Rotational velocities in large scale structure

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ROTATIONAL VELOCITIES IN LARGE SCALE STRUCTURE


By

JAMES H. COONEY















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2004

















For my parents


Digitized by the Internet Archive
in 2011 with funding from
University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation


http://www.archive.org/details/rotationalveloci00coon














ACKNOWLEDGMENTS

First, I wish to thank my advisor, Dr. Jim Fry, without whose wisdom and

patience this dissertation would never have been completed. This project was his

idea from the start and his assistance along the way has proved invaluable. I also

wish to thank my dissertation committee members, Drs. Steve Detweiler, Bernard

Whiting, Peter Hirschfeld, Art Hebard, and Fred Hamman.

Thanks go to my parents, Peter and Adele Cooney, and my brother Dan for

their support over the years in every way that I needed it. My gratitude also goes

to my friends in the department, including Kevin McCarthy, David Red, and Brian

Thorndyke, for sharing in the fun, and my SMCM chums, affetionately known as

The Herd, for many years of great times.














TABLE OF CONTENTS
page
ACKNOWLEDGMENTS ............................. iii

LIST OF TABLES ................................. vi

LIST OF FIGURES ...... ...... ..... .... ............ vii

ABSTRACT ................ .............. ..... ix

CHAPTER

1 INTRODUCTION .... .......................... 1

1.1 Overview .................. ......... ...... 1
1.2 Motivation and Outline .......................... 3

2 FOUNDATIONS ............................... 5

2.1 Gravitational Instability ....................... 5
2.2 Statistics ........... ............. ...... .. 12
2.3 Small-Scale Considerations ..................... 15

3 VELOCITY MODES IN PERTURBATION THEORY .......... 19

3.1 First Order Solutions .................... ..... 19
3.2 Second Order Solutions in Real Space ............... 20
3.3 Second Order Solutions in Fourier Space ............. 22
3.4 Third Order Solutions in Real Space ............... 26
3.5 Evolution of Vorticity .................... ..... 30

4 THE CMB AND THE SUNYAEV-ZELDOVICH EFFECT ........ 32

4.1 Introduction ..... ........... ......... ..... 32
4.2 Anisotropies in the CMB ........ ... ............ 34
4.3 Primary Anisotropies ......................... 34
4.4 Secondary Anisotropies ........................ 36
4.5 Thermal and Non-Thermal Sunyaev-Zeldovich Effects ...... 40
4.6 Kinetic Sunyaev-Zeldovich Effect ................. 41

5 KINETIC SZ ANGULAR POWER SPECTRUM INGREDIENTS .... 42

5.1 Introduction ....... ... ... ... ........ ...... 42
5.2 Linear Power Spectrum ............ ... ........ 42









5.3 Nonlinear Power Spectrum . . ... 44
5.4 Angular Power Spectrum Machinery . ... 49
5.5 Ostriker-Vishniac Angular Power Spectrum . ... 57
5.6 Nonlinear Angular Power Spectrum . ... 58

6 SZ POWER SPECTRUM INCLUDING ROTATIONAL MODES .... 63

6.1 Introduction ................... ......... 63
6.2 First-Order Results .......................... 63
6.3 Non-Linear Results .......................... 64

7 CONCLUSION .................... ............ 84

APPENDIX

A CONSERVATION OF ANGULAR MOMENTUM IN AN EXPANDING
UNIVERSE .......................... ...... 86

B SPHERICAL COLLAPSE AND THE DERIVATION OF 6c ....... 88

C PHYSICAL CONSTANTS AND COSMOLOGICAL PARAMETERS 93

D CODE FOR CALCULATING ANGULAR SZ SPECTRUM ....... 94

REFERENCE LIST .................. ... ...... ..... 105

BIOGRAPHICAL SKETCH ........................... 110














LIST OF TABLES
Table page

5-1 Linear Power Spectrum Parameters ................... 43

C-1 Physical Constants and Cosmological Parameters . .... 93
















LIST OF FIGURES


Figure

4-1

4-2

4-3

5-1

5-2

5-3

5-4

5-5

5-6

5-7

5-8

5-9

5-10

5-11

6-1

6-2

6-3

6-4

6-5

6-6

6-7

6-8


Cosmic Blackbody .................

WMAP Angular Power Spectrum .........

Assorted Angular Power Spectrum Data .

Input Linear Power Spectrum ...........

a(M) vs. M . ..... ...

Bias Parameter Versus Mass . .

Power Spectra for NFW Halo Profile .

Power Spectra for Moore Halo Profile .

Power Spectra for Mixed Halo Profile .

Breakdown of Non-linear Power Spectrum .

Distance vs. Redshift ..... ...........

Ostriker-Vishniac Angular Power Spectrum .

Nonlinear Angular Power Spectrum .

Kinetic and Thermal SZ Effects . .

Ce's for First Order Rotational Velocities .

Moore Profile Mapping ...............

NFW Profile Mapping ...............

Moore Collapse Transforms .............

Moore Collapse Fit .................

NFW Collapse Transforms .............

NFW Collapse Fit . . .

Comparison of Density-Density Power Spectra .


6-9 Different Halo Profiles and the Density-Density Power Spectrum


page

. 33

. 37

. 38

. 45

. 48

. 50

. 51

. 52

. 53

. 54

. 56

. 59

. 61

. 62

. 65

. 67

. 68

. 74

. 75

. 76

. 77

. . 79









6-10 Angular Spectrum Due to Rotational Mode . ... 81

6-11 Angular Spectrum Due to Rotational Mode Comparison ...... ..82

6-12 Linear and Nonlinear Rotational Mode Contributions ... 83

B-1 Radius and Time in Spherical Collapse . ... 89

B-2 Densities in Spherical Collapse . . 92














Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

ROTATIONAL VELOCITIES IN LARGE SCALE STRUCTURE

By

James H. Cooney

August 2004

Chair: James N. Fry
Major Department: Physics

In perturbation theory studies of the large-scale structure of the universe, it

is common procedure to ignore the effects of an initial rotational velocity field.

This has been justified by the fact that in the linear regime, rotational velocities

decay with the expansion of the universe, a simple result of conservation of angular

momentum. In this dissertation, we explore the consequences of retaining this mode,

based on the assumption that gravitational collapse should lead to the return of

appreciable rotational velocities. We extend the classical perturbation theory to

include rotational modes up to the quasi-linear third order and find that these modes

contribute in a non-trivial manner. We then include these modes in the non-linear

regime, incorporating them into the halo model. We study a specific application of

non-linear collapse, the calculation of the kinetic Sunyaev-Zeldovich effect, with the

inclusion of rotational velocity modes. We find that these modes do not contribute

significantly at any angular scale available to modern observation.














CHAPTER 1
INTRODUCTION

1.1 Overview

Cosmology, the study of the nature and evolution of the universe as a whole,

has emerged in the last one hundred years as a mature science. Though cosmol-

ogy encompasses a great many fields of study, we will primarily be interested in

what is called large-scale structure, the study of the general distribution of matter

and energy in the universe. To understand this distribution today, we must first

understand what processes underlie its evolution.

