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Missing energy in the universe: Quintessence and the microwave background

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MISSING ENERGY IN THE UNIVERSE: QUINTESSENCE AND THE
MICROWAVE BACKGROUND
Rahul Dave
A DISSERTATION
in
PHYSICS AND ASTRONOMY
Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of
the Requirements for the Degree of Doctor of Philosophy
2002
r
-----
Paul J. Stein h ^ dt
Supervisor of Dissertation
Randall D. Kamien
Graduate Group Chairperson
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UMI Number 3043862
Copyright 2002 by
Dave, Rahul Surendra
All rights reserved.
UMI’
UMI Microform 3043862
Copyright 2002 by ProQuest Information and Learning Company.
Ail rights reserved. This microform edition is protected against
unauthorized copying under Title 17, United States Code.
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COPYRIGHT
Rahul Dave
2002
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DEDICATION
To the one and only Bindu
iii
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ACKNOW LEDGEM ENTS
I’d like to thank:
Paul Steinhardt for being a patient and nurturing advisor,
David Coulson, Rob Caldwell, Greg Huey, Limin Wang and Ivaylo Zlatev for fun
conversation and collaborations,
Paul, Miriam, Phil, Sid, Nigel, Bob, Ira for the great courses I took from some of
them, and for their support,
Pavlos, for all the help and friendship over the years, and for teaching me that energy
is indeed conserved,
Farrukh, Mihir, Anish, Tycho, Ahmed, for conversation, many a dinner, and general
insanity,
Mom, Dad, Dada, Auntie, Harish, and Renuka for being there,
And more than anyone else, for all you have so selflessly given me, my sweetheart
and soulmate, Bindu.
iv
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A BSTR A C T
MISSING ENERGY IN THE UNIVERSE: QUINTESSENCE AND THE
MICROWAVE BACKGROUND
Rahul Dave
Paul J. Steinhardt
If the Universe is flat, as predicted by inflationary cosmology, then the vast majority
of the cosmic energy density is non-baryonic. One possibility is that a substan­
tial fraction of this missing, non-baryonic energy consists of quintessence, a timedependent and spatially inhomogeneous component whose equation-of-state differs
from th at of baryons, neutrinos, dark m atter, or radiation or cosmological constant.
An example is a scalar field evolving in a potential, but our treatment is more gen­
eral. Including this component as a replacement of the cosmological constant alters
cosmic evolution in a way that fits current observations well. Unlike the cosmologi­
cal constant, it evolves dynamically and develops fluctuations, leaving a distinctive
imprint on the microwave background anisotropy and mass power spectrum.
This work describes the effects of quintessence on the Cosmic Microwave Back­
ground Radiation (CMBR) anisotropy. It shows that the evolution of fluctuations in
this component, and the resulting CMB anisotropy, is independent of the initial con­
ditions on the fluctuations. The microwave background anisotropy power spectrum
in models which include quintessence is numerically computed and analytically exv
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plained, and the imprints left by quintessence are categorized by epoch and physical
effect.
There exists a fundamental degeneracy which will prevent near-future CMBR
satellite experiments from discriminating between quintessence and alternative can­
didates for the missing energy by themselves. However, combining these measure­
ments with those from high redshift supernovas will enable us to set loose constraints
on the relevant parameters. If quintessence is discovered, there may be fundamen­
tal implications for the cosmological constant problem, and there will be a lot of
interesting low energy particle physics to explore.
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Contents
1 Introduction
1.1 Finding out about the universe
1
................................................................
1
1.2 The problem of Missing Energyand a possible answer: quintessence .
3
1.3 Outline of this w ork.......................................................................................
9
2 Prelim inaries
13
2.1 F u n d am en tals.................................................................................................
14
2.1.1 The space time metric and equations of m o t i o n .......................
15
2.1.2 Characteristic scales in the universe..............................................
18
2.2 Thermal origin of the universe: the Big Bang M o d el.............................
19
2.2.1 The thermal universe: Standard Big-Bang M o d e l ....................
21
2.2.2 M atter content of the universe in the big-bang m o d e l
22
2.2.3 Problems with the Big Bang M o d e l..............................................
25
2.3 I n fla tio n ..........................................................................................................
27
2.3.1 The scalar field mechanism for inflation........................................
28
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viii
2.3.2
Inflation gives rise to perturbations in the m e t r i c ...................
29
2.3.3
The power spectrum of the perturbations...................................
30
2.4
The Gravitational Instability scenario of structure fo rm a tio n ............
32
2.5
The anisotropy power s p e c tr u m ...............................................................
34
2.6
Perturbations, the “Missing Energy problem”, and quintessence . . .
37
2.7
S u m m a r y ......................................................................................................
39
3 Classical Cosmological signals of missing energy
40
3.1
Open Universe? or Missing e n e rg y ? .........................................................
41
3.2
Modeling missing e n e r g y ............................................................................
42
3.3
Observations based on distance m easurem ents......................................
42
3.3.1
44
The age of the universe....................................................................
3.3.2 The apparent magnitude of high redshift supernovas
..............
46
3.4
Gravitational Lensing S ta tis tic s ...............................................................
48
3.5
Dynamical measure indications of missing energy
................................
49
3.5.1 The bend in the m atter power s p e c tr u m ....................................
49
3.5.2 The abundance of high redshift quasars and clusters.................
51
3.6
Evidence for missing energy from CMBR anisotropy measurements
.
52
3.7
S u m m a r y ......................................................................................................
53
4 Calculating the CM BR anisotropy in quintessence models
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55
4.1
Modeling quintessence as a scalar f i e l d ...................................................
56
4.2
Relationships to previous w o r k s ................................................................
58
4.3
Dynamics of the scalar f i e l d ......................................................................
59
4.3.1
Background dynamics of the scalar field
....................................
60
4.3.2
The perturbed Einstein e q u a tio n s.................................................
62
4.3.3
Dynamics of quintessence fluctuations...........................................
63
Results from the Boltzmann codes.............................................................
67
4.4.1
67
4.4
Normalizing the CMBR power s p e c t r a ........................................
4.4.2 Models with constant equations of state
....................................
70
4.4.3 Models with changing equation of s t a t e ........................................
75
4.4.4 Anisotropies in models with monotonically evolving equations
of state
4.5
.............................................................................................
78
4.4.5 Anisotropies in non-monotonic m o d e l s ........................................
81
S u m m a r y ......................................................................................................
83
5 Understanding quintessence fluctuation evolution and the sensitiv­
ity o f C M B R spectra to this evolution
85
5.1
Why constant w approximates well a large range of potentials . . . .
86
5.2
Solving fluctuation equations for constant w ...........................................
91
5.3
Sensitivity to Initial C o n d itio n s ....................................................................101
5.4
Stability of Q-field fluctuations to gravitational e v o lu tio n .......................108
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X
5.5
S u m m a r y ................................................................................................ 109
6 Processes affecting the C M B R anisotropy in quintessence models 112
6.1
From linear perturbations to radiation a n is o tro p y ......................... 113
6.2
Anatomy of the CMBR power spectra
6.3
Evolution of the gravitational p o te n tia l.............................................126
6.4
Early contributions: decay befor recombination, and amplification after 130
6.5
Parameter dependence the positions and heights of the doppler peaks
......................................................120
132
6.5.1 Horizon W idth-the positions of the doppler p e a k s ........................ 132
6.5.2 The heights of the doppler peaks
6.6
.....................................................134
Anisotropies in the recent p a s t .............................................................136
6.6.1 The late ISW e ffe c t...............................................................................136
6.6.2 The direct effect of the Q fluctuations at late t i m e s ..................... 137
6.7
Large angular scales: the cumulation of anisotropy..........................139
6.8
Large angular scales: the evolution of anisotropy.............................140
6.9
S u m m a r y ................................................................................................ 142
7 Comparing predictions from quintessence and observational results:
com patibility and degeneracy
154
7.1
CMBR anisotropies in Q-field models: the observations so far . . . .
155
7.2
Measurements of the m atter power spectrum and <r8 ...................... 158
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xi
7.3
The anisotropy degeneracies and their width: a new statistical index . 159
7.4
Application of the index: Inherent degeneracies in distinguishing Qfield models from A models.............................................................................. 162
7.5
S u m m a r y .........................................................................................................166
8 A ttem pting to lift the anisotropy degeneracy in quintessence mod­
els
168
8.1
Combining the CMBR anisotropy with other m easurem ents.................. 169
8.2
Combining supernova and CMBR constraints to investigate spatial
curvature
..........................................................................................................173
8.3
Combining supernova and CMBR to investigate Q ...................................177
8.4
Summary:Present limits on parameter e s tim a tio n ...................................178
9 Conclusions, and the significance of this work
183
A Com putational details on the Boltzm ann codes
187
A .l
Equations of motion in different g au g es...................................................... 187
A. 1.1 Background E v o lu tio n ...................................................................... 187
A.1.2
Evolution of F lu c tu a tio n s ................................................................188
A. 1.3
Changing variables for implicitly definedp o te n tia ls .................... 191
A.2 Formulae for constant w and time varying potential m odels.....................192
A.2.1
Constant Equation of State Model
................................................192
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A.2.2 Time-Varying Equation of State
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List o f Tables
XIll
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List o f Figures
2.1 The CMBR anisotropy power spectrum as a function of multipole
for the Standard CDM model (Qm = 1, h = 0.5, Qg/i2 = 0.0125,
n = 1). The theory predicts only the shape of the power spectrum,
not the normalization. The spectrum is normalized using the COBE
measurement. The band in the figure is the cosmic variance band,
with the central curve inth at band being the ’average’, the £(£ + 1 ) ^ .
3.1 Contours of age in the
36
— w plane in a flat universe. Note how the
age can be used to rule out the less negative tu’s on the assumption
that it is at-least 11 Gyrs. If w ~ —1 one may derive lower limits on
fig or Q \ ...........................................................................................................
3.2
45
The m atter power spectrum in a standard CDM and a ACDM model
are plotted against wavenumber in this figure. Note that the turnover
of the spectrum depends on the epoch of m atter radiation equality.
xiv
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.
50
XV
3.3
Here we have plotted combined data obtained from an analysis of
almost all the experiments that have reported CMBR anisotropies,
along with the Standard CDM model, an open universe model, and
a A model, both with Q m < 0.3. One can see th at the observations
are at a variety of angular scales. There is clear indication of a large
intermediate scale
= 200) anisotropy rise. The open universe model
is inconsistent with the data.........................................................................
4.1
54
This figure shows CMBR anisotropy spectra for a set of constant equa­
tion of state QCDM models with Q-field energy density Qq = 0.6, for
different values of the w .................................................................................
4.2
71
This figure shows CMBR anisotropy spectra for a set of QCDM models
with constant w = —1/6, for different values of the Q-field energy
density.
4.3
..........................................................................................................
72
The feature-fullness of the low I anisotropy in Q-field models. Notice
in particular the extremely low multipole dip in the w = —1/3 model.
This is a consequence of the direct effect of the Q-fluctuations. No­
tice also the bump in the model where the Q-field has a mass; this
bump occurs at the angular scale corresponding to its Compton wave­
length. The w = 0 models shows a relatively steep and steady rise, a
consequence of the merger of the late and early ISVV effects..................
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74
XVI
4.4
This figure shows the E-polarization anisotropy spectra for a set of
constant equation of state QCDM models with Q-field energy density
Q q = 0.6, for different values of w ...............................................................
4.5
75
This figure compares the CMBR anisotropy spectra for three models
with the same present-day values of the equation of state and Q-field
energy density, but with different values in the past. Shown are a
constant w = 0 model, and an exponential and cosine potential with
w (Vo) = 0. The equation of state of the exponential potential started
at u; = —1 at early times, and has evolved monotonically towards
w = 0 by the present. For the cosine potential, the equation of state
started at w = —1 at early times, evolved monotonically towards
w = 0 by a redshift z ~ 1, and then has oscillated from w = +1 back
to w = 0 by t o d a y . .......................................................................................
4.6
80
This figure shows CMBR anisotropy spectra for a set of QCDM models
with an exponential potential where the equation of state has evolved
monotonically from w = —1 at early times, towards
w (tjq)
= 0 at the
present, for different values of the Q-field energy d e n s i t y .....................
4.7
81
This figure shows CMBR anisotropy spectra for a set of exponential
potential QCDM models with Q-field energy density Q q = 0.6, for
different values of the final value of w f o ) ..................................................
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82
XVII
4.8
This figure shows the evolution of the equation of state for a sequence
of three quadratic potential models. For increasing values of the mass,
relative to the present-day inverse Hubble length, the Q-field begins
to oscillate earlier. All models have Q q = 0.6........................................
4.9
This figure shows the CMBR anisotropy spectra for the same sequence
of three quadratic potential models. For increasing values of the mass,
relative to the present-day inverse Hubble length, the energy density
in the Q-field becomes comparable to the CDM density earlier. Con­
sequently, a feature in the CMBR spectra develops at angular scales
corresponding to the apparent size of the horizon when the Q-field
first began to oscillate. All models have Q q = 0.6, h = 0.65, and
Qbh2 = 0.02...................................................................................................
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xviii
5.1
In the left panel, we see the evolution of the equation of state for a set
of potentials, all with Qq = 0.6 and w = —1/3 today. The evolution of
the ratio of the quintessence energy density to the critical energy den­
sity is shown in the right panel. In both panels, the upper, solid curve
represents a constant w = —0.55 model, while the lower solid curve
represents a constant w = —0.66 model. The w = —0.55 model has
been chosen to best-fit approximate the exponential potential, while
the w = —0.66 model has been chosen to best-fit approximate the
quadratic and quartic potentials. Notice the similarity in the energy
density evolutions............................................................................................
5.2
87
Here we plot the ratio of the power spectrum in the models from Figure
5.1 to the power spectrum in the corresponding best-fit constant w
model. The fractional cosmic variance with respect to the best-fit
model is also shown (outer thin lines). The ratio for each model falls
well within this variance envelope at most of the multipole moments,
thus, the predicted anisotropy is observationally indistinguishable.
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89
XIX
5.3
The evolution of the equation of state for a pair of oscillatory QCDM
models is shown, both with
Qq
= 0.2. The exponential potential which
approaches the attractor solution w = 0 by z ~ 2 has been chosen to
approximate the oscillatory quadratic potential which completes its
first oscillation near the same redshift........................................................
5.4
The evolution of the ratio of the energy density to the total energy
density,
with
Qq
Q q (z ),
for a pair of oscillatory QCDM models is shown, both
= 0.2 at the present day. The fractional energy density for
each model first approaches its asymptotic value by z ~ 2.....................
5.5
90
91
The CMBR anisotropy spectra for a quadratic potential oscillatory
QCDM model is compared to the spectra for an exponential potential
QCDM model with the same average equation of state history (see
figure 5.4) both with
Qq
= 0.2 at the present day.
.............................
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92
XX
5.6 We compare the evolution of quintessence and m atter fluctuation en­
ergy density for a long wavelength fc = 10-4 Mpc~l mode in three
different models with equations of state w = —1/3, w = —2/3, and
w = -0 .9 , and with Qq = 0.6, h = 0.65, and Q eh2 = 0.02. The three
lower curves are the quintessence fluctuation evolutions at the differ­
ent equations of state, while the three upper curves, all very close to
each other, are the corresponding m atter fluctuation evolutions. No­
tice th at the energy density in quintessence fluctuations changes with
equation of state, but remains much smaller than the energy density
in m atter fluctuations.....................................................................................
93
5.7 The figure in the left panel shows the evolution of the ratio of energy in
Q fluctuations to that in m atter fluctuations
at k = 10~4Mpc~l
for both smooth initial conditions (inhomogeneous solution) and adi­
abatic initial conditions at w = —0.9. The figure in the right panel
shows the corresponding CMB power spectra as a function of multi­
pole moment. Plotted below the power spectrum is the percentage
residual of the power spectrum for adiabatic initial conditions from
smooth ones, compared to the fractional cosmic variance (100 x
plotted as a black line). The anisotropy change in going from smooth
to adiabatic initial conditions is well below the variance
...................... 103
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xxi
5.8
The figure compares the evolution of the ratio (J^ -)2 for both adia­
batic (F = 1) and artificially amplified (F = 104 and F = 105) initial
conditions for the w = —0.9 model from Figure 5.6. The ratio is plot­
ted for wave number k = 10~AM pc~l. We also plot a solid horizontal
line to indicate a ratio of magnitude unity. Notice that the amplifi­
cation prolongs the domination of the homogeneous solution, and the
resultant closeness of the energy density ratio to u n i t y . .......................... 104
5.9
The figure depicts the CMB power spectrum as a function of mul­
tipole moment for two of the models of Figure 5.6 with w = —2/3
and w = —0.9. The power spectra are plotted for a series of cases
with artificially amplified initial conditions, and for the correspond­
ing model with adiabatic initial conditions. Also shown in the lower
panel is the absolute value of the percentage residual of the amplified
cases from the adiabatic case, as compared to the fractional cosmic
variance (black line). At w = —2/3, an amplification of the adiabatic
initial conditions even by F = 105, is not enough to make an observ­
able change in the CMB power spectrum. On the other hand, in the
w = —0.9 case, the power spectrum for the same value of F is markedly
different................................................................................................................ I l l
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XXII
6.1
We have plotted certain salient features of the CMBR anisotropy, or
“milestones” in four separate models. These are the Standard CDM
model, a A model (with Q \ = 0.6), and a open universe model, with
Q\[ — 0.3. The fourth model is a QCDM models with constant w =
—1/6 and Qq=0.6. In the latter three models fifl/i2 is fixed at 0.02,
while h=0.65. The CMBR power spectra at large angular scales make
a plateau with a increase in the anisotropy at the lowest multipoles
in the missing energy or curvature dominated(today) models. The
plateau rises to the doppler peaks in a smooth fashion due to the
“Early ISW effect” and the aliasing of power from higher multipole
moments. The doppler peaks represent oscillations in the baryonphoton fluid before the last scattering surface, and are sampled at
small angular scales due to baryon drag..................................................
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xxiii
6.2
The evolution of gravitational potential $ for four different wavenumbers is shown for three different models: The solid line represents a
fluctuating Q-field model with Qq = 0.6 and w = - 1 /3 , the dot­
ted line an unphysical smooth Q-field model with the same param­
eters, and the dot-dashed line a model with cosmological constant
(Q\ = 0.6). The wave numbers are, from the top-left panel moving
clockwise, 10~4Mpc~l , 10_3A/pc_ l, 5 x 10-2A/pc_ l, and 10~2A/pc_1
respectively. Notice the slow change for a —►1 of the solid line rep­
resenting the fluctuating Q-field model in figure with k = 10-4A/pc-1
(low wavenumbers), and the rapid change in the potential in the A
model in the same figure. Also notice in the bottom panels the low
a difference between the Q-field and A models’ potentials at highwavenumbers....................................................................................................... 145
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xxiv
6.3
The ratio of energy density in an individual species to the critical
energy density today as a function of time. Shown here are three
models, each with Qq = 0.6, and with equations of state w -0.33,
-0.05, and 0. Note th at in a w = —1/3 model only pn and
comparable at the surface of last scattering. For w=-0.05
pm
pq
are
is as
large as pa at last scattering, making a significant effect on the CMBR.
When w = 0,
pq
is always higher than
pm
and the universe never gets
m atter dominated. The large fraction of Q at last scattering ensures a
rise in intermediate scale anisotropy through resonant oscillations and
the early ISW effect.......................................................................................... 146
6.4
CMBR anisotropy as a result of resonant acoustic oscillations and the
early ISW in models with Qq = 0.6. The height of the peak does not
change much from w = —1 to w = —0.3 while the position changes
mildly due to changes in the angular diameter of the sound horizon.
There is greater change from w = —0.4 to w = -0 .1 , at which value Q,
radiation, and m atter contribute roughly equal amounts to the total
energy density at last scattering , and a steep increase in the height
fromw = —0.1 to w = 0 , in part of which range Q always dominates
over m atter.......................................................................................................... 147
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XXV
6.5
Position of the first, second and third peaks as a function of w. Notice
how the curves on the upper panel turn back up for w > —0.1................ 148
6.6
Peaks heights as a function of w when the power spectra are normalized
to C O B E .............................................................................................................. 149
6.7
Peak heights with the normalization as output by the code. Since
this normalization is the same for all models as opposed to the COBE
normalization, we can isolate the physical effects th at contribute to
the trend. Notice how all the curves are flat until w becomes large.
For (w >= —0.1), the Q fraction at last scattering increases to be
comparable to the radiation fraction, leading to a rise in anisotropy.
As w gets very close to 0, the Q fraction even dominates the m atter
fraction, and there is a steep rise in the heights of the doppler peaks.
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150
xxvi
6.8
This figure shows the CMBR power spectrum code and COBE nor­
malized for a Qq = 0.6 model with w = —1/3, in the physical case
with fluctuations turned on and responding to gravity, and in the un­
physical case of their being turned off which violates the equivalence
principle. Note the strong difference in low multipole behavior, the
unphysical smooth Q-field model showing an upturn in the anisotropy
power spectrum at those scales, while the fluctuating Q-field model
shows a slight dip as a result of the direct effect of the Q-field fluc­
tuations. Q-field fluctuations feed the gravitational potential with
energy, leading to this rather large difference from the smooth model.
On COBE normalization, the change shows up at the doppler peak.
6.9
151
The figure shows the progression of low multipole CMBR anisotropy in
smooth and fluctuating Q-field models with w = —1/3 and Q q = 0.6,
and in a A model with Q \ = 0.6. The lower and upper curves in
each of these cases are the CMBR anisotropy power spectra seen by
an observer at an epoch when the ratio of Q (A) to m atter density
was 0.1 and today (ratio=1.5) respectively. Note the large change
in the unphysical smooth Q-field model, the appreciable change in
the fluctuating Q-field model, and the unspectacular change in the A
model....................................................................................................................152
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xxvii
6.10 CMBR anisotropy low multipole power spectra at different epochs in
a fluctuating Q-field model with w = - 1/3 and Qq = 0.6. The epochs
are marked by the ratio of Q energy density to m atter energy density;
in this model, a ratio of 0.005 corresponds to a redshift of 299.
. . . 153
7.1 In this plot we add to the curves in figure 3.3 two Q-models: one, a
high Q q , high w constant equation of state model, and two, a model
with a massive scalar field sloshing about in a quadratic potential. We
see th at the constant w models fits the observationaldata well.
7.2
...
156
Here we simply change the previous plot by focusing onthe bandpowers from a few observations that we believe to be the best in terms
of the quality of their data. VVe once again see that the constant w
models fits the observational data well.................................................157
7.3 (a) Variation of mass power spectrum for some representative QCDM
examples, (b) The variation of crs with Q q . For ACDM, Q q is Q a The suppression of ct8 In QCDM compared to Standard CDM makes
for a better fit with current observations. The grey swath illustrates
constraints from x-ray cluster abundance......................................................158
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xxviii
7.4
The CMBR degeneracy problem: Each dashed curve in (a) represents
a family of QCDM and ACDM models with indistinguishable CMBR
anisotropy power spectra. The width of the curve is 5-10 % the values
on it. For example, Panel (b) shows two overlapping spectra for the
A (square) and quintessence (circle) models indicated in (a). Models
beyond the dotted line in (a) (e.g., the triangle) are distinguishable.
8.1
167
The CMBR anisotropy constrains models to a particular degeneracy
curve and, independently, provides tight constraints on ns, Qmh2 and
Qfc/i2. The latter constraints, along with other observational limits
discussed in the text, fixes an allowed range of fim and w (the shaded
region using the example discussed in the text). The combination de­
termines the best-fit models. The degeneracy region becomes smaller
as we include multipoles up to t = 2000 due to the inclusion of non­
linear gravitational tensing effects on the CMBR a n is o tro p y .
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170
xxix
8.2
The degeneracy contours of the four fiducial models being considered
are shown: ( n M,fiA) = {(0.35,0.65), (0.9,0.1), (0.35,0.45), (0.6,0.6)},
all with h = 0.65, ns = 1.0 and Q^/i2 = 0.02. The contours are cut off
above the flA = 0.75 and to the left of the
= 0.2 lines. Also shown
are dotted lines radiating from (—0.145,1.040) and passing through
the fiducial points, which are marked by a triangle. Note that the
degeneracy lines are not perfectly straight lines........................................... 174
8.3
These figures show how the apparent magnitude of a supernova at
z=1.5 may be used to constrain the values of flA and
We generate
2 percent error contours on the luminosity distance (dotted lines) and
see how they cut the CMBR degeneracy contour in four different fidu­
cial universes. From left to right and top to bottom the figures corre­
spond to
(nm,nA) =
(0.35,0.65), (0.35,0.45), (0.90,0.10), (0.60,0.60).
The point corresponding to the fiducial universe is marked by a trian­
gle. Note th at we can obtain a ~ 10% determination of fim..................... 180
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XXX
8.4
The magnitude-red shift relation may be an approach for distinguish­
ing A models (thick solid curve) from the family of quintessence models
(dashed curves) along the degeneracy curve. Am is the difference in
the predicted magnitude of a standard candle for a given model and
an open universe (fim —> 0, middle dotted curve). The dashed curves
are QCDM models with w = —5/6, —2/3, —1/2, —1/3 from top to bot­
tom, respectively. Type 1A supernova data is from Garnavich [5]. For
reference, an Q \ = 1 (upper dotted) and Qm = 1 (lower dotted) flat
model are shown................................................................................................. 181
8.5
2% and 5% error contours for apparent magnitude(luminosity dis­
tance) measurements made on a z = 1 supernova, with the fiducial sky
chosen to have been derived from a flat universe with Q q = 0.6 and
w = —0.5 (marked with a circle in the shaded region). Notice that the
contours, while not entirely perpendicular to the degeneracy curves
of figure 7.4, may be used in conjunction with the above mentioned
CMBR degeneracy curves to set 10%-20% limits on the Q parameters. 182
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Chapter 1
Introduction
1.1
Finding out about the universe
The study of the skies is one of the oldest scientific endeavors. For centuries people
have asked fundamental questions like: how big is the universe, what is it made of,
how old is it, how did it come into being, and how much longer will it continue to
exist?
For most of this time, much of cosmology has been in the realm of theology, as
there was little observational evidence to back up theories. Starting at the beginning
of this century, the pioneering efforts of observers like Edwin Hubble have transformed
it into a science. As technology has progressed, we have been able to look out to
greater and greater distances. We have built optical, radio and microwave telescopes
on earth and also launched them into space on satellites, and have collected signals
from billions of years in the past.
Now, on the verge of the 21st century, as a result of the dramatic increase in
1
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2
our technological capabilities, we stand on the threshold of a wealth of experimental
data. In the next decade or so, researchers will measure the distribution and velocities
of galaxies, dark m atter, and the anisotropy in the Cosmic Microwave Background
Radiation (CMBR) to unprecedented accuracy. The quantity of the data we obtain
will far exceed our entire present databases, and the results will possibly constrain
all our present theories of the evolution of the Universe. If some of the data is
inconsistent, we may well be forced to come up with new paradigms to describe the
nature and the history of the universe.
Measurements of the cosmic microwave background anisotropy m il be among
the most decisive cosmological tests because the microwave background probes the
oldest and farthest features of the Universe. Anisotropy measurements will provide
a spectrum of precise, quantitative data that, by itself, can significantly improve the
determination of cosmological parameters. In fact, the bounds on human capability
to explain the Universe are likely to be decided by what is discovered in the microwave
background during the coming decade.
The study of cosmology, and of the distant past of the universe through the
CMBR anisotropy in particular, offers us the most im portant experimental window
on fundamental physics at high energy scales. All fundamental theories of particle
physics must pass through the cosmological wringer: they must predict physics that
produces the correct amount of large scale structure today, the correct amount of
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3
CMBR anisotropy in the past and the correct expansion history of the universe.
We shall, in this work, using the anisotropy of the CMBR, address two key chal­
lenges that our understanding of the universe faces: the determination of the total
energy density of the Universe, and the identification of its physical composition.
These issues are essential to understanding the evolution of the Universe and may
have profound implications for fundamental theories of physics.
1.2
The problem o f M issing Energy and a possible answer: quintessence
The standard Big Bang model, which is the mainstay of our view of the universe,
has been enormously successful at explaining key features of our universe such as the
Hubble expansion, the light element abundances, and the existence of the Cosmic
Microwave background Radiation (CMBR). It is, however, silent on the issues of the
incredible flatness and homogeneity of the universe, and on how the initial conditions
necessary to produce the large scale structure we see today came about.
Inflation is a modification of the standard Big Bang picture which explains these
mysteries as a consequence of a series of events occurring in the first 10-35 seconds or
so after the big bang. It predicts a flat universe and provides a method of generating
small inhomogeneities from quantum fluctuations. These inhomogeneities then evolve
by gravitational collapse to form the structures we see today.
There is a critical value of the total energy density of the universe which ensures,
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4
from Einstein’s equations of gravity, that the universe is flat. From the predictions
of big bang nucleosynthesis, luminous (baryonic) m atter accounts for at most 10% of
the critical density[1]. The standard cold or hot dark m atter models describe a flat
universe with the remaining energy density supplied by this dark m atter. However,
growing evidence suggests that the remaining dark m atter density is substantially
less than the critical value, as well. This evidence is mounting every day(see [2]):
recent results from galaxy lensing surveys have estimated the m atter fraction to be no
more than 0.55(3], and high redshift supernovas surveys point to the m atter fraction
being no more than 0.4([4],[5]. Given the currently favored measurements of the
Hubble constant are in the range 65- 75 kms- l Mpc- l ([6]), requiring the universe to
be 10-12 billion years old at the very least(from globular clusters, see [7],[8]) forces
m atter to make up no more than 40% percent of the universe. There is also other
evidence from velocity flows, redshift surveys, etc([9],[ 10],[2]) which along with the
measurements quote here all point to a deficit of energy in the universe as compared
to the critical value required to make it flat.
