0f af3+ y%5t^ + sr^f013+ f aP5T00=
o.
( c . 21 )
We can examine this piece by piece. The first term is
(C.22)
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103
T he portions proportional to the perturbed m etric
cancel in the second and
th ird term s, leaving only
\ h - 2 + ~ ( 6 P + UP),
2 a2 a2 a
where h = T
(C.23)
r = J T ha. The last two term s follow in a similar way
M + 4 ^ .
2 a 2
a a2
(0-24)
P u ttin g these all together and multiplying by a2 gives
5p + 3-(<5/o + SP) + (p + P ) ^ Vi + \ { p + P)h = 0.
CL
£
Last, wecan make the definitions 5 = 5p/p, 6 = dlVi and make use
(C.25)
of the
identities w = P / p and d i n P / d r = —3(1 + w)a/a (the latter is strictly true only
when dw/dr can be neglected). W ith these in place, dividing the previous equation
by p reveals
^ + 3 a (<hP ~~ W) + ^ +
^ + 2^ =
(C.26)
U ndertaking a similar exercise with the other com ponents of the energy-momentum
conservation equation gives the equivalent expression for 0.
C .3
C o n n ectio n b etw een tw o form s o f th e Fisher
m a trix
The general form for the Fisher m atrix of m ultiple angular power spectra is
given in 5.9 as (absorbing the coefficient factors of \ f l into CcnqT1)
c w ^ c o v r '80'
dsi
dsj
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(C.27)
104
For the special case where the Cov^ are 2 x 2 matrices, the above equation can be
rew ritten as in chapter 4 as
nr.l
I,J = (T ,E ,C )
AnJ
dsi
ds-;
‘
£
(C.28)
) i j -8 0 1
3
where now the covariance inform ation is contained in a 3 x 3 m atrix D . The form
for specific entries of D was also given in chapter 4. In this appendix I will explicitly
connect the two forms.
Analogous to the process in chapter 5, th e covariance m atrix for tem perature
and E-mode polarization spectra can be w ritten as
/
1
Cov^ =
yj
C
c Jt T‘ ++ N ij
C
c JrE
r'TE
C
JE
r~iEE
_i_ N
atef
C fE +
+ ^/sky
(C.29)
/
Noise is presumed to be uncorrelated between T and E modes, however this is not
im portant to the derivation here. For brevity I will use the notation C xx = C'fx+ N f ,
and C xx = d C f x j d
Using this notation, the inverse of the covariance m atrix is
|) /s k y
C ov„1
r i T T r< E E
u£ u£
^ Lx
riEE
o
(C.30)
r^T E 2
u£
c*TE
^
LxQ
r iT E
ue
y
fiT T
'-'£
The full expression for the i , j - th entry of th e Fisher m atrix can be com puted w ith
some algebra
cr,cr1cr,+icr-pv'cr,
/sky
p iT T p iE E _
U£
Ui
/o T £ 2V
J
rys~iTT
P iE E s~iT E /~iT E
U
^ £ ;
~
,
s~tTT n T E 2 n E E
+
°
—
rs/~iTE
cy/^iEE
z u e
-
j
cy/~iTE
ZL /£
, on T E
+.
P iT T r ^ T E r^iE E
f i E E /~iTE yoT T
;i
Ue
U£
P iT T f i E E n T E
;iW
+
pp
s~iEE
° £
L/ £
-j
s~iTTz r ^ E E
.i Lpi
Kj q
P iT T /~vTE /~iTE
-i u £
u £
-j
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;j
105
T he expression in square brackets can be rew ritten in m atrix form
/
\\ T /
(jT T
r*TE
9 r^E E riT E
cf
9nE E
-ZU£
r iE E
\
\ C* *J
c:t e
nTE
9 ( r iT T (^ tE E , s t T E 2\
Zl
U£ U£ -f-U£
I
2
\
(jT E 2
riT T riT E
~9
L Lx
d \~yn
( (J T T
n s^iTT/~<TE
q
U£
a
<rprpZ
\
TE
riE E
V6 * ; i /
(C.32)
T he 3 x 3 m atrix in the middle, along w ith th e coefficient from th e previous equation,
is the m atrix D -1 from equation C.28. After inverting and canceling several factors
of { C j T C f E - C J E2),
/
q
TT2
fiT T riT E
D =
+
ue
|) /s k y
a
\
L'e
n T T s~iTE
\y o Kyo
1 (r^T T r iE E
TE2
\
cilT E 2
, s~<TE2 \ r ^ T E /~iE E
/-iT E r> EE
J L'e
(C.33)
L/i
C£
/
which corresponds exactly to the expressions given in chapter 4.
