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THE LARGE SCALE ANISOTROPY OF THE COSMIC MICROWAVE BACKGROUND RADIATION (INFLATION)

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8517051
S c h a e f e r , Robert Karl
THE LARGE SCALE ANISOTROPY OF THE COSMIC MICROWAVE
BACKGROUND RADIATION
Brandeis University
University
Microfilms
International
Ph.D.
1985
300 N. Zeeb Road, Ann Arbor, Ml 48106
Copyright 1985
by
Schaefer, Robert Karl
All Rights Reserved
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
THE LARGE SCALE ANISOTROPY OF THE
COSMIC MICROWAVE BACKGROUND RADIATION
A Dissertation
Presented to
The Faculty of the Graduate School of Arts and Sciences
of Brandeis University
Department of Physics
In Partial Fulfillment
of the Requirements of the Degree
Doctor of Philosophy
by
Robert Karl Schaefer
April 1985
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This dissertation, directed and approved by the candidate's
Committee, has been accepted and approved by the Graduate
Faculty of Brandeis University in Partial fulfillment of the
requirements for the degree of
DOCTOR OF PHILOSOPHY
Iteaff, Graduate School/ E/Arts
and Science*^
MAY 1 9 1985
Dissertation Committee
Laurence F. Abbott (Chairman)
% H A M i
ugh/N. Pendleton, if
David H. Roberts
Steve Rosenberg (Mathematic^)
ii
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Copyright by
Robert Karl Schaefer
1985
iii
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ABSTRACT
The Large Scale Anisotropy of the Cosmic
Microwave Background Radiation
(A Dissertation Presented to the Faculty of the
Graduate School of Arts and Sciences of Brandeis
University, Waltham, Massachusetts.)
by Robert Karl Schaefer
Although the predicted moments of the distribution of the cosmic
microwave background radiation in an inflationary universe have previ­
ously been calculated, the sensitivity of these moments to small changes
in the conditions required by inflation had not been explored. In this
thesis, formulae are presented for calculating the moments of the
background radiation due to an arbitrary perturbation in the gravitation­
al field of the early universe. It is found that only scalar (energy
density) and tensor (gravitational wave) perturbations are important for
this effect. Then numerical calculations are performed; first, to
confirm the previous inflationary calculations which require a critical
cosmological density and a Harrison-Zel'dovich scale invariant spectrum
of perturbations, and second, to compute the values of the moments in
universes with non-critical energy densities and different spectra (k* )•
In all cases we keep the assumption that the perturbations are caused by
some random process which follows a Gaussian distribution, as this seems
to be a reasonable feature of inflation to keep even in non-inflationary
universes.
The previously predicted values of the inflation induced moments are
confirmed, which imply that £.H< 4 * 1 0 _t and the value of the quadrupole
to dipole ratio must be two orders of magnitude smaller than the current
upper bound, unless gravitational waves are present. If this latter pos­
sibility is true then the moments higher than dipole must follow the
1-dependence predicted for gravitational wave perturbations. The values
of the moments were found to be sensitive enough to distinguish a k+l or
a k“* spectrum from a Harrison-Zel'dovich spectrum in a critical density
universe, (but not in a universe which had only a few tenths of the crit­
ical density). The moments higher than dipole for a Harrison-Zel'dovich
spectrum were not very sensitive to differences in the energy density but
the quadrupole-to-dipole ratio was. We find that the values of the mulipole moments are indeed a good test of the inflationary model. In the
event that an alternative model is found to inflation, the equations pre­
sented here are general enough to be used to calculate moments from this
alternative model.
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Acknowledgement
I am greatly indebted to Larry Abbott first for suggesting a prob­
lem which would have interesting results no matter what they were, and
second, for his personal kindness and patience throughout ray graduate
career.
I am also thankful to my family for the unfailing support they
gave to me for the duration of my seemingly endless stay in institu­
tions of higher learning.
I am also grateful to the Physics Department
secretaries, especially Mimi, for her patience and help in getting me
to this stage despite her ongoing balancing act in the Physics office.
iv
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TABLE OF CONTENTS
Page
1. Introduction
2.
1
The Sachs-Wolfe Effect and Moments of the Distribution
of the Cosmic Background Radiation
5
1. The Generalized Sachs-WolfeEffect
5
2. Gaussian Random Variables andMultipole Predictions
A.
8 Ta/T0
12
B.
Multipole Moments and Their Expectation Values
12
3. Time Evolution and Radial Dependence of thePerturbations
1. Scalar Perturbations
A.
15
17
8T0 /To
17
B. Evolution of € . ( T )
21
C. The Harmonic Functions Q (x)
22
2. Vector Perturbations
25
A.
8T0 /T0
25
B.
Evolution of
26
C.
The Harmonic Functions
O'
Q (x)
3. Tensor Perturbations
27
28
A.
8T0/T0
28
B.
Evolution of H^?(Y )
28
C.
The Harmonic Functions
Qj- (sD
4. Predictions of Moments
1.
11
30
31
Dipole and Quadrupole Bounds
31
A.
eu and the Dipole Moment
33
B.
The Quadrupole to Dipole Ratio
35
v
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Table of Contents - continued
Page
2.
Higher Multipole Moments
37
A.
The Harrison-Zel'dovich Spectrum
37
B.
£ H= X ^ tl
38
C. Gravitational Wave Perturbations
5.
Conclusions
39
41
APPENDICES
A.
Spatial Harmonic Functions
B.
Growth of Density Perturbations During the
Radiation Dominated Era
44
64
REFERENCES
70
BIBLIOGRAPHY
73
FIGURE CAPTIONS
76
FIGURES 1-25
78
vi
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CHAPTER 1
INTRODUCTION
In the past couple of decades the marriage of the fields of elemen­
tary particle physics and cosmology has proved to be a fruitful union.
Cosmology has provided some constraints on particle physics models unat­
tainable through other methods (e.g., the limit on the number of neutrino
species.1) As particle physics looks to a unification of the fundamental
forces of nature at energy scales far beyond the reach of particle
accelerators, physicists are increasingly appealing to cosmology for
direction.
An unexpected benefit of this research occurred when particle
physics returned the favor by providing a possible solution of many
cosmological puzzles with the invention of the inflationary cosmology.2-1*
This model of the very early universe solves in one clean sweep the
horizon, flatness, and monopole problems and also provides a natural
origin of the density perturbations necessary to form galaxies, and
eventually, us.
With all its theoretical successes, this model of the
universe is as yet observationally untested.
It is the purpose of this
thesis to explore a possible observational test of this theory through
observations of anisotropy in the cosmic microwave background radiation.
The inflationary cosmology leads to two predicted properties of the
universe which we will examine.
First, that the power spectrum of energy
density perturbations (and gravitational waves) is scale-invariant.5-8
This is known to astrophysicists as a Harrison-Zel'dovich spectrum after
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2
the authors who pointed out the need for such a spectrum long before the
invention of the inflationary model.9>10
Second, the energy density of the
universe must be equal to the critical energy density which is the value
critically between an open and closed Robertson-Walker universe.
We will
use the standard variable for characterizing the energy density fl = p/ptr
(energy density)/(critical energy density), which equals 1 for an .infla­
tionary universe.
Attempts have been made to measure £ \ directly which
provide a narrow range of values for IT , 0.1 <£l< 4.
worth noting here:
Two points are
first, /I = 1 cannot be completely accounted for with
normal baryonic matter.
In fact another result from the crossfertilization
of particle physics and cosmology is that the total energy due to baryons
must be less than the critical energy density by a factor of 1/5-1/711 in
order for primordial nucleosynthesis to produce the observed light element
abundances.
Thus the inflationary cosmology (or any cosmology with jfl >
0.2) requires that the energy density of the universe be dominated by
some non-baryonic component.
This assumption is not unfounded as some
type of "dark matter” is needed to explain observed properties of
galaxies.12
The best candidates for this non-baryonic matter are those
which fall in the category of cold dark matter (e.g., axions,13-15 photinos,15*17 etc.).
The second point worth mentioning is that measure­
ments of/I from clusters leads to /I £ 0.5 in clusters18-20; hence,
although we will not use this assumption explicitly, we point out that
for values of fl > 0.5 we assume galaxies are not good tracers of the
mass distribution of the universe.
This assumption has some supporting
evidence in that it seems to help explain the velocity correlations of
galaxies in clusters.21
In the inflationary cosmology perturbations are produced in the early
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universe on all scales according to the Harrison-Zel'dovich spectrum.
The long wavelength density and gravitational wave fluctuations22-25 will
produce gravitational redshifts of photons in the background radiation
leading to anisotropies in the temperature distribution of these photons.
The values of multipole moments of the temperature of the photon dis­
tribution have been previously calculated and thus provide a test of
inflation.22-27
The question is, how good a test is this?
words, can we tell that f\ = 1 and not say 0.2?
In other
Can we tell if the
spectrum is really a Harrison-Zel'dovich spectrum and not something close
to it?
In chapter 4 of this thesis we will try to answer these questions
by comparing the predictions of the inflationary cosmology with those for
other spectra and for open and closed universes.
One key element of the inflationary cosmology will be kept in all
cases considered— the assumption that the source of the perturbations is
a Gaussian random process.
is separately testable.28*29
It is interesting to note that this hypothesis
We include in our analysis the effects of
this uncertainty on the final moment predictions to see under what con­
ditions the inflationary predictions can be distinguished from those of
different spectra and non-critical energy densities.
This element of our
analysis is unique and has not been used in previous anisotropy calcula­
tions for universes with spatial curvature.30-32
We will explain the
implementation of this assumption in chapter 2.
Also in chapter 2 we
present the derivation of the general formula for finding fluctuations in
the temperature of photons from an arbitrary perturbation in the metric of
a Robertson-Walker universe (the Sachs-Wolfe formula).
Any arbitrary perturbation in the metric (and hence the gravitational
field) can be decomposed into 3 types according to their behavior under
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4
spatial transformations in the background space-time: scalar, vector,
and tensor.
In chapter 3 we use the elegant gauge-invariant formalism
of Bardeen33 for treating each specific class of perturbation in the
Sachs-Wolfe formula to find the effects these fluctuations have on the
microwave background radiation.
Finally we will draw a few conclusions
from the small catalogue of moment predictions displayed in chapter 4.
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5
CHAPTER 2
THE SACHS-WOLFE EFFECT AND MOMENTS OF THE DISTRIBUTION
OF THE COSMIC MICROWAVE BACKGROUND RADIATION
The main effect of small energy-density perturbations on the cosmic
microwave background radiation is the gravitational redshift they produce.
This redshift has become known as the Sachs-Wolfe effect after their
original work for a flat Robertson-Walker universe.31* Since this is the
main source of anisotropy in the cosmic microwave background we present
the full derivation of the generalized Sachs-Wolfe formula in section 2.1
for all Robertson-Walker-Friedmann universes.
In section 2.2 we show how
we define multipole moments from the Sachs-Wolfe effect.
We also explain
how we implement the assumption that the fluctuation amplitudes are
gaussian distributed.