Thus we may be forced to abandon inflation and its attendant solutions to the
problems of the Big Bang Model, and admit that the Universe is open (since its
total density is less than the critical density) or consider the existence of a hitherto
undetected additional constituent in the universe besides baryons, photons, neutrinos
and dark m atter. Present observations are mostly sensitive to the lowered m atter
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5
fraction, and not geometry, and thus it is hard to discriminate between open universe
models and models with a missing energy component.
There is some tentative evidence against open models coming from high redshift
supernovas and CMBR anisotropy data[5, 11]. While this evidence is not yet conclu­
sive, owing, in part, to uncertainty in systematics, it has heightened interest in miss­
ing energy models in conjunction with the strong theoretical motivation(inflation)
for these models.
This missing constituent has been traditionally assumed to be the cosmologi­
cal constant (A), an all-pervasive spatially uniform vacuum energy density which
was supposedly left over as a remnant of super-symmetry breaking in the early uni­
verse. Cold dark m atter models with a substantial cosmological constant (ACDM)
are among the models which best fit existing observational data.[10] Since this fit is
mostly sensitive to the m atter fraction, it does not distinguish whether the missing
energy consists of a cosmological constant or some other constituent.
The cosmological constant is not very well motivated from fundamental physics.
If it is indeed a remnant of super-symmetry breaking, why is its energy density today
comparable to that of matter? W hat dynamical processes would induce a spatially
uniform vacuum energy density produced at the Planck scale to traverse 120 orders of
magnitude to present day energy densities. We could insist that the vacuum energy
is this small from the start, however, this would necessitate an incredibly fine tuned
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6
cancellation of bosonic and fermionic energies at super-symmetry breaking, leaving
us with an ugly and unnatural picture of the early universe.
The fundamental idea of this work is to explore replacing A with a dynamical,
time-dependent and spatially inhomogeneous constituent whose equation of state is
different from baryons, neutrinos, dark matter, and radiation, and whose inhomo­
geneities evolve in time. The equation of state of the new component, denoted as
w , is the ratio of its pressure to its energy density. This contribution to the cosmic
energy density, which we call “quintessence” or Q-energy, is broadly defined, allow­
ing a spectrum of possibilities corresponding to negative equations of state that are
constant, uniformly evolving or oscillatory. Examples of a Q-component are funda­
mental fields (scalar, vector, or tensor) or macroscopic objects, such as a network of
light, tangled cosmic strings.[12] The analysis in the present work applies to any com­
ponent whose hydrodynamic properties can be mimicked by a scalar field evolving
in a potential which couples to m atter only through gravitation.
There has been previous work on models with fluids possessing negative equations
of state. [13, 14]. However, these efforts have not considered fluctuations in the
fluids because it was believed th at these fluctuations are not stable to gravitational
evolution because they have an imaginary sound speed, implying that pressure and
gravity act in the same way, leading to the fluctuations imploding on themselves. In
this work we show th at the fluctuations are in-fact stable, and remarkably insensitive
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7
to the initial conditions set on them. This will enable physicists to concentrate on the
fundamental physics which leads to quintessence rather than the phenomenological
details, and provide a structure in which models in which no fine-tuning of initial
conditions is required are developed.
Quintessence allows us to bring fundamental dynamical fields into the study of
cosmology, possibly providing us with a mechanism to bridge the huge gap in energy
scales th at plagues the theory with the cosmological constant. It is not a given that
such a model can be found. Our contribution is to make available a framework in
which the dynamics of such a model can be studied.
Quintessence (0 > w > —1) and cosmological constant (A, w = —1) have qualita­
tively similar effects on cosmology. Both components increase the expansion rate of
the Universe relative to a flat, Universe composed only of baryonic and cold dark matter( CDM). Both increase the age of the Universe. Both result in earlier formation of
large-scale structures than in CDM models. Both agree better with current cosmo­
logical observations than CDM models do [15, 13, 16, 10, 17]. They fit the current
constraints from supernovas, gravitational lensing, structure formation, and CMBR
anisotropy. [16, 18, 19, 17] As in A models, the change in the late time expansion
history of the universe suppresses the growth of perturbations in the metric, predict­
ing more structure formation at deep redshifts, in accord with recent observations of
quasars. [9]
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8
Hence, the differences in classical measurements between these two possibilities
for cosmological measurement are quantitative - the differences are found only in the
details. Nevertheless, differentiating between the two, even in classical cosmological
tests, is of paramount importance because the physics that underlies the two possi­
bilities is so different. In particular, if the cosmological measurements establish that
quintessence accounts for the missing energy of the Universe, it may unveil a new
regime of ultra-low energy phenomena to be explored.
The major difference between the two possibilities is that quintessence is con­
stant neither in space nor in time. It thus develops fluctuations. These fluctuations
affect the CMBR anisotropy in models with quintessence directly; they feed energy
into the gravitational potential. This leads to very distinctive features in the CMBR
anisotropy at intermediate and large angular scales. Further, the CMBR power spec­
trum normalization is changed, and this affects the fit to the observed mass power
spectrum([20, 2]). In this work we will describe how the physics of quintessence leaves
its mark on the CMBR anisotropy in a variety of models. We will parameterize the
models by their equation of state histories, and study the effect of the variation in
these histories on the anisotropy power spectra. We will finally address the question
of whether the distinctive features imposed on the CMBR power spectra by the new
physics of the Q-field fluctuations is enough to lead to its discovery from CMBR
anisotropy measurements alone, and from a combination of CMBR anisotropy mea­
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9
surements with other ones.
1.3
Outline of this work
In chapter 2 we introduce the various concepts prerequisite to a understanding of
quintessence cosmology. We describe the relevant concepts from the Big Bang model
and Inflation. We describe the process that produces a scale invariant spectrum of
perturbations, and further describe the gravitational instability scenario for large
scale structure formation. We introduce the CMBR power spectrum as a measure of
the microwave background anisotropy.
In chapter 3 we describe the experimental and theoretical motivations for
quintessential cosmology as related to the the m atter density being much smaller
than critical. In this chapter we concentrate on the kinematic effects of a energy
density component with negative equation of state. We end the chapter by sur­
veying the present observational picture of the CMBR anisotropy, accumulating in
the process even more evidence for the existence of missing energy, in preference to
curvature.
In contrast to the cosmological constant, the Q-field develops inhomogeneities
which evolve with time, and directly affect the evolution of perturbations in the
metric. We introduce in chapter 4 the equations for the dynamics of the Q compo­
nent, and the modifications of the Einstein equations for cosmological models with
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10
a mixture of cold dark m atter and quintessence 4.4. In particular, we develop a for­
malism which describes the field and fluctuation evolution in terms of the equation
of state history of the field. We integrate Boltzmann equations using this formal­
ism, and describe the resulting CMBR anisotropy for a slew of models with different
potentials.
We shall, in chapter 5, show that quintessence field and fluctuation evolution in
models with light fields can be described a simple constant equation of state ap­
proximation. This approximation enables us to solve analytically for the fluctuation
evolution at large wavelengths. We use these solutions to show that the fluctuations
are stable and very insensitive to initial conditions. Should particle physics provide
us with a viable quintessence model that can be described by constant w , it will have
well behaved fluctuations fitting current day observations([19]) without any need to
fine-tune initial conditions.
We embark on a detailed, epoch-by-epoch study of how the physics of quintessence
affects the CMBR anisotropy power spectrum in chapter 6. We show that the key
signatures of an inflationary model - a flat power spectrum at large angular scales, a
doppler (acoustic) peak at 1 degree scales, and acoustic oscillations at small angular
scales[21] - remain features of QCDM models. However, we also identify subtle
modulations of the spectral shape and doppler peak positions in QCDM models that
distinguish them from CDM, open, or ACDM models, and hitherto undiscovered
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11
direct effects of the fluctuations in the Q-component on the CMBR anisotropy at
large angular scales. One of the key points of this work is that this direct effect leaves
very distinctive imprints on the behavior of the CMBR anisotropy power spectrum.
VVe discuss in detail the variations of the heights and positions of the doppler peaks
with quintessential parameters, the intermediate scale plateau and rise to the first
doppler peak, and the unique low multipole behavior in these models.
In chapter 7 we introduce a new statistical analysis comparing predictions of
QCDM and ACDM models, including uncertainties in all cosmic parameters. We use
this methodology to ask if these models may be distinguished from the cosmological
constant models by their CMBR anisotropies alone. We will find that this is not
possible for a large range of equations-of-state and quintessence energy densities due
to fundamental degeneracies in the CMBR anisotropy.
In the last chapter we show that the degeneracy with the cosmological constant
is part of a greater parameter degeneracy(in h, ftg/i2, Q,\, etc) that makes it diffi­
cult to distinguish the equation of state and cosmic density of the missing energy
for some parts of the quintessence density-equation of state (Qq — w) parameter
space. We attem pt to resolve these degeneracies using observations of high redshift
supernovas, based on work done by us on models with arbitrary geometry and cosmo­
logical constant. O ur methodology results in some payoff: as we shall see, combining
the anisotropy measurements with distance measurements from high redshift super­
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12
novas will allow us to lift the degeneracy somewhat and obtain loose limits on the
cosmological parameters.
Chapters 4 and 6 represent work done in collaboration with Paul Steinhardt and
Rob Caldwell, building on what was published in [16]. The results in chapters 7 and
the last chapter and the previous one are the results of a collaboration with Paul
Steinhardt, Robert Caldwell, Limin VVang, and especially Greg Huey, as published
in [19].
Our systematic study of the effects of quintessence provides methods by which
fundamental physics may be tested out in the laboratory of the universe. The im­
prints of different particle physics models may be distinguished by studying, for
example, the observed CMBR anisotropy power spectrum. These imprints may well
be at a level which allows us to rule out certain classes of models. If this turns out
to be indeed the case, we will have narrowed the range of answers to a lot of the
fundamental questions we posed earlier in this chapter.
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Chapter 2
Preliminaries
We shall, in this chapter give a brief introduction to the study of the kinematics
and dynamics of the universe. It serves as a “hitchhikers” introduction to the study
of those concepts of cosmology pertinent to this work, and sets up equations and
notation th at we shall freely use in the following chapters.
Observations and the theory of general relativity together motivate the so-called
Big-Bang model of the universe. We describe the evolution of the universe’s and
its constituents in this model. The problems of the model, and their resolution by
inflation are discussed. Inflation provides a natural mechanism for the generation
of perturbations that span cosmological scales, and imposes a evolution on these
perturbations which is crucial to our development of perturbation theory in models
with quintessence. We describe the cosmic microwave background radiation (CMBR)
power spectrum of the fluctuations resulting from inflation. Finally, we introduce the
“missing energy” problem th at quintessence attem pts to resolve.
13
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14
2.1
Fundamentals
It has been found, from looking at various electromagnetic backgrounds at far dis­
tances, and from deep galaxy counts, that the large scale universe, as viewed from our
unique vantage point in it, is quite close to being isotropic. Assuming this isotropy
in the large scale ought to hold from all other observational points in the universe,
rendering the universe identical in the laws of physics at each such vantage point, or
homogeneous. These assumptions of isotropy and homogeneity have been historically
known as the “cosmological principle” .
This conclusion is borne out by the homogeneity of the large scale mass distri­
bution and element fractions in the universe as measured by the experiments like
the COBE satellite: at scales corresponding to and larger than the size of the causal
horizon (4000 Mpc, where a parsec is an astronomical unit roughly equal to the dis­
tance light travels in 3 years), the deviation from average is no more than 1 part in
105.
It is further clear from observation that galaxies are receding from each other
at a speed proportional to their distance apart. Given that gravity at the scales at
which the Newtonian theory applies is an attractive force, this is clear evidence for
a expanding of the universe. One can make the analogy to a rubber sheet with a
grid drawn on it. If the grid is stretched, we find that the grid lines themselves move
further apart, and that pins, placed at the intersection of horizontal and vertical lines
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15
will find themselves moving apart from each other. The speed of this “motion” is
proportional to the distance the pins are apart; this is the only “motion” allowed by
the imposition of homogeneity. The constant of proportionality is called the Hubble
Constant, and its value today (denoted Ho) has been found to lie in the range 5080 K m /s/M pc. It is usual to define a dimensionless number h, which takes values
between 0.5 and 0.8, such that Ho = 100h K m /s/M pc.[6]
We need a way to mathematically describe this expansion, without attempting
to answer questions such as what the universe is expanding into. This framework
is the General theory of relativity. It gives us a mathematically consistent and
observationally correct picture of the expanding universe, and we shall use it now to
set up a vocabulary for the rest of this chapter.
2.1.1
The space tim e metric and equations of motion
For the purposes of this work, we will assume that the universe is described by the
Friedman-Robertson-Walker (FRW) space-times. The space-time metric is given, in
the (- + + + ) signature that we will use, by,
ds2 = —dt2 + a l {dx1),
(2.1)
where a(t) is the expansion scale factor whose value today a0 will be conventionally
taken to be 1, t is physical time and the space-like co-ordinates are the so-called comoving co-ordinates, in which the expansion of the universe has been factored out.
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16
These are the co-ordinates measured by the lines of the grid drawn on the rubber
sheet in our analogy: despite the sheet being stretched, the third horizontal line from
the origin is still the third line afterwards. More physically, we can say that if no
external forces are acting, a particle at a set of given co-moving co-ordinates will stay
there.
The metric equation can be rewritten as
ds2 = a2(ri)(—dT}2 + dx2)
(2.2)
where // is the conformal time. This is a convenient parameterization, as we can see
from equation 2.2 that, in these variables, the FRW metric is conformally equivalent
to a Minkowski metric. Note that the real time differential is dt = adr] . We will
denote derivatives with respect to real time by dots, and derivatives with respect to
conformal time by primes (').
The space-like part of the metric can be written out in polar co-ordinates as
df2 = i
+ r2rffi2
<2-3)
where dQ is the solid angle differential, and where k characterizes the global geometry
of the universe We can restrict the global geometry to be that of a flat universe by
choosing A: = 0. O ther choices are open and closed, as given by a non-zero value of
k ( -1 and +1 respectively) in the radial part of the space-like metric.
At the long distances that cosmology concerns itself with, the only pertinent force
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17
of nature is gravity. Gravity is described by Einstein’s equations:
Gpt/ = 8;xG T ^
(2.4)
where Gpu is the gravitational tensor, an expression of the deviation of the local
curvature from the global value, and Tpv is the stress-energy tensor, a collection of all
the physical agents which are able to affect the geometry: energy, pressure, rotation,
and shear. Thus the expansion of the universe is governed by the properties and the
interactions of the m atter that constitutes it.
If we assume an ideal fluid, TM„ is diagonal with its time-time component the
energy density p, and Tspacet3pace = ~P- Then, the time-time Einstein equation is
a = - ^ Y ~ { p + 3p)a = ~ ^ H 2{1+ Zw)a,
(2.5)
where G is Newton’s constant, p is the energy density, p is the pressure, and H is
the Hubble parameter which is defined by the other independent Einstein equation,
also known as the Friedman equation:
H2 =
-
a
=
8 ttG p /3 - A
ai
(2.6)
The Hubble parameter describes the fractional expansion of the universe and is thus
the proportionality parameter between the relative velocity of receding galaxies and
the distance between them (Its value today is the so called Hubble constant). The
ratio w = p /p defines the equation of state of the universe. It is 0 for a universe filled
with only dust, and 1/3 for one filled with radiation.
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18
The p from Friedman equation 2.6 represents the sum of contributions from var­
ious species which make up the universe like neutrinos, protons, electrons and neu­
trons, photons, baryons, gluons, etc However, there is considerable astrophysical
evidence for other, non-observable types of matter([10, 2]).
We define Q to be the ratio of the total density of the constituents in the universe
ZH2
to the critical density pc = ^
required to render it flat. It is easy to show from
equation 2.6 that
n - l - A .
(2.7)
Thus, a value of Q < 1 corresponds to a universe with negative curvature, i.e. a
hyperbolic or open universe. A value of Q > 1 implies a closed or positively curved
universe.
2.1.2
Characteristic scales in the universe
The present value of the Hubble parameter is traditionally measured by observing
the redshift of galaxies which are receding from us with speeds much smaller than
that of light, after correcting for the local velocities of our and the other galaxy.
The inverse of the Hubble constant represents the time scale for an appreciable
expansion, since the Hubble parameter is the fractional expansion of the universe
per unit time. Its value today is H q 1 = 9.76h~l billion years (the subscript “0” on
quantities is used to denote the present value). This number provides an estimate
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19
of the age of the universe, and when multiplied by the speed of light, yields a value
of 2998h~l Mpc, giving us an estimate of the size of the observable universe, or the
causal horizon today. Thus H ~ l is the fundamental length/tim e scales in cosmology.
Another characteristic scale which we only touch on briefly in this work is the cur­
vature scale in a non-flat universe. This is the scale below which the universe can be
thought of as essentially flat. We can easily see from Friedman’s equations(2.5,2.6,2.7)
th at the ratio of the Hubble scale to the curvature scale is the fractional departure of
the density of the universe from the critical density required to make it flat. That is,
the smaller the Hubble length is compared to the curvature scale, the more difficult
to detect the curvature.
2.2
Thermal origin o f the universe: the Big Bang M odel
We have so far established two cornerstones of the Big Bang Model: the large scale
homogeneity and isotropy of space, and the expansion of the universe. Let us now
add to these the third cornerstone: the universe expanded from a hot dense state
where its energy density was dominated by black body radiation.
The word “Big Bang” is a misleading term for such an initial state as it seems to
convey an explosion localized in space and time. It actually refers to a homogeneous
expansion(or contraction) with no distinguishable center. The Einstein equations,
pulled back into the infinite past predict a singularity; it is assumed th at new physics
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20
takes over at about the Planck scale. The word “Big Bang” signifies only that this
new physics leads to a hot and extremely dense universe at this early epoch which
expands uniformly to form the universe which we live in today. The expansion leads
to cooling, and consequently, to the binding together of elementary particles to form
light element nuclei.
At wavelengths in the mm-cm range, the extra-terrestrial electromagnetic radi­
ation background is dominated by an isotropic component called the Cosmic Mi­
crowave Background Radiation(CMBR), which was first discovered by Penzias and
Wilson in 1964 at the Holmdel radiometer in New Jersey. This isotropy suggests a
sea of radiation uniformly filling space. The cooling universe scenario suggests that
the CMBR is a relic from an earlier epoch when the expanding universe was hot and
dense enough to have relaxed to thermal equilibrium. Thus we would expect the
radiation to have a thermal blackbody spectrum.
In 1989, the Far Infra-Red Absolute Spectrophotometer (FIRAS) experiment on
board the Cosmic Background Explorer (COBE) satellite measured the spectrum of
the radiation and confirmed the big bang predictions. W ith exquisite experimental
precision, the spectrum was shown to be perfect black-body with a temperature of
T0 = 2.726 ± 0.010 K [22].
It is difficult to come up with a mechanism that could have induced the microwavetransparent present universe to produce a thermal spectrum in the CMBR. Thus the
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21
CMBR is considered to be one of the key predictions of the Big Bang theory. A
blackbody radiation spectrum will remain blackbody if the interaction of the radia­
tion with m atter is miniscule, and if the universe is homogeneous and adiabatically
expanding. VVe see today a thermal spectrum at the 3K temperature, with the wave­
length of the radiation having been red-shifted into the microwave.
2.2.1
The thermal universe: Standard Big-Bang M odel
The knowledge of the present CMBR temperature has enabled the straightforward
computation of the thermal history of m atter. According to the Big-Bang model,
the CMBR is radiation emitted some 100,000 years after the Big-Bang. Prior to
that time, the universe consisted of a hot, dense gas of free electrons and nuclei in
equilibrium with photons. Three quarters of the nuclei formed were hydrogen(free
protons), and a quarter helium, with small abundances of other light nuclei. Know­
ing the CMBR tem perature accurately today has enabled us to peg down the epoch
of nucleosynthesis, and has explained the ubiquitousness of the helium fraction in
nebulae and the field, since the prediction is that most of the Helium was formed primordially due to the extremely high temperatures that existed in the early universe.
After 100,000 years or so, the universe had cooled enough for the free electrons
and nuclei to combine into neutral atoms(13.6eVr). Thus the baryon-photon fluid
which existed in the universe gets decoupled. This time is called the decoupling
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22
epoch. Once the universe cools a bit further, the photons, which interacted only very
weakly with the newly neutralized medium, began to freely stream in all directions.
The space-like hyper-surface(really a hyper-volume, since the process was not instan­
taneous) at which the photons start to free stream is known as the last scattering
surface(LSS). Initially, the spectrum was perfectly black body with a temperature
of nearly 10,000 K. Since the energy density in the radiation is too low to ionize
hydrogen or to undergo appreciable Thomson scattering after the decoupling epoch,
the radiation spectrum remains blackbody. Over the subsequent 10 billion years,
the photon distribution red shifted into the microwave due to the expansion of the
universe.
The isotropy of the universe at the scales of the Hubble horizon(3000Mpc) is
bounded by measurements made by the COBE satellite to be 1 part in 105. This small
anisotropy is however very important, as it is the small departures from homogeneity
in the universe which lead to the formation of structure in the universe by the process
of gravitational instability.
2.2.2
M atter content of the universe in th e big-bang model
We describe in brief the various constituents of the universe in the standard big-bang
cosmology. These species dominate the universe’s energy density at different epochs
due to the different dependence of their energy densities on the scale factor.
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23
For any species th at contributes to the composition of the universe, let us define
Qspedea as the ratio of the energy density in that species to the critical energy density
today. We describe some of the species below:
• Radiation
Photons are massless particles, and all their energy is in their motion. They
have an equation of state w = 1/3, and provide pressure against gravitational
collapse. They do not contribute much to the energy density today (fir ~ 10-5).
However, in the distant past, when the universe was a hot black body, radiation
dominated the energy of the universe.
• Baryons
Baryons as well as other form of matter, exert no pressure, and hence have
w = 0. Baryons are formed out of the quark-gluon plasma th at existed in
the early universe. The m atter density scales as the inverse third power of
the scale factor (p\i = pc^ ) , which is why it dominates the universe more
than radiation today. The turnover point at which radiation domination gives
way to m atter domination is called matter-radiation equality and occurs at the
redshift z = (£ — 1) = ^ - - 1 . Luminous baryons account for no more than
15% of the critical density of the universe (QB < 0.15).[10]
• Dark Matter
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24
The rotation( velocity versus distance from center) curves of galaxies when
computed using the luminous m atter found in the galaxies ought to be monotonically decreasing from the core outwards, according to Newton’s law of
gravity which applies on these scales.
However these curves are flat, lead­
ing us to postulate the existence of baryonic and non-baryonic dark matter,
which we did not take into account in the calculations of these curves. Such
baryons may be in non-emitting gases, compact halo objects, or dead stars.
Candidates for Non-baryonic dark m atter include the super-symmetric part­
ners of normal fermions and bosons (Cold Dark Matter, or CDM), and massive
neutrinos (Hot Dark Matter, or HDM). Observation lead us to believe that
— &dm + Db < 0.4.[10, 17]
• Matter with negative equation of state
Observations (see above) have shown that m atter accounts for no more than 40
% of the critical density in the universe. Cosmologists([10, 14, 13]) have thus
invoked the cosmological constant or fluids with negative equations of state to
account for the remaining energy necessary to make the universe flat.
The energy density in such a fluid, provided w is constant, scales as pw =
pc£lw/a 3( 1 + w). Such a constituent has the property that, if it dominates the
universe, for w < - 1 /3 , the universe undergoes an accelerated expansion. This
follows from the Einstein equation 2.5. As we shall show in the next chapter,
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25
constituents with negative equations of state increase the age of the universe
and better fit current observations as compared to those with positive equations
of state. We will concern ourselves with the dynamics of such a component in
this work.
The notion of a constituent with equation of state w ~ —1 is very important
physically, as such a situation can be easily achieved by a slow rolling scalar
field:
D
If
+ V
~ 0 , then the field has very slow dynamics and w ~ —1. The limiting
case w = —1 corresponds to a field with no dynamics.
If such a compo­
nent is spatially homogeneous and time-independent, it is called a cosmological
constant, invoked earlier. Secondly, the Einstein equation (2.5) in a universe
dominated by such a scalar field with w = —1 shows accelerated expansion, or
inflation, which is the topic of the next section.
2.2.3
Problem s w ith the Big Bang M odel
The Big Bang Model does not explain some very critical observations of our present
universe. We list here three of them pertinent to our work:
• Flatness Problem: Why is the universe so flat? fi = 1 is an unstable equilibrium
of the Friedman equation, i.e., if the initial conditions of the universe are such
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26
that tt # 1 but very close to 1, then the universe will move further away
from flatness as time evolves. Thus, it would require the initial departure from
flatness to be dialed to less than one part in 1060 [23] for
to be between
0 and a few today, as confirmed by [5]. This is an enormous amount of fine
tuning.
• HorizonfHomogeneity) Problem: Why is the universe so homogeneous? The
perturbations in the CMBR of 1 part in 105 mean th at the universe is, on
length scales much larger than the present Hubble horizon, incredibly homo­
geneous. Since the entire visible could not have arisen from a region causally
connected in the past, what physical process set the initial conditions to be
very homogeneous on a super causal region?
• Inhomogeneity (Initial Conditions) Problem: The Big Bang model provides no
explanation for the initial conditions which lead to the formation of structure
in the universe today.
Any structure formed today would have had to be
obtained from the growing modes of the perturbations in the m atter density.
To restrict these perturbations to 1 part in 10° at scales corresponding to the
Hubble horizon, the initial conditions on these perturbations would have to be
incredibly finely specified. There is no mechanism in the standard Big Bang
model to specify such conditions or to naturally generate them; its time to
supplant the model with some “new” physics.
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There is a second horizon problem associated with the existence of CMBR
anisotropy on cosmological scales. Given that the entire visible universe could
not have arisen from a patch which was causally connected in the distant
past, why are there perturbations on scales larger than the size of one of these
patches. How were they laid out on the initial hyper-surface?
2.3
Inflation
Inflation is an incredibly simple and beautiful idea which takes away the above (and
other) problems with the standard big bang model. It is an accelerated expansion of
the universe in the first 1 0 '35 seconds of its existence, a short-lived period in which
the universe expands by at-least 30 orders of magnitude, possibly much more.
An expansion of this magnitude smoothes out any initial curvature the universe
may have had, thus providing an initial space-like hyper-surface which is flat enough
to explain the value of fV
This solves the flatness problem associated with the
standard big-bang model. Inflation sends the value of Q arbitrarily close to 1 and
thus a flat universe with Q = 1 is favored by most theorists.
The incredible expansion also wipes away any inhomogeneities that may have
existed prior to inflation. Since the expansion is itself superluminal, the “initial”
hyper-surface at the end of inflation is super-causal, and hence we solve the hori­
zon problem, explaining the origin of the incredible homogeneity in the large scale
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28
universe.
W hat will produce the large scale structure we see in the universe today? The
standard big bang model just laid down an initial spectrum; by invoking inflation we
have wiped this spectrum out. As we shall see, the physics which leads to inflation
is a quantum process, and it is the quantum fluctuations inherent in the Heisenberg
principle which give rise to the structure we see in the universe today!
2.3.1
The scalar field mechanism for inflation
If we look back at the Einstein equation (2.5) we can see that if the universe is
dominated by a species with equation of state w < - 1 /3 , then the expansion of the
universe accelerates. Thus inflation can be induced by any physics which leads to
large negative pressures.
The usual example is a scalar field (the “inflaton”) which is moving very slowly
along the ridge of a rather flat potential:
- ; P - f| 0 r2 +r V-
(2-9>
where V is the effective potential for the inflaton. If the potential energy of the scalar
field (the “inflaton”) dominates its kinetic energy, and as can be seen from the above
expression, it is very easy then to get w < —1/3.
An inordinate amount of expansion ensues, and it can be shown that an expansion
of the universe by a factor of e60 solves the flatness and horizon problems of the
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29
standard Big-Bang model by stretching causally connected regions of space very far
apart. Such a factor is very easily achieved, and inflation will continue until the
inflaton rolls into a minimum of its potential. It then sloshes about in its minimum,
converting potential to kinetic energy. The kinetic energy is damped by expansion,
and at the same time, converted to m atter and radiation through the coupling of the
inflaton to m atter and radiation fields.
2.3.2
Inflation gives rise to perturbations in the metric
The inflaton and other light fields naturally experience quantum De-Sitter fluctua­
tions at causal scales. It is these fluctuations which seed the formation of large scale
structure by creating perturbations in the stress-energy tensor of the universe, and
thus the metric. The expansion stretches the fluctuations and the accompanying fluc­
tuations in the metric and other fields to cosmic (and acausal) scales. Thus inflation
wipes away any pre-inflationary initial conditions on the inflaton and metric, and
creates new perturbations at all length scales. We can thus specify any reasonable
initial conditions on the fluctuations of the inflaton and the other fields. Thus the
Homogeneity or Initial Conditions problem is solved.
The De-sitter fluctuations of the inflaton and other light fields may be studied by
expanding them into fourier modes and considering the wavelength of each fourier
node as a scale in the above discussion. As the universe expands during inflation,
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30
the physical wavelengths exceed the size of the horizon; this happens at a different
time for each mode, the longest wavelength modes exiting first as they have the least
’catching-up’ to do with respect to H j l. The amplitude of such a fluctuation mode as
it leaves the horizon is given as H j xflit. The additional stretching does not change
this amplitude since causal processes are unable to act over distances greater than
the horizon. Since the universe is in slow roll, most waves leave the horizon with
almost the same amplitude.
After inflation ends the Hubble length re-catches up to these modes one-by-one
(the modes “re-enter the horizon”), even though they are still being stretched by the
expansion of the universe. The longer wavelength modes re-enter later, and some are
still re-entering today. The modes re-enter with the same amplitude they left the
horizon with, thus resulting in the approximate scale invariance of the amplitude of
the perturbations. In this way modes whose wavelengths were at microscopic scale
before the end of inflation are now expanded to cosmic scales and can act as seeds
for large scale structure formation.