C .4
Specific len sin g n oise estim a to rs
Okamoto and Hu [49] give the general expression for th e noise in a lensing m ap
th a t has been reconstructed from cosmic microwave background tem perature and
polarization maps in equation 39 of their paper. I will give a few specific expressions
for the most useful cases. If one only has a tem perature map, th en the noise in the
reconstructed lensing m ap is given by
N (ee)(ee) = L ^L + 1^ 2L + ^
^ 3^
-i
x
^Pifo
2 C *f1e C «f2e
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106
where C f e are the tem perature multipole moments as m easured by experiment
(including noise), and the tilde indicates the unlensed multipoles w ithout noise.
The expressions for the noise if reconstructed from th e E-mode polarization m ap
or the T E cross-correlation map alone are quite similar, w ith the former being
N
(-E E ) ( E E )
— L{L + 1)(2L + 1)
( riEE jp
_V_______
x
(C.35)
, r^EE jp
/even
9 riEEr’EE
L't2
and the latter
N\
{ OE) ( e E)
— L ( L + 1)(2L + 1)
fiQE jp
E
hl2
x
(C.36)
i
-1
fiQE jp
(^e1 2^t2Lh + (-'e2 0 ^ h U 2
ryriQEri&E
where “even” indicates th a t the expression in parentheses is non-zero only when the
sum L + £i +£ 2 is even. The 0 © and E E expressions are w hat was used to produce
figure 5.1 of chapter 5. To fully determ ine the noise for a lensing m ap reconstructed
from both tem perature and E-mode polarization d ata, one also needs to compute
(@e ) ( EE)
N-.
, which is a significantly more complicated expression.
The coefficient sF ^ l ^ appears often and is related to the W igner-3j symbol in
the following way
s F e
1i e 2 —
[ L ( L
/
x
(C.37)
+ 1) + -^2 (^ 2 + 1) —^1 (^ 1 + 1)]
( 2 L + l ) ( 2 l j + l ) ( 2 l 2 + 1)
167T
h l t2
s
0
—
s
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107
The W igner-3j symbol only has an exact analytic expression if s = 0, in which case
( ix L
\
0 0 0
.1 )5 /2
(5/2)!
(S/2 — £i)\(S/2 — L ) \( S /2 — ^2)!
(C.38)
/
x
' ( 5 - 2 ^ ) ! ( 5 - 2 L ) ! ( 5 - 2 £ 2)!
(5 + 1)!
where S = t \ + L + i 2. At high multipoles, however, there is an approxim ation for
the s = 2 case when the sum S is even
2 0 -2
1 (L 2 - l \ - t
2
l\l\
1 lx L
0 0 0
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(C.39)
108
A p p en d ix D
C ode
E volve
This code numerically integrates the equation of m otion for a single scalar field
in order to find the detailed shape of the prim ordial power spectrum of density
perturbations. The program is w ritten in C and includes an interactive command
line for choosing the potential, setting initial conditions, and producing plots of
the output. Custom inflationary potentials are relatively easy to add. The code is
released under the BSD license (see th e license file for details), and can be found
at the URL
h t t p : / /bubba.u c d a v i s . e d u / ~ g o l d / e v o l v e .t a r .gz
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109
P erl scrip ts
I have w ritten several scripts useful for processing the outp u t of C M B F A S T
and other codes. These are periodically u p d ated and collectively made available
for free and public use at the URL
h t t p : //b u b b a . u c d a v is . e d u /~ g o ld /p e r l/in d e x .htm l
Included are scripts for diagonalizing and sorting the eigenvectors of Fisher m atri­
ces (or any symmetric m atrix), and a script th a t implements th e marginalization
technique described in appendix C .l. Also available is a module for treating d ata
files of a certain form at as objects upon which various m athem atical operations
can be performed, e.g. two d ata files can be added together w ith autom atic spline
interpolation as necessary.
Several of these scripts make use of th e Math: :M atrix R eal package, available
from CPAN at
h t t p : //www. c p a n . o rg /
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