2.1)
The Generalized Sachs-ffolfe Effect
A well known consequence of General Relativity is that the
distribution of matter-energy in the universe determines the geometry of
space-time.
Different distributions will be associated with different
metrics describing their manifolds, and thus their respective geodesics
will also be different.
If we consider small perturbations in the matter-
energy density of the universe we must expect small variations in the
geodesics as well.
This fact implies that photons reaching us from
different directions in a perturbed expanding universe may have redshifts
slightly different from the standard redshifts resulting from these (null)
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geodesic variations.
This phenomenon is known as the Sachs-Wolfe effect.
The original work of Sachs and Wolfe considered perturbations only
in a "flat" Robertson-Walker universe.
Their work was extended by Anile
and Motta35 to include open (K = -1) and closed (K = +1) Robertson-Walker
metrics, although they worked in a different gauge than Sachs and Wolfe.
Of course, the final formulae will be gauge invariant and it doesn't
really matter which gauge is chosen, provided one is careful enough to
completely specify the gauge.
For completeness we will present the
derivation of the Sachs-Wolfe formula for all three cases (open, closed,
and flat) in the Sachs-Wolfe gauge.
The choice of gauge will become
irrelevant when we rewrite the formula in terms of Bardeen's33 gauge
invariant variables.
This will also eliminate any worries of unphysical
"gauge mode" solutions36-39 appearing as solutions to the perturbation
equations.
We will find it to be convenient to use the conformal time 'f defined
by
where R(t) is just the usual Robertson-Walker scale factor which describes
the general expansion of the universe with time.
This gives us the
(unperturbed) Robertson-Walker metric
(2 .1 .1)
where
= (1 + Kjjj*), x° = 'X , and:
K= +1
K=
0
K= -1
closed universe
4 ^
flat universe
open universe.
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7
At this point we also note that latin indices are summed from 1 to 3 and
greek indices are summed from 0 to 3 and that c = 1 in our units.
We now wish to perturb the metric so that
and we remember that /)£='£, not the usual time, t.
We will assume that
the universe is a perfect fluid with a four velocity U 01 . We will choose
our coordinates so that the fluid is at rest (U *■ = 0) and also choose
h oo = 0.
These four conditions constitute our ( and Sachs' and Wolfe's )
choice of gauge.
The restriction U* U,* = -1 now means that
C /s e n
We must now find the null geodesics in our perturbed space.
These
perturbed null geodesics will give rise to an anisotropy in the cosmic
background radiation.
To make the task simpler, we only need to find the
null geodesics of d s l , where
J U *
=
S
\ r ) d A
K
because the geodesics of two conformally related spaces are parallel.
The affine parameters describing geodesics in these two spaces do not nec­
essarily coincide,35 but they are related.
p*
So, for example, the vectors
tangent to the null geodesic in the full space are simply related
to the tangent vectors "p*4 in the barred space (specified by dl*' ) by
J3
=
b
f J
(2.1.3)
Using y as our affine parameter along the null geodesic in the barred
space, the tangent vectors "p * are just
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The equation for this null geodesic is
=
<ty,
O
After performing the variation and doing an integration by parts we get
the Euler-Lagrange type differential equation
which we will solve perturbatively.
We start by breaking up the tangent vector pT* into the tangent
vector pT* for the Robertson-Walker background and a small perturbation
‘p < :
—
■f
A')
(2.1.4)
-4- JF)*
- f J0>) + f m
To zeroeth order the geodesic equation is
fw jr
x
3 ^ f&> f '«
or rewriting
dq
+ a t** I f w +
tv,*)fl ft4 = 0
Instead of finding the general solution to this equation we will solve only
for radial geodesics.
If we put ourselves at the origin of our coordinate
system (r = 0), then the only photons which we can observe from external
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sources must travel along radial geodesics.1+0
The tangent vector to these
unperturbed radial geodesics is easily found to be
( - M s)
where e
<2-1,5a)
is the unit vector in the radial direction "e
= r*/|?|.
We
then get
_
dr
,
<h
_
Jbo
v
If we arrange y=0 to be the origin at the point of observation ( /t'o >0»0>0)>
then
t= r. - r
K=-|
C2.taMl(%)
< 2 - 1 ' 5 b )
K= o
k = +i
/L = )
(_ a t « / > v ( H O
Note that because we are considering photons received at the origin dr/d
is negative.
MW
We can use the zeroeth order solution p, . to obtain a first order
equation for p*4 :
CO
or:
-
JP,!’
±
l
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<2.1.6)
10
We will only need the zeroeth component pc°
[
dr A d
f^)
If we define "e^( o)= 0, the solution of (2.1.6) is
<?M0
o
a
3r
Z.JL Q
JT\P
(2.1.7)
f< » - f®
We will use this solution to find the redshift Z from the relation1*1
(2 .1 .8 )
Quantities to be evaluated at the time of emission %
will be denoted by
an e subscript and quantities to be evaluated at observation time 7*o will
be denoted with an o subscript.
Using eqs. (2.1.2), (2.L.3), and (2.1.4)
in eq. (2.1.8),
+z =4*
!\S
'e
is
-Se
—
f c o
Also using eqs. (2.1.5) and (2.1.7),
-%'X r
l+Z
(2.1.9)
l+*i
This is our generalized Sachs-Wolfe formula.
It tells us how the redshift
is related to perturbations in the metric, which in turn are related to
the state of the matter-energy density of the universe through Einstein's
field equations.
Our next task will be to rewrite equation (2.1.9) in
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11
terms of Bardeen's gauge invariant variables.
We can then use this
generalized Sachs-Wolfe formula in the determination of the temperature
distribution of the cosmic microwave background radiation.
We will do this
by predicting the expected values of multipole moments of the distribution
of the background radiation.
We will also assume the perturbations are
caused by some Gaussian process.
This assumption of gaussian fluctuations
will have to be put into the boundary conditions of the perturbation
equations, so we will first examine the implementation of the gaussian
assumption and how 1l will affect the prediction of multipole moments,
then we can fully recast equation (2.1.9) in terms of Bardeen's variables
with the proper boundary conditions already enforced.
2.2)
Gaussian Random Variables and Multipole Moment Predictions
A key element in the analysis presented here is the assumption of
a Gaussian distribution of the perturbation amplitudes.
The inflationary
cosmology in fact does not predict the amplitudes of the fluctuations,
but rather gives the width of a Gaussian probability distribution for the
perturbations.
We will include this Gaussian probabilistic description in
our non-inflationary predictions as well, because there is evidence to
support this assumption in the observation of the correlations of rich
galactic clusters.28-29
The uncertainty in our predictions will be realized by making the
fluctuation amplitudes proportional to a Gaussian distributed random
variable.
We will examine this in more detail once we explain how the
moments are derived.
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12
2.2A)
8Te /Te
We would like to predict the observationally relevant quantity
S T6 /T0 , the fluctuation in the observed temperature of the photon
distribution.
These variations can come from two sources: a) the Sachs-
Wolfe effect and b) fluctuations in the density of matter when photons
are last emitted (i.e., recombination, or photon decoupling time).
Any perturbation in the density of matter which exists just before the
photons decouple from matter will be subsequently have its image "frozen"
in to the photon distribution at decoupling, which for long wavelengths
can be taken to be an instantaneous process.1*2
perturbation
recombination.
STg
This causes a
in the average temperature of photons Te
This initial temperature Tg
+
8Te
at
distribution gets
further distorted by the gravitational Sachs-Wolfe effect.
Thus the
temperature observed today is
(2 .2 .1 )
where 1+Z is given by eq. (2.1.9).
Equation (2.2.1) is our master
equation for predicting fluctuations in the temperature of the photons
today.
2.2B)
Multipole Moments and Their Expectation Values
We can manipulate eq. (2.2.1) to get an expression for 8 T& /T0 .
Once this is done, we can expand
STj, /T0
in terms of the spherical
harmonics to get the amplitudes of the various moments
(2 .2 .2 )
The Gaussian nature of the perturbations is taken care of by making the
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13
coefficients a.
Xrrr>
proportional to the random variable
unfortunate choice of letter, but this
the literature.)The/a^/m(k)
sl.
JLrm
(k) (an
is the notationappearing
parametrize the amplitude
individual fluctuation mode is excited.
in
to which each
The expectation value of
satisfies
f
k = -|
( 2 - 2 - 3)
$
where k
=
k=+i
- (R+1)K, R is the rank of the type of perturbation
under consideration (scalar, vector, or tensor), and
p
>
0
/3 =
K =
0 , - 1
K =
+ 1
Our final result will be to predict the rotationally invariant
quantities
I
(2.2.4)
rn\- ~x
which we can now do using eq. (2.2.3).
o
the expectation value of a^
Jim
We can think of eq. (2.2.4) as
averaged over an ensemble of universes.
We can determine a probability distribution for the values a^ , namely
p
f.
\_
f i r
!__________ { Q a L 1 * 3 -S’— ( a l W ) ( ( £ ) ^ +l C
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(2.2.5)
14
where
0"
r
JL
(2.2.6)
QJL+1
Using this probability distribution we can determine "error bars”
corresponding to one standard deviation which we will present in the
results section.
This is an important feature in that the error bars
will tell you whether a given set of multipole moments for one model can
theoretically be distinguished from another set, regardless of how small
observational errors can be made.
We see that we are really expanding
the fluctuations in terms of the modes £^^(k)
We have now
shown how to incorporate the Gaussian assumption into predictions of the
multipole moments of the temperature variations in the cosmic microwave
background radiation.
This leaves only the radial dependence and the
(conformal) time dependence which will be discussed in the next chapter.
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15
CHAPTER 3
TIME EVOLUTION AND RADIAL DEPENDENCE
OF THE PERTURBATIONS
The most elegant method for treating cosmological perturbations is
the gauge invariant formulation of Bardeen.33
In this chapter we apply
this formalism to solve for the time evolution of these perturbations.
Bardeen's approach uses an expansion in spatial harmonic functions to
describe the purely spatial dependence.
In curved space this is the
equivalent of a Fourier transformation of the Einstein equations, which
then yields an ordinary differential equation to solve for the time
dependence.
The solutions of the time evolution equations will be
presented here, but we only show the pieces of the spatial harmonic
functions which are necessary for our calculations.
The complete spatial
harmonic functions along with some of their properties will be presented
in gory detail in Appendix A.
At various points in the derivation we will also need to know how
the Robertson-Walker scale factor depends on the conformal time 'Y. The
Friedmann Equations in terms of 'Y become
(3.1)
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16
where /O and P are the background energy density and pressure, and a dot
indicates a derivative with respect to '7'.
Since we will be concerned
with times later than the time of recombination, which is later than the
time of matter domination, we will need the solution to these equations
for P = 0, which is
(3.2)
AX/r^
where ^
and S 0 are the values of the present energy density and scale
factor.
Perturbations in various quantities can be classified as scalars,
vectors, and tensors, according to how they transform under spatial
coordinate transformations in the background spacetime.
The spatial and
temporal evolution of the three types will be governed by a separate set
of equations for each type.
Any completely general perturbation of the
gravitational field can be written as a linear combination of the three
types, with each type evolving independently of the others.