2.3.3
The power spectrum of the perturbations
The primordial power spectrum is determined by the amplitudes on re-entry. This
spectrum is the initial condition for post-inflationary Einstein and Boltzmann equa­
tions of motion. Since the amplitude of modes which grew longer than the Hubble
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31
length during inflation is frozen, these modes start experiencing causal physics start­
ing at those very amplitudes on re-entry.
Since the fluctuations all left the horizon with nearly the same amplitude, we can
predict the fluctuation spectrum (calculated at re-entry) to be nearly scale-invariant:
if one expands the energy density field p(x) in a sum of fourier modes with amplitude
S(A), then the amplitude of a mode is nearly independent of its wavelength A.
If the spectrum is parameterized by a spectral index, n, defined by £(A) ~ A(l-r*)/2,
then a precisely scale-invariant spectrum corresponds to n = 1. In inflationary models
this index can be in the range 0.7 < n < 1.2, the deviations from pure scale invariance
at large scales measuring the speed of the slow-roll around 60 e-folds before the end of
inflation, since the largest scale fluctuations exit the horizon at the start of inflation.
This spectral index, has been measured by COBE([24|) to be in the above range,
consistent with the predictions of inflation. The amplitude of this power spectrum
is one of the key determinations that physicists hope to make from observations of
large scale structure and the cosmic microwave background, for it gives us a window
onto the coupling constants which define the effective inflationary potential.
Inflation also generates similar fluctuations in other light fields, but most of these
signatures are erased by the interactions that these fields undergo. A notable ex­
ception are the gravitons, which, by the very nature of their weak coupling escape
erasure by re-heating and interactions.
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32
2.4
The Gravitational Instability scenario o f structure formation
The Cosmic Microwave Background Radiation (CMBR) spectrum was shown by the
FIRAS satellite to be perfect black-body with a temperature of T0 = 2.726 ±0.010 K
[22]. This has confirmed the picture of the early universe as a dense and hot baryonphoton plasma which cooled as the universe expanded.
It was nearly 30 years after the discovery of the CMBR before any non-uniformity
in the CMBR temperature across the sky was detected. The temperature variation is
so tiny that instruments with ^Kelvin sensitivity had to be developed. This first suc­
cessful detection was by the Differential Microwave Radiometer experiment aboard
the COBE satellite (1992) [25, 24]. The satellite measured the amplitude and slope of
this spectrum by probing the low i CMBR anisotropy, and found it to be consistent
with inflation.
The root-mean-square fluctuation in temperature, found to be roughly 0.001%,
is an im portant cosmological parameter for understanding the formation of galaxies.
How did the universe evolved from being highly homogeneous at the 100,000 year
mark, as imaged by the CMBR, to being highly inhomogeneous today, as shown in
recent maps of the distribution of galaxies? And, what can the pattern of CMBR
anisotropy on the sky tell us about the composition of our universe?
The CMBR anisotropy characterizes the effects left on the photons at and since
the so-called last scattering surface(when Hydrogen was formed) by the metric fluc­
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33
tuations. Most of these effects happen at epochs when these fluctuations were linear
in perturbation theory. This means th at the metric fluctuations at these epochs were
still small compared to the unperturbed metric, i.e., the fluctuations hadn’t started
to collapse on themselves under gravity the way a closely spaced group of particles
would under their mutual attraction.
Once fluctuations re-enter the horizon, they evolve causally. Fluctuations can
remain stable as long as they are longer than the Jeans length, which is the product
of the sound speed and the characteristic gravitational time-scale. Once the Jeans
length catches up to them [23], gravitational collapse can occur faster than the time
required for a pressure wave to fight it. Thus “gravitational instability” sets in and
causes the scales to collapse on themselves. This leads to over-dense and underdense regions in the universe. There is then an amplification of inhomogeneities
caused by gravity drawing additional m atter into over-dense regions and away from
under-dense ones. The fluctuations go non-linear (i.e. linear perturbation theory is
no longer valid) and form the structures we see in the universe today.
This process is called the gravitational instability picture of structure formation,
and we believe th at it is this process, whose humble beginnings lay in small quan­
tum fluctuations of the inflaton th at has led to the inhomogeneous universe we see
around us today. We may track this evolution by comparing the fluctuations in the
m atter density today (which tell us the whole non-linear story) with the anisotropy
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34
of the Cosmic Microwave background radiation, which were imprinted in the linear
epoch. It can be shown that the inhomogeneity must have been 0.001%/Q at the
100,000 year mark in order for gravitational instability alone to explain the inhomo­
geneity seen today. The COBE observation has provided important support for the
gravitational instability concept provided Q is not much smaller than one.
2.5
The anisotropy power spectrum
The microwave background comes at us from all points on the celestial sphere. Vari­
ations in the energy density result in varying gravitational potentials that red shift
or blue shift the CMBR photons by different amounts across the sky, and in bulk
flows th at doppler shift the photon frequency, producing apparent CMBR tempera­
ture differences across the sky. To probe the angular dependence of the anisotropy,
we need to correlate the anisotropy coming from different rays on the sky. Given a
map showing the tem perature variation A T / T (x) of the CMBR across the sky, we
define the auto-correlation function of the anisotropy as:
C (a) = ( ^ ( x ) ~ ( j O )
*0
JO
= £
x .x '—cosa
i
a^fMcosa)
(2.10)
47T
where (...) is an all-sky average over every pair of directions separated by angle
a . The auto-correlation function is expanded on the basis of Legendre polynomi­
als. The values of the aj's, called multipole moments, measure roughly the average
tem perature variation between two directions separated by ~ 100/f degrees (ir/l
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35
radians).
Any theoretical prediction in an inflationary model, however, cannot predict a
precise value of oj within our Hubble Horizon, but, rather, the average a] and its
variance over an ensemble of Hubble horizon patches. (This is a fundamental re­
striction of cosmology: we are observers of the universe, not experimenters, since we
do not have the means to measure over an ensemble of causally connected patches).
Hence we must repeat the above all-sky averaging procedure in every causally con­
nected patch of the universe (in every map of the universe that we simulate) and
make an ensemble average over the obtained a f’s:
C, = (a?)
(2.11)
Inflation also predicts the temperature fluctuations have gaussian distributed har­
monics. Hence, the a}'s, for each £, follow a 2£ + 1 dimensional ^-distribution, with
average C* and variance 2^r[C}, which is called cosmic variance. The gaussianity of
the fluctuations is a consequence of random fluctuations generated during inflation
by quantum mechanics: each aj is a sum of contributions of many of these random,
uncorrelated fluctuations from distinct Hubble patches, since the relevant scales cor­
responding to the lower £ span many Hi's. The cosmic variance must be added on
to the experimental variance when reporting a measurement; it signifies the band of
theoretical predictions we are comparing the observation to.
A plot of £(£ + l)f£ vs. £ is called the CMBR anisotropy power spectrum. It is a
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36
6.0
x
4.0
I 2.0
0.0
100
1000
multipole moment (I)
Figure 2.1: The CMBR anisotropy power spectrum as a function of multipole for
the Standard CDM model (Q*/ = 1, h = 0.5, f is h 2 = 0.0125, n = 1). The theory
predicts only the shape of the power spectrum, not the normalization. The spectrum
is normalized using the COBE measurement. The band in the figure is the cosmic
variance band, with the central curve in th at band being the ’average’, the £(£+1 ) ^ .
tell-tale fingerprint that can be used to distinguish competing cosmological models.
The characteristic power spectrum predicted by inflation (2.1) has a plateau for
£ < 100 (large angular scales) and a series of acoustic peaks (misleadingly called
doppler peaks) for £ > 100.
If the energy density fluctuations are precisely scale-invariant, the variations they
induce on the CMBR temperature are too, resulting in a power spectrum independent
of i. This is true for those long-wavelength energy density fluctuations that have not
evolved since inflation, which are the fourier modes relevant to £ < 100. These modes
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37
have wavelengths A
100,000 light-years, and so no redistribution of m atter could
have happened across those length scales by the time the CMBR radiation first began
free-streaming. Some of them are entering the horizon in the present epoch, and since
the longest returning wavelengths represent those modes that grew larger than the
horizon earliest during inflation, they give us a window on the physics operating at
the time of inflation.
The smaller wavelength modes enter the horizon at or before last scattering and
begin acoustic oscillations. The over-densities seeded by the fluctuation modes draw
the baryon-photon plasma into them, and radiation pressure keeps the m atter from
collapsing on itself, thus setting up oscillations in the fluid which imprint themselves
on CMBR photons by causing sinusoidally modulated blueshifts and redshifts. The
comparative fraction of radiation at last scattering ensures that the gravitational po­
tential decays through last scattering, imprinting additional anisotropy on the CMBR
by contracting photon wavelengths and resonantly strengthening compressions with
respect to rarefactions in the baryon-photon fluid.
2.6
Perturbations, the “M issing Energy problem” , and quintessence
We shall discuss observations in the next chapter that show th at m atter accounts for
no more than 40 % of the critical density in the universe(for e.g. [10, 17]). This has
led people to speculate that the universe is either open or is flat and has some other
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38
mysterious constituent which makes up the deficit to the critical density.
The cosmological constant (A) is one such species. It is believed to be a remnant
of super-symmetry breaking in the early universe. It has equation of state w = —1
and its energy density remains constant. If it dominates the universe it drives an
accelerated expansion of the universe. Its value would have to be of the order of the
critical energy, at the electron volt (meV'4) scale, far removed from the much higher
supersymmetric scales at which it was formed. Thus it is poorly motivated from
fundamental physics.
It is the object of this work to pursue another option for this “missing energy” : a
dynamical fluid called quintessence (Q ) with negative equation of state w and positive
sub-horizon sound speed, which we shall model by a scalar field. Our investigations
are not without precedent, Silvera and Waga[14], and Turner and White[13] tackled
the problem of time varying cosmological constants and smooth fluid models, both of
which are unphysical, and R atra and Peebles tackled the problem of the cosmological
consequences of a time varying scalar field. Our approach however is more general,
and is the first one to recognize the cosmological importance of fluctuations in the
scalar field dynamics. VVe will show that these fluctuations are both stable and
insensitive to initial conditions.
We shall study how fluctuations in the Q field
leave their mark on the mid to large angular scale CMBR anisotropy, and explore
the possibility that we may use high precision CMBR anisotropy measurements to
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39
distinguish between “missing energy” constituents in the universe.
2.7
Summary
In this chapter, we introduced the basic cosmology concepts that we will use through­
out this work. VVe described the process of inflation and the formation of structure
through gravitational instability from fluctuations generated during inflation. The
peculiar act of superluminal stretching and reentry of perturbations generated in the
process of inflation imprints upon the primordial power spectrum a scale invariance
born from the potential energy domination needed to drive inflation.
In the next chapter, we will present the observational evidence which shows that
a m atter only provides a fraction of the critical density required to make the universe
flat. Thus we must either abandon inflation and accept significant global curvature,
or invoke the cosmological constant or a hitherto undiscovered missing energy.
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Chapter 3
Classical Cosmological signals of missing energy
In this chapter we describe some of the observations that are used to obtain con­
straints on the energy density in the universe. VVe first describe the theoretical
prejudice for a flat universe arising from the nature of Q = 1 as an unstable criti­
cal point and the incredible efficacy of inflation in washing out all initial curvature.
We then report on observational results from classical cosmological measurements
based on distance measures, from quasar and cluster abundances and from m atter
power spectra. All these observations point to m atter(both luminous and dark) not
accounting for more than 40 % of the critical energy density. We will show that
the evidence marginally favors missing energy models over models with curvature.
We then show th at results from present-day CMBR anisotropy measurements point
to the existence of missing energy rather than curvature. Thus, we are forced to
consider models in which a substantial part of the critical energy density is in some
hitherto unobserved constituent.
40
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41
3.1
Open Universe? or M issing energy?
We have seen in section 2.2.3 that ft = 1 is an unstable critical point of the Friedman
equation. T hat is, to obtain fi0 between 0.1 and 2 today, it had to be very close
to 1 when the initial conditions were laid out. This fine tuning problem was solved
by incorporating inflation into the standard Big Bang scenario, as inflation ensures
that ft will be arbitrarily close to 1 at its end. In fact the requirements of sufficient
inflation (60 e-foldings) guarantee that ft0 will also be arbitrarily close to 1, unless
one indulges in extreme fine tuning (as some have considered, see for example [26])
of the number of e-foidings of inflation. Such fine-tuning defeats the very spirit of
inflation, and is to be strongly discouraged.
There are consistent and converging experimental findings (see [10] and [17]) that
all of m atter, luminous and non luminous, can account for at most 40 % of the critical
density. These include measurements of classical distance measures such as age of the
universe from globular clusters, ([7],[8]) and high redshift supernovas([27],[5]), and
dynamical measures such as large scale structure([28],[29]), baryon fractional],[30]),
and cluster evolution([31]). There are also hints from the CMBR([11]) that the
m atter fraction must be smaller than unity.
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42
3.2
M odeling missing energy
What then, is the nature of this missing energy? We will adopt a general approach
which is the beginning of the study of quintessence, in that we shall parameterize this
m atter by a negative equation of state (w ) between 0 and -1. We shall consider in this
chapter both quintessence with negative equations of state between -1 and 0, and the
cosmological constant. For reasons of simplicity in calculation and parameterization,
only constant equations of state will be considered.
The cosmological constant is included amongst these models as a fluid with w =
—1. Although it is physically poorly motivated as described earlier (see sections 1.2
and 2.2.2), the cosmological constant has been studied for a long time, and models
based on it have a good track record of fitting experimental data([10],[17]).
We shall not consider the dynamical effects of quintessence in this chapter, thus
postponing a discussion of quintessence proper until the forthcoming chapters. Here
we are interested only in kinematics, and the unphysical smooth fluid models which
have been proposed by others([14],[13j) suffice.
3.3
Observations based on distance measurements
There are certain classical cosmological tests of the expansion and composition of
the universe which rely on plain kinematics. These include deep redshift supernova
searches, the Sunyayev Zeldovich effect, and the gravitational lens time delay.
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43
The key quantity that all the above observations probe is the distance-redshift
relation, that is, the physical or co-moving distance to a high redshift object, as a
function of the redshift. Since the distance is a function of h and the cosmological
mix, we can use it to constrain various components of the energy' density of our uni­
verse. This distance may be expressed and measured in various forms: the luminosity
distance measured by a decrease in the intensity of light using the inverse square law,
the angular diameter distance measured as a ratio of the physical size of a object to
the angle it subtends at our eyes, or the age of the universe or a high redshift object,
a measurement of the distance light would have traveled since the Big-Bang.
We will consider here the calculation of two different distance measures, the age
of the universe, and the angular diameter or luminosity distance to a distant object
(in the guise of the distance to a high redshift supernova).
Since the density of quintessence pq scales as an inverse power of the scale factor
a between 0 and 3 for w between -1 and 0 respectively, one may expect that it
will come to dominate the energy density of the universe anytime between matterradiation equality (w = 0) and a redshift of ( ^ - ) 1/3(when w = —1). The expansion
history of the universe will then be changed, and this will have an effect on all distance
measures. This translates into a high redshift dependence on parameters in distance
measures, which may then be used to fit experimental data for the aforementioned
parameters[18].
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44
3.3.1
The age of the universe
The age of the universe from the beginning of time to now can be calculated from
dt = adr) as
t = f (■“ )da
Jo a
(3.1)
where ' denotes derivative w.r.t. conformal time. A lower limit on the age of the
universe is obtained from estimates of the age of the oldest stars in globular clusters
are thought to include some of the oldest stellar populations in the universe.
The age of the universe in a model which includes a Q component or a cosmolog­
ical constant, together parametrized by w and Qw, can be obtained by substituting
the Friedman equation in equation 3.1:
r l da
t = / ------- ,
J° a H qJ S I m / o? +
(3.2)
n iu/ a 3 0 + u') +
(1 —
Qnf
— f i w)/a?
This expression holds for arbitrary geometry of the universe.
Measurements of the Hubble constant have settled to be between 65- 75
kms- l Mpc_ l[6]. From equation 3.2, we can show that the age of the universe works
out to be, for a flat m atter dominated universe, t = (2/3 )H q X= 9.7G yr(65///0), and
for a open one (fi > 1.2): t < .85H q 1 = 12.5Gyr(65/f/o)- Notice that the universe
need to be less than 10 billion years old for the standard m atter dominated model
to explain its age.
When compared to the curves in figure 3.1 (for Ha = 65fcms~lMpc“ l) , the above
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45
'age-g.daf —
15
13
11
—
...........
—
15 Gyrs
0.5
11 Gyrs
i
-
0.5
0
equation of state w
Figure 3.1: Contours of age in the Qw —w plane in a flat universe. Note how the age
can be used to rule out the less negative urs on the assumption th at it is at-least 11
Gyrs. If w ~ —1 one may derive lower limits on Qq or i n ­
equations tell us th at (a)if the age of the universe is less than 11 billion years, no
constraints may be placed on its m atter content, but (b) if the age is greater than
llG y rs, strong constrains may be placed on both the content and the equation of
state, and further, (c)if the age is greater than 12.5Gyrs, a open universe with Q > 0.2
is ruled out too. The last restriction becomes less onerous if the Hubble constant is
found to be larger.
As recently as 1996, the lower limits on the age of the universe were set at 12Gyrs
[7]. This strongly disfavors a flat, matter-only universe, and while the open universe
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46
model is still consistent with the observations, the data pushes in the direction of
flat “missing-energy” models. This estim ate has subsequently been revised by some
observers to a open-universe consistent 9.8Gyrs(95% lower limit)[8]; however the data
is still too preliminary to put too much weight on.
3.3.2
The apparent magnitude o f high redshift supernovas
The apparent magnitude of a high redshift supernova measures the luminosity dis­
tance to that supernova. This luminosity distance figures into the attenuation of
brightness of the light coming from the supernova. The apparent brightness( appar­
ent magnitude) is related to the actual brightness(absolute magnitude) by the inverse
square law in the luminosity distance.
The deviation of the magnitudes of standard-candle supernovas vs. redshift from
the Hubble law at high redshift depends on
CIq (or Sl\ )> and 1 — Q(0< (for a
universe with arbitrary geometry). One extracts a measure of luminosity distance
vs. redshift from the observed magnitude vs. redshift of high-redshift supernovas
as follows [33]: An observation of a standard candle supernova at redshift z yields
the apparent bolometric magnitude m. This is related to the absolute bolometric
magnitude M by:
m = M + 5 log(df(2 )) + 25
(3.3)
where di is the luminosity distance, the effective distance determined from the ob­
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47
served brightness as if there were no Hubble expansion. Defining Di = hdt (which is
independent of h) we can rewrite the above as:
m = M + 5 log(A )
(3.4)
where M = M — olog/i + 25, and D/ = a(z)hr(z) is as defined in [33] up to an
irrelevant overall constant. The scale factor at the time of the supernova is 0 ( 2 ), and
r{z) is the conformal distance to the supernova:
(3.5)
where, k = h2(l — Qtot) and Arj(z) is the conformal time since the supernova:
H q\J Q \ i / a?
For very low redshift
(2
+ ftu,/a3(1+u') -f (1 —
(3.6)
— Q.w)/<
< 0.1) Di oc cz, independent of the underlying cosmo­
logical model. Thus, apparent magnitude measurements at low redshift will enable
us to fit for M . Di has traditionally been expressed in terms of qQ, but the standard
relation ([23]) is accurate for low redshift only
(2
< 0.3). For high redshift
(2
> 0.4)
Di is sensitive to ftm and Qw as elaborated by Steinhardt in [34].
Perlmutter et.al and Garnavich et. al. [33, 27, 5] have set constraints at the
95% confidence level on the existence of a non-zero cosmological constant (with
ft a = 0.6—0.7)in a flat universe. Their measurements have ruled out closed universes
at the same level of confidence, and have thus confirmed the missing energy problem
(or openness of the universe).
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48
Garnavich et. al. have very recently ([5]) even set limits on quintessence models.
They find —0.4 > w > —1 at the 95% confidence level regardless of open or flat
geometry, and for any
the limits improve to w < —0.55 if we assume a flat
universe. We will discuss these results in greater detail in a later chapter(7).
3.4
Gravitational Lensing Statistics
Evidence against A dominated flat universes comes from the statistics of the grav­
itational lensing of quasars by galaxies with a best fit value of Q0 = 0 found by
Kochanek[35], with limits fiA < 0.65 at the 95% confidence level. However, this
measurement is still not settled, recently Krauss et. al.([3]) find that a maximum
likelihood analysis shows that the data favors a low 0.25 < n matter < 0.55 flat cosmo­
logical constant dominated universe, as opposed to a flat m atter dominated one, and
even with respect to a low mass open one. Their understanding of the systematic
effects on the data is still poor making any conclusions premature. Further observa­
tional work will be required to settle the question, but the important conclusion to
take away from these observations is that there is no definitive result that rules out
a cosmological constant dominated universe.
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49
3.5
Dynam ical measure indications of missing energy
In the previous section we looked at measures of the mass of the universe which
depended upon physical distances to high redshift objects. In this section we report
findings on measures which use perturbation evolution to estimate the m atter content
of the universe.
3.5.1
The bend in the m atter power spectrum
The bend in the m atter power spectrum(see figure 3.2) parameterizes the epoch at
which the universe changed from being radiation dominated to being m atter dom­
inated, since in a radiation dominated universe, the gravitational potential decays,
and the growth of structure through gravity cannot be sustained.
We can measure the angular correlation of galaxies at different scales in high
redshift surveys. The d ata can be used to find the scale at which the growth of
structure turns over. Since the scale parameterizes the epoch of m atter radiation
equality, it carries a signature of the m atter density of the universe. Data compiled
from these surveys([28], [29]) tells us that
0.25 < T < 0.35,
(3.7)
where T = fimatterh in standard CDM models, and some more complicated function
of Dmatter> h, and fi*. Since H q has been measured to lie in the range of 65- 75
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50
MASS POWER SPECTRUM
Q.
Figure 3.2: The m atter power spectrum in a standard CDM and a ACDM model are
plotted against wavenumber in this figure. Note that the turnover of the spectrum
depends on the epoch of m atter radiation equality.
kms- l Mpc-1, we must have fima«er < 1. This conclusion holds for the other models
too.
The standard CDM model also produces too much power at small scales due to
the linear growth of its potential. People have proposed Hot dark M atter models
in which streaming neutrinos slow the growth of structure. However, these reduce
the smaller scale structure formation by too much, unless some cold dark m atter is
thrown into the mix. A and Q-field models reduce the growth of the potential just
right (see figures 7.3 and 6.2) to fit the observations from the APM and other high
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51
redshift surveys.
3.5.2
The abundance of high redshift quasars and clusters
At z=4, we are seeing more quasars (> 1) than predicted by the standard CDM
model([9]), and even more than the predictions of the ACDM models. Since the
cosmological constant dominates the universe (at about
2
= 1) the gravitational
potential starts to decay and cuts down the growth of structure. In a Q-field model
the potential starts to decay much earlier, since Q domination occurs at an earlier
epoch, changing the expansion history and diluting gravitational potential growth.
Since the potential growth turned off earlier, the change in structure is less in a Q
m atter model than in a A model, and hence, given a fixed amount of structure today,
we’d expect more structure at high redshift, for fixed fIm .
Similarly, if the standard CDM model described the universe, if one were to look
at high redshifts one ought to find significantly fewer rich clusters of galaxies than
can be seen today. Bahcall et. al.[3l] have cataloged clusters out to a redshift of
0.7, and claim to have found definite evidence of the existence of such a cluster at
this high redshift. In a standard CDM model, due to strong structure growth, one
should not even find one such cluster. Open models, or flat models with A are both
consistent with this data.
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52
3.6
Evidence for missing energy from C M BR anisotropy measurements
The CMBR anisotropy power spectrum, a measure of the CMBR temperature
anisotropy at a given angular scale, is likely to be in the coming years, the treasure
trove of experimental data in cosmology. There have been more than 20 experiments
already, attem pting to map out the power spectrum (see section 2.5) at different an­
gular scales and in different regions of the sky. In figure 3.3, we show the present lay
of the land in observational CMBR anisotropy astronomy, by plotting the results of
a combined likelihood analysis from most of these observations. The data has been
obtained from a compilation by Tegmark et al [32] , who did a joint likelihood anal­
ysis on multi-year runs from many observations: COBE, FIRS, IACB, Tenerife, SP,
BAM, ARGO, IAB, MAX, MSAM, CAT, OVRO, PYTHON, VIPER, IAC, TOCO,
BOOMerang, MAXIMA, DASI, QMASK, and CBI.
Notice from this figure that the standard CDM model does poorly with respect
to the experimental data. While most of these observations are in small patches, and
have not been repeated, rendering them inherently suspect from a statistical and
systematic viewpoint, they represent what can be achieved by making measurements
from earth, and show the unmistakable signs of a reasonably tall first doppler peak,
as well as the existence of subsequent peaks.
The present data seems to be pointing to an intermediate height first doppler
peak, at a £ of about 220. In the open universe model the doppler peak is shifted by
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53
the bending of geodesics to a smaller angular scale making it inconsistent with the
rise and fall in the data. From just the present data, an open universe model with
n = 0.95 can already be ruled out at the 2a level[32].) While we ought to be careful
about betting on this (we do not have a handle on all the foregrounds), there seem
to be good grounds for pursuing A-like models with missing energy.
3.7
Summary
We have described in this chapter some of the observational evidence in favor of
missing energy or a open universe. We found that both curvature and a missing
energy component are consistent with many present experimental observations. Fur­
thermore, some of the experimental observations are better fit by an adiabatic fluid
with equation of state larger than -1. We compared the CMBR anisotropy predic­
tions for a missing energy cosmological constant model against those from an open
universe model and the standard CDM model Amongst these, the A model fits the
present anisotropy data the best. It thus seems that there is rather strong evidence
for missing energy in the universe, and the rest of this work is devoted to the study
of a candidate for this missing energy, quintessence.
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e r4
0
10.0
•
—
OWeivational data
Standard COM
mm
•
n = 1 ,h s0 .6 5 , Q^sO.7
op«n (na1.15)
8.0
6.0
4.0
2.0
100
1000
Multipole Moments: /
Figure 3.3: Here we have plotted combined data obtained from an analysis of almost
all the experiments th at have reported CMBR anisotropies, along with the Standard
CDM model, an open universe model, and a A model, both with Q\r < 0.3. One can
see th at the observations are at a variety of angular scales. There is clear indication
of a large intermediate scale (£ = 200) anisotropy rise. The open universe model is
inconsistent with the data.
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Chapter 4
Calculating the CM BR anisotropy in quintessence models
Introducing quintessence as a dynamical energy component of the universe is at least
as well motivated by fundamental physics as introducing a cosmological constant (A).
In fact, the theoretical prejudice based on fundamental physics is that A is precisely
zero; if it is non-zero, there is no conceivable mechanism to explain why the vacuum
density should be comparable to the present m atter density, other than arguments
based on the anthropic principle. By all rights, the energy scale of the cosmological
constant should be nearer to the Planck scale or SUSY breaking scales rather than
those associated with the present m atter density in the universe.
On the other
hand, dynamical fields abound in quantum gravity, supergravity and superstring
models (e.g., hidden sector fields, moduli, pseudo-Nambu-Goldstone bosons, defect
condensates), and it may even be possible to utilize the interaction of these fields
with m atter to find a natural explanation why the Q-component and m atter have
comparable energy densities today([36]).
55
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56
In this chapter we will model quintessence as a scalar field, henceforth called
the Q field. VVe briefly compare our efforts with past work on negative equation of
state cosmologies. We develop a powerful formalism to describe the background and
fluctuation equations-of-motion of the Q field, and their coupling to the perturbed
Einstein equations, in terms of the equation of state of the Q field. This formalism
is used in a Boltzmann code to obtain the CMBR power spectrum. The results from
numerically calculating the anisotropy for various models are reported on.
4.1
M odeling quintessence as a scalar field
One may ask, given the tantalizing possibility of finding the cosmological significance
of fundamental quantum fields, why a concerted attem pt to investigate contributions
from such fields to the energy density of the universe has not been made. The answer
is twofold: firstly, the energy scales associated with these fundamental fields and of
the present energy density of the universe are more than 40 orders of magnitude
apart. There needs to be a physical mechanism to bridge these scales (this is also
true of A, however the explicit lack of such a mechanism in that case has stopped
no one from investigating it!). Secondly, experimental evidence([10],[17],[19]) favors
negative equations-of-state, and in an ideal adiabatic fluid with w < 0, the sound
speed would be im a g in a ry ^ = w < 0), and hence small wavelength perturbations
would be hydrodynamically unstable. [37] This has been a reason for not considering
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57
dynamical negative equation of state models. The motivation for our models really
comes from field theories and collective phenomena. It is natural in such theories to
achieve real sound speeds, as we shall show.
We model these Quids by the simplest possible field that is capable of producing
a negative equation of state, a real scalar field. The sound speed in a scalar field
is a function of wavelength, increasing from negative values for long wavelength
perturbations and approaching
= 1 at small wavelengths .[38] We shall show later
on in this chapter by considering a universe filled with solely such a scalar field that all
modes of interest, i.e., those inside the horizon have real sound speeds. Consequently,
small wavelength modes remain stable, and we can thus overcome the objections of
some previous investigators and forge ahead with a perturbation theory for these
fields.
We should however stress([16, 39]) th at our analysis is not restricted to scalar
fields only. Any component whose hydrodynamic properties can be mimicked by a
scalar field evolving in a potential which couples to m atter only through gravitation
will make itself amenable to our treatment.
W ith this motivation, we shall now make a systematic study of Q-field models,
with respect to the dynamics of the scalar Q-field and its fluctuations, the equation
of state evolution, and the effect of Q-field on the expansion history of the universe.