The first part of the twentieth century was marked by a great number of ad-

vancements in the physical sciences, not the least of which was Albert Einstein's

development of the theory of relativity. This is a theory in which time and space

are no longer absolute, and the physics of Newton and his followers no longer ap-

plies. One of the strangest predictions of the theory was that the universe may not

be static, that is, that the universe as a whole may be expanding or contracting.

If it were expanding, then at some point in the past, everything must have been

much closer together, a tiny speck that expanded into everything in the universe

today. This state of affairs became known as the big bang. Einstein himself held

this possibility in distaste, and when it was shown that a static universe was, in

fact, necessarily unstable, he added another term into his equations, the cosmolog-

ical constant, to allow for a stable universe. Not long after, the astronomer Edwin

Hubble made the discovery that galaxies far away from our own are receding from us,

in fact the farther away they are, the faster they are receding. The only conclusion

to be drawn was that the universe is expanding. So the distribution of matter and

energy in the universe is first influenced by the general expansion of the universe.








The other process which is believed to play a major role in shaping today's

universe is gravity. It is believed that tiny irregularities early in the universe grew,

as the universe expanded, by gravitational attraction. A volume with a slightly

higher density than the surrounding region would grow by gravitational accretion

until it had gathered enough to form the stars and galaxies we see today. In actuality,

however, what we see is but a tiny fraction of all that is out there. The universe is

dominated by mysterious "dark matter" and even more mysterious "dark energy."

Dark matter is matter that we cannot see, but it is influenced by gravity in the same

way as normal matter. One of the primary goals of cosmology today is to determine

the nature of dark matter. Dark energy is even less well understood, but its presence

is implied by a number of current intriguing observational and theoretical findings.

It is believed responsible for the fact that the rate of expansion of the universe seems

to be increasing. Curiously, it can be seen as a realization of the very cosmological

constant that Einstein called his greatest blunder.

With these two ingredients we are in a position to describe the evolution of

both fluctuations in the density field of the universe and fluctuations in the peculiar

velocity, the quantity of central interest in this dissertation. The peculiar velocity

is simply the velocity of some cosmological object with the velocity due to the

expansion of the universe factored out.

One way to get a handle on the nature of the distribution of matter and energy

today, as well as far into the past, is to look at fluctuations of the temperature of the

cosmic microwave background (CMB). The CMB is a steady stream of microwave

light that reaches us after a journey of more than 13 billion years. The light comes

from a period of time called recombination, just a few hundred thousand years after

the big bang, so it is able to indirectly give us information about the state of the

universe at that time. It also contains information about the universe of the more

recent past and present because some of it has interacted with galaxies and clusters








of galaxies on its way to us. The peculiar velocities of these foreground objects

cause distinct fluctuations in the temperature of the CMB, a process known as the

kinetic Sunyaev-Zeldovich (kSZ) effect. The particular goal of this dissertation is to

determine the role rotational peculiar velocities play in this effect. It is hoped that

near-future astronomical missions can provide observations of these fluctuations to

verify the predictions made by the powerful theoretical framework.

1.2 Motivation and Outline

The initial conditions for formation of large-scale structure are imprinted at

the time of the decoupling of matter from radiation. The peculiar velocity field of

matter at that time can be decomposed into two modes. The first, called the lon-

gitudinal mode, is a purely radial velocity. The other, called the rotational mode,

is, as the name suggests, a pure rotation. In most cosmological applications it has

been common procedure to disregard the rotational mode of the peculiar velocity

because it is seen to decay as the universe expands, a simple consequence of the con-

servation of angular momentum. While this is indeed true, conservation of angular

momentum demands the return of appreciable rotational velocities upon gravita-

tional collapse of cosmological objects. This means that for applications where

nonlinear (large density fluctuations) scales are important, the rotational velocities

may make important contributions. One application where this may be important

is in the aforementioned kSZ effect, where photons from the CMB are scattered by

moving, collapsed, foreground objects. It is hoped that the next generation of CMB

anisotropy observations will be able to see this effect.

The basic model for understanding the evolution of densities and velocities, the

gravitational instability picture, will be discussed in chapter 2. Chapter 3 will ex-

plore the perturbation theory approach with the rotational velocity modes included.

In chapter 4 we will introduce the physics of the CMB and its anisotropies. Chap-

ter 5 will discuss the ingredients that go into a calculation of the expected angular






4

power spectrum of CMB temperature fluctuations and then show the results for

the quasi-linear and nonlinear regimes, both without inclusion of rotational velocity

effects. In chapter 6 we will build a model which includes rotational velocity effects

and calculate the expected angular spectrum with this inclusion. Chapter 7 will

contain concluding remarks.














CHAPTER 2
FOUNDATIONS

2.1 Gravitational Instability

One of the great cosmological puzzles of the last half century has been ex-

plaining the nature and evolution of inhomogeneities in today's universe. Einstein's

cosmological principle makes the claim that on the largest scales the universe is

homogeneous and isotropic, and this assumption is strongly supported by numerous

observations.' On smaller scales, however, the universe is clearly quite inhomoge-

neous, as matter is clustered into stars, galaxies, clusters of galaxies, and beyond,

with structure extending to the great walls and voids seen on scales of a few percent

of the observable universe.

The leading picture for understanding the growth of these inhomogeneities from

the early universe to today is known quite broadly as the gravitational instability

model. The basic assumption of this model is that gravitation alone is responsible for

the growth of inhomogeneities from their seeds in the early universe to their present

state. The exact origin of these seeds is still something of an open question, but

it is fairly well accepted today that they are inflationary amplifications of quantum

zero-point fluctuations.2 Quite generally, perturbations can be of two orthogonal

modes, adiabatic or isocurvature, or a linear combination of the two. Adiabatic per-

turbations are those where volume elements are uniformly contracted or expanded,

causing equal changes to the number densities of matter and radiation. This obvi-

ously causes perturbations in the energy densities, and, in fact, affects the energy

density of matter differently than that of radiation. The orthogonal mode, consist-

ing of isocurvature perturbations, describes entropy density fluctuations where the








energy densities remain smooth. Number density fluctuations in one species are bal-

anced by those of the others. At times well earlier than the era of matter-radiation

equivalence, both modes are distinct and take on important roles. At and after the

matter-radiation equivalence era, we are principally interested in sub-horizon sized

fluctuations, and sub-horizon sized isocurvature perturbations become adiabatic as

pressure differences lead to actual density fluctuations. Thus beginning at the era of

equivalence we can effectively treat all perturbations as adiabatic and use Newtonian

fluid dynamics to follow their evolution.

The use of Newtonian mechanics in place of general relativity in describing

the evolution of density fluctuations is justified so long as the characteristic size

of the perturbations is small compared to the horizon size and the magnitude of

the density perturbations is not large, thus ensuring nonrelativistic gravitational

potentials. To find the equations that govern such evolution we begin by writing

down the basic equations for an ideal, pressureless fluid. The first is the continuity

equation, expressing conservation of mass.