Therefore,
our analysis will be broken into three sections, one for each class of
perturbation.
It is also worth mentioning that the equations given here
are valid only for adiabatic perturbations.
In each of the three
sections we will first rewrite the expression for
/T0
the Bardeen variables and the appropriate spatial harmonics.
in terms of
Then we will
describe the functional form of the evolution of the Bardeen variables
and the spatial harmonics.
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17
3.1)
Scalar Perturbations
3.1A)
STb /T0
Scalar perturbations are characterized by the scalar spatial
harmonic functions Q (1?) which satisfy the Helmholtz equation
D^Gt + l l Q - 0
where D
3.
= D
t
D'
(3.1.D
and D ^ is the three dimensional covariant derivative
in the spaces of (2.1.1).
We will write Q (3) in the form
Q(#)=r
( 3 - i - 2)
j-)/*i
and so we will label individual modes by k, 1, and m.
For each mode we
define perturbations using Bardeen's notation
X, ■ - B W Q iU )
-
(3.1.3)
aHLWQ(?)f., +aHT(T)Q,I»)
and the perturbations in a (perfect fluid) stress energy tensor with
adiabatic perturbations are given by
ST>S j° = (pto+
p
with
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<3.i.4)
18
(3.1.5a)
(3.1.5b)
where 6 ^
is the three space metric in eq. (2.1.1).
variables A, B, HL , HT , and 8
In terms of the
Bardeen defines the gauge-invariant
combinations:
0l A =
m
HL+ iH T+i. (l)^'je(f)H t
o
-1- u
£
M
=
=
5+
Vs = V "
(3.1.6)
)N -b)
±Hr
Our goal here is to rewrite eqs. (2.2.1) and (2.1.9) in terras of
these gauge-invariant variables.
We first note that the temperature
fluctuation in the emitting plasma previously mentioned in section 2.2A
can be written as
(3.1.7)
where ^ 0 ^ =
since
is proportional to T*
Using this and
the definitions of eq. (3.1.5) in eq. (2.2.1) we find
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19
S L
c r\ I
- 1
- fV* 0
(3.1.8)
where the perturbations are evaluated at values of T
terms of y by eq. (2.1.5b).
and x given in
Using the identity for an arbitrary vector
V;
jf J
y '
L
0 ' vl
(3.1.9)
along a radial null geodesic and the fact that
4
= - &
+ jrei^
from eq. (2.1.5b), we can rewrite eq. (3.1.8) as
(
¥
■ “i
q p
- Ai Mf
I r
R \ ~
W -T S -T e .
+
(3.1.11)
"i-
-V7c
L
'0
3
H
X
t
A i'TJ
-r ^t n
q
1/
The first term in (3.1.11) only contributes to the raonopole moment of
S Te /Tp
so we will absorb it into our definition of T0 . The next step
is to re-express (3.1.11) in terms of the gauge-invariant variables of
eq. (3.1.6).
Here a slight complication arises.
energy density fluctuation 8^ appears.
gauge invariant variables
and v|
In (3.1.11) the photon
Therefore we must introduce
in analogy with (3.1.6),^**
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20
P
(3.1.12)
v
>
s= v „ ' i h t
Eq. (3.1.11) is valid in the gauge A =
in Bardeen's notation.
=0, the Sachs Wolfe gauge
From this we can rewrite ST0 /T0
using gauge-
invariant variables as
ST.
/I
£a+ T ' a ( ^ +(j)v/))q
11
X
*i(v,wv,vaOQi’:r
-f
-
(3.1.13)
i k
We now use the evolution equations for the gauge-invariant variables1*1^ 5
which come from energy and momentum conservation and from the Einstein
equations.
For a perfect fluid in a matter dominated universe these give
<|a = - | H
(3.1.14)
i
l u s
A
v x
= t [ (dk K)/(X-3K)] (e - 1e
A - s K ^
This leads to an expression for
/T0
involving £ y as well as the
total energy-density fluctuation variable £ . However, for the long
wavelength modes which dominate the multipole moments we will evaluate,
€ y is proportional to £ , £ y = 4/3£ . This can be verified using the
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21
multifluid extension of Bardeen's formalism1*1**115 and is a result of the
fact that these long wavelength fluctuations are more massive than the
maximum Jeans mass.
Therefore, we can write (3.1.11) purely in terms of
£ as
I^ = r e-r0
Tc ~ J C - 3 K 'I
I
+
Q '^=0
(3.1.15)
4-3
To obtain our final results we must sum and/or integrate over the mode
variables k, 1, and m and project out the appropriate moment.
We do this
numerically and present the results in chapter 4.
3.IB)
Evolution of £ (T)
The gauge invariant variable £
Is
€
-
obeys the evolution equation33
0
(3.1.16)
This can be most easily solved by writing (3.1.16) as a function of S/S
which in turn is a function of T •
Defining
W= f
(3.1.17)
eq. (3.1.16) has the solution
( r ) < K ^ M
= V / ( w a+ K )
which is a shrinking mode and can be ignored.
(3.1.18)
The growing mode is
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22
J v y 3’
| w
( w
a- i ) £ u d M i A ( w )
k
i(w)- < w
^
l+ I
= - i
|^= 0
| W(w^-t) Oa A
a u
We will characterize the amplitude of £
of horizon crossing (i.e., when S/S = k).
(3.1.19)
K~+l
(vJt)
by its value at the time
Thus we write
(3.1.20)
where aA/m is the Gaussian random variable satisfying (2.2.3).
of a
The role
now becomes clear; while the Einstein equations specify how each
fluctuation mode must evolve in time, they say nothing about the initial
amplitudes of the modes.
The a ^ are a way to enforce that each mode
is initially excited to an amplitude which follows a Gaussian random
probability distribution, whose width scales as €^.
Since each mode
independently follows this distribution there can be no correlation
between modes.
This is the information of eq. (2.2.3).
ATI' is introduced so that
The factor of
will agree with the definition commonly
used in inflationary cosmology.5>2 7 3
In the case of inflation
is
a constant but for non scale invariant spectra it is a function of k.
The expression (3.1.20) is to be substituted into eq. (3.1.13) to obtain
3.1C)
The harmonic functions Q(x)
The harmonic functions QO?) appearing in (3.1.11) have been expressed
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23
.1 .2 1 )
J{/m
The radial functions
are given by the following expressions.
For
K = 0
<jvV) =
jjiiM)
(3, 1 .22 )
where j^(kr) is a spherical Bessel function which satisfies the
orthonormality relation (no sum on 1)
{
jx ( M p (Jm
The radial functions
) =
§
(A - M )
«•
1.23)
(j)^ for K ^ 0 have been given by Harrison.1*6 We
have changed the normalization slightly to make their orthogonality
properties more like those of j^(kr).
—
For K = -1
--- ' r - v - W
(3. 1.24)
where Pj£ is an associated Legendre function,
A y
1.25)
and
P
A
/n = o
In analogy with (3.1.23) these functions satisfy the orthonormality
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24
condition (no sum on 1)
,-£ & ). * V ) 4 » ,V ) = g i 5 ( ^ )
Further properties of
c .1.26)
(j)^ are given in appendix A.
For K = +1
(tr*r
/ ^
V ‘—
z6
D
c .1.27)
QjcA-f^
-J- + /S
with
=
=
W&F
c .1.28)
/s = 3 ^ ^ - -
and
n?=
i if-*')
These functions satisfy the orthogonality relation (no sum on 1)
.
co
f
i
f
e
*> ) ^
v
Further properties are also given in appendix A.
.1.29)
Note that in any case
k and ^3 are related by
.1.30)
The orthonormality condition can then be summarized as
y£Jtb- .
J (I+Kff ^
Of
space
_ Tc, f
K=0,-1
^ W -2/3°
.1.31)
^
K=-H
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25
3.2)
Vector Fluctuations
3.2A)
&T0 /Te
We now repeat the analysis of section 3.1 for vector fluctuations.
These are the least physically interesting as they can only shrink with
time.
They are presented for completeness.
The perturbations are
characterized by a vector harmonic satisfying
p a Q(p + j w =
o
and
D 1Q
f
=
o
Defining
Q i f -
su* (Pi Q f +
Dj
Q i ' )
(3-2-2)
we can write the fluctuation hyuy in the form
Ju= o
(3.2.3)
X
,
=
Jiij =
- B 0)( r )
2
Q .‘ < W
H?(r) Q?j &
)
and perturbations in the stress energy tensor as
5 V = 51/= o
(3.2.4)
&T-“=if’
b>))q(;Y^)
We next rewrite eqs. (2.1.9) and (2.2.1) in terras of these variables to
obtain
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26
STa
To0
0
”
<4[i
Q'i’e‘e3-*B<0Q"el]
Integrating by parts and using the identity (3.1.9) this becomes
a ,ji
$
[ta r-
(3.2.6)
For vector harmonics Bardeen33 defines the gauge-invariant variables
i(I)
_L
|/- BU
)-j
1
/j CO
Hr
(3.2.7)
Vc - v l° - B0>
and
However in the gauge we have chosen (V^ =0), so eq. (3.2.6) can be
rewritten in terms of the gauge invariant variables as
-j
lo
3.2B)
(3.2.8)
Evolution of Perturbations
For a pressureless, perfect fluid the gauge invariant variables andy
V^.
satisfy the evolution equations
1
tf-a-K. y
=
p\Jc
s
(3.2.9)
and
V.. = ' j Vc
The second equation can be solved immediately to give only a shrinking
mode
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27
Vc
°<
(3.2.10)
ys
There is no growing mode for the vector perturbations.
There is also no
growing mode in a radiation dominated universe.
3.2C)
Vector Harmonics
Because the unit vector ~e appearing in eq. (3.2.8), which points
from the observer out to the source, is in the radial direction we only
need the radial component of the vector harmonic 7$^ .
are given in appendix A.
The radial component
Other components
is given by
(3.2.11)
where
(3.2.12)
The functions (j)^
and the variable Y are as defined in section 3.1C,
with P> now given by
(3.2.13)
for K = 0, ±1.
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28
3.3)
Tensor Perturbations
3.3A)
8T0 /T0
caO
Tensor perturbations are characterized by tensor harmonics Q;.
1
satisfying
+
o
Qf] = Q'tl
D 1q
= o - Q ?1
^
For tensor perturbations the fluctuation variable of relevance here is
Ht
as
A.. - A . i
-
o
and
st:
(a)
- o
is already gauge invariant so we immediately obtain a gauge
invariant form for
CT
Q lo
C
\
Ye ~ ~ J
3.3B)
H
WO
&TC /Te
.
n
ci^,
from eqs. (2.2.1) and (2.1.9),
I iCT) /*\ Ctt
Ht
i
!
Q4 e ei
C3.3.3)
Evolution of HL*
satisfies the equation (for a perfect fluid)
+ ( i) f i r + (a i i +
H r r °
For a matter dominated universe we have a shrinking mode solution
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(3.3.4)
29
Ht *• ^
(2
)
which can be ignored, and a growing-mode solution
(3.3.5)
K N y ^ f ^a|&(Z)
where
(j>1 are the scalar radial functions given in section 3.1C,
>
a=
(3*-3K
(3.3.6)
and Z is given by
K = - 1
2/
a
Zr
c <h l
K ~
(3.3.7)
o
£
As in section 3.IB we characterize the size of the fluctuations at the
horizon crossing time by dividing out the value of
S/S = k.