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58
4.2
Relationships to previous works
A number of studies[15, 13, 14] have assumed a time varying cosmological constant
or a “smooth” (spatially uniform), time-dependent component with arbitrary equa­
tion of state (sometimes called xCDM). It is important to note that the smooth and
Q-scenarios are not competing models. Rather, a smooth, time-evolving component
is an ill-defined concept that has no physical manifestation. As we shall show, the
smoothness condition cannot be maintained without violating the equivalence prin­
ciple. Quintessence, a fluctuating, inhomogeneous component is the only valid way
of introducing an additional energy component which approximates a time varying
cosmological constant. The inhomogeneity has its own consequences, as we shall see
later.
There have also been some independent discussions in the literature of an energy
component consisting of a dynamical, fluctuating cosmic scalar field evolving in a
potential,[40, 41, 42]. In the case of a scalar field, the CMBR anisotropy and power
spectrum were computed by Coble et al for a cosine potential,[40] and by Ferreira and
Joyce for an exponential potential.[42] Here we go beyond these isolated examples
to explore the range of possibilities and the range of imprints on the CMBR and
large-scale structure. We characterize these models in terms of the features of their
equation of state histories. This enables us to delineate the contribution to the CMBR
anisotropy of different epochs of Q-field evolution. This is the first such systematic
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59
study of the parameter space of these models.
4.3
Dynam ics o f the scalar field
In this section, we develop a formalism for studying fluctuations in QCDM (Qenergy+CDM) models. The dynamical Q-component affects both the background
and perturbation equations, so that the computational implications are widespread.
We find that, not only does the Q-component alter the cosmological expansion rate,
it also leaves a direct imprint on the anisotropy spectrum.
Since the fluctuations in these QCDM models are gaussian in character, it is
sufficient to compute their power spectrum only. Consequently, we have adapted
standard algorithms to evolve the linearized Einstein equations from early in the
radiation era, up to the present time, including the effects of Q-field in the stressenergy tensor of the universe. We evolve the Q field and its fluctuations according to
the perturbed equations of evolution obtained from the scalar field Lagrangian. In the
following, we discuss the various aspects of this computational problem, prerequisite
to our later, deeper analysis of the CMBR anisotropy pattern in QCDM models.
Detailed equation manipulations in multiple gauges are left to the appendix. Here
we limit ourselves to the salient points of the analysis, carried out in the synchronous
gauge.
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60
4.3.1
Background dynamics o f the scalar field
The formal starting point for the investigation of Q-field is the Lagrangian describing
the m atter content of the cosmological scenario.
L = L flu id
-I-
L
(4.1)
q
The fluid refers broadly to all species of particles and fields - including baryons,
photons, cold dark m atter, and neutrinos - apart from Q-field. VVe model the Q-field
as a classical, self-interacting scalar field with negligible couplings to other matter.
Hence, we write
Lq
=
— V(Q). It is then straightforward to obtain the
system of equations for the evolution of the fluid and Q-field under Einstein gravity.
Using the assumption that the background cosmology is described by the FRW
metric, the homogeneous, background energy density and pressure of the Q-field are
(4.2)
where the prime(') represents d/dr]. The time evolution, determined by the equation
of motion
(4.3)
is governed by the particular form of the potential V(Q). T hat is, by specifying the
functional form of the potential, along with initial conditions Q, Q' at a time T)i n i t , the
subsequent evolution is determined for all times 7/ >
An equivalent formulation
is obtained by specifying the equation of state of the Q-component, w =
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P q /p q ,
as
61
a function of time or scale factor. In this case, the energy density and pressure are
given by
PQia) =
exP (3t loS ~ +
fa
Y w(«)])»
Pq(°) = w (a)pQ(a)
(4.4)
where a0, Hq are present-day values of the expansion scale factor and the Hubble
constant, and Qq is the present-day Q-field energy density as a fraction of the critical
energy density.
Given the evolution of w{a), we may reconstruct the equivalent potential and
field evolution, V'(Q[a]), using the parametrized system of integral equations
y (a) =
m \
I1 ~
[ z HSQq f a
Q{a) = V_8?G_L
exP (3[ lQg Y + I a
J 1 + w(a)
^3
a0
/•“odd
...,\
T+h T “'(a)1)-
.
<4-6>
Note that we implicitly require the Einstein-FRVV equation for H(a) to evaluate
Q(a), so that the form of the potential which yields a particular equation of state
depends on the full cosmological fluid. As a consequence of equations (4.4-4.6) our
investigation of QCDM scenarios is not restricted by the particular choice of the
scalar field potential. The formulation of the background field equations in terms
of the equation of state allows us greater freedom in exploring the possible range of
behavior of the Q-component, thus enabling us to make a systematic study of a large
part of the quintessence space.
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62
4.3.2
The perturbed Einstein equations
We perturb the Einstein equations to linear order in the fluctuations in both the syn­
chronous and conformal gauge, following the treatment in Ma and Bertschinger.[43]
We reproduce the perturbed Einstein equations and energy momentum conservation
in the synchronous gauge here for convenience. The conformal gauge equations are
in the appendix. (A.9)
The synchronous gauge is defined by the condition that the time-time and timespace part of the metric does not carry a perturbation term. The perturbed metric
is given as:
d s 2 = a 2(r}){—dTf2 + (<$„ + hij ) d x ld x i } .
The metric perturbation
(4.7)
can be decomposed into a trace part h = hti and a
traceless part consisting of three pieces, h)^, hf-, and hfj, where hi} = hSij/3 -I-
+
hf-j+hfj. We shall only concern ourselves here with the trace, whose fourier transform
is denoted as h(k). We will henceforth work with the trace and other perturbation
quantities in fourier space, unless specified otherwise.
W ith this notation, Einstein’s linear order perturbed equations (reproduced from
[43]) are given as:
k2r ] - ] - - h
2a
=
k2r) =
47rGa2£r°0(Syn),
(4.8)
47rGo2(p + P)0(S yn),
(4.9)
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63
h. + 2 ^ h - 2 k 2T} =
h + 6fj + 2 ^ (h + 677) - 2k2rj =
—87rGa2<ST*j(Syn),
(4.10)
- 24n Ga2(p + P)er(Syn).
(4.11)
The variables ^(velocity perturbation) and ^(anisotropic shear) are defined as
(i>+p)» = ik‘s r ‘j ,
{i>+P)o = -(kik , - l- 6ij)Vj ,
(4.12)
and S j = T'j —5lj T kk/3 denotes the traceless component of T j.
In synchronous gauge, the perturbed part of energy-momentum conservation
equations
T
= d j ^ + r % T Q^ + r Q^ T ^ = 0
(4.13)
in fc-space are:
i =
0 =
4.3.3
+
- “ ( l - 3 u ; ) 0 - —— 0 + -5 P ^ - k 25 - k 2a .
a
l+ w
l+ w
(4.14)
'
Dynam ics of quintessence fluctuations
The cosmological perturbation equations for the dynamical Q-component are ob­
tained by expanding the scalar field equations about the homogeneous background.
In the synchronous gauge, small fluctuations 6Q( t), x ) in the scalar field obey the
equation
a!’
cPV
1
SQ" + 2 - 6 Q + (a2— - g'’d,d,)6Q = - - H Q .
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(4.15)
64
Here, h is the trace of the spatial metric perturbation, as developed in the previous
subsection and in [43].
We may immediately make several observations about the behavior of SQ. There
are two qualitatively different types of solutions to 4.15. The source-less, homoge­
neous solutions belong to a two-parameter family specified by the initial conditions
for SQ, SQ'. Fourier decomposing the spatial dependence of SQ, we see that the
fluctuation amplitude evolves in time like a scalar field:
SQ" + l -aS Q ! + (a20
+ k2)6Q =
(4.16)
Oscillatory or damped behavior results, depending on the relative magnitudes of H
and a2V'QQ + k2. The inhomogeneous solutions, driven by the source term —^h'Q
are due to the response of the Q-field to the fluctuations in the surrounding medium.
Hence, primordial fluctuations in cold dark matter, for example, will generate fluc­
tuations SQ. It is interesting to consider the case of Q-field as A, corresponding to a
scalar field with w = —1, so that Q' = 0. The source term vanishes, so no fluctuations
develop; by definition it is a cosmological constant. In contrast, a dynamical field
invariably has Q' ^ 0 for some period of time, so th at it must develop fluctuations.
The energy density, pressure, and momentum of the Q-fluctuations in the syn­
chronous gauge are given by
SPQ =
1 Q'SQ' + V qSQ
CL
8Pq = \
q '6 Q
' - V q8Q
CL
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65
[Pq +P(j){vQ)i = - - f Q ' i S Q ) ; .
(4.17)
These quantities contribute as sources for the scalar metric perturbation equations
(see equations 21a-c of [43]), used to evolve the momentsof the photon distribution
in a Boltzmann
code for the computation of the CMBR anisotropy powerspectrum.
As in the case of the evolution of the background equations, we may specify a
specific potential V(Q), or alternatively, the history of the equation of state w(a).
Given the time evolution of w, the equations
* v*
=
« - 18>
1
r
wn
w"
+i
It T^
r - +
l + w 14(1 + w)
2
,a!
i
w '-(3 w + 2)1
a
'1
(4.19)
v
'
may be used in 4.15-4.17.
If we define Sijj = SQ/ ^ 1 +
w (t )
and if/ = 0 7 A +
w (t ),
we can use Eqs. 4.18
to convert the field (Eq. 4.3) and fluctuation (Eq. 4.16) equations for the Q-field
into a form in which the potential is implicit:
* . + ( (1 + 3
^ - 2
^ ) f
= 0,
and
sr + (2^+
+ (*2 - 5(1 - “ > [ 7 - ( | ) 2( j + 5 “ )] +
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(4.20)
66
= ~^tiil/y/T+w{A.2l)
The change of variables enables us to remove the dependence of the second derivative
of the potential on w" from the fluctuation equation. It also avoids the numerical
difficulties th at occur when the equation of state reaches the value w = —1 (see the
appendix). Thus, given a well sampled table of the equation of state history w(a), we
can numerically integrate this equation in a Boltzmann code to obtain the evolution
of the quintessence field and its fluctuations.
At this point, we can re-address the study of adiabatic fluids with negative equa­
tions of state as done by some researchers[15, 13, 14]. These fluids were assumed
to have no fluctuations, i.e., they were smooth, and affected the CMBR anisotropy
only by changing the expansion history of the universe. However, even if we specify
smooth initial conditions for a given gauge, it is unphysical to ignore the response of
the new component to the inhomogeneities in the surrounding cosmological fluid. It
can be clearly seen from equation 4.15 that an initially smooth field will not remain
smooth due to the source term, in which the synchronous gauge metric perturbation
trace h is also fed by the fluctuations in the other constituents of the universe, as
can be seen from equations 4.8 and 4.14. Thus the equations of motion demand
that smoothness of the Q-field cannot be preserved. This is the critical flaw in those
studies, and claiming th at the Q m atter does not fluctuate is identical to claiming
a violation of the equivalence principle; the Q m atter must respond to gravity in a
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67
way similar to the response of other constituents.
4.4
R esults from the Boltzm ann codes
We modified two separate Boltzmann codes, the lightning fast CMBFAST([44]) and
the slower but more accurate LINGER([43]) to make sure our results were consis­
tent. The results were obtained in the synchronous gauge, and cross-checked by
recomputing them in the conformal gauge.
4.4.1
Normalizing the CM BR power spectra
The CMBR anisotropy power spectrum may be described in terms of its shape and
its normalization. The shape encodes the detailed physics that the universe has
undergone around and since last scattering, and the relative amounts of its various
constituents. The normalization, on the other hand identifies the amplitude of the
anisotropy at the epoch when observations are made, i.e., today. Thus the initial
fluctuation amplitude for every model must be potentially adjusted to obtain the
observed values.
One can turn the tables and normalize at a fixed initial hyper-surface. This inputpower normalization corresponds to the normalization obtained when the coefficient
of proportionality in P(k) ~ A k n is set to a constant, say 1. The input-power
normalization allows us to concentrate on the physical effects of the post-processing of
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68
fluctuations caused by the evolution of the universe through the Boltzmann equation.
This is the normalization the code uses; hence we also call it code normalization.
The COBE observation, which was sensitive only to the long-wavelength modes
(multipoles with t < 30) constrained the spectral index n to be n = 1.2 ± 0.3 [24],
and it measured the CMBR anisotropy to be (STt)2 = (1.05 ±0.19) x 10- l ° [24] The
COBE satellite, in the four years of its operation has collected enough data to obtain
a sample variance limited, minimally post-processed knowledge of the primordial
power spectrum. By using the amplitude found by COBE we can normalize the
power spectrum by anisotropy at those large scales which have escaped most of the
evolution of the universe by dint of having wavelengths larger than the causal scale
for most of time since inflation. This translates to the largest angular scales, and
hence lowest £. (By sample variance we mean the cosmic variance associated with
the finite region of sky samples, proportional to 1 /y/A where .4 is the area of the
sky covered. The limited area is a result of having to cut out the galactic plane
from the fields of observation. Thus this area is unlikely to improve significantly in
future experiments. This means th at another experiment with the same operational
parameters would do better over the same range only if it had more sky coverage.)
Normalization to COBE is usually carried out after the code runs by using the
method given by Bunn and White [45], which has been applied to the spectra pre­
dicted for a variety of tilted, open, and cosmological constant CDM cosmologies. The
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69
procedure reduces to fitting the power spectrum to a quadratic in log101 over the
range of angular scales observed by COBE. Hence, we compute
D
=
£(e+ l ) Ct ,
x = logl01
(4.22)
to obtain the normalized multipole moment from the approximation formula supplied
in [45]
10llCio =
0.64575 + 0.02282D' + 0.01391(D')2 - 0.01819D"
—0.00646D 'D " + 0.00103(D")2 .
(4.23)
The estimated error in this procedure, due to the parameterization of the power
spectrum by a quadratic, is ~ 1% for the range of usual CDM models. Note that
this does not include the 13.8%, 1-ct statistical uncertainty in C io, or any systematic
errors in the COBE measurement.
It turns out th at we can employ the Bunn-White method to normalize QCDM
models. We have found that the large angular scale power spectra of QCDM models
may display more features than usual CDM , or even ACDM models, due to the direct
effect of the Q fluctuations (see section 6.6.2). In this case, we might expect that the
Bunn-White normalization procedure, a quadratic fit centered at C = 10, would be
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70
inadequate. As a simple check, we fit the power spectrum to a cubic polynomial in
log10 £, to better describe these features in our parameterization, and computed the
new D', D" for use in equation (4.23). We found that the new fit changes the value
of Cio by at most ~ 2%, for the range of QCDM models presented in this work. The
reason for this lack of sensitivity is that the relevant quantity for the normalization is
really the integrated band-power over the COBE window function, and the QCDM
spectra is well fit by a quadratic over the corresponding multipole moments, when
these multipoles are weighted with the COBE window function. This is consistent
with the results of Bunn and White[45] who note that although certain models are
not well fit by a quadratic, especially near the lowest multipole moments, the error
incurred by the quadratic fit is small, £ 1%.
It is the COBE normalized power spectra which correspond to what is observable
on the sky, and it is these Bunn-White normalized QCDM models that we mostly
present in this work. However, we will revert to the input-power normalization to
explain the physics behind the results, especially in Chapter 6.
4.4.2
Models w ith constant equations o f state
Let us first look at the anisotropy obtained in models with the simplest possible
equation of state, a constant, equation of state.
In Figure 4.1 we show a series of constant equation of state models with Q-field
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71
energy density Q q = 0.6 for different values of w. We notice firstly that in this COBE
normalized case, the height as well as the peak multipole of the first Doppler peak
increases as w is decreased to -1. Secondly, the rise to the peak is steeper as w —> —1,
and the large angular scale anisotropy flatter. Thirdly, the very large angular scale
anisotropy shows an interesting dip at w = —0.6, a trend which asymptotes towards
the to = 0 model having a continuous increase in power to the doppler peak. Each
of these features illustrates a different aspect of the physics of CMB anisotropy in
quintessence models, and we shall return to them in detail in Chapter 6.
8
w=0
w=-1/6
w=-1/3
w=-1/2
ACDM
6
4
2
:
0
1
10
100
4
1000
multipole moment: I
Figure 4.1: This figure shows CMBR anisotropy spectra for a set of constant equation
of state QCDM models with Q-field energy density Q q = 0.6, for different values of
the w.
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72
In Figure 4.2 we show a series of COBE normalized constant equation of state
models with w = —1/6 for different values of the Q-field energy density. For grow­
ing Q q , the amplitude of the acoustic peaks decreases relative to the large angular
scale plateau. Furthermore, the slope or effective spectral index of the large angle
anisotropy is for all practical purposes independent of Q q .
6
0
1
10
100
1000
multipole moment: I
Figure 4.2: This figure shows CMBR anisotropy spectra for a set of QCDM models
with constant w = —1/6, for different values of the Q-field energy density.
We shall show in chapter 6 th at the increase of peak height with decreasing Q q
and w in both these figures is an artifact of the COBE normalization.
The large angular scale anisotropy in Q-field models is very feature-full, as a result
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73
of effects, both of the changed expansion history and the Q-field fluctuations, at late
times, when Q starts to dominate the energy density of the universe. One of the most
interesting physical aspects of these quintessence models is the “direct” effect that
the Q-field fluctuations have on the CMBR anisotropy. This direct effect, the physics
of which are described in detail in section 6.6.2 arises due to strong pumping of power
into metric fluctuations by the velocity fluctuations in the Q-field, leading to a slower
decay and possibly even an increase in the gravitational potential at late times. This
effect, as can be seen in figure 4.3, for the Q-field models with constant w, in the
contrast with the ACDM model, shows up at the lowest multipoles, or the highest
angular scales. Notice in the figure the lowering of the CMBR power spectrum at
the lowest multipoles with respect to the £ = 10 multipole in the w = —0.33 and the
w = 0 models. In the A model, the lowest multipoles are actually higher, due to the
Late ISVV effect, while in the high w models, this direct effect has pulled down the
lowest £ multipoles.
Secondly, notice steep and monotonic rise of the spectrum in the w = 0 model.
In such a model, Q-field plays a dominant role in the universe at all epochs, since
last scattering, and thus there is no plateau, but rather a continuous ascent to the
doppler peak. Finally, notice the effect of the mass in the power spectrum of the
quadratic potential model. The change in the expansion history and the direct effect
of the fluctuations in the oscillating field leave an unmistakable signature at the scale
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74
corresponding to the horizon when the field started to oscillate.
1.8
=7 1.0
0.6
10
Multipole Moments: I
Figure 4.3: The feature-fullness of the low ( anisotropy in Q-field models. Notice
in particular the extremely low multipole dip in the w = - 1 /3 model. This is
a consequence of the direct effect of the Q-fluctuations. Notice also the bump in
the model where the Q-field has a mass; this bump occurs at the angular scale
corresponding to its Compton wavelength. The w = 0 models shows a relatively
steep and steady rise, a consequence of the merger of the late and early ISW effects.
In Figure 4.4 we show the E-polarization anisotropy spectrum for a series of
constant equation of state models with Q-field energy density Q q = 0.6 and different
values of w. Just as the pattern of acoustic oscillations in the temperature anisotropy
is subject to the variations in the Q-model, so is the pattern of the oscillations in the
small angular scale E-polarization anisotropy. (For a pedagogical review of CMBR
polarization, see [46].) Note that the predicted amplitude is two orders of magnitude
lower than the moments of the temperature anisotropy. Future experiments may
have the necessary sensitivity to detect the polarization [47].
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75
—
w=-1/3
w=-1/2
ACDM
500
1000
1500
multipole moment: /
Figure 4.4: This figure shows the E-polarization anisotropy spectra for a set of con­
stant equation of state QCDM models with Q-field energy density Qq = 0.6, for
different values of w.
4.4.3
Models w ith changing equation o f state
Our development of the perturbation formalism in terms of w and w' enables us to
investigate models with a pre-specified arbitrary equation of state history. To this
end, we have considered some models with a specific potential V(Q) :
’ 5 ? 7 l2 Q 2
V(Q) =
m4[cos(/3Q) + 1]
„m4 exp {~PQ)
quadratic,
cosine,
exponential.
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(4.24)
76
Each of these are typical of the potentials which arise in various fundamental particle
physics models, the most obvious of which is the quadratic potential, as occurs for a
massive field. The cosine potential may arise in models of pseudo-Nambu-Goldstone
bosons, proposed as a simple mechanism to generate ultra-low mass scalar fields
[48, 49, 50]. The exponential may arise as a consequence of dimensional reduction in
Kaluza-Klein theories [51], for example. Certainly, this is not an exhaustive list of
potentials, but gives an accurate sample of reasonable models. While none of these
particle-physics potentials are “new” , this is the first survey of the range of possible
behavior of a scalar field as Q-field.
To construct cosmologically interesting Q-field scenarios, we make several require­
ments of the potentials. To obtain an equation of state in the range —1 < w < 0, the
potential must dominate over the kinetic energy, similar to the slow roll condition
for inflationary fields. (The usual slow-roll parameters used to describe inflation are
not generally applicable, as they implicitly assume th at the inflaton energy density
dominates the cosmological fluid.) As well, we require that the energy density in the
Q-field be comparable to the total energy density at late times. For the quadratic
potential, these conditions are easily met if the Compton wavelength of the field is
comparable to or longer than the present-day Hubble length:
= I K w l '1' 2 *
(4.25)
In this case, the evolution of the field is damped, with w ~ —1, until late times
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77
when the field oscillates freely in the potential, and the time-averaged equation of
state is (w) = 0. This behavior is typical of all three potentials listed above: earlytime, vacuum-dominated evolution with w ~ —1 so that the Q-field behaves as a
cosmological “almost-constant”; late-time behavior in which the equation of state
approaches and may oscillate about w = 0.
As an aside, it is possible to reconstruct the potential which leads to a particular
equation of state. In the presence of pressure-less dust, the Q-field potential for a
constant equation of state is
using equations (4.S-4.6). In the limit of w —>0, we obtain
(4.27)
which corresponds to a special case of the exponential potential in the presence of
dust, with m4 = 3H qQq /16 itG and 0 = ^24 ttG/Q q . The Q-field with an exponential
potential has an attractor solution for a certain range of parameters (m,/3), in which
w evolves towards and locks onto the equation of state of the remaining cosmological
fluid. Properties of this scalar field system, and the cosmological scenario, have been
investigated in [41, 42, 52, 53]. In the case at hand, the Q-field is pressure-less, with
w = 0. However, it is important to stress that the fluctuations do not behave as
CDM particles. Referring back to equation 4.15, we see that the dispersion relation
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78
is (csk)2 = (k2 + cl2Vqq)(?
so
th at on scales smaller than the Q-field Compton
wavelength, the perturbations free stream relativistically with cs ~ c. Much like hot
dark m atter, the Q-field does not cluster. This is different from a scalar field with a
quadratic potential, oscillating at the bottom of the well. If the field has Aq «C H ,
then the oscillations are so rapid that (w ) = 0, averaging on time scales £ Aq. If
this time scale is smaller than any other astrophysically relevant time scale, then the
Q-field behaves identically to cold dark matter.
4.4.4
Anisotropies in models w ith monotonicaliy evolving equations of
state
In this section we compute the CMBR anisotropies in models in which the equation of
state is increasing or decreasing monotonicaliy. Unless stated otherwise, in both the
sections, all spectra are COBE-normalized using the Bunn-White procedure (4.4.1,
[45]), and use spectral tilt n s = 1, Hubble parameter h = 0.65, and baryon density
n 6h2 = 0.02.
We first consider a set of QCDM models with the same present-day values of the
equation of state and the energy density, but different values of w ( t ) ) and
P q {tj)
in the
past. The resulting CMBR spectra are shown in Figure 4.5. This figure illustrates
the importance of the time history of the properties of the Q-field. While these
models have similar low redshift behavior, as may be probed by classical tests such
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79
as number counts, the high redshift behavior is obviously different, as manifest in
the CMBR. Since the anisotropy spectra look further back in time as compared to
other cosmological measurements, they will be a critical tool used to elucidate the
nature of the missing energy in the Universe.
In Figure 4.6 we show a series of exponential potential models with the same
present-day equation of state w{ijo), for different values of the present-day Q-field
energy density. In each of these models, the equation of state remains at w ~ —1,
until late times when Aq ~ H ~ l, after which it rapidly evolves towards w = 0. The
rapid change in the equation of state leads to a shift in the expansion rate, which
may contribute to a feature in the large angle anisotropy spectrum through the ISW.
VVe also observe that the locations and heights of the acoustic oscillations are affected
by the Q-model.
In Figure 4.7 we show a series of exponential potential models with Q-field energy
density Q q = 0.6 for different values of the present-day equation of state
w ( t)q).
In
each of these models, the equation of state remains at w ~ —1, until late times when
it evolves towards
( ). For decreasing w(rjo), the nature of the Q-field switches
117 770
from highly dynamic to nearly static, and the models approach ACDM.
In all of the above graphs, the peak heights seem to be increasing with decreasing
w(rjo) and Q q ( t]q). We shall once again show in chapter 6 that this is entirely an
artifact of the COBE normalization.
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80
—
—
1
constant, w=0
exponential, w(ti0)=0
cosine, w(ri0)=0
10
/ \
j r %\
1 / \l
100
1000
multipole moment: /
Figure 4.5: This figure compares the CMBR anisotropy spectra for three models
with the same present-day values of the equation of state and Q-field energy density,
but with different values in the past. Shown are a constant w = 0 model, and an
exponential and cosine potential with w(rfo) = 0. The equation of state of the expo­
nential potential started at w = —1 at early times, and has evolved monotonicaliy
towards w = 0 by the present. For the cosine potential, the equation of state started
at w = —1 at early times, evolved monotonicaliy towards w = 0 by a redshift z ~ 1,
and then has oscillated from w = +1 back to w = 0 by today.
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81
1
10
100
1000
multipole moment: /
Figure 4.6: This figure shows CMBR anisotropy spectra for a set of QCDM models
with an exponential potential where the equation of state has evolved monotonicaliy
from w = —1 at early times, towards w ( t ] 0 ) = 0 at the present, for different values of
the Q-field energy density
4.4.5
Anisotropies in non-monotonic models
Up to this point, in most of the QCDM models we have examined the Q-component
evolves monotonicaliy. There arises unusual and interesting behavior when the field
oscillates. In Figures 4.8-4.9 we show the evolution of the equation of state and
the CMBR anisotropy spectra for a sequence of quadratic potentials with increasing
values of the mass. As described in the previous section, choosing initial conditions
so th at the Q-field starts with an equation of state w ~ —1, the field will continue
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82
w(ti0)=0
w(r|0)=—1/6
w(ri0)=-1/3
w(t}0)=-1/2
ACDM
10
n
a
itYJ
100
1000
multipole moment: /
Figure 4.7: This figure shows CMBR anisotropy spectra for a set of exponential
potential QCDM models with Q-field energy density Qq = 0.6, for different values
of the final value of w f o ) .
to evolve slowly and monotonicaliy until Aq ~ H ~ l , after which it may begin to
oscillate.
The sequence displayed illustrates this behavior, as the onset of oscillatory be­
havior becomes earlier with increasing mass. A feature develops in the anisotropy
spectrum on angular scales corresponding to the angular size of the horizon when
the field began to oscillate.
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83
.2
CO
(0
i
“5i
o.o
c
o
(0
3
-0.5
0)
O’
-
1.0
10
1
redshift: 1+z
Figure 4.8: This figure shows the evolution of the equation of state for a sequence of
three quadratic potential models. For increasing values of the mass, relative to the
present-day inverse Hubble length, the Q-field begins to oscillate earlier. All models
have Qq = 0.6.
4.5
Summary
We have modeled quintessence, a spatially homogeneous and time varying candidate
for the missing energy in the universe, as a scalar field. We developed a simple but
powerful formalism to study the Q-field and its fluctuations in terms of its equation of
state history. Any constituent which mimics the hydrodynamic property of a scalar
field will show similar behavior(a defect condensate, for example) if it has a similar
equation of state history. This formalism allows us to make systematic studies of
the effect of Q-field on cosmology, separating out the epochs in which these effects
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84
8
6
o
"■ N
+
2
0
1
10
100
1000
multipole moment: /
Figure 4.9: This figure shows the CMBR anisotropy spectra for the same sequence
of three quadratic potential models. For increasing values of the mass, relative to
the present-day inverse Hubble length, the energy density in the Q-field becomes
comparable to the CDM density earlier. Consequently, a feature in the CMBR
spectra develops at angular scales corresponding to the apparent size of the horizon
when the Q-field first began to oscillate. All models have Qq = 0.6, h = 0.65, and
Qbh2 = 0.02.
occurred, and the exact causes of these effects.
We numerically integrated Boltzmann codes using this formalism, and obtained
a slew of results for different equation of state evolutions: constant, monotonic, and
oscillatory. This is a wide parameter space of possible evolutions, and in the next
section we shall see how constant w models are sufficient to capture the field and
fluctuation dynamics of the monotonic cases.
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Chapter 5
Understanding quintessence fluctuation evolution and the sensitivity of
CM BR spectra to this evolution
In this section we show that models with constant equation of state can be used to
mimic the field and fluctuation dynamics in a large class of models with monotoni­
caliy changing equations of state and even some with oscillatory equations of state.
We show that for this class, the constant w approximation enables us to analytically
solve the fluctuation equation at large wavelengths. We use the solutions to explain
the time evolution of the quintessence evolution, and show that, for all physically
motivated initial conditions on the quintessence fluctuations, their evolution is both
stable and insensitive to those initial conditions. This translates into the insensitiv­
ity of the CMBR anisotropy to these initial conditions, and thus the robustness of
anisotropy predictions in quintessence models.