O( ) + V, (P) = 0, (2.1)
orr

or

S+ Vr = -PVr u. (2.2)

Here p is the density, r is the proper distance, and u its time derivative. We can

also write down the Euler equation, expressing conservation of momentum,

d = -(0 + Vu -= Vr(. (2.3)
dt & r P

Here p refers to pressure, which we will set to zero for the remainder of this discus-

sion. When density contrasts become large, as in the collapse of a cluster of material,

pressure again becomes important, but at that point the perturbative approach we

are now exploring fails. The ( in the above equation is the gravitational potential.








The final necessary fluid flow equation is the Poisson equation

V = 47rGp, (2.4)

where G is Newton's gravitational constant. Now we wish to introduce a change of

variables to "co-moving" coordinates, x = r/a(t). a(t) is just the cosmological scale

factor, as it appears in the most general possible line element for a homogeneous,

isotropic, expanding universe, the Robertson-Walker line element3

2 C2dt2 2(t dx2
ds -cdt + t)1- 2) + x2 (2.5)

The change of variables to co-moving coordinates is essentially a "subtracting out"

of the effects of the expansion of the universe. With this definition of x we find

u = r = ax+xa

= Hr + v, (2.6)

where H = a/a. H is known as the Hubble constant, which is a function of time.

It is normally expressed as H = 100hkm s-Mpc-1, with the present value of the

dimensionless constant being h = 0.71 0.04.4 The velocity in these coordinates, v,

is called the peculiar velocity, and is just the velocity of the material in excess of its

expansion velocity. It is the "perturbation" to the background smooth expansion.

Before continuing our look at the fluid flow equations, we should pause to take

a closer look at the behavior of a(t) and its time derivatives. To begin we express the

pressureless Euler equation in terms of the new co-moving coordinates, noting that

in the unperturbed case, dx/dt = 0, and that the solution for (o in the unperturbed

Poisson equation is 4o = (2/3)rGpor2. This gives us

d2 V 2
dt (ax) = IrGpOa2 2, (2.7)
dt2 a 3








which leaves, (as V(x2) = 2x),

4
a = --7rGpoa. (2.8)
3

Here the V refers to differentiation with respect to x. We now have the behavior

of i, which we will need later in this discussion. We gain further information by

integrating the above equation once over time. This gives


( -)2 Gpo + -. (2.9)
a 3 a

The constant of integration, C, is seen to be proportional to the k in the Robertson-

Walker metric, which is a measure of the overall curvature of the universe. The most

recent findings4 suggest that the universe is flat, so we set C = k = 0 and ignore

the contribution of curvature to the overall energy density of the universe for the

remainder of the discussion.

Now, back to the discussion of the fluid flow equations. Since we are interested

in examining the evolution of the density fluctuations, we write the density as p =

po[l + 6]. Our goal then is to determine the evolution of 6 and v. po is the mean
density, which satisfies po oc a(t)-3 and therefore has a time evolution fio = -3poH.

Finally, we note the identity that for any function f under this change of coordinates


S- Vf where V -- (2.10)
O& "t a a

Now we are in a position to rewrite the fluid flow equations in terms of the

co-moving variables and the perturbations, 6 and v. The continuity equation now

looks like
( V) po[l + 6]+ PoV [(1 + 6)(&x + v)] = 0, (2.11)
a / a
which simplifies to
S+ -V [(1 + 6)v] = 0. (2.12)
a








Poisson's equation is now

-V2b = 47Gpo(1+ 6). (2.13)

To remove the unperturbed part we write

S= + -7rGpoa2x2, (2.14)

where the second term on the right hand side is the solution to the unperturbed

Poisson's equation, as above. Using this for 4 we have

V24 = 47Gpoa26. (2.15)

At this point we should note that in the above discussion we have assumed the

density to consist only of mass density. This is certainly not at all the case, as fully

seventy percent of the universe's energy density is contained in a mysterious "dark

energy" or cosmological constant that provides a negative pressure and is thought to

be fueling the acceleration of the expansion of the universe. As the name suggests,

however, the contribution to the energy density from the cosmological constant is

smooth and does not influence the evolution of density fluctuations; thus the terms

involving the cosmological constant that belong in the full fluid flow equations cancel

when we reduce to the perturbation evolution equations.

Finally, we look at the Euler equation under the change of coordinates. With

the pressure term and the cosmological constant term left out we have

f a- a- V (ax+v)+ (axz + v) V(aax + v) = -V --nGpoa2x. (2.16)
t a a a a3

Using the result of equation [2.8] for the second time derivative of a(t), this simplifies

to
Ov e 1 1
+ -v + -(v V)v = -V. (2.17)
9t a a a








This equation, along with eqns.[2.12] and [2.15] describe the evolution of 6 and v.

By combining equations [2.12] and [2.17] we can write one second order differential

equation for the evolution of the density contrast.

823 &0 3 1 1
2 + 2aOt 4rGpo = 47rGpod2 + V0 V5 + Vj [(1 + 6)vJ. (2.18)

This equation is written so that the left hand side has only terms of first order

in the perturbed quantities, while the right hand side has the higher order terms.

Assuming the perturbations are small (i.e. 6 << 1) we can, to good approximation,

set the right hand side equal to zero. When 6 approaches unity, the above equation

can be used to generate contributions to second order or higher. Analyses of these

terms fall under the heading of "weakly non-linear" calculations and have been often

explored.58 At 6 > 1 perturbation theory breaks down and different methodologies

must be pursued.

In order to solve for the first order behavior of 6 we need to go back and look

at the time dependence of a(t). Equation [2.9] provides a differential equation for

a(t) which we can solve, noting again that po oc a-3. The solution is a(t) oc t2/3, so

that a/a H = 2/3t. Our differential equation for 6 is now, to first order,

082 4 08 2
+ 6 = 0 (2.19)
t2 + 3t O 3t2

The two solutions are 6 c t2/3, the growing mode, and 6 oc t-1, the decaying mode.

We will take a closer look at these solutions in the next chapter, as well as solutions

to higher order in perturbation theory.

With these solutions in hand we can now begin to look at the behavior of the

quantity of central interest in this dissertation, the peculiar velocity. We begin by

noting that by observation we can write the general solution to the linearized version








of equation [2.12] (ignoring the V (6v) term) as

F(x)
v(x, t) = -aVV-(x, t) + F() (2.20)
a

where F is an arbitrary function which satisfies V F 0. We will pursue the

higher order terms in the next chapter. Here the V-2 operator simply refers to
the solution of the Poisson equation, V-26 = 4/4rGpoa2. It is the second term in
this expression for the velocity which will be of central importance for the majority
of this dissertation. We can see that it decays as a-1, and thus is typically (and
often justifiably, for the large scales we are dealing with in the perturbation theory
regime) ignored. The decay of this term is expected as a result of simple conserva-
tion of angular momentum in an expanding universe. Length grows by a factor of

a(t), so the product of length and rotational velocity should remain constant. See
Appendix A for a more rigorous derivation of conservation of angular momentum in

an expanding universe. Of course, when objects collapse, on small scales, the same
conservation of momentum demands the return of appreciable rotational velocities.
We can write this solution more explicitly by writing 6(x,t) oc D(t) where
D could be either the growing or decaying time dependence, and defining f -