We will also use an arbitrary constant H h
at
such that
to get the form
H„/^S
(3.3.8)
where
(3.3.9)
Ail) =
(zM
T such iliod
§■-jfz.
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30
3.3C)
Tensor Harmonics
As in the case of the vector harmonics, we will only need the radial
component of the tensor harmonics since "e-in eq. (3.3.3) points in the
radial direction.
This component can be written as
(3.3.10)
where
K=-)
(3.3.11)
K= o
N?=
K=+ l
and <j>* is as given in section 3.1C where now /S is
K~=
(3.3.12)
/3a - 3 K
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31
CHAPTER 4
PREDICTIONS OF MOMENTS
4.1)
Dipole and Quadrupole Bounds
In the preceeding sections we have derived formulae for calculating
the 1 = 1 , 1 = 2 , and higher moments of the distribution of the cosmic
microwave background radiation by using the appropriate functions in
eq.
(2-2.1) for 8Te /Te . We would like to examine some physically
interesting cases and see what information we can first get out of the
predicted dipole and quadrupole alone.
Remember that the amplitude of our density fluctuations was
proportional to £ H (see eq. 3.1.20).
If inflation provides the correct
picture of the early universe, we could predict the value of €.H if only
we knew the correct unified field theory of elementary particle physics.
Unfortunately we cannot say what the correct unified field theory is, so
we do not know a priori what £ h
what limit the dipole imposes on
is.
Abbott and Wise1*3 have calculated
in a purely inflationary universe.
The dipole has also been considered by others1*7~l
*9 in a similar manner.
Here we would like to consider the more general cases in which we have
initial conditions similar to those naturally provided by Inflation, but
are produced by some now unknown mechanism.
A Harrison-Zel'dovich spec­
trum (or a spectrum very close to it) of Gaussian density perturbations
is required for successful galaxy formation whether or not inflation is
correct.12
If we could extract Information about the value of the energy
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32
density jfL (=
or about deviations from a Harrison Zel'dovich
spectrum of perturbations we could observationally test the inflationary
universe, perhaps excluding inflation from models of the early universe.
We believe such is possible and we will discuss this more after first
presenting bounds on
.
For a fixed value of £ H , the predictions for the size of the of the
dipole and quadrupole moments vary with SI . We can thus use the moments
to give us information about
as a function of X I .
In order to make
our results even more general, we will consider small deviations from a
Harrison-Zel'dovich Spectrum by parametrizing
as
where n = 0 gives the Harrison-Zel'dovich spectrum and X is a constant
to be evaluated when the horizon size is equal to the wavelength of the
mode under consideration.
(The letter X is used to signify that it is
the amplitude when the fluctuation crosses or "Xs" the horizon.)
In
non-scale-invariant spectra the value of X becomes proportional to some
scale length to the power n/2, X
sionless.
(size) ^ , so X is no longer dimen-
But if, as these spectrum deviations require, inflation is
not the correct model of the universe we don't know how density pertur­
bations can form in the first place, and so the power law spectrum for
k is just a guess that requires minimal information, and also has been
used in models of galaxy formation.
We will now look at the bounds on X
imposed by the measured anisotropy of the microwave background radiation.
We can obtain a weak bound on X from the present limit on the
quadrupole moment50*51
<
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33
This gives a 90% confidence bound on X
)
X ^ (fl/euj X | 0
for -1 < n < 1 and 0.05 <
XI < 2
(see figure 1).
However, we can get
much stronger bounds by considering the dipole moment.
This is not as
straightforward as imposing the quadrupole bound and we will discuss the
complications in the next section.
4.1A)
and the Dipole Moment
Unlike the other moments of the background radiation distribution,
the dipole is sensitive to short wavelength fluctuations.
This is most
easily seen through the fact that the value of the dipole depends on the
velocity of the observer.
In fact the bulk of the observed dipole5°>51
seems to be due to infall velocity of the Milky Way galaxy.52
Thus we
cannot simply set our predicted dipole equal to the observed value of the
dipole and determine X, because the short wavelength fluctuations have
grown too big to be treated by linear perturbation theory.
Calculating
the behavior of these short wavelength modes is quite complicated and
model dependent, so we have adopted a different strategy.
We have cal­
culated the contribution of only the linear modes to the dipole moment.
Since the modes are uncorrelated the total dipole moment must be larger
than just the contribution from linear modes.
Thus we can demand that the
value of X must be small enough to make the linear contribution at least
as small as the total observed dipole moment.
Of course, the possibility
exists that the linear part of the dipole is anti-aligned so the two
effects could cancel, but the probability of this occurring is very
small, much smaller than the probability that the dipole is outside our
90% confidence range.
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34
We must cut off our integral over modes of different wavenumber at
some ky^ in order to prevent including non-linear modes.
The largest
scale of structure in the universe seems to be the supercluster scale
which is the order of 60 Mpc today, so we will pick our kmax correspond­
ing to a wavelength of 60 Mpc today.
bound on X.
This will give us our best possible
This kp,^ corresponds to a value of 1 for the expectation
value of the function
of Peebles53 for a flat universe (K = 0)
and a Harrison-Zel'dovich spectrum.
As an added complication we find that a fluctuation of wavenumber
kmox crosses the horizon (wavelength = horizon size) while the universe
is still dominated by radiation (not matter).
The growth of perturbations
during this phase is very different from that of the matter dominated era
A check of the equations for multiple uncoupled perfect fluids4*1* in non­
flat Robertson-Walker universes reveals that we areconsldering values
of XI for which we can use the equations for K = 0 to an accuracy of 1%.
Thus we can use the treatment of Abbott and Wise1*3 to describe the
behavior of 6. in section 3.1.
The technique used by Abbott and Wise,
along with a check of the accuracy of using the (K = 0) equations is
presented in Appendix B.
Using our 60 Mpc cutoff then we get the bounds
on X shown graphically in figures 2, 3, and 4.
As can be seen from figures 2, 3, and 4 the dipole moment provides
quite a strong limit on X.
n = 0, we see that X
For example, in the inflationary case,_Q. = 1,
must be less than 3.6 x 10
is just barely big enough to allow galaxy formation.
the dipole c< k mftx, so our upper bound on € ^
of inflationary models, the value of £ h
(see figure 2), which
Note that forXI~l,
In the context
is significant in that it is not
easy to arrange for a potential in a particle theory which can both pro­
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35
vide for a successful inflationary phase and predict such a small value
of
Another popular set of parameters is n = 0 and Xh = 0.2, for which
must be less than 5.4 * 10
allowed by inflation.
, a value about twice as big as that
If one wants to make galaxies in his/her favorite
early universe scenario, figures 2, 3, and 4 provide information about
the initial amplitudes of the density perturbations.
While we have provided some constraints on the conditions of the
early universe, we have not learned anything about the value of n
or n.
Next we will consider the value of the ratio of the quadrupole to dipole
moments which may in fact give us such information.
4.IB)
The Quadrupole to Dipole Ratio
An annoying feature of the previous discussion of results is that
the predictions of the moments are proportional to an unknown constant X.
We would like to get information from the cosmic background radiation
which is independent of the value of X.
By dividing the quadrupole by the
dipole we can cancel the factors of X in each individual moment.
By the
same reasoning used in the previous section, we see that our predicted
value of the quadrupole to dipole ratio (Q/D) must be larger than the
observed value of Q/D because we have calculated only part of the dipole
moment.
Since we have at present an upper bound on Q we have only an upper bound
on Q/D (see previous discussion of quadrupole bound on X).
Thus all we
can say now is
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36
As the measurement of the quadrupole gets more precise, the value of
(Q/D)|obs will decrease.
When the quadrupole measurement is accurate
enough to yield a positive result, we will be able to rule out all
theories with a predicted (Q/D)^
< (Q/D)^ . The results of our
calculations are presented in figure 5.
The curves represent the 90%
confidence upper bounds on the values of these ratios.
When the
quadrupole is found the observed value of Q/D will be represented by a
horizontal line, which will eliminate all theories with values of Q/D
below that line.
For convenience we have multiplied the predicted Q/D
ratio by the observed value of the dipole so that the value of the
quadrupole can be read directly from the vertical scale on the right
side of figure 5.
Looking at figure 2 we can see that in order for inflation to
survive, the observed value of Q ( = <(aa)a'> ) must be over 2 orders of
magnitude smaller than the present upper bound, not a pleasant thought
for observers.
If the quadrupole is found very near to its present
upper bound, however, this could spell death to both the inflationary
models and the Harrison-Zel'dovich spectrum.
The only way for Inflation
to evade this bound would be if copius amounts of gravitational waves
are produced by the inflationary era.
This possibility and observation­
al constraints on gravitational wave influences on the microwave back­
ground will be discussed at the end of the next section.
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37
4.2)
Higher Multipole Moments
In the previous section we gave some hope that we may get strong
constraints on our models of the early universe from the quadrupole to
dipole ratio.
However, if the quadrupole is found to be very small
compared to its present bound, the Q/D ratio will not give much infor­
mation about modelling the early universe.
If this is indeed the case,
we would like to discover if different models give distinguishably
different sets of values for the higher moments.
Again, if we knew what
was and whether it depended on k, we could tell which model was
correct on the basis of the quadrupole moment alone.
Since we do not
know £ h a priori, we must use the quadrupole moment to fix the value of
X, (the constant amplitude of the fluctuations at the time they cross the
horizon).
The question then becomes, "Once the quadrupole is measured,
how many higher moments (octupole, etc.) will need to be measured before
we distinguish one model from another.
ing cases in the following sections.
We will consider a few interest­
In all cases we will plot the
predicted values along with their theoretical error bars.
These error
bars arise from the fact that the physical perturbing process is assumed
to be Gaussian.
Thus, we can determine upper and lower limits corre­
sponding to 1 O' error bars (see reference [43]).
Because of this
feature, it may be impossible to distinguish two sets of multipole
moments regardless of how small observational errors can be made to be,
because of the intrinsically random nature of the underlying physical
process.
4.2A)
The Harrison-Zel'dovich spectrum
We present the values of the first few moments up to 1 = 9,
calculated using the Harrison-Zel'dovich scale invariant spectrum of
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38
perturbations in the matter density of the universe.
In figure 6 we show
the values of the moments for 3 values of the parameter XL.
In the next
section we will show how these moment predictions change with small
deviations from the Harrison-Zel'dovich spectrum for a fixed XL.
In the
the three spectra under consideration, the behavior of the moments as a
function of XI is similar.
As we can see from figure 6, the distribution of moments for XL = 1
and XL = 0.2 are nearly identical, even if the moments are measured up
to 1 = 9.
Thus we do not realistically expect that higher moments will
be able to help us distinguish between the important cases XL = 1 and
XI < 1 for a Harrison-Zel'dovich spectrum.
If XT. really is less than 1,
we will have to rely on the hope that the Q/D ratio test will tell us
this.