These conclusions were originally mentioned in our first paper on quintessence
models [54], and also by Viana and Liddle [52], who noted that the amplitude of the
85
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86
energy density perturbations in the background fluid are largely independent of the
initial conditions in the scalar field, provided the initial energy contrast is less than
unity. Our present work is consistent with these earlier conclusions. However, we
go further here by examining in detail the behavior of the scalar field perturbations,
and the causes for the insensitivity.
5.1
W h y c o n s ta n t w a p p ro x im a te s w ell a larg e ran g e o f p o te n tia ls
For quintessence models described by a scalar field with potential V(Q), the equation
of state varies with the scale-factor, depending on the initial conditions on Q, and
the form of V(Q). In certain cases, especially for large mass fields (e.g. harmonic
potential with large mass field, m > > H0), the evolution is oscillatory. In most cases,
however, the mass of the quintessence field is smaller than or comparable to the
Hubble parameter (m & H0), and the evolution of w(a) is monotonic. Observations
are consistent with models in which the equation of state evolution is monotonic and
slowly varying [55, 36].
In Figure 5.1 we consider equation of state histories in a group of models with
different potentials and initial conditions such that w(a) increases monotonicaliy.
The examples plotted are quadratic, quartic, and exponential potentials [51]. We
compare the evolution of the cosmological energy density in quintessence, p<?(a), in
these models with that from models with constant equation of state. The parameters
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87
0.6
5
w constant
V-exponential
V-quartic
V-quadratic
-0 .4
0.4
•S
^5
(A
J-
01
- 0 .6
'«
c
w=-0.66
©
c
o
♦5
a
3
r=-0.55
-
0.8
-
1.0
0.2
w =-0.66
©
c
©
r
0.1
0.0
0.1
a
•o
>.
S»
a
Figure 5.1: In the left panel, we see the evolution of the equation of state for a set
of potentials, all with Q q = 0.6 and w = —1/3 today. The evolution of the ratio of
the quintessence energy density to the critical energy density is shown in the right
panel. In both panels, the upper, solid curve represents a constant w = —0.55 model,
while the lower solid curve represents a constant w = —0.66 model. The w = —0.55
model has been chosen to best-fit approximate the exponential potential, while the
w = —0.66 model has been chosen to best-fit approximate the quadratic and quartic
potentials. Notice the similarity in the energy density evolutions.
and initial conditions have been chosen so th at each case produces the same presentday values for the equation of state and total energy density. The left panel shows
the evolution of the equation of state, while the right panel shows the evolution
of the energy density. In both panels, the upper, solid curve represents a constant
w = —0.55 model, while the lower solid curve represents a constant w = —0.66 model.
The w = —0.55 model has been chosen to best-fit approximate the exponential
potential, while the w = —0.66 model has been chosen to best-fit approximate the
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88
quadratic and quartic potentials. For each of the potentials,
p q
{ cl )
is a monotonicaliy
increasing function of the scale-factor. Although the time-history of the equation of
state is quite different between the constant w and evolving potential cases, we can
see from the figure that the evolution of the energy densities is closely comparable.
Quintessence affects the CMB anisotropy chiefly through effects which depend on
Pq (o ),
th at change the expansion history of the universe [16]. Thus, these potentials
predict nearly identical CMB anisotropy to the best-fit constant w models with w =
weff . The effective equation of state is empirically obtained as the pq weighted
average value of w(a) [56]:
f d a w ( a ) p Q(a)
= ~ * ,* ,( „ ) '
^
(ol)
In Figure 5.2 we plot the ratio of the multipole moments of the CMB power
spectrum in the exponential, quartic, and quadratic potential models to the power
spectrum of the corresponding model with constant equation of state w = wef f . Also
plotted is the fractional cosmic variance. (Cosmic variance is the intrinsic theoreti­
cal uncertainty for any model prediction based on adiabatic gaussian perturbations.)
The ratio in each of the cases falls within the cosmic variance uncertainty for almost
all multipoles. Hence, the power spectrum in each case is observationally indistin­
guishable from the corresponding constant w model.
The category of models with oscillatory equations of state is a much larger one,
since the parameters of the potentials can be fine tuned to obtain almost any desirable
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89
1.2
V-exponential
V-quartic
V-quadratic
1.1
^
c
(0
w
8
1.0
5
s
0.9
0.8
1
10
100
1000
multipole moment: /
Figure 5.2: Here we plot the ratio of the power spectrum in the models from Figure
5.1 to the power spectrum in the corresponding best-fit constant w model. The
fractional cosmic variance with respect to the best-fit model is also shown (outer
thin lines). The ratio for each model falls well within this variance envelope at
most of the multipole moments, thus, the predicted anisotropy is observationally
indistinguishable.
combination of periods and values for the equation of state. Thus, it is much more
difficult to make generalizations on the behavior of such models. Nevertheless, in
oscillatory models as depicted in Figures 4.8-4.9, we might hope to find some simpler
model which captures the important features: early time evolution with w ~ —1,
followed by a transition after which the equation of state oscillates about w = 0.
In fact, an exponential potential model in which the Q-field reaches the attractor
solution w = 0 may serve as an approximant. An example is shown in Figures 5.3-
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90
5.5. However, arranging such a model requires just as much fine tuning as the original
oscillatory model. For this reason, we will not pursue an approximation scheme for
the oscillatory models.
—
5
0.5
©
V-exponential
V -quadratic
<5
to
i
0 0.0
1
c
o
to
3
y -0.5
-
1.0
10
1
redshift: 1+z
Figure 5.3: The evolution of the equation of state for a pair of oscillatory QCDM
models is shown, both with Qq = 0.2. The exponential potential which approaches
the attractor solution w = 0 by z ~ 2 has been chosen to approximate the oscillatory
quadratic potential which completes its first oscillation near the same redshift.
Certainly, these two categories do not cover all possible Q-field models. We may
conceive models in which the history of the equation of state is not monotonic, but
is hardly oscillatory. Such models need to be tackled on a case by case basis. We
shall not do that here, as we are more interested in generic properties of the Q-field
models and the physical explanations for these properties.
Thus, for a large class of models with monotonicaliy changing w, the evolution
of quintessence and its fluctuations are described, to within cosmic variance, by a
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91
0 .4
—
V-exponential
V-quadratic
S 0.2
T3
>»
0.0
redshift: 1+z
Figure 5.4: The evolution of the ratio of the energy density to the total energy
density, Q q { z ), for a pair of oscillatory QCDM models is shown, both with Q q = 0.2
at the present day. The fractional energy density for each model first approaches its
asymptotic value by z ~ 2.
constant effective equation of state.
In this paper we restrict ourselves to these
models, and additionally require the sound speed in the quintessence fluctuations,
or the group velocity of the fluctuations,
to be ~ 1 at sub-horizon scales. We do
not deal with oscillatory equations of state or models in which the kinetic energy is
non-canonical and cj < 1 at smaller wavelengths, such as k-essence models [57].
5.2
Solving fluctuation equations for constant w
To study the evolution of quintessence fluctuations, let us consider numerically
evolved fluctuation and Einstein equations in a QCDM model with Qq = 0.6,
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92
6
V-exponential
V-quadratic
+
0
1
10
100
1000
multipole moment: /
Figure 5.5: The CMBR anisotropy spectra for a quadratic potential oscillatory
QCDM model is compared to the spectra for an exponential potential QCDM model
with the same average equation of state history (see figure 5.4) both with Q q = 0.2
at the present day.
h = 0.65, and Qeh2 = 0.02. In Figure 5.6 we plot the quintessence and m atter
fluctuation energy density obtained in this model for a mode with wavelength larger
than the horizon today (A: = lO~4Mpc~l). The lower three curves are the fluctuation
evolutions at three different equations of state, w = —1/3, w = —2/3 and w = -0.9.
We see in the figure th at for all the equations of state, the fluctuation amplitude first
oscillates and decreases, reaches a minimum, and then ultimately starts to increase.
The decrease of the amplitude is sustained for a longer time, and the subsequent
increase is sharper, as w becomes more negative, i.e., closer to -1. The upper three
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93
k»10"* M pc'1
.•21
/ -
,Z !
1-30
.-3 3
Figure 5.6: We compare the evolution of quintessence and m atter fluctuation energy
density for a long wavelength k = 10-4Mpc~l mode in three different models with
equations of state w = —1/3, w = —2/3, and w = —0.9, and with Qq = 0.6.
h = 0.65, and Qfl/i2 = 0.02. The three lower curves are the quintessence fluctuation
evolutions at the different equations of state, while the three upper curves, all very
close to each other, are the corresponding m atter fluctuation evolutions. Notice that
the energy density in quintessence fluctuations changes with equation of state, but
remains much smaller than the energy density in m atter fluctuations.
curves, which are all almost on top of each other, are the corresponding m atter fluctu­
ation evolutions. The change in the quintessence fluctuation evolution as a function
of w does not impact the m atter fluctuation evolution at all.
To understand the nature of this evolution, we look for analytical solutions to
the quintessence fluctuation equation. We collect together the other constituents
of the universe into an adiabatic background fluid denoted by the label B \ with
equation of state
wb
and energy density pa- We can then solve the fluctuation
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94
equation analytically in two limits: (a) the energy density in the quintessence field
is negligible compared to that in the other constituents of the universe (pq «
pB),
and (b) the energy density in quintessence dominates the energy density in the rest
of the constituents (pq »
pB). In these limits we can solve for Sip deep inside the
radiation- and m atter- dominated epochs where w B is a constant, and deep inside the
quintessence-dominated epoch respectively. In both cases, the fluctuation equation
(Eqn. 4.21) then simplifies to the form of a forced harmonic oscillator with constant
coefficients and a force term dictated by the coupling of quintessence to the perturbed
metric:
drSip
dSip
+ v .n
d(loga)2
d(loga)
o\e
1 a
+ m2)6ip = —-n/1 +
2
w
dh
dip
— ------- .
d(loga) d(loga)
._
(5.2)
Here k is the co-moving wave number,
v = ^(1 - w B)
(5.3)
is the damping coefficient of the oscillator equation, and
3 .
m = - y ( l —w){2 + w + w B)
(5.4)
is the Compton mass of the field Sip, all expressed in units of the co-moving Hubble
parameter (£ ).
There are two qualitatively different types of solutions to this fluctuation equa­
tion. The homogeneous solutions belong to a two-parameter family specified by the
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95
initial conditions Sipinit and Sip'inil, and are unaffected by the fluctuations in the
background cosmological fluid. The inhomogeneous solutions, on the other hand,
arise as the response of the quintessence field to the fluctuations of the background.
The evolution of quintessence fluctuations as a function of scale-factor is determined
by the combination of these solutions. The solutions are different in the radiation-,
m atter- and Q-dominated epochs, and must be matched by continuity at the bound­
ary between radiation and m atter domination, and between m atter and quintessence
domination.
Solving the inhomogeneous equation requires knowledge of the time evolutions
of the quintessence field and the metric perturbations. The former evolution can be
obtained from the equation of motion for the field (Eqn. 4.20):
dip
d(loga)
l3H$SlQa2
8ttG a' V
V
’
while the latter dependence can be obtained from the Einstein equations [43]:
d?h
t ( 1 - 3 wB) dh
(6p + 8p^_ J p B ,,
,
+
O
x = - 3 ( ----------) — ~ 3 ----- (1 + WB),
d(loga)2
2
d(loga)
p
pB
where ^
(5.6)
is the fractional energy density in the background fluctuations. The be­
havior of ^
at scales larger than the Jean’s length, the largest scale at which the
collapse of a fluctuation through gravitational instability can be counteracted by
the propagation of mechanical disturbances in the baryon-photon fluid, in the syn-
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96
chronous gauge, is well known [58]:
SpB
( CL \ §
v P b 1 H'
Pb
Here
\ PB / H i
(5.7)
is the fractional background energy density and aH. is the scale-factor
when the fluctuation mode under consideration exits the horizon during inflation
{Hi). The density power spectrum at horizon crossing is taken to be a scale invariant
Harrison-Zeldovich spectrum with a COBE normalized amplitude D b ,
(5.8)
with p = 4 deep in the radiation dominated epoch, and p = 2 deep in the m atter
dominated one [59].
VVe use Eqn. 5.5 and the solution of Eqn. 5.6 to solve the fluctuation equation
(Eqn. 5.2) for the evolution of metric perturbations, and consequently for the inho­
mogeneous solution in the radiation and m atter dominated epochs. At wavelengths
much larger than the Jean’s length, we find that:
=
- c i Dsaffia*,
(5.9)
where p = p + 3{wb — w), and
cr =
\ / l + u;
p2 + 2pv + 4m2
(5.10)
Notice th at the inhomogeneous solution depends on two separate epochs, a//,, and
a0 = 1, the latter entering the equation through the dependence on the evolution of
the quintessence field.
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97
The magnitude of the inhomogeneous solution is proportional to the amplitude
D b of the Harrison-Zeldovich spectrum and is thus determined by the COBE nor­
malization. Since fi > 0, the solution increases with increasing scale-factor for all
equations of state. This behavior corresponds to the gravitational amplification of
large wavelength quintessence fluctuations due to CDM potential wells. Furthermore,
the inhomogeneous solution at a given scale-factor is smaller for values of w closer to
-1, since the a t scaling and coefficient c/ are both smaller for more negative values of
w and the ajj* term is almost independent of w. The reduced amplitude reflects the
smaller coupling to the background in the source term of the fluctuation equation
(Eq. 5.2).
The homogeneous equation has the form of a damped harmonic oscillator with
constant coefficients. Thus the solution at all wavelengths in each epoch is simply:
Si)H = cHa.-%d(a,k,m,v),
(5.11)
where c/j is the amplitude of the solution and where 6 is an oscillatory function
of order unity. Since v > 0 in all epochs, the oscillation envelope decreases as a
power law of the scale-factor. The amplitude
ch
must be determined by the initial
conditions on the quintessence fluctuations at the initial hyper-surface far outside
the horizon, deep in the radiation dominated epoch (we chose ainit ~ 10-8 in our
simulations and analysis).
The inhomogeneous solution scales as a positive power of a, and is hence negligible
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98
(< 10-20) at the initial hyper-surface. The initial fluctuations in quintessence are
thus entirely due to the homogeneous solution {8ip(init) = 8ipH(i„it))- Hence, a change
in initial conditions affects the homogeneous solution only. To determine c # , we
consider the initial conditions predicted by inflation. Inflation creates a nearly scaleinvariant primordial spectrum of adiabatic density perturbations in all light fields.
Since the quintessence fields of interest in this paper are also light fields, the entropy
perturbation for the entire fluid, just after inflation, vanishes:
rj
T 6 S = 6 p - ^ 6 p = 0.
(5 .1 2 )
This condition gives one equation between the initial fluctuations Stpmit and
A
second constraint is obtained from the observation that long wavelength fluctuation
modes are frozen outside the horizon, and thus we set:
init) = 0-
(5.1 3 )
We solve the constraint equations for the amplitude of the homogeneous solution:
6+3w
1 / I Pc ^
^
®mit a Hi r i
/- i
(o-i4 >
The declining power law scaling (a- ?) of the homogeneous solution is independent
of w. Thus, the equation of state dependence of the homogeneous solution comes
only from its amplitude c«. Consequently, the value of the homogeneous solution at
a given scale-factor is larger for w closer to -1.
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99
Having obtained the approximate solutions of the fluctuation equation as a func­
tion of scale-factor and equation of state (Eqns. 5.9 and 5.11), it is now possible to
understand the numerically obtained long wavelength evolution shown in Figure 5.6.
Firstly, note th at the scale of both the solutions is determined by the amplitude of the
m atter fluctuation at horizon re-entry, and consequently the COBE normalization.
Secondly, we can see from the equations that that the homogeneous solution decreases
as fl- ^, while the inhomogeneous solution increases a s a i Thus the amplitude of
the fluctuations decreases until the inhomogeneous solution becomes comparable to
the homogeneous solution, and then it starts to increase. The scale factor at which
the solutions become comparable,
CHO-Hi
(5.15)
increases from ar ~ 10-5 at w = —1/3 to ar ~ 1.7 x 10-3 at w = —2/3 to ar ~
4 x 10-2 at w = —0.9. Thus, as can be seen in the figure, the homogeneous solution
is comparable to the inhomogeneous one for the w = —1/3 model at last scattering
(a ~ 7 x 10 4), while it dominates the inhomogeneous solution at both the more
negative equations of state, w = —2/3 and w = —0.9.
The magnitude of the homogeneous solution at a given value of the scale factor
increases as w approaches -1. By contrast, the magnitude of the inhomogeneous
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100
solution decreases. Additionally, since
5pq <x if}1 oc
a
(5.16)
2
the energy density in the homogeneous solution at a given scale-factor further in­
creases with decreasing w for all w < —1/3 [56]. Consequently, in models with w
closer to -1, such as w = —0.9, the energy density of the quintessence fluctuations is
initially larger and decreases more gradually. Thus, the evolution remains dominated
by the homogeneous solution until a later time.
The power law decline of the homogeneous solution is independent of wavelength.
On the other hand, the amplitude of the inhomogeneous solution for small wavelength
modes is suppressed compared to amplitude for large wavelength modes. For wave­
lengths smaller than the co-moving free streaming scale for the quintessence fluid, Lf„
the fluctuations free-stream from over-dense to under-dense regions. Thus, modes
smaller than L /s experience oscillations and the damping of the power law growth of
the inhomogeneous solution due to the competing effects of gravitational amplifica­
tion and pressure support from free streaming.
VVe found in this section that for w closer to -1, the homogeneous solutions domi­
nate the inhomogeneous ones until later in the evolution of the universe. In the next
section we study the sensitivity of the CMB anisotropy to initial conditions. VVe
show that this longer lasting domination at values of the equation of state closer to
-1 determines the extent to which the initial conditions must be changed from the
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101
case of perfectly smooth initial conditions to affect the CMB anisotropy.
5.3
Sensitivity to Initial Conditions
We have seen in the last section that initial conditions affect only the homogeneous
solutions of the fluctuation equation. For models with w closer to -1 such as w = —2/3
and w = —0.9, the homogeneous solution is larger and dominates the inhomogeneous
solution longer. In particular, the homogeneous solution dominates at last scattering,
and a change in initial conditions can propagate forward in time to a change in the
total fluctuation energy density, and consequently, to a change in the temperature
anisotropy. Since the power law decline of the homogeneous solution is independent
of wavelength, and the amplitude of the inhomogeneous solution is suppressed at
smaller wavelengths, any conclusions on sensitivity to initial conditions drawn at
larger wavelengths will continue to hold at smaller ones.
At long wavelengths, an expression for the effect of the fluctuations in pm and
pq
on the metric perturbation can be obtained in a very simple form from the Einstein
equations [43}:
A ? L h<^ 4irGa28pm(2 + ^ ) .
2q
8pm
(5.17)
The fluctuations in quintessence produce an effect which depends upon the ratio
8pQ/8pm. This ratio must become comparable to unity at last scattering for there to
be any distinguishable effect on the CMB anisotropy. As can be seen in Figure 5.6, in
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102
the case of adiabatic initial conditions, quintessence fluctuations are sub-dominant to
m atter fluctuations by many orders of magnitude for all w. While the ratio
S pq/Spm
will increase if one amplifies the initial conditions, we will see th at it is still too small
at last scattering for most w in order to have a distinguishable effect on the metric
perturbation and consequently the CMB anisotropy.
To study the effect of changing initial conditions on the CMB anisotropy, we
start with the simplest possible initial conditions, smooth initial conditions, where
the values of the fluctuation amplitudes Sip and Sip' are set to zero on the initial hyper­
surface. Smooth initial conditions have the unique property that the quintessence
fluctuation evolution is determined solely by the inhomogeneous solution of the fluc­
tuation equation. To test sensitivity, we compare to the case of adiabatic initial
fluctuations in Q. This corresponds to mixing the homogeneous solution into the
inhomogeneous one.
In the left panel of Figure 5.7, we compare the evolution of the ratio
Spq/Spm
at
large wavelength (k — 10-4A/pc-1) for smooth and adiabatic initial conditions, at
w = —0.9. We see that the evolution of the ratio for the smooth case tracks the power
law rise of the inhomogeneous solution to its present-day value. The magnitude of
the ratio a t last scattering
(a
~ 10-3) is much larger in the adiabatic case than in
the smooth case, corresponding to the dominance of the homogeneous solution over
the inhomogeneous one. Still, Spq/Spm remains far below unity in all epochs for the
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103
©
■
m
♦
100
i
I
to*
100
1000
Figure 5.7: The figure in the left panel shows the evolution of the ratio of energy
in Q fluctuations to that in m atter fluctuations
at k = 1Q~AMpc~l for both
smooth initial conditions (inhomogeneous solution) and adiabatic initial conditions
at it; = -0 .9 . The figure in the right panel shows the corresponding CMB power
spectra as a function of multipole moment. Plotted below the power spectrum is
the percentage residual of the power spectrum for adiabatic initial conditions from
smooth ones, compared to the fractional cosmic variance (lOOx
plotted as a black
line). The anisotropy change in going from smooth to adiabatic initial conditions is
well below the variance
adiabatic case.
VVe plot the effect on the CMB anisotropy at w = —0.9 in the right panel of the
figure, for both the smooth and adiabatic initial conditions. Below the spectra is
plotted the absolute value of the residual, or the percentage difference of the power
spectrum in the model with adiabatic initial conditions compared to the model with
smooth initial conditions. VVe also plot the fractional cosmic variance. Residuals
smaller than the variance cannot be observationally measured. As can be seen from
the figure, the anisotropy for the adiabatic case at w = —0.9 is not observationally
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lOtf
10s
— *
10 xadatotffcinits
v10°
u
€ io-5
1 0 '°
10’15
10"*
IO-®
10"*
10-3
a
10*
10''
10°
Figure 5.8: The figure compares the evolution of the ratio ( j j ^ ) 2 for both adiabatic
(F = 1) and artificially amplified (F = 104 and F = 105) initial conditions for the w =
—0.9 model from Figure 5.6. The ratio is plotted for wave number k = 10-4Mpc~l .
We also plot a solid horizontal line to indicate a ratio of magnitude unity. Notice
that the amplification prolongs the domination of the homogeneous solution, and the
resultant closeness of the energy density ratio to unity.
distinguishable from the smooth case within cosmic variance. While the addition of
the homogeneous solution to the inhomogeneous one does increase the ratio 5pq/Spm
by many orders of magnitude, the increase is not enough, even at w = —0.9, to alter
the CMB power spectrum.
If there is to be a distinguishable imprint of a change in initial conditions on the
CMB anisotropy, the energy density in quintessence fluctuations must increase dras­
tically so th at 8pq/Spm is of order unity at last scattering. Let us estimate how large
the initial amplitude of 8pq / pq must be in order to have a distinguishable effect by
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105
artificially multiplying adiabatic initial conditions by a factor F, and then comparing
the result to the results for smooth initial conditions. VVe want to show that F must
be quite large in order to have any detectable effect on the CMB anisotropy. In
Figure 5.8 we display the evolution of SpQ/Spm in the w = -0 .9 model, for initial
conditions that are adiabatic (F = 1), and for initial conditions F = 104 and F = 105
times the adiabatic initial conditions. The ratio is plotted at a wavelength longer
than the horizon today, k = 10_4A/pc_ l. In the case of amplified initial conditions,
the homogeneous solution is larger, and thus it takes until very recent epochs for the
inhomogeneous solution to become comparable to the homogeneous solution. The
amplification of the initial conditions by F = 104 makes SpQ/Spm larger than unity
initially, and the dominance of the homogeneous solution keeps it close to, but smaller
than unity at last scattering. An amplification by F = 105 makes the ratio to be of
order unity at last scattering, which will leave an imprint on the CMB anisotropy.
To understand these effects from the perspective of the value of
we need to
obtain the value of the m atter fluctuation at the initial hyper-surface (a ~ I0-8).
Since COBE normalization sets the amplitude of the m atter fluctuation on re-entry
to be &
-P£s~ 10-5, we find from Eqn.
5.7 th at \
m
^
Pm )
init ~ 10-16 in the synchronous
gauge. The ratio of ( ^ ) m„ to (£> ■ )_ f is set either by imposing smooth or adiabatic
initial conditions. In the former case we have Spq / pq = 0 and so the ratio is zero.
In the latter case, the ratio can be obtained by combining Eqns. 5.12 and 5.13 with
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106
the scale-factor dependence of pm and pq. The absolute value of this ratio ranges
from ~ 10“ 1 at w = 0 to 1022 a t w = —0.9. In other words, for w closer to -1, ^
PQ
is
initially quite large. Yet there is no observable difference in anisotropy, as can be seen
from Figure. 5.7, between the cases of smooth and adiabatic initial conditions. The
amplitude of the homogeneous fluctuations swiftly declines and both the above ratio,
and consequently Spq/8pm are much smaller than one by last scattering. Thus there
is no observable change in the CMB anisotropy. It is only when the initial conditions
are amplified by F = 105 (so that ( ^ )
1011 at the initial hyper-surface) th a t the
steep decline cannot offset the initially large value by the epoch of last scattering,
that there is any observable effect on the CMB. Of course, this large value of F is
physically unrealistic, many orders of magnitude greater than what is expected from
inflation, for example. Also, for such extreme values of F, the linear approximation
used in CMB analysis is invalid. This exercise shows clearly th at we can ignore the
initial conditions on the quintessence fluctuations for all reasonable models.
In Figure 5.9, we plot the numerically computed power spectra at w = —2/3
and w = —0.9 for adiabatic and amplified initial conditions. The amplification is
by factors of F = 102, 104 and 105 times the adiabatic initial conditions. Below the
spectra, we plot the residuals with respect to the adiabatic model.
We see th at at w = —2/3, the power spectrum in the amplified models is identical
to th at in the adiabatic model, even for F = 105. For w = —0.9, an amplification by
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107
a factor of 102 leads to no distinguishable changes in the anisotropy. On the other
hand, an amplification by F = 104 weakly suppresses the Doppler peak and creates
changes in the anisotropy at some multipoles. The ratio of energy in quintessence
fluctuations to energy in m atter fluctuations for F = 104 is of the order ~ 10-210-1 at last scattering, as can be seen in Figure 5.8, and the residual anisotropy is
barely smaller than the cosmic variance. Thus, the effects on the CMB are not large
enough to be observationally distinguished. Amplification of the initial conditions
in the w = -0 .9 case by a factor of 105 raises the amplitude of the homogeneous
solutions sufficiently that J^- ~ 1 at last scattering. For equations of state even closer
to -1, smaller amplification factors are required to make a measurable difference in
the CMB anisotropy. However, the amplification is still large and unphysical. For
example, even a t w = —0.999, an amplification by F = 103 is required to create an
observable effect.
Hence we can simply apply smooth initial conditions in the synchronous gauge
throughout this work without any significant loss of generality. The fluctuations SQ,
for all realistic initial conditions, have no memory of those initial conditions. The Qfield instead develops fluctuations in response to the spectrum of primordial density
perturbations in the surrounding CDM, baryons, photons, and neutrinos. In other
words it tracks the other components.
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108
5.4
Stability of Q-field fluctuations to gravitational evolution
In a scalar field the sound speed is a function of wavelength, increasing from negative
values for long wavelength perturbations and approaching
= 1 at wavelengths
smaller than the horizon size. Thus there arises concerns about the stability of the
perturbations.
The middle two terms on the left side of equation 4.16
2 - 6 Q ’ + ( a ^ + k*)SQ.
a
oQ*
(5.18)
Over the characteristic expansion time of the universe, the first term scales as
1/^ort-on while the second as —1IL \omvtm + a/A 2 where a is some positive con­
stant and A is the wavelength of the mode. The L's represent co-moving scales
and can be compared to understand the behavior of the equation. For modes with
wavelengths smaller than the Compton length(suitably rescaled for a), the second
coefficient in the equation is positive and fluctuation evolution is stable. For modes
between the Compton and horizon scale the second term may be negative but the
damping from the first term overwhelms any instability th at may arise. Modes with
wavelengths larger than the horizon would be unstable, but since they cannot prop­
agate or oscillate, they do not affect the dynamics of the universe.
To prove the above statements mathematically, let us consider the case of a
universe filled only with a scalar field. A long but simple calculation shows the
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109
equivalence of the fluctuation equation of motion 4.15 to equation 38 in Grischuk[38]:
(5.19)
where p, = —| ^= { h ' + a'yh), a(rj) = £ , and
7
(rj) =
1
-
Using the above
equation and these definitions, the sound speed can be found to be(see equation
44
.,
Grischuk[38]):
(5.20)
2a^ah
At low frequencies one may show from 5.19 and 5.20 that
approaches w. In the
high frequency limit, the above expression is dominated by the p' term, since equation
5.19 is dominated by its oscillatory solution. Consequently small wavelength, or high
frequency modes have real sound speeds approaching th at of light, and hence these
modes remain stable. The transition from a negative to a positive equation of state
takes place a t the horizon, thus hiding long wavelength instability from any physical
effects. The addition of matter-radiation plasma or m atter to this mixture can only
increase the stability, as the sound speed in these constituents is strictly non-negative.
5.5
Summary
VVe studied in this chapter a large class of quintessence models with light fields and
sound speed
~
1
at small wavelengths, which have the property th at they can be
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110
well approximated by constant equation of state, w. The evolution of the fluctuations
in these models was obtained by numerical integration and explained by approximate
analytic solutions to the fluctuation equation at large wavelengths.
VVe found th at the CMB anisotropy in such models is insensitive to initial con­
ditions on the quintessence fluctuations for smooth and adiabatic initial conditions
. For w = -0 .9 , the CMB anisotropy is insensitive in the large range of initial con­
ditions ( ^ ) in t < 1011 (F = 10s) for ( ^ f-) ~ 10-5 at horizon re-entry. Secondly,
the sensitivity increases as w approaches -1. At w — -0.999, the range reduces to
< 109 (F = 103). However, physically reasonable models such as those
based on inflation and ekpyrosis do not produce such large values of
and the
ratio of energy in quintessence fluctuations to that in m atter fluctuations is much
smaller than unity. We also showed that the Q-field fluctuations are stable against
gravitational collapse because c*s > 0 on sub-horizon scales.