(a/D)(dD/da). If we only retain the growing mode for D, and we can ignore the
relativistic background, as we already have, then f can be approximated by a simple
function of Q, f(fQ) = Q0.6.1 Additionally, the Poisson equation is familiar from,
among many physical applications, electromagnetism, and its solution is

0 = -poGa2 d3x, 6(x') (2.21)
j 1x' x\

So, ignoring the divergence-free part of the velocity for now we can write

S= Haf vJ d 36() 1 (2.22)
47r J aX1








We see that without the rotational term the velocity is a potential flow. This

fact is used to great advantage in many applications, including a powerful method
of reconstructing the three dimensional velocity field from redshift measurements,

known as POTENT.9'10

Finally we note that the velocity is sometimes written in terms of the peculiar

gravitational acceleration, defined by g -Vo/a. We can then write

Hfg
S= fg (2.23)
4rGpo

The disregard of the rotational term means that the peculiar velocity and accelera-

tion are parallel.
2.2 Statistics

Now that we have some idea of how density fluctuations and peculiar velocities

should act (at least on large scales, for now) we need to be able to compare these

predictions to observations or to N-body simulation results. Of course it is not the

density or velocity fluctuations at any one point that are of primary interest, but

the statistical distributions of said quatities. The distributions of such quantities are

defined by their moments. For a one dimensional variable, say x, with a distribution

f(x) the moments are defined by

(xn) = dxf(x)xn. (2.24)

This is easily generalized to fields, and the first few moments of the density, p(x),

are written

(p(x)) = po, (2.25)

(p(x1)p(x2)) = po2(1 + 12), (2.26)


(p(xi)p(x2)p(x3)) = po3(1 + 612 + 13 + 623 123).


(2.27)








Here ( is the irreducible two-point correlation function and C the irreducible three

point correlation function. Of course this hierarchy can be continued, but the two

and three point functions and their Fourier transforms are the most commonly used.

The two point correlation function, (, is also seen to be the second moment of the

density perturbation 6,

(PlP2) PO2 ((Pl Po)(p2 Po))
12 ---- = (6162), (2.28)
2P Po

where the subscripts 1 and 2 indicate a function of x1 or x2 respectively. In the

second line we have used the fact that (61) = (62) = ((l1,2) -P)/Po = 0, by equation

[2.25].

The physical meaning of ( is easily appreciated when we look at its definition

in terms of a discrete quantity, such as the number counts of galaxies. Assuming no

bias between mass distribution and galaxy distribution, we can write the probability

of finding a galaxy, which we assume to be a point-like object as far as the scales we

are interested in are concerned, in an arbitrary infinitesimal volume as dP = p(x)dV.

Then the probability of finding one galaxy in each of two different volumes is

dP = (p(x)p(x2))dVldV2 = p02[1 + 1(X12)]dVidV2. (2.29)

We can see then that (, as its name suggests, is simply the correlation between the

two galaxies. That is, if ( is positive for some x12 then galaxies are clustered at

that length scale. Given one galaxy there is a greater chance of finding another at

that distance than would be found by chance, as determined by the average density
of the universe. Having said that we assumed no bias between mass distribution

and galaxy distribution, we must at least acknowledge that such a bias may exist.
At large scales there appears to be no bias, so the two correlation functions are

equivalent, but on small scales a significant bias may exist. This is the likely source








of differences between correlation functions calculated from observations and those
from N-body simulations and analytical calculations.1
It is often useful to examine density fluctuations and their associated statistics
in Fourier space. In fact, the second moment of the transform of the density pertur-
bation, the power spectrum, is much more often calculated than ( itself. We define
the Fourier transformed density fluctuation

(k)= d 3(x)eik'x, (2.30)

then the power spectrum is defined by

(J(kl)J(k2)) = [(2r)36D(kl + k2)]P(k), (2.31)

where P(k) is the power spectrum and 6D is the Dirac delta function. Here k is
the magnitude of either wave vector (as the Delta function ensures they are equal.)
Isotropy demands that the power spectrum is a function of magnitude only. With
these definitions, the power spectrum can be written in terms of the two point
correlation function like
P(k) = d3(x)jo(kx), (2.32)

where jo(kx) = sin(kx)/kx is the zeroth spherical Bessel function. The same proce-
dure can be undertaken for the higher moments.5'12
Finally, we wish to examine the moments of the peculiar velocity field. As we
already have a relationship between the peculiar velocity and the density fluctua-
tions, it is straightforward to find the moments of the (irrotational) velocity field in
terms of the correlation functions. For example,
(V2) = df x d (x')V I ] dx"r(z" ")V X3

= (Haf)2 gy/y.3
= (H f dfy(y)/y. (2.33)
47r








gives the second moment of the peculiar velocity in terms of the two point correlation

function. Many authors will instead calculate moments of a variable proportional to

the velocity divergence, 0 (1/I)V v. This procedure obviously loses its utility if

the rotational, or divergence-free, mode of the velocity is kept. Many examples can

be found in the literature.8,12,13 Another interesting set of moments to calculate are
those of the difference between the velocity of two galaxies. For example the first

moment of one component of the density weighted pair velocity is

v ((1 + 61)(1 + 62)(1 V2)
(v(2 P = (2.34)
((1 + 61)(1 + 62)) (2.34)

where the subscripts refer to functions of xl or x2.

One reason these are interesting is that they can be easier to calculate from

actual data, due to difficulties in determining distances to galaxies independent of

redshift.13

2.3 Small-Scale Considerations

The previous perturbative analysis works well as long as the density perturba-

tions are small. (6 < 1.) When 6 becomes large we enter the non-linear regime and

different analytic methods are required. This non-linear regime represents objects

which have collapsed, which we can recognize as galaxies or clusters of galaxies with

their associated dark matter halos. We typically do not speak of individual stars

as collapsed objects in this sense, as much more complicated physics contributes

at those scales. There are two main ingredients in the description of non-linear

density perturbations. First one must know something about the distribution of
sizes of the collapsed objects. This has traditionally been accomplished using a

method known as the Press-Schechter approximation,14 which has been improved

upon since its original derivation.15'16 To understand the Press-Schechter approach,

we first remember that the density contrast, 6, is a Gaussian random field, or at least

is assumed to be in this derivation, and we define its standard deviation, o (62)1/2.






16

We can then write the probability that the density contrast in some randomly placed

window exceeds a critical density for collapse, Sc, as

1 "
P =- d6e-62/2 (2.35)
Sa(27r)1/2 c 2 )

where a is a function of the window size, or of the contained mass, M. This can also

be viewed as the fraction of perturbations within the window whose mean density

exceeds 6,. Press and Schechter themselves realized that this would only take into

account the mass contained in the overdense regions of the universe, and since the

mass in the under dense regions will eventually accrete onto the collapsed objects,

they simply multiplied their result by a factor of two. Now, to find the number

density of collapsed objects in the range dM (actually, it is typical to write the

result in terms of logarithmic interval d In M) we differentiate the above probability

and multiply by the inverse of the volume, po/M, (which is simply the number

density of collapsed objects of that size if all mass were contained in objects of that

size.) This gives us the result

dn (2 1/2 dn po 2 ln
dlnM = ) M d InM (

where v 6c/a. The value of the critical density contrast for collapse is c6 = 1.686.