On the positive side, these moments may in fact give us our best
upper bound on X I , as it can be seen that XT. = 2 is quite distinguishable
from XI = 1.
The moments for a variety of other values of XT. ( 0.05, 0.5,
0.75, and 1.5) are presented in figures 7 through 10.
4.2b ) eH=
XX f
We would also like to test whether the Harrison-Zel'dovich spec­
trum is discemable from some weak deviations from it.
we are considering the two cases
= X k+l and
with the Harrison-Zel'dovich spectrum,
parisons for n
= X.
Specifically,
= X k 1 contrasted
We present these com­
= 0.2, i, and 2 in figures 11, 12, and 13 respectively.
Figure 11 shows us that with X L = 0.2, it is not easy to separate
the moments for different powers of k.
Only if the observational errors
are much less than the theoretical Gaussian fluctuations can we hope to
distinguish the different powers of k dependence, and then only after
many moments are measured.
If XT. = 1 (or even XT. = 2) The observers
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39
should have an easier job as the distributions of moments for the differ­
ent powers of k separate after only a few moments, as we see in figures
12 and 13.
In figures 14 through 17 we present moments for other values
of /I ( 0.05, 0.5, 0.75, and 1.5).
From the figures we can see that
the values of the higher moments (high 1) decrease relative to the lower
moments as we decrease the exponent of k in the spectrum.
This is
because when we decrease the exponent of k we put more power into the
long wavelength fluctuations to which the lower moments are more sensi­
tive.
4.2C)
Gravitational Wave Perturbations
If the measured value of the quadrupole moment turns out to be
too large to allow Inflationary models (according to figure 6), there
is still a chance to save the inflationary picture.
It has been shown
elsewhere22-25 that an inflationary epoch in the early universe will
produce long wavelength gravitational waves in a Harrison-Zel'dovich
spectrum.
Because of the tensor character of the gravitational waves, no
dipole anisotropy is produced.
Thus, even if the quadrupole moment is
found to be close to its present upper bound, it could simply mean that
the quadrupole is dominated by the gravitational wave contribution and
inflation would still be allowed.
If the gravitational waves dominate
the quadrupole moment, they will in fact dominate all the moments,
because the 1 dependence of the moments caused by density perturbations
and gravitational waves is similar.
So for inflation to escape our Q/D
ratio test requires the moments higher than the dipole to behave as if
they were caused by a universe filled with gravitational waves.
If this is indeed the case we would like to know if it is possible
to tell this through the higher moments.
For example, can we now
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40
distinguish our inflationary (.0- = 1) density perturbation moments from
those of produced by inflationary long wavelength gravitational waves?
To see if this is feasable, we have plotted the values of this case
in figure 18, where once again we have normalized the values of the
quadrupoles to 1.
We see that the values of the moments in the two cases
are indistinguishable, so the distribution of higher moments in the
inflationary universe is basically
the same whether they are caused by
density perturbations or gravitational waves.
If the value of the Q/D ratioallows fl
figure 19 shows that the values of
= 0.2 and forbids .d. = 1,
the higher
moments (if measured
accurately enough) can make or break inflation.
For completeness, we present the values of the moments in the cosmic
background radiation due purely to gravitational waves in open, flat, and
closed Robertson-Walker-Friedman universes in figures 20 through 24.
Harrison-Zel'dovich spectrum is assumed in all cases.
A
As before, we have
used the values of fi. = 0.05, 0.2, 0.5, 0.75, 1.5, and 2.
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41
CHAPTER 5
CONCLUSIONS
Formulae have been derived for calculating the moments of the
anisotropy of the cosmic microwave background radiation for an arbitrary
perturbaton of the gravitational field in all three types of RobertsonWalker universes (open, flat, and closed).
It has been found that
the amplitudes of perturbations which transform like vectors can only
shrink with time and thus are not important for the Sachs-Wolfe effect
on the microwave background.
We have also explained how to implement the
assumption that the process which causes these perturbations is random
with a Gaussian distribution.
Actual computations of the moments due to density and gravitational
wave perturbations have been performed for the inflationary universe as
well as a range of relevant alternatives to an inflationary cosmology for
comparison leading to the following conclusions:
1)
Confirming previous work, references [25,27,43], the microwave
background radiation anisotropy provides some tough constraints for the
inflationary cosmology.
A)
An upper bound on
( € M is related to the underlying
elementary particle physics theory of inflation and also equals the
amplitude of density perturbations when their wavelengths are equal to
the horizon size).
We find that with 90% confidence
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42
r
B)
^
n
I wq ve/engTh of cutoff j
—
T *
I
GO
I
The value of the observed quadrupole must be at least two
orders of magnitude smaller that the current upper bound if the anisot­
ropy is caused by density perturbations.
C)
If the quadrupole moment Is much larger than the value pre­
sented in figure 5, then the quadrupole moment could be signalling the
presence of gravitational waves.
The higher moments must then follow
the 1 dependence presented in figure 18.
2)
The multipole moments provide a useful test of the Harrison-
Zel'dovich spectrum as we can distinguish between it and some small
deviations from it in a critical density universe.
3)
If the Harrison-Zel'dovich spectrum proves correct then the
multipole moments will probably not give us any information about the
value of the energy density.
If inflation flunks the quadrupole test then
we must first rule out the presence of gravitational waves in order to say
that the energy density is not the critical value.
If inflation does pass
the quadrupole test, the values of the higher moments are not much help in
distinguishing values in the range 0.2 <.f"L< 1, where the true value is
believed to be anyway.
4)
As distasteful to theorists as it may be these multipole moments
can provide ways of killing the inflationary model.
This could happen in
two ways:
A)
as much as k**.
The spectrum deviates from a Harrison-Zel'dovich by at least
This can be discovered through the quadrupole-to-dipole
ratio in combination with the higher moments.
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43
B)
If I"! is outside the range 0.2 to 1.3 with a Harrison-
Zel'dovich spectrum then inflation can be ruled out.
If inflation is killed by either of these possibilities, then the
Q/D ratio and the 1 dependence of the higher moments presented in chapter
4 can be used as a diagnostic tool for pointing the way towards the
correct model of the universe.
If a better model is found, the equations
in chapter 3 are general enough to be used to calculating moments to test
this new model.
In any event, it is quite amazing to consider the fact that, as
Larry Abbott has pointed out to me, we are considering measurements of
phenomena on the largest scales of the universe possible which ultimately
will give us information about phenomena about physics on the smallest
scales being considered by particle physics.
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44
APPENDIX A
SPATIAL HARMONIC FUNCTIONS
We present the derivation and explicit forms of the spatial har­
monic functions which are solutions of a generalized Helmholtz equation.
We will separate the discussion into a section for each of the three
types of harmonics:
scalar, vector, and tensor.
We will present the
derivation of the scalar harmonics <*&) in detail since they are the
basis for the vector and tensor harmonics as well.
The treatment of
the scalar functions follows that of Harrison1*6 and the treatment of
the vector and tensor functions follows that of Tomita.51*
Al)
Scalar Harmonic Functions
We want to find the eigenfunctions Q(o0 of the covariant Laplacian
in a general Robertson-Walker space.
We need to solve the Helmholtz
equation
where D
*5.
= D
I
D t and D
is a spatial covariant derivative of our space
defined by the spatial part of our metric,
<JU? =
with
S
( T ) [ ~ d r :L 4-
(Al.2)
= 1 + K — . With this metric we can see that only the radial
dependence of the m )
stant K.
Tp-Jfc'd!#]
will change with the value of the curvature con­
In flat space (K = 0) eq. (Al.l) becomes
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45
( V a- +
A ’-jC H a )
=
O
<a i .3)
which is easily solved by Q (x) = exp(ik*lc).
We can express the Q (?)
in terms of spherical coordinates by remembering the relation
J!)fyn
with the
defined by
This form of the Q (?) is more useful for our purposes because we want to
expand our formulae in terms of the
to get the multipole moments.
It
is worthwhile to note that the equations for Q (1?) and for the pertur­
bation amplitudes depend only on the magnitude of k and not on its
direction.
The directional part of k here is introduced purely as a
convenience so that we can write the solutions as exp(i^.?), a function
which is simple and easy to think about.
However, we can dispense with
the directional parts of k along with the accompanying factors of i and
4If to write Q (?) as
Q ( * )
=
Y jL m
which is perfectly acceptable.
w i.4 )
Thus the general form for our solutions
will be
where (p^~ = CQ j
(kr) for K = 0,
= k 3- + K, and C q
is a normali-
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46
zation constant.
As previously stated, only the radial equations differ
in the curved spaces and the
are eigenfunctions of the angular part
of the covariant Laplacian for any value of K.
The radial equation be­
comes
0
(Al.6)
For K=0 we see that this reduces to The spherical Bessel equation for the
argument kr.
We will next consider separately the equations for K = +1
and K = -1.
For K = +1, let
a
s & jT
L
*
(A1.7)
/l£ =
This y is the same y which is used as the affine parameter along the null
geodesics described in section 2.1.
The radial equation (A1.6) then
becomes
j g + i )
[ x - j
With C^-
the equation for
A
d
AUr* fij.
J
o
o
(A 1' 8)
J T - 0
(A1.9)
=
is
X ( \ + 0 -
17 J
with
For K = -1, let
-
A A n Jb
/k
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(Al.10)
47
and just as in K = +1 we note that y is the parameter describing the null
geodesics of section 2.1.
The radial equation is now
(Al.11)
In this case define
• The equation for TT(<i^) is
AlrtJn*
then
^
"TT-0
[x(x+iV ^
If we define the variable Cj = K
(ai. 12)
we can turn both radial equations
into a single equation
i
d
JL
■
where now \ ( \ + 1) = Kk1^
X
x
Since V*
- O
(Al.13)
(cos( ^ )) with
has two solutions
= - £ ± ( , + k ^
= P(^ , or P ^
(Al.14)
= P^^we will use V = X *
integer but JU. is not an integer we can use P^
solutions instead of P^*- and
the origin.
it
, and now has solutions which are associ­
ated Legendre functions P^(cos(^)) and
and V-= \.
(&$£
Also sinceyU ± V is an
and P^
. We will demand that
as independent
be regular at
This boundary condition eliminates P J
vl+ ’A^as a solution and
"Jb-'h
leaves us with P^
as the only solution we need to consider.
When talking about these eigenfunctions and their eigenvalues, it is
more convenient to redefine k
in terras of the variable
. For K = +1
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48
we define
(A1.15)
r
=
f
-
Then the radial function becomes
<f>p
= C+|
In order that (fp^
A PyJ fi (CXL^)
be single valued,
(A1-16)
must be an integer, i.e.,
It can be seen that the values of ft = 1 and ft = 2 correspond to modes
which are pure gauge terms.33*36
Thus our spectrum of eigenvalues for
K = +1 are
J N
/3* - 1
,3=3^
S y ..
j3 > £
wi-w
For K = -1, we will define |3 such that
Jl
= fi* 4
|
(A1.19)
so the eigenfunctions are
Because space is open there are no periodic boundary conditions to
satisfy, so ft can take on any positive real value.