This stability of the fluctuations and their lack of sensitivity to initial conditions
guarantees th at candidate light field particle physics models for quintessence will
possess stable, well behaved, m atter tracking Q fluctuations.
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I ll
adiabatic inits
o— o ioJ X adiabatic
10* X adiabatic
t - - * . to1 X adiabatic
100
1000
10
multipole moment I
Figure 5.9: The figure depicts the CMB power spectrum as a function of multipole
moment for two of the models of Figure 5.6 with w = —2/3 and w = —0.9. The power
spectra are plotted for a series of cases with artificially amplified initial conditions,
and for the corresponding model with adiabatic initial conditions. Also shown in the
lower panel is the absolute value of the percentage residual of the amplified cases
from the adiabatic case, as compared to the fractional cosmic variance (black line).
At w = —2/3, an amplification of the adiabatic initial conditions even by F = 105,
is not enough to make an observable change in the CMB power spectrum. On the
other hand, in the w = —0.9 case, the power spectrum for the same value of F is
markedly different.
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Chapter 6
P ro c esses affectin g th e C M B R a n iso tro p y in q u in tessen ce m o d els
In this chapter we study in detail the features of the CMBR power spectrum in
constant equation of state Q-m atter models. VVe first review the physical processes
affecting CMBR anisotropy as a function of the epoch in which they occur. VVe use
this understanding to describe the anatomy of the anisotropy power spectrum in
a distinctive quintessence model as a function of angular scale. VVe then correlate
this anatomy to the evolution of the gravitational potential in quintessence models.
The effects affecting the anisotropy come into play at different epochs: an increased
radiation and Q fraction at last scattering, a different rate of background evolution
during Q-field domination, and the d ire c t effect of Q fluctuations on the metric
perturbations. We zoom in on the dependence of these features on the quintessence
energy fraction and equation of state. VVe finally turn to the the direct imprint of
the quintessence fluctuations on the large angular scale (low I) anisotropy.
112
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113
6.1
From linear perturbations to radiation anisotropy
Let us recall the process by which fluctuations in all the constituents of the universe
affect the CMBR anisotropy. Once the fluctuation modes re-enter the horizon, they
begin to participate in the dynamics of the universe. Due to competition between
gravitational instability and radiation pressure until last scattering, oscillations are
set up in the cosmological fluid. Modes which are still linear at and after last scattering(because they just re-entered the horizon or were supported by radiation pressure)
leave their imprint on the large and intermediate scale CMBR anisotropy. It is this
anisotropy that we shall be concerned with in this chapter.
The fourier components of the linear perturbations, S^(t), grow independently
and may be related to the primordial power spectrum via the “transfer function,
T{k)” :
Sk(t) = T i W & i ) ,
(6.1)
where <%(£*) is the primordial spectrum at horizon re-entry, usually taken to be
P(k) =
= A k n8&,
(6.2)
where the angled brackets represent an averaging over all horizon sized patches.
Notice that modes corresponding to different k ’s are un-correlated.
The transfer function incorporates all deformations undergone by the fc-mode per­
turbation and t may be any time t > U. The transfer function is typically computed
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114
by a Boltzmann code. The Boltzmann code evolves the radiation fluctuation in time
by evolving simultaneously the Boltzmann equation for the photons, the linearized
Einstein perturbation equations, and the Friedman equation for the unperturbed
constituent fraction(really the unperturbed Einstein equation). Numerically, divid­
ing out the final amplitude of a mode by its initial value gives us the transfer function.
The evolution is done in k space since the fourier modes are un-coupled. Since we
want to measure anisotropy on the celestial sphere, however, the evolved fluctuations
are converted to multipole moment(£) space. This involves expanding the plain waves
onto a spherical harmonic basis. The coefficients of this expansion are spherical Bessel
functions je{k). This means that we can consider a £ space transfer function Tt (k)
such th at
(2« + l)C ,/4 x =
J
T?(k)P(k)d(\nk).
(6.3)
The £ space Transfer function includes all the evolution from the initial conditions
of that mode until today, and constitutes essentially a convolution of the various
sources of anisotropy with the appropriate spherical Bessel functions:
T,(k) ~ I dT)jttk(r) - rjtoday}) £ Sources.
(6.4)
Physically this means that anisotropy at a given angular scale(given £) will have
contributions not only from the physical scale that subtends the given angle, but
also from all the smaller scales(higher Ar’s) due to the oscillatory nature of the Bessel
function. In other words, power is aliased down to smaller multipole moments from
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115
effects which imprint the higher ones.
Let us consider the different sources, or effects which contribute to the CMB
temperature
AT
anisotropy:
• Sachs and Wolfe [60] obtained the expression relating fluctuations in the grav­
itational potential on the last scattering surface to the CMB temperature
anisotropies on large angular scales ( 0 > 1 deg). The equation is:
--------- ^ H------------3 + 3w
3(1 -I- w)
-
/ f0 ( * - * ) ( * (t),0,
where w is the effective equation of state of the universe at last scattering, 0 for
the m atter dominated case, and 1/3 in the radiation dominated one. $ and $
are the perturbations defined in the longitudinal gauge(see appendix). $ can
be interpreted as the gravitational potential.
is the velocity fluctuation in
the baryon-photon fluid.
• The first term is Sachs-Wolfe effect (SW), in which the photon suffers a redshift
from its intrinsic temperature as it climbs out of a potential well at last scatter­
ing. In the displayed term it adds to a constant part evaluated at the present
epoch which does not contribute to the observed anisotropy. The intrinsic
temperature if the universe is m atter dominated is two-third the gravitational
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116
potential at the last scattering surface(LSS), and one-half the potential if the
universe is radiation dominated. When a photon climbs out of the potential
well at last scattering, it suffers a redshift equal to its potential in the well,
and this decreases its temperature. The final temperature thus depends on the
composition of the universe on the LSS.
• The second term represents the potential decay at the last scattering surface.
It adds to the temperature imprinted upon the photon by dint of its position
in the potential well (D ) which includes the contribution from the density
oscillations of the fluid (third term), and from the velocity oscillations of the
baryons in the baryon-photon fluid (fourth term).
• Finally, there is the last term, usually called the integrated Sachs-Wolfe effect
(ISW). It is the integral along the line of sight of the gravitational potential
from the last scattering surface to us. Photons traveling towards us encounter
potential wells on the way. If the potential decays while a photon is crossing
it, the photon suffers a different amount of redshift and blueshift, resulting in
a net increase in its frequency (since the potential well at exit is shallower),
which translates to a temperature rise. Additionally, since the time-scales for
these decays are of the order of the expansion time, a few oscillations of the
Bessel function modulate the source term in equation 6.4, leading to aliasing
of anisotropy from higher I to lower £(transfer of power from smaller to longer
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117
length scales by addition) in the ISW effect.
The integrated Sachs-Wolfe term is non-zero both at early times(Early ISW
effect) at last scattering, if the cosmological fluid contains a appreciable fraction
of radiation, and at late times in missing energy or open models (Late ISW
effect), when the missing energy or curvature contributes an appreciable portion
of the energy density of the universe and dominates its dynamics.
The late ISW effect occurs primarily at the lowest multipoles. The reason for
this is that most of the contributions at these angles come from scales large
enough to be comparable to the size of the present day horizon. Modes with
wavelengths comparable to the horizon, decay on the time-scale of photon pass
through. Thus the photon will gain a net blueshift. On the other hand, if
the mode has a small wavelength, the photon will, during the expansion timescale, pass through many such modes and suffer no net temperature change.
Mathematically this can be viewed as multiplying the source in equation 6.4
with a high frequency oscillation of the Bessel function, leading to an incoherent
addition and no total change in temperature. The end result is that the late
ISW effect enhances only the lowest multipole moments, and is damped at
smaller angular scales.
In a universe dominated by radiation at last scattering, the early ISW effect
cancels part of the Sachs-Wolfe effect at the larger angular scales and brings
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118
the tem perature anisotropy down to the level of the Sachs Wolfe effect in a
m atter dominated universe(which is smaller than th at the value in a radiation
dominated universe by two-thirds). At smaller angular scales it increases the
anisotropy intensely by imprinting blueshift on the photons: this raises some­
what the height of the first doppler peak, and leads to a steep rise from the
Sachs Wolfe plateau, so that the Sachs-Wolfe anisotropy is matched to that of
the doppler peak.
Since the effect occurs closer to us and later in time than the acoustic oscil­
lations, it shows up primarily in the power spectrum a t angular scales larger
(lower 1) than what would be the first doppler peak, possibly shifting the peak
to a larger angular scale and increasing its height. Because the time-scale of
the potential decay is no higher than the order of an expansion time, there is
significant transfer of power to higher angular scales. Fluctuations at a given
scale get projected to lower multipoles, leading to a gradual rise to the peak.
• Even if the universe is m atter dominated at Last Scattering, there is a sig­
nificant radiation constituent in the universe, leading to potential decay that
enhances tem perature fluctuations then in two ways. Firstly, the decay con­
tributes strongly to the driving force for the acoustic oscillations in the baryon
photon fluid. Since the potential decays monotonically compressions in the
baryon photon fluid are stronger then subsequent rarefactions, an effect en­
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119
hanced by the inertial drag of the baryons. The modes lock in to the source
term and there is almost a resonant effect on the oscillations from the potential.
Secondly, photons blueshift inside a potential well as a decaying potential con­
tracts the photons’ wavelengths. Both these effects are sensitive to the fraction
of radiation and Q-field in the cosmological fluid at last scattering, and provide
tem perature rises which raise the heights of the doppler peaks.
• Adiabatic oscillations in the baryon-photon fluid lead to the prominent features,
known as the acoustic or doppler peaks. The oscillations in a given mode begin
when the wavelength falls below the Jeans length (i.e., pressure dominates over
gravity). The Jean’s length near recombination is roughly 2ncsH ~l, where
cs ss l / \ / 3 is the sound speed. Infall is followed by rebound as gravity is
opposed by radiation pressure, and the density of space time regions oscillates
about a zero-point density whose value is set by the baryon drag and the
sound speed. Photons suffer redshifts and blueshifts in a sinusoidally modulated
fashion. The net blueshift or redshift depends upon the stage of the oscillation
at which last scattering occurs. The first and other odd-numbered peaks in the
power spectrum diagram correspond to compressions and the even-numbered
peaks correspond to rarefactions; thus the first two peaks roughly correspond to
the epoch when the first adiabatic oscillation has just completed. The driving
effect of potential decay raises the heights of the peak crests via its resonant
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120
contribution.
• W ith all the bulk movement of m atter due to compressions and rarefactions,
there are velocity oscillations set up 90 degrees out of phase with the den­
sity. They contribute anisotropy through the doppler effect. Peak maxima and
minima in C ivst correspond to maxima and minima of the acoustic density
oscillations; the minima do not extend to zero because they are filled in by
these out of phase velocity contributions ([21]).
• The subsequent peaks after the first are imprints of modes which have un­
dergone more than one adiabatic oscillation. The imperfect coupling of the
baryons to the photons in the cosmological fluid leads to a scattering of pho­
tons at the smaller scales corresponding to the subsequent peaks, and thus
damps the anisotropy at higher multipole moments. As a result, the doppler
peaks are suppressed at the higher Vs.
6.2
Anatom y o f the CM BR power spectra
We now use our understanding of the processes affecting the CMB anisotropy to
describe the features of the CMB power spectrum. We restrict ourselves here to
models with only scalar anisotropy. We consider the Standard CDM model, a A
model (with
= 0.6), and an open universe model, with Q m = 0.3. In the latter
two models Qa/i2 is fixed at 0.02, while /i=0.65. To emphasize the features imprinted
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121
by the Q-field on the CMBR anisotropy spectrum, we choose for our Q model a
universe with with w = - 1 / 6 and Qq=0.6, with Qb^2 = 0.02, and h — 0.65.
There are four distinct regions in the power spectrum as labelled in figure 6.1: the
low and intermediate I plateau, the intermediate £ steep rise to doppler peaks, the
first doppler peak(which has a very dominant signature), and the subsequent high £
damped doppler peaks. For each region of the spectrum we will contrast the chosen
models, and also indicate the effects that changing cosmological parameters will have
on the features present. The models are plotted in figure 6.1 along with the COBE
measurement.
The distinctive features of the fingerprint are (reading 6.1 from left to right):
The plateau at large angular scales (i & 100/ is due to fluctuations in the gravitational
potential on the last-scattering surface (the Sachs-Wolfe effect[60]). For a precisely
scale-invariant (ns = 1) spectrum of scalar fluctuations, the Sachs-Wolfe contribution
to Ct is proportional to !/(£(£ + 1)), and thus the power spectrum is flat. A more
detailed computation shows a slightly upward slope at small-intermediate £, due
to the leakage of power(small £ contribution) from the higher k modes (also called
aliasing), and the early ISW effect. In all the models displayed, the slope of the
low £(large angular scale) plateau is not very sensitive to the value of h, Qb> R \ or
other cosmological parameters. Notice the slightly increased anisotropy at the lowest
multipoles in both the open and A models but not in the standard CDM model; this
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122
is due to the Late ISW effect. The anisotropy increases with
in A models. In
open models, the rise is shifted to higher multipoles, due to the fact th at in an open
universe, geodesics are curved, and thus a given physical scale subtends a smaller
angle than in a flat universe. Thus we would expect to find all notable features
moved to higher multipoles in the open model.
In the Q model, the plateau is undulated. The low multipole anisotropy is sup­
pressed and the Late ISW effect is moved to higher multipoles. The latter is a
consequence of earlier dominance of the universe by Q-field, while the former is a
consequence of the “direct” effect of the Q-field fluctuations. Velocity fluctuations
pump energy into the gravitational field suppressing the decay of the gravitational
potential and even possibly leading to its growth. The consequence is greater redshift
imparted to photons traveling along the line of sight, and thus interference with the
existing anisotropy.
The COBE normalization attem pts to give both the QCDM and standard CDM
models roughly equal power around I — 10 when convolved with the window function.
If the models had not been COBE normalized, but instead normalized by the code or
input power spectrum, the Q-field power spectrum would have been higher than in
the standard CDM model, with the doppler peaks at roughly the same height. W ith
that assumption, the earlier epoch of Q-dominance shows itself via the ISW effect
in the raising of the plateau. The detailed effect of normalization on peak height is
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123
described later in this chapter.
The First Doppler Peak at £ ss 200 probes wavelengths smaller than the horizon at
last-scattering (£> 1°). The first peak represents an oscillation which has not had
enough time to undergo more than half a cycle. The value of t at the maximum
[61] of the first Doppler peak is £peak « 220/ \/Qtatat ■ Since the location is relatively
insensitive to the value of h or Db> measuring £peak is a means of roughly measuring
Qtotai in simple models. This is not terribly interesting in flat universe models, but
in an open universe model with ft ~ 0.4 the doppler peak is moved much over to the
right in t space. We may then ask: what geometry does the experimental CMBR
data prefer? (See section 3.6 for the present answer)
The height of the Doppler peak depends on the power spectrum amplitude, the
scalar spectral index (n,), the expansion rate, and the pressure. In a A model, in­
creasing flA increases the heights of the peaks due to the decreased m atter fraction
in the universe. The higher radiation to m atter ratio means th at matter-radiation
equality is pushed further ahead in time. This means that potential decay can con­
tinue for a longer time, and thus we obtain larger blueshifts, and higher peaks are
obtained. The odd peaks are particularly raised by the resonant driving force due to
this decay. The same effect occurs in a open universe model, and raises the doppler
peak higher due to stronger decay in the potential.
If the spectrum is fixed at large angular scales by COBE DMR, smaller values
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124
of ns imply decreasing primordial amplitudes on smaller angular scales and, conse­
quently, a smaller Doppler peak. Increasing the Hubble constant (expansion rate)
pushes back matter-radiation equality relative to recombination, thereby increasing
the adiabatic growth of perturbations. Photons escaping from the deeper gravita­
tional potential suffer greater redshifts. Thus, increasing h suppresses the Doppler
peak. Increasing the pressure (by decreasing D eh2) also decreases the anisotropy
since the fluctuations stop growing once pressure dominates the gravitational in-fall.
For fixed fig , increasing h thus produces to competing effects. The pressure de­
crease effect wins out for Q0 & 0.1.The height of the first Doppler peak is relatively
insensitive to whether the dark m atter is cold or a mixture of hot and cold dark
matter.
The rise to the doppler peak from smaller multipole moments is more gradual and
continuous in the QCDM model than in the standard CDM model: this is because
of the increase in anisotropy at scales larger than the peak from the early ISW effect.
The large fraction of Q today implies less m atter in the universe at last scattering, the
deficiency being made up by increased Q and radiation fractions, leading to stronger
potential decay.
Second and Subsequent Doppler peaks are due to the modes that have undergone
further adiabatic oscillations. The peaks are somewhat periodically placed in t space,
with deviations due to time-variation in c,.
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125
The acoustic density and velocity oscillations are out-of-phase with one another,
with he acoustic density oscillation contribution dominating, and the doppler velocity
effect filling in at the troughs of the acoustic oscillation’s contribution.
Gravity tends to enhance the compressions and suppress the rarefactions, es­
pecially noticeable at low pressure (high Qah2) where the even-numbered peaks
are greatly suppressed or absent altogether. The subsequent peaks are sensitive
to whether the dark m atter is cold or a mixture of hot and cold.
The detailed shape of the first and subsequent peaks depend sensitively on cos­
mological parameters. Precise measurements of these peaks, and other features of
the power spectrum, especially when combined with more conventional astronomical
observations, will attem pt to determine the energy densities in baryons and var­
ious species of dark m atter, the vacuum energy density or the energy density in
quintessence, and the Hubble expansion rate.
Notice in the Q model th at the odd numbered peaks are higher, as opposed to
the trend in the standard CDM model. This is as the greater potential decay due
to the decrease in m atter content and the presence of Q at last scattering(in models
with w —►0) enhances the acoustic oscillation compressions at the expense of the
rarefactions.
Damping at Ek, 1000: CMB fluctuations are suppressed by photon diffusion and
by the fact that at these small wavelengths, the last scattering surface is thicker and
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126
destructive interference of contributions occurs.
6.3
Evolution of the gravitational potential
To correlate the processes associated with the CMBR fluctuations described above to
the quintessence fluctuation evolution, and to contrast them with their counterparts
in the cosmological constant case, let us consider the evolution of the gravitational
potential in three different models: a model with cosmological constant (w = —1)
which has
= 0.6, a fluctuating Q-field model with
Q
q
= 0.6 and w
=
—1/3 , and
a unphysical “smooth” {J-field model with the same parameters as above but with
the fluctuations artificially removed. In all the models, h = 0.65 and Qb/i2 = 0.02.
The sole purpose of the smooth Q model is to separate the effects on the CMBR
anisotropy of the evolution of the background equations of motion from those of the
fluctuations. We label effects which depend upon change in the expansion history
and relative m atter content of the universe “Indirect Effects” .
ds2 = a2(r)[—(1 + 2'ff)dr2 + (1 —2<f>)'yijdx'dx*].
(6.6)
The gravitational p o tential, denoted as # where # = -<£ in the absence of shear,
is that quantity whose limit in weak-field general relativity is the Newtonian poten­
tial. We shall concern ourselves with its magnitude (thus a decay in the potential
means a decrease in its magnitude). It is constructed from the synchronous gauge
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127
variables h and 7? as
* = - / ? + 2 ^ a (/l, + 67?,)’
(6 J)
where T) here is the synchronous gauge variable, not the conformal time. The evo­
lution of
in the three models for four different wavelengths is shown in Figure
6 .2 .
A few important observations can be made from the figure.
• At the lowest wavenumbers (k ~ 10-4/A/pc), as shown in the upper left panel,
the late time potential evolution dip in the fluctuating Q-field model is less
in magnitude than observed in the A model, and much smaller and shallower
than in the unphysical smooth model. This points to the existence of a “direct
effect” of the high wavelength fluctuations in Q on the gravitational potential
which serves to compensate for the decay caused by the change in expansion
history and deepen the gravitational potential wells. This effect is caused by
gravitational pumping from velocity fluctuations in the Q field, and is only
observed at late times. (See section 6.6.2)
• Notice also, that in all the panels, in both the smooth and fluctuating Qfield model, the departure of the potential curves from their flat mid-time(a ~
5 x 10-2) profile starts at an epoch earlier than in the corresponding A model.
The combined effect of the above two observations is that the potential decay in the
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128
A model is short in duration and sharp when compared to th at in the fluctuating Q
model. The gradual decay means that, for a fixed amount of large scale structure in
the sky today, there is a greater amount of high redshift structure than in a A model,
in better accord with deep redshift images. Thus QCDM does better in fitting data
from high redshift structure formation surveys.
The comparatively shallow potential decay in the fluctuating QCDM model also
means that the Late ISW effect is suppressed in these models as compared to A
models. The earlier decay means that the effect is also shifted to an earlier epoch.
The combined effect on the power spectrum in a QCDM model is that the Late
ISW enhancement of anisotropy is less than that in the corresponding A model, and
shifted to a higher multipole. The direct effect of the Q fluctuations then acts at the
lowest multipoles to create a suppression of anisotropy, with respect to that in a A,
and even a standard CDM model.
We can make some further observations from the figure:
• In all of the panels, the smooth and fluctuating model curves hug each other
at all wavenumbers. However, as seen in the bottom panels, the early time
evolution of the potential in either of these models is rather different from that
observed in the A model at high wavenumbers (A; ~ 10~2/M pc). This is caused
by small but im portant changes in the ratio of radiation to m atter content
(with respect to the radiation content at last scattering in the A model). This
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129
combines with the increase in the Q fraction at last scattering to affect the
background evolution of the universe. The added presence of Q and radiation
at the expense of m atter at the last scattering surface causes strong potential
decay. The effect is larger with w approaching zero as the epoch of Q domi­
nation is pushed back close to m atter radiation equality. As we dial w from -1
towards 0, the Q content moves towards becoming comparable with the radi­
ation content at last scattering, leading to a rise in the height of the doppler
peaks. It leads to a larger source term in the equations for the oscillations of
the baryon-photon fluid before last scattering, and a stronger imprint on the
free-streaming photons after, as compared to what is observed in the corre­
sponding A model. These “indirect” effects are described in more detail in the
next section (section 6.4).
• Finally, notice that at the smaller wavelengths, the potential decays strongest
in the fluctuating Q model, particularly at recent epochs. However, at the
epochs when Q m atter gets to dominate the energy density, the horizon is much
larger than the wavelengths, and photons travel through many wavelengths of
the perturbations and gain no net frequency shift. This strong decay is thus
mostly irrelevant. Its only impact is to create weak late ISW increases in CMBR
anisotropy at intermediate scales (and comparatively old epochs).
We can summarize the potential evolution as follows: At early times the universe
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130
is radiation dominated. There is a tightly coupled baryon-photon fluid. Last scatter­
ing occurs and the photons free stream towards us. If there is a fraction of radiation
left over and/or Q m atter in the cosmological fluid somewhat before and/or after last
scattering, the gravitational potential decays differently giving rise to different metric
perturbations. When m atter dominates(if ever) the potential stops its decay, only
to resume at recent epochs when the Q-field starts to dominate. The fluctuations
in Q now become im portant, and the direct effect of Q-field fluctuations counteracts
and possibly overwhelms the effect of the altered late-time expansion history of the
universe, leaving its imprint on the low multipole CMBR anisotropy.
6.4
Early contributions: decay befor recombination, and amplification
after
In Q-field models, as in models with a cosmological constant, the m atter density in
the universe is reduced at last scattering. This means that a larger fraction of the
energy density at last scattering is in radiation and Q than in standard models (See
Figure 6.3). For example, for w = —0.1, the matter, radiation and Q contribute
roughly equal amounts at last scattering. As a result the gravitational potential $
is not constant, and this evolution leaves an imprint on the CMBR power spectrum
before and after last scattering.
As described in 6.1, the decaying gravitational potential enhances temperature
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131
fluctuations before last scattering in two ways. Firstly, it contributes strongly to
the driving force for the acoustic oscillations in the baryon photon fluid. Secondly,
it provides a blueshift to photons inside a potential well as a decaying potential
contracts the photons’ wavelengths. After last scattering, the Early ISW effect kicks
in, and also enhances the anisotropy at angular scales larger than the first doppler
peak.
As w is moved towards 0 the Q-field dominates at earlier and earlier epochs,
becomes comparable to and surpasses the radiation content, and may even dominate
the universe earlier than m atter (See Figure 6.3). This causes greater potential decay
and leaves an additional imprint on the CMBR anisotropy through the blueshift the
decaying potentials impart on the photons. If w is very close to 0, we begin to see
the effects of the Q field at very early epochs, and the resultant CMBR anisotropy is
a combination of the direct effect of the Q fluctuations and the merger of the early
and “late” ISW effects (section 6.6.1).
These effects lead to a very high first doppler peak (under primordial power
normalization), and a linear decline(over a large range of angular scales, £ = 30 —►
200) in power from the peak at i ~ 200, as can be seen in figure 6.4. The linearity is
a consequence of the increase in power at the doppler peak, the shifting of the late
ISW effect to smaller scales, and the suppression of anisotropy by the direct effect
of the Q-field fluctuations at larger wavelengths. Finally, fluctuations at a given
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132
physical scale get projected to higher angular scales or lower multipoles, increasing
the linearity of the decline. The power spectrum may also show a slight bump(see
6.4) at the scale at which the direct effect takes over.
6.5
Param eter dependence the positions and heights of the doppler peaks
6.5.1
Horizon W idth-the positions of the doppler peaks
The positions of the first peak (in I space) is given roughly by the ratio of the distance
to the sound horizon to the length of the sound horizon at last scattering, i.e., the
inverse of the angle subtended by the sound horizon. The others peaks positions
may be obtained from this position using the periodicity of the oscillations. Two
opposing effects determine these positions in Q-field CDM models . Changing w
from -1 towards 0 decreases the distance to the last scattering surface, pushing the
doppler peaks to the left with respect to the peaks of a A model. However, the smaller
ratio of m atter to radiation at last scattering results in a smaller sound velocity at
last scattering, which in turn leads to a smaller sound horizon. In this way the
increased radiation fraction tries to move the peaks to smaller angular scales, or to
higher £.
Let us consider the peak position as a function of w for fixed Qq . For values of w
for which Q is not an appreciable part of the cosmological fluid at last scattering, the
first effect wins out, moving the peak to lower £, as compared to the corresponding
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133
A model. For w —►0 the contribution of Q to the energy density dominates th at of
radiation at last scattering and makes the second effect win, and the doppler peaks
should move to higher £'s again. This is however seen clearly only in the third doppler
peak, in the upper panel in Figure 6.5, where the position of the third doppler peak
reverses its downward trend and approaches the position found in a pure CDM model
(with same h and
Q ft/i2 )
as w —> 0, independent of the value of Q q . The reason that
the effect is not seen in the first doppler peak, and only partially seen in the second
doppler peak (lower and middle panels respectively, Figure 6.5) is the early ISW
effect, which moves the peaks to the left by adding in power at higher angular scales
than those associated with the sound horizon. The reversal can be seen is at its
clearest in the third peak, which is the one least affected by the early ISW effect.
The second interesting trend occurs on fixing w and varying Qq; the highest Qq
models suffer the most rightward shift for w near -1 and the most leftward shift for w
near 0. The turn-over point seems to be near w = —0.3 from the third doppler peak
which is least adulterated by the Early ISW effect. The reason for the turnover has
to do with a change in the dependence of the sound horizon on Qq; for w < —0.3,
the sound horizon decreases in size with increasing
Qq ,
while for w > —0.3 the trend
is reversed. The critical value of w at which the turnover occurs depends on the
fractions of energy density in radiation and Q at last scattering.
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134
6.5.2
The heights o f the doppler peaks
The heights of the doppler peaks are relatively insensitive to the value of w, until
w gets close enough to zero. The additional potential decay (as compared to a A
model) in w —> 0 models due to the presence of an appreciable amount of Q-field
before last scattering (since Q-field dominates earlier and earlier as w —>0) leads to
amplification of temperature fluctuations through the source terms in the Boltzmann
equations and to a blue-shifting of photons captured in a decaying potential. The
effect leads to a rise in the peak heights, visible clearly on the upper panel in Figure
6.7, in the third doppler peak. The same effect after last scattering, the early ISW
effect, raises the peaks and moves them to higher angular scales by contributing
anisotropy at those scales. It adds to the pre-last scattering contributions and raises
the height of the power spectrum; this can be seen clearly in the bottom panel of
Figure 6.7 where the rise of the peak value as w -> 0 is rather more marked than in
the other panels. Increasing Q q increases the fraction of Q and decreases the m atter
content in the universe at around last scattering making both the pre- and post- last
scattering effects even more marked, thus increasing the anisotropy further.
If we fix Qq instead, and change w, we find that the peak heights in the un­
normalized spectra increase as w —> 0. This is a result of the extension of Qdomination to earlier and earlier epochs. This again translates to a smaller m atter
fraction at last scattering, as compared to models with more negative w’s, and con­
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135
sequently to a stronger potential decay, with all of its attendant effects.