(See Appendix B for a derivation and discussion of this result.)

Though the Press-Schechter mass function has been the standard model used

since its inception, it seems to predict too many low mass halos, whereas N-body sim-

ulations and astronomical data suggest that the low mass objects are more strongly

clustered and thus should properly be classified as higher mass objects. There have

been many recent efforts to improve upon the Press-Schechter formalism, including

an analytical attempt, based on the Zeldovich approximation15 (using an ellipsoidal

collapse model instead of the spherical collapse used by Press-Schechter), and fits

to N-body data.16 Sheth and Tormen,"1 find a mass function, valid for N-body








simulations of a variety of cold dark matter cosmologies, given by

dn Po dln c' 1\ /'2\ 1/2
2A 1 + e- '2/2 (2.37)
dlnM M Md lnM v'2q 2 '

where i/ = a-1/2, a = 0.707, q = 0.3, and A = 0.322. Note that the values

a = 1, q = 0, A = 1/2 reproduce the Press-Schechter function.

The second ingredient in describing non-linear perturbations is a knowledge of

the shape and nature of the objects that have collapsed. This has been possible

only through the analysis of N-body simulations, where (possibly) universal profiles

of collapsed objects have been extracted. The first successful universal profile for

dark matter halos was presented by Navarro, Frenk, and White (hereafter NFW) in

1997.17 Their study of N-body results led to the suggestion of a universal density

profile of the form
p(r) -a
p(r) -(2.38)
Pc (r/r,)(1+r/r,)2'
where Pc = 3H2/87rG is the critical density for closure of the universe, 8a is a

characteristic density, and r, is a scale radius which is different for each individual

cluster. (The scale radius is found by N-body simulations to be a rough function of

cluster mass, a fact we will use in chapters 5 and 6.) More recent results of numerical
simulations, performed by Moore and collaborators, with higher resolution have

indicated that the universal profile may instead be18

p(r) __
(2.39)
Pc (r/r,)15[1 + (r/r,)15]

As we have often made mention of the results of N-body simulations, it would
be appropriate at this point to make a few remarks regarding them. Because exact

solutions to non-linear equations of motion are difficult to come by, we are led to try
to use computer simulation to gain insight. These N-body simulations approximate

the continuous density field by a set of discrete particles. The simulations simply








calculate the gravitational force on each particle due to all the other particles, ac-

celerate and move the particles appropriately by a small amount, then iterate the

procedure. Of course this procedure quickly becomes intractable when the number

of particles becomes large, so many methods have been developed to shorten the

calculations. Typically, as in the most popularly used codes, called particle-particle-

particle-mesh, or P3M codes, exact forces are only calculated for particles close by,

with the remaining particles binned onto a grid.19

The use of discrete particles to represent an essentially continuous density cre-

ates its own problems. The most serious of these arises when particles pass close to

one another and scatter at wide angles. Since each particle actually represents a huge

number of smaller particles spaced far apart, such collisions should not take place.

In order to correct for this, forces are "softened" over appropriately small length

scales. This effect, along with the complicated physics that arises out of the colli-

sions that do take place, such as shock heating and radiative cooling, significantly

limits the resolution of N-body simulations.

One of the key ingredients in determining the outcome of the simulations is the

selection of the initial conditions. These conditions depend on the cosmology which

one is trying to simulate, but typically the particle positions are distributed in a

Guassian fashion and the initial velocities are projected onto the growing, radial,

mode.













CHAPTER 3
VELOCITY MODES IN PERTURBATION THEORY

3.1 First Order Solutions

In this chapter we wish to systematically include rotational velocity terms in

the structure of perturbation theory. We begin by rewriting the equations for the

evolution of 6 and v from the last chapter. Equation [2.12] is written with the first

order terms on the left and the second order source term on the right,

1 1
+ V -v = -V (v). (3.1)
a a

We also remember equation [2.17], which we divide by the scale factor a,

0 v v4 2 v V
+ + = V -26 = V -, (3.2)
t Va a 3t 3t2(a a

where we have used the fact that a oc t2/3 and eqns. [2.9] and [2.15]. As we saw, the

first order solution for 6 is

61 = A(x)t2/3 + B(x)t-1 (3.3)

where A and B are arbitrary functions of position which represent the density per-

turbations at some initial time. Here and in the rest of the chapter we adopt the

notation used by Hwang.20'21 The first order solution for v is obtained by setting

the right hand side of equation [3.1] equal to zero and putting in the above solution
for 6. We thus have

= -VV-2(3) + F
a a2
= VV-2 At-1/3+ Bt-2 + t-4/3, (3.4)
3 1








where V F = V C = 0. The last term is the rotational mode. For algebraic

ease we will set B = 0 as this is the decaying mode in the first order solution for
6. The function A then is the density perturbation at some initial time, A(x) =

6(x, to) o0(x). The vector C is the initial rotational velocity, C(x) = vt(x). As
A and C describe initial conditions, they are typically taken to be random variables
with Gaussian distribution.
3.2 Second Order Solutions in Real Space

In order to find second order solutions for 6 we plug the first order solution into
the right hand side of equation [3.1],

1 1
+ V2= -- V (v161). (3.5)
a a

We then have

62+ 2 = V- At2/3 -2 2 At-1/3) + C-4/3
a a 3
= Et + E2t-2/3, (3.6)
3

where
El = V (AVV-2A) and E2 = -V (CA) = -C VA. (3.7)

Doing the same with equation [3.2] we have

a v2 v2 4 2 2
) +- + -VV'62
at a a 3t 3t2

3. 1 V) 3 "
= ([(-VV-2At-1/3) + Ct-4/3] v) (VV2At-1/3 + Ct-4/3)

= t-2/3F + 2t-5/3F2 + -8/3 53, (3.8)
9 3








where

F = -(VV-2A) V(VV-2A)

= V(VV-2A VV-2A) (3.9)
2
F2 = (VV-2A V)C+ C. V(VV-2A) (3.10)

F3 = -(C .V)C. (3.11)

Taking the divergence of equation [3.8] and inserting equation [3.6], we obtain one

second order differential equation for 62,

4 2 2
82 -+ -262
3 3
t-2/3( E1 V F) + t-5/3( E2 F2) 8/3(. F3)]
L 9 3 3
S0, (3.12)

which admits a solution of the form

52 = Ht-2/3 + It1/3 + Jt43, (3.13)

where H, I and J are arbitrary functions of position. Plugging this in and matching

powers of t, we obtain the result

1 3 9
62 = (5E1 2V F)t4/3 + (-E2 + V F)t/3 + (V F3)t2/3. (3.14)
7 2 4

The first term is still the dominant one, and has no contribution from rotational

modes. We will revisit this term in a moment, when we look at its Fourier coun-

terpart. The second term describes the mixing of the rotational and longitudinal

modes. An important feature is the fact that this mode does not decay with time
buts actually grows, meaning that even an initially small rotational component will

have an impact on the future longitudinal mode.The third term is due to the purely

rotational mode, and it decays.