Thus the spectrum
for K = -1 is
'k
I N ~~ /3a-H
/ 3^> 0~
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(Al.21)
49
We have yet to discuss our normaliztion conditions so we can fix the
values of the constants C0 , CL| , and C4, . Our condition is that
rr
js
(g ^ L
(q (*)),,-
W
(A1-22)
all1 space.
where y9 " = X + K and
8 (f^/3')
is the delta "function" with respect
to the measure ^(k)
5T/3,/2.^)= ^(p')
]Sjx(K)
(Al.23)
The measure JU(k) is defined differently for different curvatures in
Table 1.
Table 1.
The k-space measure in curved spaces
Mm ')
K
-1
- Dirac delta function
0
~ Dirac delta function
+1
T- ft-
(5=3 1
& ,
P><2>
- Kronecker delta
where we will use the definition
(A1.24)
for the rest of the section on scalars.
We have now specified the nor­
malization in which our functions now take the form (with their normali­
zation constants evaluated)
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50
ir* ~
_^_^ A>.—^ )
7 /
?T
2.|3 AX/Kj /\^_
i^
K= + l
where
N l'
X
-
v r
"1=°
i f +
Jt
1 ^ = fr
n (f
1n
2.
/n- ^
o
'
r f )
(A1.26)
- r^)/
after Harrison1*5 and y is defined as (A1.7) for K = +1, (A1.10) for
K = -1, and y = r for K = 0.
While equation (A1.25) provides a neat well known form for the
functions
> it is of little value for numerical computation.
will present some properties of the (p^
We
useful in calculating which are
derivable from the tabulated properties55-57 of the Bessel and Legendre
functions.
First of all we can rewrite the derivative of the radial functions
which appears in the scalar formula for
Ve ‘ d£ Q(2) - 1)1^
L
^
8T0 /Tp
eq. (3.1.15) as
^Tn.
Since we will need to calculate the functions
<A1-27)
anyway, it is useful
to know that we can express this derivative in terms of them.
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51
X cotky $
-
o3 (j)"' K= -1
i £+1
eh
dn'P
K- o
a y t ^ tpf - J p - c x + n * <j)f
This leaves us with the task of calculating the
(Al.28)
|<- + 1
(j)^-s.
We can do this recursively using the relations
f
(it-1)jr $'* - fr (j)Jl-x
%s
K =r o
<A1-29)
K= + 1
C
and exact formulae for two of the c±p^
s, e.g., the 1=0 and 1=1 modes
K=-l
l/nr
K =
0
(Al.30)
K -+1
and
K= -I
*;= ^
x<
Kj p P T
[ o
-
p
t
to -
K= o
p
K -+1
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52
A closed form for any other (p^
can be found from the generating
functions
K--I
( - i ^ 1
i+i
4£=
)<rO
(A1.31)
( - 0
v.
K=+l
/syTTf
For small values of ky it is better to use the Taylor series for
(M- since the recursion relations would then involve subtractions of
A*
numbers very close in magnitude, resulting in the loss of significant
figures.
r
The Taylor series are
—
~~
\H ' ‘
(j1
(M+Oi!
c
(_|) (.|y
K=-i
1 *rr2\~1
I
fA
_ X _
(A1.32)
K ^ o
P (ax+i)!l
^
m -1
1s ' - * - ” ®
K - +1
where
v(/3a+a+j)1'i k--i
am
ft
K=0
D„ = T T - (!•+-> +!t)
r-1
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53
In this form it is now clear that at small distances and short wave­
lengths (small y and large k), we recover the flat space limit
&
4 ^ =
(ai-33)
as we must since space is locally flat.
A2)
Vector Harmonic Functions
We now want to find the solution of the vector Helmholtz equation
(A2.1)
where
t
is divergenceless
D‘Qt>= o
<42'2)
As in the scalar case we will decompose the Q ^
into modes defined by k,
1, and m
Ot\
Q?(Z)=L
(Q'c
/if*
(A2.3)
^
1
Counting the degrees of freedom we see we have two degrees (3 components
- 1 constraint =2).
In flat Euclidean space we usually think of this as
two choices of polarization vector in the transverse plane.
coordinates the two solutions are modes with parity (-1)^
In spherical
and (-l)^+l.
Thus any arbitrary transverse vector field can be expressed as a linear
combination of even parity and odd parity modes.
In deriving these
solutions for the curved spaces (K = ±1), it is easiest to use the
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54
coordinates described by the three space line element
"k
+
<A2- 4 >
where
K
~ K
h. \ "
A>>
y
Notice that we can recover the flat space limit
These coordinates are useful because we can treat both closed and
open universes simultaneously and the connections take on a simple form.
We present only the non-zero coefficients.
~ - A C n ^ c o o . 1^
Q
r
% -
1 I'i.
r 3" -
co*
*ai
=
&
5
r = '*»
r 4= t o * E^
M'i
where x 1 = ^ , xa
Q
1
r s=
'-S'JL
= (? , and x 3 = (J) . We now present separately the
solutions of the even and odd parity modes,
a)
Solutions of even parity
When we use eq. (A2.2) in (A2.1) we find we get an equation which
involves solely the radial component
1
(3).
We can expand Qt(* 0t) in
i
terms of spherical harmonics
(A2-6)
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55
to get the equation for the radial function of
(A2.7)
and taking the limit K-*0, cos£-> 1, and
hV * $
V
+ 0 ? -
K^sin£-*' y
XJ^
h
l‘ »
Equation (A2.7) has the solution
v
. [
K = i l
'‘ " U f
where
and
(A2.8)
K= °
(p-* are the solutions of the scalar equation defined in (A1.25)
TP
is defined here as
/ 3
and Ne
* =
X
+
a
K
<A2- 9>
is a normalization constant.
We get only one solution to this
second order differential equation because we enforce the same boundary
conditions as in the scalar case.
and (A2 =2),
Using (A2.6) and (A2.8) in £4 3• (A2.1)
we can find the other components of
. We present the
solutions in terms of the variable y which is more useful than ^
$Te /T
formulae
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for our
56
where
M
I)
"
y3>
"
and
r
/UndhsAs^
AJLnr*
(^s)filLw Mt+f) Vlib 3<t>
K*-i
X
(A2.ll)
‘t’")
with
(j)/^
K-
~ I
(A2.12)
vl(b‘ = <
K - 0
/*»
A
£
K - + I
They obey the normalization condition
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57
(A2.13)
f
s(M-)
all Space.
where
£(^(2,0
is the appropriate Dirac delta or Kronecker delta as
defined in eq. (A1.20).
b)
The solutions of odd parity
The radial equation (A2.7) does not admit a solution of odd
parity and thus S®
=0.
The remaining components are then solved for
using eqs. (A2.1) and (A2.2).
The solution is as follows
Q? = I N.
(A2.14)
cm
j£,/m
'
where
N o
=
a*
i
)J%p
and
& ) * „
A / 1
v 3 (^>
AX tk O-
—
Y
IJim
(A2.15)
s > ° ) /him
with
K = -I
nf
K =
>IAV
K - +
o
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(A2.16)
58
The odd parity harmonics have the same normalization as the even
harmonics
a/1 Space
A3)
Tensor Harmonic Functions
We now want to find the solutions of the generalized tensor Helmholtz
equation
( D* + X ) Q u = 0
(A3-U
d
subject to the constraints
DlQ1?. = O
i
(A3.2)
QSj = Qfc
and
Q*u = o
\&) C
i
As in the vector case there are two degrees of freedom which correspond
to even and odd parity modes.
Any transverse traceless symmetric tensor
can be expressed as a linear combination of even and odd parity modes,
a)
Solutions of even parity
Using conditions (A3.2) in eq. (A3.1) we find that the equation
for the radial-radial component decouples from the other components an
can be expressed as
Q» =X)(tv\
L'XU%)XJW)
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59
resulting in an equation for the radial function “Xft,
(using the coordi­
nates of eq. (A2.4))
(A3.4)
Of- ^
+
)-A=° K=O
These are solved by
K4
o
(A3.5)
y;=
\
where the
K= o
%
are the radial part of the scalar harmonic functions
defined in eq. (A2.25), N&
is a normalization constant, and
is now
defined by
/6*= X +•3 K
(A3.6)
Once this component is known we can use eqs. (A3.1) and (A3.2) to find
relationships between the radial-radial component and the other compo­
nents.
We present the results
Q n) = Z Ne (Ge(A3.7)
ki<.«_ lu+xtii-t-1)jl
a-
o
e '/
We remember j3 is defined as in eq. (A3.6).
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60
The G &
are given by
K=-i
I
(
‘*4
K- o
K =
l&e
\
^xx
30 Y
'lim
= t
+
'
\
2(i>
-Y
\&-J/bJL/m
d(J>
‘i/rr,
(A3.8)
I£
Gt
v>.
I 35*
YIjL
/m
C
G,
43
= T,
L - “ t o >» X
s fi
2
<^
\G
i
I X
+
1
o
Y.
je*Y3
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61
and
r
x
aj
I d
t:=JLU+OdyW1
jU+OYfi* ' i
±_
rh*
K-°
Kr+i
Cot
v.
r
( y^AAL/K AA 1
d
T
*
•5p.
=
K- -1
l
J
(~ n
(x*i)u+iu(£-r)
Co-d. <tiKi
(A3.9)
I cC- i - X
V^
/4-^kA,7-
’)/y(UmAZ
K=
K -&
U
K-
i- 1
a
+1
'*O X i v ^
K~~l
T Hp* '= JC+k-X
J—
(cH1,4
V
V
^lAv1
; "
'1
(ft
((&"+'"
K-0
K-M
V
g
and the G^- s satisfy the normalization condition
o
(A3.10)
o
V e ' / V>
lVlUo*\
kfLW^t U ^
all spoxc
where
is the appropriate Dirac or Kronecker delta as defined in
eq. (A1.20).
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62
b)
The solutions of odd parity
There is no odd parity solution to the radial-radial component
equation (A3.4), so G*
= 0, just as there is no odd parity solution of
the vector radial component.
The odd parity solutions are then
pSL(rC\
(A3.11)
„y^-i)o±ji_u±3i
where
G°
=o
^ l\
/
fn°
y,„\
(A3.12)
l&.
ItaY - -c*c &xJLm
J
L
r(
\
.
(j.a'b
tp
(>9.m
M ™ ' 0r3 e ' - X ™
* ^ ( - S e h X ^ c*t°r*'h)
with
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I
63
16/a
K
-r'A
I
d ,
1
(A3.13)
K - o
U-f
K =+ l
and have the same normalization as the even parity modes,
r dfy
(A3.14)
ctU space.
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64
APPENDIX B
GROWTH OF DENSITY PERTURBATIONS
DURING THE RADIATION DOMINATED ERA
To get the dipole bound in chapter 4 we had to cutoff our integral
(or sum) over /3 modes at a wavelength which corresponds to 60 Mpc today.
This wavelength fluctuation crosses the horizon while the density of the
universe is still dominated by radiation.
Fluctuations which cross the
horizon during this time grow at a different rate than that of the
matter dominated era presented in section 3.1.