Normalizing to COBE reverses the apparent trend at w —►0, hence reducing the
heights of the doppler peaks in those models. The less negative w models have their
low t < 10 multipole anisotropy suppressed (see section 6.6.2) with respect to their
intermediate t anisotropy as compared to models with more negative w, and the
rise to this intermediate power is earlier and less gradual. At the same time, due to
the constant decay in the potential, the overall anisotropy level in the code is much
larger than that in more negative w models (See the w = 0 and w = —0.1 models in
Figure 6.4). The models are normalized to COBE via the Bunn-White prescription
(see section 4.4.1 and [45]) which ensures roughly equal power around I = 10 for the
band corresponding to the COBE window function (I & 30). To compensate for the
intrinsically larger low i anisotropy level and the earlier rise at intermediate scales,
the very low-1 multipoles (t «
10 ) in w —>0 models are pulled far below their more
negative w peers; this brings the doppler peaks much lower, resulting in the decrease
in height with both increasing fig and increasing w, for w £ 0.3. Thus we obtain
the counter-intuitive trends in the curves that we previously, in figures 4.2 and 4.1.
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136
6.6
Anisotropies in the recent past
6.6.1
The late ISW effect
After Q-field domination at late times, the potential once again decays, and this
contributes to the low multipole anisotropy. The contribution from the decay at a
given epoch is largest for modes which have just entered the horizon at that epoch. It
decreases with increasing i as the photons have to travel through many wavelengths
of the modes corresponding to larger multipoles, hence suffering redshift-blueshift
cancellations.
Compared to lambda models, the increase is shifted to the right in £, or to smaller
angular scales. This happens due to two reasons. Firstly, universes with Q-field
reach Q-domination earlier and earlier as w changes from -1 (A) to 0. This results in
potential decay and thus temperature change earlier than in A models, which can be
seen clearly in the lower wavenumber panels of Figure 6.2. The second reason is the
direct effect of the Q-fluctuations which causes a suppression in the highest angular
scale (lowest i) anisotropy, moving the ISW rise to the right.
The Q fluctuations retard the change in potential as compared to in a A model
for the lowest wavenumber (k ~ 10-4/Mpc) wavelengths (see figure 6.2); this leads
to a suppression of temperature fluctuations at the corresponding angular scales
{t < 8). For the same wavenumber the gravitational potential in the smooth model
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137
can be seen (figure 6.2) to rapidly plummet, thus leading to a very strong late ISW
effect. As we go to higher wave numbers we see that the change in potential for
the fluctuating Q model gets larger. At k = 10-3 the drop just starts overtaking
the drop in the smooth model. This wavenumber contributes to more intermediate
£'s (£ > 8), and thus the intermediate t s are boosted via the late ISW effect with
respect to the low ones. At even higher wave numbers the drop in the fluctuating
model potential strongly dominates th at in the smooth one; however for these smaller
wavelengths (which have entered the horizon much earlier) the photons undergo a
series of successive redshifts and blueshifts resulting in phase cancellation and no net
effect on the temperature anisotropy.
As w —►0, Q dominates the energy density of the universe earlier and earlier, and
thus the late ISW effects move to lower angular scales. At the same time, the early
ISW effect moves to larger angular scales, and the two effects merge. The anisotropy
power spectrum at scales larger than the first doppler peak appears sharply slanted,
and ever increasing, as in Figure 6.4, and as described in earlier sections.
6.6.2
T h e d ire c t effect o f th e Q flu c tu a tio n s a t la te tim es
Most of the physical effects we have talked about are common to physical, fluctuating
Q models and to unphysical, smooth Q models, since they are related to the changes
in the background evolution due to the increase in radiation and Q fraction at last
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138
scattering and the late time change in expansion history as a result of Q-domination.
There are, however, impacts on the CMBR anisotropy from the “direct effect” of the
Q fluctuations. It serves to suppress the large angular scale anisotropy, leading to
the appearance of a downward bending notch in the power spectrum.
Consider the case in Figure 6.8. In the top panel in this figure we see two Qfleld models, primordial power normalized, one smooth and the other fluctuating.
The predictions of the CMBR power spectrum are identical for I > 100 but diverge
strongly at larger angular scales. The smooth model shows a strong late ISW boost
of the low t anisotropy while the fluctuating model shows a mild suppression as a
result of the weaker potential decay described in section 6.3.
There is a strong time evolution of the velocity perturbations of the Q-field which
pump energy into the gravitational field thus preventing rapid decay of and possibly
amplifying the potential. The strength of this effect varies with w\ for w = —1(A)
there is no fluctuating field and thus no direct effect. As can be seen from equation
4.16, which is reproduced here for convenience,
SOT + 2a- i Q
+ (a20
+ k')SQ = - i h'C?,
(6.8)
and from the change of variable in the appendix, the right hand side source term
depends on w in such a way that it is strongest for w = 0, and continuously reduces
in value to 0 at w = —1. A larger source term implies greater fluctuations in the
Q-field, and thus larger velocity fluctuations feeding the r( term in the perturbed
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139
Einstein equation (4.9, along with 4.17):
k2ff = (pQ + pQ)(vQ)i =
.
(6.9)
This leads to a strong change in 7/ which in turn feeds the energy into the gravitational
potential, arresting its decay as can be seen from equation 6.7. It is worth noting
that the effect is largest for small k , or long wavelengths.
6.7
Large angular scales: the cumulation o f anisotropy
The change of the CMBR low £ fluctuations in the above two models and in a A
model with Q.\ = 0.6 can be seen in Figure 6.9. The difference in anisotropy between
a energy density ratio r = pQ/pmatter of 0.1 and 1.5 is much more marked in the
fluctuating Q model than in the A model. The change in a smooth Q model is even
greater. The reason for that is the following: Since the evolution of the conformal
time is different in a Q model than in a A model the late time steep decrease in the
gravitational potential in the lambda model (see figure 6.2) is projected with a small
weight by the low £ (< 10) spherical Bessel functions onto the celestial sphere, and
thus the overall addition of anisotropy is not very spectacular. The more gradual but
ever-present low wavenumber potential dip in the fluctuating Q-field model ensures
that the potential curve hits the peak of the projection function at some epoch, and
thus the time integrated anisotropy increase is much larger, even while the lowest
£ anisotropy is suppressed with respect to the £ = 10 anisotropy. This effect is
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140
enhanced by the fact th at the Q-field model accumulates these anisotropies over a
much longer time (since Q domination occurs earlier), as compared to the A model
(For example, referring to figure 6.9, the Q-field model there attains 10% of the
critical energy density at z = 14, while the A model achieves the same fraction at
z = 1.5).
6.8
Large angular scales: the evolution of anisotropy
VVe are now able to describe the complete time evolution history of CMBR fluctua­
tions at low £ or large angular scales in a fluctuating Q-field model. VVe consider a
model with the values Q q = 0.6 and w = —1/3, and track the multipoles from when
the ratio r of Q-field energy density to m atter energy density is 0.005 (corresponding
to the epoch z = 299 in this model) to today, when the same ratio is 1.5. VVe show
this evolution in figure 6.10, where the numbers(l-6) labelling the curves indicate, in
ascending order, the epochs at which the Q-field to m atter ratio’s are 0.05, 0.1, 0.3,
0.7, 1.1, and 1.5(today) respectively.
VVe first notice (curves 1 and 2 in Figure 6.10) that the anisotropy falls from
the flat Sachs-VVolfe level for small t s (£ < 10) while it actually rises for larger
£'s (£ > 20). The increase at smaller angular scales is a result of late ISW effect
kicking in at smaller angular scales. The decrease at higher angular scales occurs for
the largest wavelength modes which pump energy into the gravitational potential,
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141
arresting its decay. This reduces the blueshift of the photons enhancing the negative
interference between the SW and ISW effects, at the cost of the power from the
ISW effect.
At about
r
= 0.3(curve 3), the low £ anisotropy also turns around
and starts to increase, but the higher i (£ > 6) anisotropy rises much faster. From
r
= 0.3(curve 3) to
r
= 0.7(curve 4) the trend of increase in anisotropy, faster at
lower angular scales, continues. This leads to a “flattening” of the power spectrum
around I — 10. The lower multipoles (i < 5) are sharply below the flattening due
to the direct contribution of the Q-field fluctuations retarding the potential growth.
The late ISW effect then
(r
> 0.7, curves 4-6) starts to affect multipoles around
£ — 6, leading to the familiar Late ISW low multipole anisotropy rise, but at smaller
angular scale as compared to the corresponding A model. The effect of cancellations
due to photons traveling through many wavelengths can be seen through the slowing
of the rise of the higher multipoles. Thus the £ = 6 anisotropy is now boosted with
respect to the multipoles immediately to its right. Meanwhile the £ < 6 anisotropy
also rises but never gets higher than the £ = 6 anisotropy as a result of the same direct
effect. These effects combined lead to the final appearance of the low multipole power
spectrum in this model: a suppression at £ < 6, a bump around I = 6, a subsequent
slight decrease until £ = 20, and then the rise to the first doppler peak.
The features imprinted on the low £ multipoles by this evolutionary process com­
bines with the change in the morphology of the doppler peaks that we described
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142
earlier to make the radiation power spectrum in Q-field models qualitatively differ­
ent from th at in A models. While these features are harder to discern quantitatively
thanks to the cosmic variance, there are models in which the difference is enough for
the next generation of satellites to be able to constrain the Q parameters, as we shall
show in the next chapter.
6.9
Summary
We have studied in this chapter the physical effects which leave an imprint on the
CMBR anisotropy. In particular, we have noted the effects of the potential decay
as a result of the increased
Qq
in the universe both at late and early times. We
have seen how the fluctuations leave a direct effect on the CMBR anisotropy by
compensating for the decay of the gravitational potential, leading to the sharp sup­
pression of anisotropy at the highest angular scales. This suppression is unique to
quintessence models, and is caused by the pumping of energy into the gravitational
field by velocity fluctuations in the Q-field.
Back in chapter 3, we saw that quintessence is consistent with many classical
cosmological measurements. In the rest of this work we present the comparison of
our anisotropy predictions to observations of the CMBR anisotropy. We investigate
whether the distinctive features imposed upon the CMBR anisotropy power spec­
trum by quintessence are sufficient to resolve intrinsic degeneracies in cosmological
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143
parameters, in combination with distance based cosmological measurements.
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144
8.0
3 — □ C 0B E 4yr
Q model, Qq = 0.6, w=—1/6
Standard CDM
model with A
Open Universe
First Peqjk
Early ISW;
6.0
<N
Damping
+ 4.0
Normalization brings
the peak down
2.0
Late ISW
Aliasing of
Power/
Low I plateau
Suppression from direct effect
0.0
10
Stronger compresssions
100
1000
Multipole Moments: I
Figure 6.1: We have plotted certain salient features of the CMBR anisotropy, or
“milestones” in four separate models. These are the Standard CDM model, a A
model (with Da = 0.6), and a open universe model, with QM = 0.3. The fourth
model is a QCDM models with constant w = - 1 / 6 and fi<3=0.6. In the latter
three models D eh2 is fixed at 0.02, while h=0.65. The CMBR power spectra at
large angular scales make a plateau with a increase in the anisotropy at the lowest
multipoles in the missing energy or curvature dominated(today) models. The plateau
rises to the doppler peaks in a smooth fashion due to the “Early ISW effect” and
the aliasing of power from higher multipole moments. The doppler peaks represent
oscillations in the baryon-photon fluid before the last scattering surface, and are
sampled at small angular scales due to baryon drag.
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145
0.9
v_
\
0.7
0.4
0.9
0.7
0.4
10
* ’
10“ 10
Figure 6.2: The evolution of gravitational potential $ for four different wavenumbers
is shown for three different models: The solid line represents a fluctuating Q-field
model with Qq = 0.6 and w = —1/3 , the dotted line an unphysical smooth Q-field
model with the same parameters, and the dot-dashed line a model with cosmological
constant (fi* = 0.6). The wave numbers are, from the top-left panel moving clock­
wise, 10~*Mpc~l, 10~3M pc~l, 5 x 10_2A/pc_1, and lQ~2M pc~l respectively. Notice
the slow change for a —> 1 of the solid line representing the fluctuating Q-field model
in figure with k = 10-4A/pc-1 (low wavenumbers), and the rapid change in the po­
tential in the A model in the same figure. Also notice in the bottom panels the low
a difference between the Q-field and A models’ potentials at high-wavenumbers.
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146
10'
m atter
radiation
Qy=0.6, h = - 1 / 3 Q model
Qu=0.6, tv=-0.05 Q model
Q0=0.6, w=0 Q model
10°
0
1
10
100
1000
z
Figure 6.3: The ratio of energy density in an individual species to the critical energy
density today as a function of time. Shown here are three models, each with Q q = 0.6,
and with equations of state w -0.33, -0.05, and 0. Note that in a w = —1/3 model
only pr and pm are comparable at the surface of last scattering. For w=-0.05 pq is as
large as pr at last scattering, making a significant effect on the CMBR. When w = 0,
P q is always higher than pM and the universe never gets matter dominated. The large
fraction of Q at last scattering ensures a rise in intermediate scale anisotropy through
resonant oscillations and the early ISW effect
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147
t \
i
125.0
w=0
w = -0 .1
w=-0.4
w=-1(A)
IT 100.0
/
/
i
..
2
\
\
l
\
i
75.0
\
C
r<
\\!\ n
;i
50.0
■■ ■ > .-
25.0
0.0
i.-v
'Ja
10
100
1000
Figure 6.4: CMBR anisotropy as a result of resonant acoustic oscillations and the
early ISW in models with Qq = 0.6. The height of the peak does not change much
from w = —1 to w = —0.3 while the position changes mildly due to changes in
the angular diameter of the sound horizon. There is greater change from w = —0.4
to w = —0.1, at which value Q, radiation, and m atter contribute roughly equal
amounts to the total energy density at last scattering , and a steep increase in the
height fromu; = —0.1 to uj = 0 , in part of which range Q always dominates over
matter.
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148
M
2r*
«
8.
~ I §
i 5
s«
▼
1— r
*r
—r
T
T
M
m
O
& s.
a
a
jts
t>
a
o
Si
i
S
a , - o.i
0 , • 0.4
0, - 0.8
3
1
•
0.8
-
0.6
-0.4
•
0.2
0
Figure 6.5: Position of the first, second and third peaks as a function of w. Notice
how the curves on the upper panel turn back up for w > —0.1.
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149
»
*oa
c
E
u
O
z
1
—
o
u
— '
■ .
.....
. .
3.
■a
e
ou
t
>
----------- :
n, - 0.1
■
r~ T '
---- '- 1 -------------- T - r - v
i
\
■
V.
1
. . .
":
t
r
■ 0. 4
0. 8
i
i
|
I
i
i
i
.
.
1
. . .
■■r~"r
|
f
«
CV
I
vaa
a
a
■oo
0.8
•
• 0.6
•0.4
•
0.2
Figure 6.6: Peaks heights as a function of w when the power spectra are normalized
to COBE.
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150
t
i
r-
■'
/
O
C
Cl
o
S :
*
8:
O _|_
_1
a
■O
o
I
I
I
l_
■a , « o.t
CL
n , » 0.4
/
/ *
a , - o.e
0.8
•
-0.6
w
-0.4
-
0.2
Figure 6.7: Peak heights with the normalization as output by the code. Since this
normalization is the same for all models as opposed to the COBE normalization, we
can isolate the physical effects that contribute to the trend. Notice how all the curves
are flat until w becomes large. For (w > = —0.1), the Q fraction at last scattering
increases to be comparable to the radiation fraction, leading to a rise in anisotropy.
As w gets very close to 0, the Q fraction even dominates the m atter fraction, and
there is a steep rise in the heights of the doppler peaks.
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151
80.0
go 0
...............................
Q,j=0.6, »=-1/3, no fluctuations
Qu=0 6, w=-1/3, fluctuations in Q
40.0
20.0
0.0
6e-10
Q,_,=0.6, w=-1/3, no fluctuations
Q,,=0.6, w=-1/3, fluctuations in Q
2e-10
100
1000
I
Figure 6.8: This figure shows the CMBR power spectrum code and COBE normalized
for a Qq = 0.6 model with w = —1/3, in the physical case with fluctuations turned
on and responding to gravity, and in the unphysical case of their being turned off
which violates the equivalence principle. Note the strong difference in low multipole
behavior, the unphysical smooth Q-field model showing an upturn in the anisotropy
power spectrum at those scales, while the fluctuating Q-field model shows a slight
dip as a result of the direct effect of the Q-field fluctuations. Q-field fluctuations feed
the gravitational potential with energy, leading to this rather large difference from
the smooth model. On COBE normalization, the change shows up at the doppler
peak.
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152
45.0
Qu=0.6, w=-1 (A)
Qu=0.6, w=-1/3, Q fluctuations
Q 0=0.6, wr=—1/3, no Q fluctuations
N
gi
25.0
.u ,
u'
//
5.0
100
I
Figure 6.9: The figure shows the progression of low multipole CMBR anisotropy in
smooth and fluctuating Q-field models with w = —1/3 and Qq = 0.6, and in a A
model with
= 0.6. The lower and upper curves in each of these cases are the
CMBR anisotropy power spectra seen by an observer a t an epoch when the ratio
of Q (A) to m atter density was 0.1 and today (ratio=1.5) respectively. Note the
large change in the unphysical smooth Q-field model, the appreciable change in the
fluctuating Q-field model, and the unspectacular change in the A model.
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153
Qe=0.6, w=-1/3, model with Q fluctuations
25.0
N
O
C
p(/fVp1/f^0.005, z=299
0.1, z=14
0.3, z=4
0.7, z=1.14
1.1, z=0.36
1.5 (today)
i
5.0
100
Figure 6.10: CMBR anisotropy low multipole power spectra at different epochs in
a fluctuating Q-field model with w = —1/3 and Qq = 0.6. The epochs are marked
by the ratio of Q energy density to m atter energy density; in this model, a ratio of
0.005 corresponds to a redshift of 299.
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Chapter 7
Comparing predictions from quintessence and observational results:
com patibility and degeneracy
VVe describe in this chapter how the predictions of Q-field models studied in this work
are compatible with the results returned by observations of the CMBR anisotropy
power spectra, m atter power spectra, and rich cluster abundance. VVe then introduce
a new statistical index analogous to the log-likelihood which takes into account the
existence of cosmic variance in quantifying the overlap between two CMBR power
spectra. We apply the index to show th at there are degeneracies in the CMBR
anisotropy between Q m atter and A models, and amongst Q m atter models. We
explain the physical reasons for these degeneracies.
The work described in this chapter is the result of a collaboration with Paul
Steinhardt, Limin Wang, Robert Caldwell, and Greg Huey.
154
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155
7.1
C M BR anisotropies in Q-field models: th e observations so far
In figures 7.1 and 7.2 we show the comparison of two quintessence models to obser­
vational data. To the models in figure 3.3, we added two QCDM models, a Qq = 0.7,
w = —0.5 constant equation of state model, and a ilQ(today) = 0-7, W(today) = —0.5
model with a massive scalar field oscillating in a quadratic potential. In the first
figure we simply change figure 3.3 by adding these two models, while in the latter
figure we choose and plot the specific observations th at we believe to be of the highest
quality. While most of the CMBR experimental data is very preliminary, and not
verified by re-observation, we chose COBE, OVRO, CAT, Saskatoon, Boomerang,
Maxima, DASI and TOCO, since these results are known to be low in contamination
or have revisited the same region of the sky in successive years and verified that their
results were not systematic errors or statistical fluctuations. We can visually see
from both figures th at the CMBR power spectra predicted in the constant w QCDM
model fits the CMBR anisotropy measurements just as well as the A model.
In the model with a quadratic potential the scalar field mass far exceeds the Hub­
ble Energy scale. Thus the model has a feature in its power spectrum at a angular
scale corresponding to the size of the horizon when its equation of state started to
oscillate. It is conceivable that interesting dynamical Q models will leave such signa­
tures in their power spectrum, thus rendering themselves amenable to experimental
falsification. Future CMBR experiments probing this scale, for example, MAP, will
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I 5 6
10.0
-
Observational Data
Standard CDM
rtel. h=0.65. Qa«0l7
opan (n=l .15)
n=1. hs0.65. 0
8.0
-
/\
^ 1,w >-05
n=1. h=0.65, Q^xOJ. w#*0, m >» H,
£
<N
+
1000
Multipole Moments: I
Figure 7.1: In this plot we add to the curves in figure 3.3 two Q-models: one, a high
Oq , high w constant equation of state model, and two, a model with a massive scalar
field sloshing about in a quadratic potential. We see that the constant w models fits
the observational data well.
be able to distinguish such features from plateaus. This specific model is observationally un-interesting, since the prediction that w = 0 today is completely inconsistent
with observations, as can be clearly seen from the figure.
The coming d ata from the MAP and PLANCK satellites will revolutionize the
field by supplying us full-sky d ata at angular scales stretching from I = 30(MAP) to
i = 2000(PLANCK). The huge amount of data will enable us to close in on a lot of
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<5 ?
▲
0
□
^
—
OASI
MAXIMA
TOCO
Sukatoon-95
Standard CDM
• _
n=1.h=0.65.aA«0.7
— • open (n=1.15)
•
#
■— ■
CAT
OVRO:7-22
_
n=1. hsO.65, Qq- 0.7. «r>-0.5
. . .
n=1. ltrO.65. OQt«0.a, «*0»0, m » H ^
Multipole Moments: /
Figure 7.2: Here we simply change the previous plot by focusing on the band-powers
from a few observations th at we believe to be the best in terms of the quality of their
data. We once again see that the constant w models fits the observational data well.
fundamental parameters which characterize the universe. However some degeneracies
amongst these parameters will remain, as we shall see later in this chapter and the
next, requiring us to use other measurements to break this degeneracy.
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158
MMpc’)
Figure 7.3: (a) Variation of mass power spectrum for some representative QCDM
examples, (b) The variation of cr8 with fig. For ACDM, Qq is QA. The suppression
of c78
QCDM compared to Standard CDM makes for a better fit with current
observations. The grey swath illustrates constraints from x-ray cluster abundance.
7.2
Measurements of the m atter power spectrum and cr8
We have computed the constraints on the m atter power spectrum and <r8 in the
Q-field models(see figure 7.3). Obtaining these quantities involves calculating the
transfer function and COBE normalization for each of the models. The key feature
in the figure 7.3 is the sensitivity of the power spectrum to all parameters: the value
of w; the time-dependence of tu; the effective potential V(Q) and initial conditions;
and the value of Qq . Hence, combined with the CMBR anisotropy, the power spec­
trum provides a powerful test of QCDM and its parameters. An important effect
is the suppression of the mass power spectrum and cr8 (the RAIS mass fluctuation
at 8/i-1 Mpc) compared to SCDM, which makes for a better fit to current observa­
tions (see Steinhardt and Ostriker [10]). The grey swath in the figure represents the
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159
constraint from x-ray cluster abundance for ACDM;[62] in general, the constraint
is weakly model-dependent. Note that invoking Q-field reduces the amount of Qq
required to fit the cluster abundance constraints, on comparison to the A model, in
accordance with the recent observations of lens counts [10].
While QCDM and ACDM both compare well to current observations of CMBR
anisotropy and large-scale structure today (see chapter 3), QCDM has advantages in
fitting constraints from high red shift supernovas, gravitational lensing, and structure
formation a t large red shift (z « 5). Constraints based on classical cosmological
tests on A (w.\ = -1 ), such as supernovas and lensing, are significantly relaxed for
QCDM with w % - 1 /2 or greater[18], Another property of QCDM (or ACDM) is
that structure growth and evolution ceases when the Q-component (or A) begins
to dominate over the m atter density. Comparing QCDM and ACDM models with
Dq = Da , this cessation of growth occurs earlier in QCDM. For larger and larger
values of w, the growth ceases earlier and earlier. Hence, more large scale structure
and quasar formation at large red shift are predicted, in better accordance with deep
red shift images[9].
7.3
The anisotropy degeneracies and their width: a new statistical index
Let us turn our attention to the computation of CMBR anisotropy degeneracy be­
tween QCDM models and ACDM models, and more generally, between different
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160
QCDM models.
Degeneracy refers to the possibility that there are combinations of parameters
in the parameter space we consider(fia/i2, n3, h, Q \((CIq )) which predict the same
microwave sky?
It must be remembered that such theories need not have identical inflationary
mean predictions to be indistinguishable on the basis of CMBR measurements. This
is because of the inherent theoretical uncertainty on measurements in cosmology, due
to the cosmic variance (See sections 1.1 and 2.5). If the predictions made by most of
the simulation realizations of one theory fall in the cosmic variance band of another
, they will be indistinguishable.
Any statistical index used to determine the degeneracy between two theories must
take the cosmic variance into account. Here we present such an index and a method
of analysis, which has the attractive feature that the likelihood is simple to calculate
analytically, avoiding the need for computationally expensive Monte Carlo.
Suppose Models .4 and B are to be compared. VVe wish to estimate the likeli­
hood th at a Model .4 real-sky would be confused as Model B . Since the prediction
of Model -4 is itself non-unique, subject to cosmic variance (and, in general, experi­
mental error), we need to average the log-likelihood over the probability distribution
associated with A. Only cosmic variance error, C ijyj2 l + 1 , is assumed for each
multipole Ct and the distribution is chi-squared. In our notation, Ct s are the cosmic
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161
mean values and x* are the values measured within our Hubble horizon. Then, the
“average log-likelihood” is defined to be
A.-
(7.1)
where P({x*}|,4) is the probability of observing the set of multipoles {x*} in a real­
ization of Model .4. Since each multipole is statistically independent, P({x/}[.4) can
be written as a simple product of chi-squared distributions for each £. Substituting
the chi-square distribution for P(x*|.4), Cba reduces to
=
+
P.2)
Note th at Cba # Cab in general due to the different priors in the probability distri­
bution in 7.1. The difference is small in practice, however.
This index is to be interpreted thus: ecHi represents the odds that a sky simulated
from theory a could be observed as to have come from theory b. There is direct
calibration in terms of x 2 ; since in this case Xmm = 0(the two models are identical),
the value of the
x
2 corresponding to a given confidence level corresponds to the
averaged log likelihood. This is different from the usual log likelihood only in the
averaging procedure.
We decide distinguishability according to the m in ^ * ,, Cab)- The log-likelihood
satisfies —C > 6 for distinguishability corresponding to the 3a level or greater.
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162
VVe can add in gaussian experimental error by replacing the C* by
Dt = Ct + w - le - l2aZ,
(7.3)
where Ob is the experimental beam size, and w = {o^ixQ.pix)~i is the weight per solid
angle,
being the noise per pixel, and flptI the solid angle covered by it. This
follows from the observation that when experimental noise is added in, the quantity
distributed as a chi-square changes from being C* to being £>*.[63]
As long as only linear effects are important, distinguishability of a pair of cos­
mological models entails comparing the shapes of the two spectra without specifying
any normalization. Non linear effects require a specification of the normalization, in
which case there is the added uncertainty of the COBE measurement[24].
7.4
Application o f the index: Inherent degeneracies in distinguishing
Q-field m odels from A models.
*
If the equation of state w in a Q-field model is rapidly varying, w
2
1, the spatial
fluctuations in Q and the variation in the cosmic expansion rate significantly alter
the shape of the cosmic microwave anisotropy power spectrum[16, 39], producing
differences from A models that are detectable in near-future satellite measurements.
The degeneracy problem between A and quintessence arises if w is constant or
, 2
slowly-varying (u)
1), as occurs for a wide range of potentials {e.g., quadratic or
exponential) and initial conditions (see section 4.4.4). For w ^ —Q q / 2 , we find that
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163
the effects of quintessence on the CMBR power spectrum can be closely mimicked
by a model with A, provided the values Qm and h are also adjusted. Even for an
ideal, cosmic variance limited, full-sky measurement of the CMBR anisotropy, there
is a degeneracy in the three-dimensional parameter space of Qm, h and w.
The degeneracy curves in figure 7.4 can be produced by tracing the minimum
of the average log-likelihood described above. These curves represent the center of
a strip of models in the
—w plane which cannot be distinguished by the CMB
alone. To pinpoint the Q-models degenerate with a given A model, we fix w, and
find the minimum averaged likelihood (using the method described above) over all
other parameters (n.,, h, Qm and fi*). To estimate the width of the degeneracy curve,
we must find the 3a likelihood contours. Given such a model, we choose as the
param eter of variation either flm, and minimizing over all other parameters find the
set of models within 3a of the minimum. This procedure is then repeated for all the
other w. (VVe could have instead worked in Qm space and used w as the parameter
of variation. The width works out on the average to roughly 5% in Q m and about
10-15 % in w.
Figure 7.4 illustrates the degeneracy problem for CMBR anisotropy measure­
ments. The first panel shows the plane of Qm and w with a sequence of dashed
curves. The case of a cosmological constant corresponds to the axis w = —I and
the remaining plane corresponds to quintessence models with constant or slowly-
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164
varying w. Each dashed curve represents a set of cosmological models with a Q or
A-component whose CMBR anisotropy power spectra cannot be distinguished even
with cosmic variance limited, full-sky measurements. (Our numerical computations
extend to multipole £ = 1000.) For example, for fixed Qbh2 and n, (the spectral index
of scalar fluctuations), a model with quintessence and Qm = 0.47, w = —1/2 and
h = 0.57 (circle) produces a nearly identical CMBR power spectrum to a A model
with S'Im = 0.29, w = - 1 and h = 0.72 (square). The second panel illustrates the
two power spectra, which overlap almost entirely. If the value of h for the first model
is changed, the value of h for the rest of the models along the curve can be adjusted
so that there remains a degeneracy.
The degeneracy curves can be understood theoretically. They correspond approx­
imately to the set of models th at obey the following constraints:
• flm + f Iq = 1
• Qmh2 =constant.
• fifth2 = const ant.
• n, =constant.
• ip =same.
where ip is the multipole corresponding the position of the first acoustic (Doppler)
peak. We assume th at r, the ratio of the tensor-to-scaiar primordial power spectrum
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165
amplitudes obeys inflationary predictions[64,65]. The peak position ip (proportional
to the ratio of the conformal time since last scattering to the sound horizon at last
scattering) depends on Qmh2,
h and w. The only way to keep ip constant along
the degeneracy curve as w varies is to adjust h, since Qmh2 and fi6/i2 are constrained
to be fixed. M. W hite has independently noted similar conditions for degeneracy for
constant w models.[66] Our results are based on full numerical codes which include
the fluctuations in Q.