Putting this solution back into equation [3.6], we see that the solution for the

second order velocity looks like

S= VV-2 ( + F)t/3 + (3 E2- V F2)t-2/3+ (V F3)t5/3
a 17 3 2 2 2
+ divergence-free part. (3.15)

To find the divergence free part, we look at the curl of equation [3.8] and note that

V x Fi = 0. This implies

t ( ) + v24 -1 -5/3 + t-8/3F (3.16)
5i a a 3 3 2

where the superscript t indicates the transverse (or divergence-free) part, which is
defined by F~ = ( VaV-2V3)F n.21 Here and for all that follows, a, / represent

spatial indices and thus run from 1 to 3. Equation [3.16] admits a solution of the
form
2= Lt-2/3 + Mt-/3, (3.17)
a
where L and M are arbitrary functions of position with the caveat that V L =

V M = 0. When L and M are solved for and inserted into the total solution we

have

= --2 (-2E, + Fl)t1/3 + ( E2 v F2)t-2/3 ( V F3)-5/3
a 7 3 2 2 2
+F t-2/3 3 t-5/3. (3.18)

We note that to second order the velocity still has only one growing mode, which is

still purely longitudinal. It does, however, have several terms constant in time, one

of which is a potential flow which is coupled to the initial rotational mode and one
which is a pure rotation, coupled to the initial longitudinal mode.

3.3 Second Order Solutions in Fourier Space

It is often convenient to work in Fourier space. (One reason for this, among
many others, is that spatial derivatives become products with the wave-vector.) We








define the Fourier transform in the normal way, i.e.,

(k,t) = d3xS(x,t)e-ik, (3.19)

6(x,t) = 3(k, t)eik. (3.20)

To first order then, disregarding the decaying density term, we can write 61 = At2/3,
and
S= i At-13 + Ct-4/3. (3.21)
a 3 k2
To transform the second order density, we look at each of the terms in equation
[3.14] in order. The first term contains E and the divergence of F1, which can be
written

El = V (AVV-2A) = (VA)- (VV-2A) + A2 (3.22)
1
V- F = -2V.V[(VV-2A) .VV-2A]

= -VaV(V-2A)VVPVV-2A VV -2AVOA. (3.23)

The first term of equation [3.14] can therefore be written

2a = t4/3[ [5A2 + (VA) (VV-2A) + 2(VoVO(V-2A)VaVV-2A) (3.24)

which is a well known result.13,22,23 In Fourier space, products become convolutions,
as defined by
d-~) 3 d3ki d3k6
F,(x) F,(x) = ... D(Eki k)Fi(kl) n(F ), (3.25)

where a quantity with a tilde denotes its Fourier transform and SD indicates the
Dirac delta function.
So we have

2a = 3 f d3k d3k [(27r) 6D [(k+k2-k)]A5 + k F k2 2 (kl-k2)2]
S(27r)3 (27r)3 [(2)D(k1k2-k)](kl)(2) --- + 7 k- 2
(3.26)








The second term contains E2 and the divergence of F2, so we write

E2 = -C aVA (3.27)

V F2 = V- [(VV-2A V)C + C V(VV-2A)]

= (VCa)VaV~V-2A + (VaCC)(VVaoV-2A) + C V3A. (3.28)

Thus
S2b= t1/3 [5 C, A + (VaV3V-2A)(VCa + VaC)] (3.29)

and

J (2r)(27r)3 (2~)a) (kl 2-) k2A2 i2A1 k1-2
2b =t1/3 Jd3k d3k2 [(27r)36D(ki + k2 k)] 52el k2A2 + i2A_ k_ ki k 2 k
(3.30)

where the subscipts on A and C refer to functions of kl or k2. The third term
contains only the divergence of F3, which is

V F3 = -VaC0VCC. (3.31)

Thus
2c = t-2/3 (- VCVC) (3.32)

and
S= t-2/3 d3kl d3k2 9
= -2/ 7r3 f ()3[(2r)3 D(kl + k2 k)] (ki C2)(k2 Cl). (3.33)

The final result for the Fourier transform of the second order density is of course
just the sum of these terms,

62 = 62a + 62b + 2c. (3.34)

The Fourier transform of the second order velocity is found in similar fashion.
Looking at each of the four terms in equation [3.18] in order, we have, for the first








term


2) = VV-2 [t1/31 (-2E + V F)
-a /a 3 (+
= VV-2/3 2A2 + 2(VA) (VV-2A)
17 3


+ -
+ VVp(V-2A)V-O(V-
21


2A)] .


(3.35)


Its Fourier transform is then


k 1/3
k2


-b2a
(i2a a


J (id3 d3k2
(3 [(2)36D(k, + k2)
(27r)3 (27)3


- k)]AlA2


2 2 k2i k2 8 (ki k2)2"
7 +3 kl +21 k fk


(3.36)


This is the full solution when the rotational mode is ignored and is well known in
the literature.1,5,7,22 The second term is


= VV-2 [t-2/3( E2
-a b 2


1
- v.F2)


= -VV-2t-2/3 2CaVcA + VoVV-2A(V"C + C)] (3.37)
1 2"


whose Fourier transform is


f3 k 3 [(27r)36D(kl + k2 k)]
(27r)3(27,)3


ki [ k2
x i2CI k2A2 + iA1 k k1C t7
ki I


(3.38)


The third term is


(V2c
kac


= v-2t-5/3 (v .F3)

= VV-2t-5/3 ( 3VC VC),
(2 )


(3.39)


f2
a )b


S_t-2/3 k
~ "fV







whose transform is

()2 = -t-5/3d3k2[(2 1 )3 D(kl 2 -k) (k C2)(k2 C1)
a k2- (27r)3(27)3 2
(3.40)
The final term is the divergence free part,

2) = F t-2/3 3F t-5/3. (3.41)

Using the previous definitions, we can write this term in component notation as

(2) t-2/3 [VV-2AVOCa + CVVa V-2A
a /da
VV-2V^3(VYV-2AV'YC ) V3V-2V0(CryV V,3V-2A)]
-3t5/3[C7V"Ca VVV-2V (C7V'CO)]. (3.42)

The Fourier transform of this term is

~iz)a d=t 2/)3J (2dr)3k1 d3kl dak2
b = t-2/3 3[(2r)36D(ki + k2 k)]


Ek1 k2 k k2,

-3t I (53 )3 (2) [(2w)36D(ki + k2 k)]

x [(C k2)C2 k (C1 k2)C2] (3.43)

Written in this form it is easy to see that this contribution is purely perpendicular
to the wave-vector k.
3.4 Third Order Solutions in Real Space
Solutions to third order can be obtained in much the same way as second order
solutions. They are necessary in order to calculate quantities which are sometimes
of interest, such as the four-point function or its Fourier analog, the trispectrum.12,24