To treat the behavior of
perturbations during the radiation dominated phase, we have to solve the
equations for £(d^ and insure that the function £(V)goes smoothly over
from the radiation to matter dominated eras.
This has already been done
for critical density universes in reference [43].
We will first show
that their results can be used here for our range of non-critical density
universes, and then briefly present their results for completeness.
In the radiation dominated era we need to consider how the matter
fluctuations behave in the presence of a dominant photon component.
To
do this we need the multifluid generalization of Bardeen's formalism
which is provided in references [44,45].
We will first solve for the
evolution of perturbations in the photon density and then find out how
they influence the matter fluctuations.
To begin with we cannot use the form of S(/y) provided by eq. (3.2)
because we can no longer neglect the photon density and pressure in the
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65
Friedmann equations
ir= •-** sv
p
-> - k
(B.l)
(!)'=
•
f c
where
.« <
- l f d
s
)
/ l r - 3
/ U
I .
and
are the densities of matter and radiation and we have
used the equation of state for photons -p^. =
these equations are for
. The solutions of
T 1<C /^n-v
r
K- -I
r
y,
81r &
?>(r) n
*6
K -0
r
u o
(B.2)
K-+1
y U A v T
V
and for
^ An
A u r d . 'l ( 1 x
^
)
K * - l
Ka
a^
o
)
z(
)
K— l
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(B.3)
66
where
is the time when matter begins to dominate and is defined by
#
With these definitions we find that S and S are continuous at
•
<v t
'■y
Notice that the argument in eq. (B.3) is —
and not simply -j£
as presented in eq. (3.2).
concerned with 'Y »
This is because in section 3 we are only
and so 'T'+
The equation governing
— "Y
in eq. (3.2).
in a radiation dominated universe can be
found from Bardeen,33
0
(B.4)
Following reference [44] we recast this equation in terms of the dimensionless parameter x
which is the ratio of the sound horizon length to Tf times the fluctua­
tion wavelength.
Equation (B.4) becomes
a
£ti+ X R
+
1
e Y =
o
(b.5)
where a prime denotes an x derivative, and
K = k (|P
(b-6)
R tells us whether or not we can neglect curvature effects and has the
physical interpretation
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67
^
=
^fachus
S ^ ( t ') ~
(
U o n z o n
of
curvature
s iz e ')
*\
R
-
,X
k / f ]
‘
As long as R «
= /
V- S i
StZg.
I ra d iu s
erf
—
Y
curvature-J
cur
1 we can ignore curvature effects and just use the
equations for K = 0 (R = 0).
For our range of jfl (10 < £"1 < 0.01) the
largest value of R while the universe is radiation dominated is R — 0.002
(for X L = 0.01), so we are accurate to at least of order < 1% by neglect­
ing the curvature terms.
£ v =
f
”
A
, d
(
The growing mode solution to this equation is
-
a * * )
ik )
<b- 7 >
where A^a£j is just a constant.
We next need the equation1*5 which gravitationally couples the photon
density perturbations to the matter density perturbations during the
_ 5^
radiation dominated era. We will also use the approximation k
« K,
which is valid for any fluctuation which crosses the horizon during the
radiation dominated epoch.
(This approximation is equivalent to setting
R = 0.)
c + (|)e + a.(f)(l
As before, rewriting this in terms of our parameter x
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68
€'
(B.9)
Ignoring the R terra we get
£ "
+
^ - [ i
+
a t i
+ ^
^
' ^
e
'
(B.10)
%
i (,4 i 4 ' U - ¥ )
X %
2-
^
-
6
C !T
£(")£) is then solved for by numerically integating eq. (B.10).
have to match the solutions of €
We then
for the matter and radiation
dominated eras at the time of equal densities (
A good numerical
fit is provided by using an £(T) modified to include the short wavelength
effects of the radiation dominated epoch.
modified to
emod(T) I
\
include rad. dom.J =
/e defined
Ame(T) [ by eq. 3.1.20 j
(B.11)
0.0005-JC^]
(B.12)
era effects
and A m is given by1*3
[l 4
and Xm = X(Tm) =
' Jz
0 . 0 4 X ^ +
and k ^ is the wavenumber which corresponds
to the wavelength which just crosses the horizon at Y . This fit is good
up to about k = 13 k ^ . Notice that for fluctuations which cross the
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69
horizon during the matter dominated era A^Xl.
To get our dipole bound we just substitute our modified Gfnoi (ecl*
(B.ll)) wherever G
appeared in our equation for
£T0 /T0
(3.1.15).
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70
REFERENCES
1)
Steigraan, G., Schramm, D. M., and Gunn, J., Phys. Lett. B66 (1977),
202
.
2) Guth, A., Phys. Rev. D23 (1981), 347.
3) Linde, A., Phys. Lett. B108 (1982), 389.
4) Albrecht, A. and Steinhardt, P., Phys. Rev. Lett. _48 (1982), 1220.
5) Guth, A. and Pi, S.-Y., Phys. Rev. Lett. 49 (1982), 1110.
6) Bardeen, J., Steinhardt, P., and Turner, M. S., Phys. Rev. D28
(1983), 679.
7) Staroblnskii, A., Phys. Lett. B117 (1982), 175.
8) Hawking, S., Phys. Lett. B115 (1982), 295.
9) Harrison, E. R., Phys. Rev. Dl_ (1970), 2726.
10) Zel'dovich, Ya. B., Monthly Not. Roy. Astronom. Soc. 160 (1972), IP.
11)
Yang., J., Turner, M. S., Stelgman, G., Schramm, D. N., and Olive, K.
A., Astrophys. J. 281 (1984), 493.
12) Blumenthal, G., Faber, S., Primack, J., and Rees, M., Nature 311
(1984), 517.
13) Preskill, J., Wise, M., and Wilczek, F., Phys. Lett. B120 (1983),
127.
14) Abbott, L. F. and Sikivie, P., Phys. Lett. B120 (1983), 133.
15) Dine, M. and Fischler, W., Phys. Lett. B120, (1983), 137.
16) Cabibo, N., Farrar, G., and Maiani, L., Phys. Lett. B105, (1983),
155.
17) Ellis, J., Hagelin, J. S., Nanopoulos, D. V., Olive, K. A., and
Srednicki, M., Nucl. Phys. B238 (1984), 453.
18)
Davis, M., Geller, M., and Huchra, J., Astrophys. J. 221 (1978), 1.
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71
19)
Peebles, P. J. E., Astron. J. 84 (1979), 730.
20)
Press, W. A. and Davis, M., Astrophys. J. 259 (1982), 449.
21)
Davis, M., Efstathiou, G., Frenk, C., and White, S. D. M., Astrophys.
J. (in press).
22)
Starobinskii, A., JETP Lett. 30 (1979), 683.
23)
Rubakov, V., Sazhira, M., and Veryaskin, A., Phys. Lett. B115, (1982),
189.
24)
Fabbri, R. and Pollock, M.,
Phys. Lett. B125
(1983), 445.
25)
Abbott, L. F. and Wise, M.,
Nucl. Phys. B244
(1984), 541.
26)
Peebles, P. J. E., Astrophys. J. (Letters) 263 (1982), LI.
27)
Abbott, L. F. and Wise, M., Phys. Lett. B135
28)
Kaiser, N., Astrophys. J. (Letters) 284 (1984), L9.
29)
Politzer, H. D., and Wise, M., Astrophys. J. (Letters) 285 (1984), LI.
30)
Fabbri, R., Guidi, I., and Natale, V., Astrophys. J. 257 (1982), 17.
(1984), 279.
31)
Tomita, K. and Kenji, K., Prog. Theor. Phys. 69_(1983),
32)
Wilson, M., Astrophys. J. 273 (1983), 2.
828.
33) Bardeen, J., Phys. Rev. D22 (1980), 1882.
34) Sachs, R.
and Wolfe, A., Astrophys. J. 147
(1967), 73.
35) Anile, A.
and Motta, S., Astrophys. J. 207
(1976), 685.
36) Lifshitz,
E. M. and Khalatnikov, I. M., Adv. Phys. 12_ (1963), 185.
37)
Field, G. B. and Shepley, L. C., Astrophys. Space Scl. 1 (1968),
309.
38)
Sakai, K., Prog. Theor. Phys. 4_1 (1969), 1461.
39)
Press, W. H. and Vishniac, E. T., Astrophys. J. 239 (1980), 1.
40)
McVittie, G. C., General Relativity and Cosmology, Univ. of Illinois
Press, Urbana, 1965.
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72
41) Schrodinger, E., Expanding Universes, Cambridge Univ. Press,
Cambridge, 1956.
42)
Silk, J., in Cosmology and Particles (Proc. of the XVI Recontre de
Moriond - Astrophysics Meeting) Les Ars, Savoie, 1981.
43)
Abbott, L. F. and Wise, M., Astrophys. J. (Letters) 282 (1984), L47.
44)
Abbott, L. F. and Wise, M., Nucl. Phys. B237 (1984), 226.
45) Wise, M., private communication.
46) Harrison, E. R., Revs. Mod. Phys. ^9 (1967), 862.
47) Kaiser, N., Astrophys. J. (Letters) 273 (1983), L17.
48) Silk, J. and Wilson, M., Astrophys. J. 243 (1981), 14.
49) Silk, J. and Wilson, M., Astrophys. J. (Letters) 244 (1981), L37.
50)
Fixsen, D., Cheng, E., and Wilkinson, D., Phys. Rev. Lett. _5£
(1983), 620.
51)
Lubin, P., Epstein, G., and Smoot, G., Phys. Rev. Lett. 50, (1983),
616.
52)
Hogan, C. J., Kaiser, N., and Rees, M., in The Big Bang and Element
Creation, The Royal Society, London, 1982.
53)
Peebles, P. J. E., The Large Scale Structure of the Universe.
Princeton Univ. Press, Princeton, 1980.
54) Tomita, K., Prog. Theor. Phys. 68^ (1982), 310.
55) Magnus, W., Oberhettinger, F., and Soni, R. P., Formulas and Theorems
for the Special Functions of Mathematical Physics, Springer-Verlag,
New York, 1966.
56)
Erdelyi, A., Higher Transcendental Functions, vol. 1 of The Bateman
Manuscript Project, McGraw-Hill, New York, 1953.
57)
Arfken, G., Mathematical Methods for Physicists. Academic Press, New
York, 1966.
R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission.
73
BIBLIOGRAPHY
Abbott, L. F. and Sikivie, P., Phys. Lett. B120 (1983), 133.
Abbott, L. F. and Wise, M., Phys. Lett. B135 (1984), 279.
Abbott, L. F. and Wise, M., Astrophys. J. (Letters) 282 (1984), L47.
Abbott, L. F. and Wise, M., Nucl. Phys. B244 (1984), 541.
Abbott, L. F. and Wise, M., Nucl. Phys. B237 (1984), 226.
Albrecht, A. and Steinhardt, P., Phys. Rev. Lett. 4j3 (1982),1220.
Anile, A. and Motta, S., Astrophys. J. 207 (1976), 685.