Our computations with the index we developed confirm that the above conditions
which we use to understand our results are a good approximation to the degeneracy
curves and th at the fluctuating Q effects are too small to break up the degeneracy
if w & —Q q / 2 (to the left of the dotted line in the first panel in Fig. 7.4). The
boundary of the i max = 1000 degeneracy region is then determined by the newly
discovered direct effects of the fluctuating Q field and the large integrated SachsWolfe contribution to the CMB anisotropy, such th at models with w k —Q q / 2 are
distinguishable from ACDM models at > 3<r level, assuming a cosmic variance limited
measurement.
Thus we are left with an inherent degeneracy in the Q-field models, a situation
not too pleasing. The degeneracy affects models with w < —0.6, and in general,
models with a slowly changing equation of state. The constraint of identical peak
position then translates this degeneracy into coarser resolution of both fim and h in
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166
the range of 5% to 10% (at the 3a level) of their likelihood minimizing values.
7.5
Summary
We have shown that the Q-field anisotropy in constant w QCDM models is con­
sistent with the present data from CMBR experiments and other measurements of
fluctuation dynamics. However, on asking the question as to how distinguishable
these Q-field models are from other Q-field models and the cosmological constant
models in their predictions, we find that there exists a fundamental degeneracy for
w ^ —D q/2 in the Q\i — w plane. We did not study the effects on distinguishability
of rapidly-time-varying equations of state; such models are definitely likely to have
less degeneracy with A and other Q models due to the features of their energy density
evolution.b
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167
1.0
E
cj
0.4
0.0
-
0.8
•
•0.4
0.8
0.0
W
8
e
S!
4
2
x
O' 0
10
100
1000
Multipote Moment /
Figure 7.4: The CMBR degeneracy problem: Each dashed curve in (a) represents
a family of QCDM and ACDM models with indistinguishable CMBR anisotropy
power spectra. The width of the curve is 5-10 % the values on it. For example,
Panel (b) shows two overlapping spectra for the A (square) and quintessence (circle)
models indicated in (a). Models beyond the dotted line in (a) (e.g., the triangle) are
distinguishable.
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Chapter 8
A ttem pting to lift the anisotropy degeneracy in quintessence models
We showed in the last chapter that there is a range of parameter space in the constant
or monotonically changing equation of state models for which an ideal, full-sky cosmic
background anisotropy experiment may not itself be able to distinguish between
the cosmological constant and Q-field.
VVe show here that this degeneracy may
unfortunately remain for a certain subset of those parameters even after considering
classical cosmological tests and measurements of large scale structure.
At least one currently underway observation holds further promise in reducing the
degeneracy region: the luminosity distance to high redshift supernovas. We present
our work on recovering m atter and cosmological constant fractions in a curved uni­
verse by combining the CMBR anisotropy measurements with apparent magnitude
measurements of high redshift supernovas, and then extend this work to the narrow­
ing of the degeneracy region in the Qq- w plane (Qm - w plane).
The work described in this chapter is the result of a collaboration with Paul
168
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169
Steinhardt and Greg Huey, who calculated the sophisticated degeneracy curves shown
on plots in this chapter from my crude degeneracy estimates.
8.1
Combining the CM BR anisotropy w ith other measurements
Consider the situation in several years’ time, after data has been analyzed from
the MAP and PLANCK experiments, and possibly others. Assume that the CMBR
anisotropy measurements conform closely with one of the degeneracy curves in Figure
7.4, a possibility consistent with current observations.[67] The degeneracy means
that one cannot distinguish whether the missing energy is quintessence or vacuum
energy. Furthermore, fim and h vary along the degeneracy curve (so as to keep Qmhr
constant), such that the uncertainty in these key parameters is very large. How can
the ambiguities be resolved?
Notice from figure 8.1 that the extent of the degeneracy region can be decreased if
we include in the degeneracy analysis, multipole moments up to I = 2000. Measure­
ments of the CMBR on smaller angular scales where non-linear effects are important
can be used to break the degeneracy. Gravitational lensing distortion of the primary,
linear CMBR anisotropy by small-scale density inhomogeneities along the line of
sight [68, 69, 70, 71] has the capability to discriminate[72, 73] between quintessence
and a cosmological constant.[74] The efficacy of this phenomena, which smoothes the
peaks and troughs in the CMBR spectrum on the scales of interest, depends on the
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170
«*
-1.0
-0.8
•• •
-0.6
-0.4
-0.2
0.0
w
Figure 8.1: The CMBR anisotropy constrains models to a particular degeneracy
curve and, independently, provides tight constraints on n,, Qmh2 and
The
latter constraints, along with other observational limits discussed in the text, fixes
an allowed range of Qm and w (the shaded region using the example discussed in
the text). The combination determines the best-fit models. The degeneracy region
becomes smaller as we include multipoles up to I = ‘2000 due to the inclusion of
non-linear gravitational lensing effects on the CMBR anisotropy.
level of mass fluctuations. If the amplitude of primordial density perturbations were
anything other than 6p/p ~ 10-5, this effect would be either completely negligible
or else the dominant effect in CMBR anisotropy. At the level measured by COBE
and MAP, the lensing is a negligible effect since it only begins to become important
for I k, 1000. However, lensing effects are non-negligible for the Planck experiment
which extends to i ~ 1500, or experiments at yet smaller angular scales.
Given this narrowed degeneracy region, let us add other cosmological observa­
tions to the analysis. These may not be as precise as those of the CMBR anisotropy,
but they have the advantage that they do not share the same degeneracy.
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If
171
other observations can be used to determine separately Qm or h (or some combi­
nation of Qm and h other than Qm/i2), then perhaps the degeneracy between A and
quintessence can be broken.
We have considered the current restrictions on Qm
and h obtained by combining the best limits on age (> 10 Gyr), Hubble constant,
baryon fraction (Qbh3^2/Q m ~ 3-10%), cluster abundance and evolution,[75] Lymana absorption,[76] deceleration parameter[27] and the mass power spectrum (APM
Survey).[77] The current constraints and the techniques for combining them have
been detailed elsewhere.[78, 67] We also include the fact that the CMB anisotropy
will provide tight constraints on n s and the combinations Qmh2 and Qbh2 to within
a few percent.[79, 80, 81, 82]
Even combining all the observational information listed above, fim and h are
not highly constrained. Assume for illustrative purposes that the CMB anisotropy
converges on n, = 1, r = 0, Qbh2 = 0.02 and Qmh2 = 0.15 (reasonable values).
Then Figure 8.1 shows the shaded region in the Qm-w plane which can satisfy the
observational constraints at the 2a level. In this case, acceptable models must lie
at the overlap of the degeneracy curve picked out by the CMB anisotropy and the
shaded region.
Three possibilities emerge, as shown in Figure 8.1: (1) the degeneracy curve over­
laps the shaded region only over a limited range of w so that the ambiguity between
quintessence and A is broken and Qm, h and w are well-constrained; (2) the degener­
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172
acy curve cuts through the shaded region in such a way th at a substantial ambiguity
remains; or (3) the degeneracy curve and the shaded region do not overlap at all.
Case (3) appears at first to be a contradiction: the CMB spectrum conforms to the
predictions of a ACDM or QCDM model, but constraints from other cosmological
observations (shaded region) suggest that the Qm is too small (or too big). However,
this situation is precisely what ought to occur if one of our underlying assumptions is
incorrect: namely, flatness. By introducing spatial curvature as an additional com­
ponent (.4 7^ 1) further degeneracy arises. Associated with curve (3) is a continuous
family of degeneracy curves in the Qm-w plane each beginning from a different value
of 0 m along the w = —1 axis[79, 83], as we shall see in the next section. Making
the universe open (closed) produces CMB degeneracy curves beginning with smaller
(larger) values of fim, whereas the shaded region in Fig. 8.1 is only modestly changed.
So, for example, curve (3) in Figure 8.1 is also degenerate with an open model with
fim = 0.4, Q \ = 0.54 and h = 0.8, which is consistent with the shaded region.
Adding curvature is inconsistent with standard inflation-based models, but case (3)
exemplifies how we may be forced observationally to consider the possibility.
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173
8.2
Combining supernova and CM BR constraints to investigate spatial
curvature
In this section we present work on combining high redshift supernova magnitudes
and CMBR anisotropy measurements together to remove the inherent degeneracies
present the CMBR power spectra in universes with curvature. VVe show how these
two measurements can be used to generate tight constraints on both
(equivalently
Qq for a fixed w) and f lu in a closed, open or flat universe. The analysis can be
easily extended to Q-field model in a flat universe: it provides important insights
on how we may combine the same two measurements in a flat universe to lay down
constraints on w and Qq . Moreover, as we saw in the previous section, if missing
energy in a flat universe turns out to be an observationally inconsistent option, we
can evaluate the consequences of curvature and missing energy(A or Q) in a non-flat
universe.
Let us imagine an attem pt to measure Qm and Q.\ in a maximally optimistic
CMBR experiment such that there is negligible instrumental error. The only sta­
tistical error is the fundamental cosmic variance. In the {Qm,Q \) plane, Qtot = 1.0
corresponds to the diagonal line in figure 8.2 (the line of flatness). If one picks a
point in this plane corresponding to a fiducial universe and then compares its power
spectra to those of other
points, an exclusion contour of 99% likelihood
can be determined. We take cosmic variance as the la error of a maximally opti­
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174
mistic CMBR power spectrum experiment, and the parameters h, Qg/i2, n, and nor­
malization are marginalized out. The resulting contours for fiducial universes with
= {(0.35,0.65), (0.9,0.1), (0.35,0.45), (0.6,0.6)}, and for measurements to
Imax = 2000 are shown in figure 8.2. The type of perturbation spectrum was re­
stricted to adiabatic scale-free, with an inflationary relation between tensor and
scalar spectral indices and normalizations. These restrictions could be relaxed, but
that would enlarge rather than reduce the degeneracy region, and we are interested
in finding the ultim ate limit of parameter resolution from the CMBR.
o>
00
o
c:
CM
O
o
0
0.1
0 .2
0.3
0 .4
0 .5
0.6
0 .7
0 .8
0 .9
1
1.1
1.2
Figure 8.2: The degeneracy contours of the four fiducial models being considered are
shown: (A m i^a) = {(0.35,0.65), (0.9,0.1), (0.35,0.45), (0.6,0.6)}, all with h = 0.65,
ns = 1.0 and Qg/i2 = 0.02. The contours are cut off above the fiA = 0.75 and to the
left of the Q m = 0.2 lines. Also shown are dotted lines radiating from (—0.145,1.040)
and passing through the fiducial points, which are marked by a triangle. Note that
the degeneracy lines are not perfectly straight lines.
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175
The line of degeneracy of the CMBR anisotropy can be explained similarly to the
lines of degeneracy in sections 7.4 and 8.1: The heights of the Doppler peaks depend
upon n3, n m/i2, and Db/i2, and their I position on Dm/r , SlBh2, and the conformal
distance to the last-scattering surface. If one alters cosmological parameters with the
constraint that these quantities remain constant, then the CMBR power spectrum
remains largely unchanged. The conformal distance to last-scattering depends on the
combination of h, fim, Da (with Qtot = Dm + Da) shown in equation. 3.5. Consider
a particular value for
Suppose one wishes to move to a new value of
without altering the power spectrum. Move along the Qtot = constant line to the
new fim, changing h such that Qmh2 remains constant, and changing fl \ such that
Dm + Da remains constant.
same, the change in
However, despite the fact that the curvature is the
causes an angular scale shift in the spectrum. This can be
compensated by a slight change in curvature: trading off Q \ for Qtot. The resulting
point will have a power spectra differing only by its late-integrated Sachs-Wolfe
contribution - which is insufficient to break the degeneracy for moderate differences
in Qm - thus the long thin degeneracy contours in figure 8.2. The approximation
method used in this study to determine the late-integrated Sachs-Wolfe contribution
for open and fiat models has been tested against the exact numerical code of [84].
However, as there are no corresponding codes available for a closed universe, it has not
been so rigorously tested for these models. Thus the actual length of the degeneracy
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176
lines in the closed regime of figure 8.2 is only an estimate.
The method of breaking this CMBR degeneracy is by combining CMBR
anisotropy measurements with a measurement of the cosmic deceleration parame­
ter qo. The deviation of the magnitudes of standard-candle supernovas vs. redshift
from the Hubble law at high redshift depends on fim, Q \, and 1 — Qtot in a way
similar to the angular scale of features in the CMBR power spectrum. Thus a mea­
surement of qo, or equivalently, the luminosity distance of a high-z supernova, has
lines of degeneracy in the (fim, Qa) plane qualitatively similar, but nearly orthogonal
to those of the CMBR anisotropy due to the very different epoch of observation.
In both CMBR and supernovas observations, one is measuring conformal distance
to a given redshift. This measurement possesses the degeneracy of being able to
simultaneously change Qm and
such that conformal distance to a given redshift
remains fixed. W hat makes the combination of CMBR and supernovas measurements
so useful for breaking this degeneracy is that the degeneracy contours rotate with
redshift: those at a z ~ 1 are roughly orthogonal to those of z at last-scattering.
Note that this method does not require an independent measurement of the Hubble
constant h.
Figure 8.3 shows these contours for z=1.5 supernova overlaid on the contours for
a CMBR measurement in different fiducial universes. Note that we can recapture
fim of the fiducial universe to ~ 10% in all of these models.
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177
8.3
Com bining supernova and C M B R to investigate Q
In section 8.2 we combined measurements of high redshift supernovas and CMBR
anisotropy to be able to set constraints on the cosmological constant ( Q \ ) and m atter
energy density (Qm)- In that analysis we assumed a universe with arbitrary curva­
ture. We have added now an additional parameter
wq,
replacing Q \ by Q q . If we
do the same analysis, we would find a surface of degeneracy in three dimensions.
Let us restrict ourselves to the physically more plausible flat universe instead and
see what degeneracy may be broken by combining supernova luminosity distance
measurements with CMBR anisotropy.
Figure 8.4 shows the prediction for the red shift luminosity relation, measured
using Type 1A supernovas as standard candles[27] for the same models along the
degeneracy curve. In this case the quintessence models are more distinct from the
A model; however, it is premature to say whether observations will become accurate
enough to make this measurable. Not only will a large number of high red shift SNe
have to be observed, but the systematic errors in the magnitude calibration will have
to be reduced, to A m £ 0 .1 , in order th at a turn-over in A m is well determined.
To check if this distinctness is enough to distinguish the Q-field models from
their correspondingly degenerate A models, and from other Q-field models, on the
assumption of this large reduction in systematic errors, we repeat the analysis we
made in section 8.2, but this time in the Q m — w plane. We assume 2% and 5%
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178
measurements of the luminosity distance on a z = l supernova made for the fiducial
parameters Q q = 0.6 and w = —0.5. The percentage error bars on the measurement
yield degeneracy swathes in the Q \f — w plane which we superpose on the CMBR
degeneracy curves in figure 8.5. VVe find that we can obtain 10-20% measurements of
Q\( and w by considering the overlap of these contours with the CMBR degeneracy
curves, and to 10% if we include all the other cosmological tests described above.
Furthermore, the angle of the z ~ 1 contours make it very easy to distinguish a
Q-field model from its A counterpart on the same degeneracy curve.
If we can improve our understanding of the systematics of light curves of high
redshift supernovas, we may be able to use the measurements of their luminosity
distances along with other cosmological measurements to obtain reasonably good
constraints on Q q and w. Even so, the location in parameter space where the de­
generacy is found may not be amenable to reasonable parameter constraining. In
such a case we will have to invent new cosmological measurements to resolve the
degeneracy.
8.4
Summary:Present lim its on parameter estimation
A large class of quintessence models, those with rapidly varying w or constant
w k — Q q / 2, can be distinguished from A models by near future CMB experiments
such as MAP. However, any given A model is indistinguishable from the subset of
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179
quintessence models along its degeneracy curve. CMB experiments which probe
small angular scales where gravitational lens distortion is expected to be important,
such as Planck, can be expected to cut down the degeneracy region. Combining
the constraints which the CMB imposes on n 3, Qmh2 and Q^h2 to the other current
observational constraints sometimes, but not always, breaks the degeneracy. Adding
spatial curvature as an additional degree of freedom increases the degeneracy.
VVe showed that these degeneracies may be lifted to some extent by combining
the measurements with luminosity distance measurements from high redshift super­
novas, leading to 10% to 20% constraints on Q q and w . However, depending upon
where in the parameter space the different measurements overlap, new observational
techniques may need to be invented to break the degeneracy.
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180
oo
o
oo
o
(O
to
o'
o
')■
o
o'
CM
o
o
0
o
0 .2
0 .4
0 .6
0 .8
0
0 .2
0 .4
0 .6
0 .8
0 .4
0 .6
0 .8
.’6 0 ' 0 . 6 ' 0 )
oo
oo
o
d
to
to
o
<
o'
o
CM
CM
o
o
o
o'
0
o
0 .2
0 .4
0 .6
0 .8
0
0 .2
Figure 8.3: These figures show how the apparent magnitude of a supernova
at z=1.5 may be used to constrain the values of QA and Q m . VVe gener­
ate 2 percent error contours on the luminosity distance (dotted lines) and see
how they cut the CMBR degeneracy contour in four different fiducial universes.
From left to right and top to bottom the figures correspond to (flm, flA) =
(0.35,0.65), (0.35,0.45), (0.90,0.10), (0.60,0.60). The point corresponding to the fidu­
cial universe is marked by a triangle. Note that we can obtain a ~ 10% determination
of nm.
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181
0.5
0.0
-0.5
0.01
0.10
1.00
Red shift (z)
Figure 8.4: The magnitude-red shift relation may be an approach for distinguishing
A models (thick solid curve) from the family of quintessence models (dashed curves)
along the degeneracy curve. Am is the difference in the predicted magnitude of a
standard candle for a given model and an open universe (Qm —> 0, middle dotted
curve). The dashed curves are QCDM models with w — - 5 / 6 , —2 /3 ,—1 /2 ,—1/3
from top to bottom, respectively. Type 1A supernova data is from Garnavich [5].
For reference, an Q \ = 1 (upper dotted) and Qm = 1 (lower dotted) flat model are
shown.
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182
’ang-g.daf —
5
—
2 ....
.2 —
■5
—
u
E
OJ
«
w
Figure 8.5: 2% and 5% error contours for apparent magnitude(luminosity distance)
measurements made on a z = 1 supernova, with the fiducial sky chosen to have been
derived from a flat universe with fig = 0.6 and w = -0 .5 (marked with a circle
in the shaded region). Notice that the contours, while not entirely perpendicular
to the degeneracy curves of figure 7.4, may be used in conjunction with the above
mentioned CMBR degeneracy curves to set 10%-20% limits on the Q parameters.
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Chapter 9
Conclusions, and the significance of this work
For the last decade or so, experiments have been indicating a low m atter density
universe (fim < 1), suggesting the presence of missing energy or curvature in the
universe. The standard candidate for missing energy has been the cosmological
constant A; however there is no mechanism to explain why it has an energy* density
comparable to the current m atter density [10].
Quintessence is a dynamic, inhomogeneous and fluctuating component of the
energy density of the universe with a negative equation of state. This work is a
systematic study of quintessence that builds on the work done before on perturbation
theory in some scalar field models [41] and work on the unphysical smooth fluid (with
negative w and no fluctuations) models [13].
We used a scalar field to create a model for quintessence with negative equation
of state w but positive sound speed c^ below the scale of the horizon. Such a model
naturally arises in defect condensates and quantum field theories. We developed a
183
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184
formalism to describe the field and fluctuation evolution in quintessence in terms of
the equation of state.
VVe cataloged a large variety of these quintessence models into two broad cate­
gories: those with monotonic equations of state, and those with oscillatory equations
of state. VVe calculated the CMB anisotropy in such models by using the equation of
state formalism in Boltzmann codes in both the synchronous and conformal gauge.
VVe further showed that light field models that had monotonically changing equations
of state were well approximated by constant equation of state models in their field
and fluctuation history, and thus CMBR power spectra.
VVe showed that for constant w, the large wavelength quintessence fluctuation
evolution could be analytically obtained. Using these analytical solutions and our
numerical simulations, we found that the evolution of Q-field fluctuations were in­
dependent of initial conditions, unless the percentage fluctuation in the Q-field was
amplified by at least 104 times initial adiabatic values. This means that in the true
spirit of a model like inflation, fine tuning on the initial values of the fluctuations be­
comes irrelevant to the formation of large scale structure. Since this result is generic
to light field quintessence models, viable models obtained from particle physics are
assured stable and well-behaved fluctuations.
We used the equation of state formalism to describe in detail the physics of the
CMBR anisotropy at every epoch in the history of the universe. The large angle, or
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185
late-time anisotropy in quintessence models is particularly interesting. Fluctuations
in Q-field exert a direct effect on the CMBR anisotropy at late times(once the Q-field
becomes a dominant contribution to the critical density) by pumping energy into the
gravitational potential. This direct effect shows up as a suppression of anisotropy at
the highest angular scales on the sky (notch effect, see figures 6.10 and 6.4.)
We then turned to the comparison of observational results to predictions from
quintessence models. We showed that, unfortunately, for a large range of constant
w (with w < —0.5) models the effects of this direct fluctuation and the changes in
expansion history are not enough to distinguish these models from their cosmolog­
ical constant counterparts using the CMBR anisotropy alone. This is a result of
degeneracy in (Qmi w ,h ) space and the uncertainty in flm and h.
We finally showed that by combining the CMBR anisotropies with distance mea­
surements from high redshift supernovas, the degeneracy situation is somewhat ame­
liorated and it is possible to set loose constraints on w and fig.
The introduction of fundamental fields into the study of cosmology and the finding
th at the fluctuations on these fields do not retain memory of natural initial conditions,
has raised the hope that we can utilize the interactions of these fields to explain why
the Q-field component and m atter have comparable energy densities today. In-fact,
Steinhardt et. al[36, 57] have recently studied precisely such models in the context of
tracker fields in inflationary scenario’s and ekpyrosis. This adds new motivation for
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186
quintessence, as, provided the quintessence fields are light, such a scenario removes
the need of fine tuning in both the background field scale and the magnitude of the
initial fluctuations.
Much work remains to be done on the particle physics origin of such fields, and
on the physics of workable quintessence models in general. On the phenomenological
side, the effect of different equation of state histories on the CMBR needs to be cata­
loged. The most pressing work however, is on the observational aspects of measuring
missing energy, where a better grip on the systematics in experiments, and the in­
troduction of novel measurement strategies is needed to break the degeneracies that
exist in the CMBR anisotropy. If we are successful in such endeavors, there exists
the distinct possibility then, th at the answers to important scientific issues such as
the reason for the comparative values of m atter and missing energy density, and how
the formation of structure all came about from fundamental physics lie squarely in
the domain of observation!
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Appendix A
Com putational details on the Boltzm ann codes
A .l
A.1.1
Equations of motion in different gauges
Background Evolution
The cosmological models we consider in this work are spatially flat, FRVV space-times
which contain radiation, collision-less dust, and a scalar field. The space-time metric
is given by
(A.l)
where a is the expansion scale factor and r is the conformal time. The Einstein-FRVV
equation describing the expansion of the universe is
(A.2)
where / = d /d r. The sum over the index i includes the contribution to the energy
density by every component of the cosmological fluid: radiation, dust, and the scalar
field.
187
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188
The equation of motion, energy density and pressure are
(A.3)
(A.4)
(A.5)
We are familiar with this system from studies of inflation, where typically V is nearly
constant in Q. As a result, the energy density and pressure are dominated by the
potential, leading to De Sitter-like expansion. In the present work, we construct
models for which the energy density pQ comes to dominate the expansion at the
present epoch, with an equation of state
w q
=
p q /p q
€ [—1,0]. For a given potential,
the parameters wq and Q q specify the cosmological model.
A.1.2
Evolution o f Fluctuations
In this section we develop the equations of motion for the fluctuations in the scalar
field.
Synchronous Gauge Equations
We described and gave the interpretation of the synchronous gauge in section 4.3.2.
In this gauge, the scalar field fluctuation 8Q obeys the equation
6Q" + 2—6Q' - V 2SQ + a2V QQSQ = - h ' Q ' .
a
£
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(A.6)
189
Here, h is the trace of the spatial metric perturbation, as described in Ma &
Bertschinger [43]. Hence, we see that fluctuations 8Q will be generated in response
to the density perturbations through h'. Only if Q' = 0, as occurs for ACDM, will
the scalar field remain unperturbed. The energy density, pressure, and momentum
perturbations are
SpQ =
^ Q 'S Q ' + V,q 8Q
8Pq = ^ Q 'S Q ' - V q 5Q
(PQ + PQ){vo)i =
•
(A.7)
These quantities must be included in the evolution of h!, the total fluctuation density
contrast 8, and the total velocity perturbation v.
Conformal-Newtonian Gauge Equations
(The equations in this subsection are reproduced from Ma & Bertschinger [43])
The conformal Newtonian gauge or longitudinal gauge is a gauge in which the
perturbations of the metric are characterized by two scalar potentials rb and <f>which
appear in the line element as
ds2 = a2(r)[—(1 + 2 * )d r2 + (1 - 2<f>hijd^da^].
(A.8)
Here $ plays the role of the gravitational potential in the Newtonian limit.
In the conformal Newtonian gauge, the first-order perturbed Einstein equations
give
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190
+ 3^
=
47rGa25T°o(Con),
k2 (j> + f y
=
47rGa2(p + P)0(Con),(A.lO)
0 + -(4 ' + 2 0 ) + ( 2 - - ^ W + ^ - t f )
a
\ a a* J
3
=
^ -G a H T iiC o n ),
3
=
127rCra2(p +P)<r(Con)GA-12)
fc2( ^ - ' I ' )
(A.9)
(A.ll)
where “Con” labels the conformal Newtonian coordinates. The perturbed part of
energy-momentum conservation equations is
8
=
9 =
-(1 +
w) (d -
30) - 3^
- - ( i - 3w)$ - - ^ — 0 +
a
1+ w
- u;j
1+
w
6,
k26 - k2a + le2* .
(A.13)
'
v
The scalar field fluctuation 8Q obeys the equation
SQ" + 2 -8 Q ' - V 25Q + a2V QQ8Q = (3Q' + r ) Q ' - 2a2V V Q .
(A. 14)
CL
Again, we see the source on the right hand side of the equation above will cause fluc­
tuations in the scalar field to develop. The energy density, pressure, and momentum
perturbations is this gauge are
SpQ =
^ (Q 'S Q ’ - ^ Q a ) + V q8Q
8Pq = ^{Q '8 Q ' - * Q /2) - V q 8Q
{pQ + P Q ) M i = ~ ^ Q '{ 8 Q ),i.
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(A.15)
191
These quantities must be included in the evolution of Q,
the total fluctuation
density contrast 8, and the total velocity perturbation v. Although the form of the
equation for vq is the same in the two gauges, we caution that 8Q is gauge dependent,
so that the velocity perturbation, as well as SpQ and SpQ, has different meanings in
the two gauges.
A .l.3
Changing variables for im plicitly defined potentials
If we define Sil) = 5Q/ ^ 1 + w(i7) and 1p' = Q' / ^/l + w(r}), we can convert the equa­
tions obtained in the above subsections into a form in which the potential is implicit:
6ir
+ (4±
v
- V 2Sip
a
1 +
w q
'
- ^ ( 1 - w q ) [ ^ - - ( ^ - ) 2( ^ + ^ q ) ] ^ + 3 u ; q ^
’
=
synchronous
r
^ [ ^ ( 37(1 -
,
W
q
)
+
(A. 16)
+ 9 ' + 3Q']
conformal
The importance of the homogeneous terms in the equation can be clearly seen as
wq
—> —1. This is the reason why near-A models retain most memory of their initial
conditions.
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192
A.2
Formulae for constant w and tim e varying potential models
A.2.1
Constant Equation of State M odel
Here are some details of the constant equation of state scalar field model used in this
work. The derivatives of the potential, V’0)
specify the equation of state of the
fluctuations in the scalar field.
Choosing a fixed equation of state for the scalar field is a particularly simple
scenario, which we now describe. From the relationship between the energy density
and pressure, we may write the potential as
(A.17)
Since w is a constant in time, we may differentiate V and plug it into the equation
of motion (A.3) for Q. VVe find
(A.18)
where a* is an integration constant. Returning to the energy density (A.5), we set
the integration constant such that
PQ — W /a ) 2 — a 2(a/tt») {i+3u,<3) = Qqo, 3(l+u,)Po
(A.19)
where p0 is the present day total energy density. Thus, we have completely specified
the contribution by the scalar field, in equation (A.2), to the background cosmological
model.
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193
It is useful to look at the expression for the potential, though it is not used
explicitly in this model. From (A.17) we obtain the quantities
I
a2V QQ =
a
~ ( l - u / Q) [ ^ - ( ^ ) 2( £ + £«;<,)]
(A.20)
which are used in the evolution of Q and 8Q.
A.2.2
Time-Varying Equation o f State
The most general case is for a scalar field which obeys a time varying equation of
state. VVe may parameterize this time dependence in terms of the expansion scale
factor by
w q
(o ).
As in A .l, the potential may be written as equation (A.17). Defining
ip' = Q '/y/1 + wq, we find
da,
[■1 /■“•> fi.fl
2 Ja
Pq
= n QA , e x p [ 3 1 o g ^ + 3
i
J
+ *)]
(A.21)
^ i c Q(a)].
(A.22)
Thus, given the trajectory of the equation of state, it is straightforward to determine
the evolution of the background energy density. The derivatives of the potential are
given by
<A-23>
3,.
, ra"
/a\2 ,7
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194
1
r
«>o
wo
,d
77^ [4(1 +»,) " T +“'«7(3“''3+2>
]•
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(A.24)
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