To third order, equation [3.1] becomes


3 V3
a


= -V. [{ ( V.F3) t-23V
2V {v( a( a
= -V- V F3 t-2/ + (-E2 + V' F2)t1/3

+ I(5E 2V F)t4/3} V-2( At-/3) + Ct-4/3

-V. (At2/3 VV-2(t1/3(-2E1 + V Fj)
1 7 3
+t-2/3( E2 F) + t-5/3( 3))+ Ft-2/3- 3t-5/3]
= tPo2 + -2P, (3.44)
= tP0 + P1+ t-1P2 + t-2P3, (3.44)


where


Po = -V. (5E, 2V F1)VV-2(-A)
7 8 3
+AVV-2(-2E1 + V Fj) (3.45)

Pi = -V. (-E2+ F2)VV-2(- A) + AVV-2 E -F2)
1 2 2 2
+7(5E 2V -F)C (3.46)

P = -v. ( F3)VV-2(- A) + 3(-E2 + V.F2)C

+AVV-2(3V" F3)- 3AF (3.47)

P = -V. ( FI)C (3.48)

Po contains all pure longitudinal couplings, P1 and P2 contain mixed couplings, and
P3 contains the purely rotational coupling.
Similarly, to third order, equation [3.2] becomes

O V3 34 2 2 -2 V1 V2 V2 V
Sa 3 3 a a a
= Qo + Qt-1 + Q2t2 + Q3t-3, (3.49)








where

2 1 8
oQ = -[VV-2(--A) V1][VV-2 (-2E1+ V -F1)]
3 7 3
-[VV-2((-2E + 8V Fi) V][VV-2( A)]
7 3 3
2 1
Q1 = -[lV-2(--A) V][VV-2( E2 v F2) + F]
3 2 2
3 1 2
-[(VV-(E2 V F)+ F) V[VV-2(-A)]
2 22 3
21 8
-(C. V)[VV-2 (-2E1 + V-F1)]
7 3
21 8
-[(VV-2 [-2E1 + V FI]) VIC,
7 3
Q, = -VV-2(_ A) V]l[VV-23 .F3)- 3F]
3 2
-[(VV-2 [3V F] 3F) VI[V-2(_ A)]

-[C V][VV-2(3E2 V F2)+ F]
2 2
3 1
-[(VV-23E2 V- + -VIC,
-V. F2] Ft) V]C,
2 2
Q3 = -(C -V)(-3F ) (-3F V)C. (3.50)

As before, we combine the divergence of equation [3.44] with equation [3.49] to
produce one second order differential equation for 63. This is



1"
3 + t-1 t-23 = (-7 Po + V. Qo) + ( P4 + V Q)t-1

+(- P2+ VQ2)t-2 + (P3 + V Q3)t-3, (3.51)
3 3

which admits a solution of the form

63 = Rt2 + St + U + Vt-1. (3.52)

Plugging in this solution and matching powers of t gives us the final solution

1 7 3
63 = (- 0 + VQo)t2 + (P,+V.Q)t
4 3 23
3 1 12
S(- P2 + V Q2) ( P3 + V Q3)t-1. (3.53)
2 3 23








The first term is the purely longitudinal mode, and is again the fastest growing term.
The second term, however, is also growing, and contains a mixture of longitudinal
and rotational couplings. The third term is a constant in time and is also a mixture of
couplings. The fourth, and still decaying, term contains purely rotational couplings.
Putting this solution into equation [3.44], we find

v3 = VV-2 PO + V QO) (PI + V Q1) + t-IP2 + t-2( P3 + V 'Q3
a It 6 3 o
+ divergence-free part. (3.54)

To find the divergence-free part, we note that V x Q0 = 0 and look at the purely
rotational part of equation [3.49],

S(!) + vtt = Q t-1 + Q2t-2 + 3, (3.55)


which admits a solution of the form

S= W + Yt-1 + Zt-2. (3.56)
a

Matching powers of t, we obtain the final result

a = IV + V 2 + -1P 2 -2 3 + V'Q3)
3 3
+-Qt + 3Qt-1 -Q-2. (3.57)
41 2 2

The fastest growing mode of this solution is the purely longitudinal peice, but there
are several other growing terms as well (remembering that the a in the denominator
of the left side is proportional to t2/3.) The second term is a potential flow, grow-
ing in time, which derives from a mixture of initially longitudinal and rotational
modes. The other growing term, (3/4)Qt, is a purely rotational mode deriving from
a mixture of initially longitudinal and rotational modes. Thus it is at third order in
perturbation theory that we find growing terms in the peculiar velocity associated
with the initial rotational flow. This should be somewhat important in quasi-linear








regimes where 6 is approaching unity, although perturbation theory at third order

and higher has a fairly small window of applicability.

3.5 Evolution of Vorticity

As a corollary to the discussion above, we look at the evolution of vorticity

in perturbation theory. We define the vorticity, C, to be the curl of the transverse

velocity. Remembering the first order solution for the peculiar velocity,

Ha f 1 F(x) ot
v = 4 V d') + a Do + (3.58)
47r |z xj a a

we find that the first oder vorticity is


C( = V x v) = V x (3.59)
a a

Thus to first order, the vorticity decays as the inverse of the scale factor, a. To

explore the time dependence of the next order term, we start with equation [2.17]

from the last chapter. We then use the vector identity

v x (V x v) = V(2v2) (v V)v, (3.60)

which gives us

1a 1 1,\ 1 1
(av) + 1 -v I -vx (Vx v)+ IV = 0. (3.61)
a t a 2 ) a a

Taking the curl of this equation and writing it in terms of C, we have


(aC) = V x (v x ). (3.62)

Using the first order velocity and vorticity as the source for the second order vorticity,

we find


(aC(2) = V x (v(1) x (1))= V x (0 x ) + V x (rot X o). (3.63)






31

The first term dominates for late times and so, disregarding the second term, we see

that

(2)= V x (ro x Co) (3.64)
a
The interesting aspect of this result is that it is the interaction between the first

order vorticity and the first order longitudinal mode. This of course just echoes the

result of equation [3.14] and the fact that as an object collapses any small rotational

velocity is amplified due to angular momentum conservation.













CHAPTER 4
THE CMB AND THE SUNYAEV-ZELDOVICH EFFECT

4.1 Introduction

One possible place in which the effect of rotational velocities may be important

is in the scattering of the cosmic microwave background (CMB) by foreground ion-

ized gas. One of the major predictions of the now almost universally accepted hot big

bang theory is the existence of background relic radiation. This is the radiation seen

from the surface of last scattering. Early in the universe (z >~ 1000) temperatures

were high enough that neutral atoms could not form. At this time photons scattered

readily off the densely packed charged particles. At about z = 1000 the tempera-

ture dropped enough to allow the production of neutral hydrogen and photons were

"frozen out." It is from this epoch that we receive the background radiation. The

existence of this radiation was first predicted by Gamow, Alpher, and Herman in he

1950's, and the search for it began in earnest in the mid 1960's. It was discovered,

rather accidentally, by Arno Penzias and Robert Wilson in 1965.25 Since that time

a great many measurements of the temperature and polarization of the CMB have

been made. The intensity of the CMB has been measured very accurately over more

than three orders of magnitude in frequency and the spectrum is found to be that

of a blackbody with a temperature of 2.725 0.002K.26 Figure [4-1] shows how

well the data fit a Planckian spectrum. One very intriguing aspect of the CMB is

its near total isotropy. The temperature in every direction is identical to a part in

103. This thermal equilibrium over so many causally disconnected regions is now

reasonably well explained by theories of inflation. (In fact, it was one of the principle

motivations for the development of inflationary theories.)
