Arfken, G., Mathematical Methods for Physicists, Academic Press, New
York, 1966.
Bardeen, J., Phys. Rev. D22 (1980), 1882.
Bardeen, J., Steinhardt, P., and Turner, M. S., Phys. Rev. D28 (1983),
679.
Bluraenthal, G., Faber, S., Primack, J., and Rees, M., Nature 311 (1984),
517.
Cabibo, N., Farrar, G., and Maiani, L., Phys. Lett. B105 (1983), 155.
Davis, M.,
Geller, M., and
Huchra, J.,
Astrophys. J. 221 (1978),1.
Davis, M.,
Efstathiou, G.,
Frenk, C., and White, S. D. M., Astrophys.
J. (in press).
Dine, M. and Fischler,
Ellis, J.,
W.,
Phys. Lett.
B120 (1983), 137.
Hagelin, J. S.,
Nanopoulos,
D. V., Olive, K. A., and
Srednicki, M., Nucl. Phys. B238 (1984), 453.
Erdelyi, A., Higher Transcendental Functions, vol. 1 of The Bateman
Manuscript Project, McGraw-Hill, New York, 1953.
Fabbri, R., Guidi, I., and Natale, V., Astrophys. J. 257 (1982), 17.
Fabbri, R. and Pollock, M., Phys. Lett. B125 (1983), 445.
R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission.
74
Field, G. B. and Shepley, L. C., Astrophys. Space Sci. _1_ (1968), 309.
Fixsen, D., Cheng, E., and Wilkinson, D., Phys. Rev. Lett. f>0 (1983),
620.
Guth, A., Phys. Rev. D23 (1981), 347.
Guth, A. and Pi, S.-Y., Phys. Rev. Lett. 49 (1982), 1110.
Harrison, E. R., Revs. Mod. Phys. 39^ (1967), 862.
Harrison, E. R., Phys. Rev. D_l_ (1970), 2726.
Hawking, S., Phys. Lett. B115 (1982), 295.
Hogan, C. J., Kaiser, N., and Rees, M., in The Big Bang and Element
Creation. The Royal Society, London, 1982.
Kaiser, N., Astrophys. J. (Letters) 273 (1983), L17.
Kaiser, N., Astrophys. J. (Letters) 284 (1984), L9.
Lifshitz, E. M. and Khalatnikov, I. M., Adv. Phys. 12 (1963), 185.
Linde, A., Phys. Lett. B108 (1982), 389.
Lubin, P., Epstein, G., and Smoot, G., Phys. Rev. Lett. 5(3(1983), 616.
Magnus,W., Oberhettinger, F., and Soni, R. P.. Formulas and Theorems
for the Special Functions of Mathematical Physics, Springer-Verlag,
New York, 1966.
McVittie, G. C., General Relativity and Cosmology, Univ. of Illinois
Press, Urbana, 1965.
Peebles, P. J. E., Astron. J. j54 (1979), 730.
Peebles, P. J. E., The Large Scale Structure of the Universe. Princeton
Univ. Press, Princeton, 1980.
Peebles, P. J. E., Astrophys. J. (Letters) 263 (1982), LI.
Politzer, H. D., and Wise, M., Astrophys. J. (Letters) 285 (1984), LI.
Preskill, J., Wise, M., and Wilczek, F., Phys. Lett. B120 (1983), 127.
Press, W. A. and Davis, M., Astrophys. J. 259 (1982), 449.
R eproduced w ith perm ission o f the copyright owner. Further reproduction prohibited w itho ut perm ission.
75
Press, W. H. and Vishniac, E. T., Astrophys. J. 239 (1980), 1.
Rubakov, V., Sahim, M., and Veryaskin, A., Phys. Lett. B115 (1982), 189.
Sachs, R. and Wolfe, A., Astrophys. J. 147 (1967), 73.
Sakai, K., Prog. Theor. Phys. 41 (1969), 1461.
Schrodinger, E., Expanding Universes, Cambridge Univ. Press, Cambridge,
1956.
Silk, J., in Cosmology and Particles (Proc. of the XVI Recontre de
Moriond - Astrophysics Meeting) Les Ars, Savoie, 1981.
Silk, J. and Wilson, M., Astrophys. J. 243 (1981), 14.
Silk, J. and Wilson, M., Astrophys. J. (Letters) 244 (1981), L37.
Starobinskii. A., JETP Lett. 30 (1979), 683,
Staroblnskii, A., Phys. Lett. B117, (1982), 175.
Steigman, G., Schramm, D.M., and Gunn, J., Phys. Lett. B66 (1977), 202.
Tomlta, K., Prog. Theor. Phys. 68^ (1982), 310.
Tomita, K. and Kenji,
K.,Prog. Theor. Phys.
Wilson, M., Astrophys. J.
(1983),
828.
273 (1983), 2.
Wise, M., private communication.
Yang., J., Turner, M.
A., Astrophys. J.
S., Steigman, G., Schramm, D. N., and Olive, K.
281 (1984), 493.
Zel'dovich, Ya. B., Monthly Not. Roy. Astronom. Soc. 160 (1972), IP.
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76
FIGURE CAPTIONS
1.
Bounds on X from the upper limit on the quadrupole moment*
With
90% confidence, the allowed values of X (as a function of/I)
are below the lines for each of the three spectra as shown.
2-4.
Dipole bounds on X. With 90% confidence, the allowed values of
X (as a function of-0.) are below the lines in the figures.
Figures 2, 3, and 4 show the upper bound lines for
Harrison-Zel'dovich, k’* , and k4 1 spectra respectively.
5.
The quadrupole to dipole ratio.
Shown here are the 90% confi­
dence bounds on the linear contribution to the quadrupole to
dipole ratio.
For convenience, the values of the ratio have
been multiplied by the observed dipole moment in the left hand
scale.
Thus the quadrupole moment must be found below the value
shown for a particular spectrum and XL value to be a viable
model.
The right hand scale shows the unaltered values of the
predicted linear quadrupole to dipole ratio.
6.
A comparison of moment predictions with 1 CT error bars for a
Harrison-Zel’dovich spectrum of density perturbations for 3
different values of XI : 0.2, 1.0, and 2.0.
7-10.
The 1-dependence of higher moments from a Harrison-Zel'dovich
spectrum of density perturbations, again with 1CT
error bars.
Figures 7, 8, 9, and 10 show the moments for ft = 0.05, 0.5,
0.75, and 1.5 respectively.
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77
11-13.
Comparison of 3 different spectra: k 0 , k+l , and k”* for a
given value of XL*
Figures 11, 12, and 13 show the 1-dependence
for .0 - = 0.2, 1.0, and 2.0.
14-17.
The 1-dependence of the higher moments for the non-scale-v.
*1 o
invariant spectra « h = k X
_
for particular values of XL .
Figures 14, 15, 16, and 17 show the moments for XL values 0.05,
0.5, 0.75, and 1.5.
18.
Inflationary predictions for the 1-dependence of the higher
moments for density fluctuations vs. gravitational waves.
This
comparison can only be made in the event that the quadrupole is
found to be small enough to pass the quadrupole-dipole ratio
test.
19.
Comparison of the inflationary gravitational wave moments and
non-inflationary (fl = 0.2) Harrison-Zel'dovich density
perturbation moments.
20.
A comparison of the 1-dependence of moments from a scaleinvariant spectrum of gravitational waves for the /I values:
0.2, 1.0, and 2.0.
21-24.
Predictions of the moments due to a Harrison-Zel'dovich
spectrum of gravitational waves for certain values of £ L .
Figures 21, 22, 23, and 24 show predicted moments for X L = 0.05,
0.5, 0.75, and 1.5 respectively.
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FIGURE 1
Quadripole 90% confidence upper bound on X.
Values of X are allowed below curves.
i
7
-S '
XIo
s
o.o%
o.oS" 0.07 a.
T
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FIGURE 2
Dipole 90/S confidence upper bounds on X,
for a Harrison Zel’dovich spectrum
*H= X .
Values of X are allowed below the curve.
100
a.o
-c.
O.i
1-0
5
SX-
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FIGURE 3
Dipole 90$ confidence upper bounds on X
for a k'
spectrum.
Values of X are allowed below the curve
-S'
*|0
a.
5
M -
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FIGURE 4
Dipole 90% confidence upper bounds on X
for a k +l( = £ h/)0') spectrum.
Values of X are allowed below the curve.
loo
x{07
2
jr
JO-
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FIGURE 5
Quadrupole to dipole ratio.
bound o n (0-/Q )»bs
X 10
X 10
X /0
xlo
X
X I-
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1---- »---- 1
<»
4 ^ 4 ________________________________
• * <
|
-g
|
f
a Harrison-Zel'dovich
’’
.
■----- 1
H
*------ 1
I
<1
*-
to
FIGURE
I
to
«-
of moments
I
for
it
6. Comparison
H
spectrum
d
ci - 6
i i i it ii i i
c—.
^
Ii » < i t i i i i I i i i i fi i i I Ii i i i
*0
O
I
—
/\,S^
|
*0
—■
o
r<
*
W>
rt
'
Ii I
q
(i
1
>dy
R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission.
to
I—
•---- 1
FIGURE
7.
Harrlson-Zel'dovich
spectrum
for fl = 0.05
h-*-
I—
-
CO
lo
0
6
t
1
I
cC
/
R eproduced with perm ission o f the copyright owner. F urth er reproduction prohibited w ith o u t perm ission.
h -— I
-
<r~
do
in
o
o
|— «--1
-p
O
(D
Q
m.
JG
0
T>
"8
I—I
<D
tSI
1
lo
g
CO
----- 1
1
w
CO
I
*-
cO
C3
M
ci*'
lo
d
Uj
6
I
i
I
Or
\
R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission.
■' h - — t
So
in
c*—
U
o
<+H
•p
o
<D
CU
CO
£
o
•H
>
O
I
.----- 1
T)
(u
t-i
I
C
o
CQ
•H
**)
w
CTv
Of
o
E
v>
o
o
t
T
R eproduced with perm ission o f the copyright owner. Further reproduction prohibited w ith o u t perm ission.
Zel'dovich
spectrum
for A = 1.5
OO
FIGURE
10. Harrison
cO
o
or
4c*c>
a ox/ I N/
4
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G>-
I
t
© ---- 1
I
<D60
~ < i? t5
e xt* ^
I
d>-
y) u) Vjj
0.2
© 0 ©
I
of Spectra'for
o-
-Q-
I---- © ------- 1
-<D-
F
<*)
Qr~
cv
■^r
i i i t—i-i I i , i i I i i i i I i i i i I i t i i
in
b
0
O
nr
1
I
/\
'd 1 t /
v l y
4
R eproduced w ith perm ission o f the copyright owner. F urth er reproduction prohibited w ith o u t perm ission.
11.
& -------
FIGURE
i
Comparison
i----- © —
i
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FIGURE 14. Comparison of non-scale-invaiant
spectra for fl = 0.05
R eproduced w ith perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission.
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FIGURE 15.
3
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Conparison of non-scale-invariant
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FIGURE 17.
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Comparison of non-scale-invariant
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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FIGURE
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Gravitational wave moments
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