Wavelength (cm)
1.0


1 10 100
Frequency (GHz)


Figure 4-1.


Measurements of the intensity of the CMBR as a function of
frequency. The solid line represents a perfect blackbody at 2.73
K. The figure is taken from Smoot.26 (See references therein for
sources of data.)


10-17


N

'.



Uv'


1000








4.2 Anisotropies in the CMB

Perhaps more interesting than the problem of explaining the isotropy of the

CMB is explaining its anisotropies. These anisotropies are typically expressed in

terms of an expansion in spherical harmonics,

AT
T (0, ) = atm~Ym(, ). (4.1)
Im
We can then define an angular correlation function by

AT AT 21 + 1
C(T) = r T(,i r2), (4.2)
T T 474

where PI are the Legendre polynomials and C =- (la'12), is the angular power

spectrum. The first, and dominant, anisotropy is the 1 = 1 dipole moment. This

is simply interpreted as due to the motion of the earth relative to the "cosmic rest

frame" as defined by the CMB. Utilizing a simple special relativistic calculation,

Peebles and Wilkinson,27 first noted that the temperature measured by an observer

in a frame moving with respect to the homogeneous background, at an angle 0 with

respect to the motion is
T(1 v2 1/2
T'(0) = T( (4.3)
1 (v/c) cos 04
which can be expanded as

v 1 v2
T'(0) = T[1 + cos + cos 20 + O(v3)]. (4.4)
C 2 C

The amplitude of the dipole anisotropy, as measured by the Cosmic Background

Explorer (COBE) satellite, is AT/T = 3.372 0.014mK.28 This implies a solar

system velocity of 371 Ikm/s towards galactic coordinates (, b) = (264.14 +

0.30,480.26 0.30).

4.3 Primary Anisotropies

After the dipole anisotropy, which, as we saw, is at the AT/T 10-3 level,

is accounted for, the next level of anisotropies are found at AT/T 10-5. These








anisotropies are believed to be due to inhomogeneities in the surface of last scat-

tering. The effect of density perturbations present at that time can be divided into

three categories. The first, known as the Sachs-Wolfe effect, is due to the photons

climbing out of the gravitational potentials of the density perturbations. The mag-

nitude of the effect is really the combination of two effects. First, as the photons

climb out of the well, they are redshifted, or cooled, which gives

AT 65
T = (4.5)
T c2

The photons are also heated, due to the associated time dilation, and you seem to

be looking at a younger, hotter universe. This effect is

At 65 AT 2(.
T = 2 3- 2' (4.6)

since T oc 1/a and a oc t2/3. Thus the total Sachs-Wolfe effect is given by

AT 6_ j
= 6 (4.7)
T 3c2

The second effect related to density perturbations at the surface of last scattering

is called the adiabatic, or intrinsic, perturbation effect. It is simply due to the

fact that denser regions would recombine later, thus look hotter. This effect yields

a temperature perturbation equal to the density perturbation responsible for the

late recombination. The final important source of primary anisotropy is simply the

doppler shift of the photons coming off the moving plasma at last scattering. This

yields a temperature perturbation of

6T 6v.-
T, =(4.8)
T c

where f represents the unit vector along the line of sight. The combination of these

effects lead to the famous acoustic peaks in the angular power spectrum of the

CMB anisotropies. Before recombination, adiabatically overdense regions initially








collapse due to gravity, but are then pushed apart by radiation pressure, setting up

acoustic waves. These standing waves are frozen out at decoupling and are seen, via

the aforementioned effects, in the angular power spectrum. The first, and largest,

peak is associated with a length scale equal to the Hubble radius at the time of

recombination. The size, shape, and position of this peak, as well as the subsequent

peaks, tell us a great deal about allowed cosmologies and the values of cosmological

parameters. For example, as previously mentioned in chapter 1, the location of the

first peak at 200 strongly indicates that the universe is flat, or very close to it.

As mentioned, the first significant detection of the CMB anisotropies was by

the COBE satellite. Since that time, many ground and balloon based measurements

have been attempted. In 2003 the most significant of these missions, the Wilkinson

Microwave Anisotropy Probe (WMAP) completed its first year of data acquisition.

Figure [4-2] shows the angular power spectrum measured by WMAP. The first

acoustic peak is very clearly seen. The second peak is also clear and the beginnings of

a third are hinted at. Figure [4-3] shows the results of a number of other experiments

and how they relate to the WMAP data. The agreement among the variety of

different measurements shows the robustness of the results. WMAP provides a very

accurate picture of the anisotropy spectrum from about = 10 to about = 800.

The large error bars at very low i are due to the fact that we have only one universe

to observe, and at large angular scales one cannot average over significantly many

independent pieces of sky. (This is equivalent to saying that there are only 2e+ 1 m's

to average over.) The limits at high e's are due to resolution limits for the WMAP

satellite. The European Space Agency plans launch of the PLANCK satellite in or

around 2007, which hopes to extend measurements to perhaps f ~ 1200.

4.4 Secondary Anisotropies

The primary temperature anisotropies, due to the nature of the surface of last

scattering, dominate the angular power spectrum at small and medium V's, but
















Angular scale
2


(deg)
0.5


0.2


10 40 100 200 400 800 1400
Multipole moment I


Figure 4-2.


The points represent the data obtained from the first year run
of the WMAP satellite, along with la error bars. The line rep-
resents a best fit ACDM model, and the gray shading represents
the la error due to cosmic variance. Taken from Hinshaw et
al.29


7000

6000


^5000


S4000


S3000
2000
" 2000


1000

0





















7000

6000


-. 5000


p 4000


S 3000
+
"2000


1000

0



Figure 4-3.


Angular scale
2


(deg)
0.5


0.2


10 40 100 200 400 800 1400
Multipole moment I

A collection of recent data from various CMB anisotropy mea-
surement experiments. The line is again the best fit ACDM
model based on the first year WMAP results. Taken from Hin-
shaw et. al.29







they are by no means the only contributors to the spectrum. At smaller scales,

or larger 's, secondary anisotropies are expected to dominate the angular power

spectrum. These effects include the Integrated Sachs-Wolfe (ISW) effect, which,

although it is a secondary effect, is expected to only be seen at large angular scales,

and the thermal and kinetic Sunyaev-Zeldovich (SZ) effects,30,31 which should be

the major contributions to the spectrum at 's greater than about 1200. The ISW

effect is the secondary analog to the primary Sachs-Wolfe effect. When a CMB

photon falls into a potential well on its journey to us it is heated, or blueshifted,

then cooled, or redshifted, as it climbs back out. If the depth of the well remains

constant while the photon traverses it, then the redshift exactly cancels the blueshift

and no trace is seen in the CMB. However, if the depth of the well changes over

the time that the photon takes to traverse the well, then there is a net change in

the photon temperature. The photon may traverse many such regions on its path

to us, thus the word "integrated" in its name. The ISW can further be split into

two contributions, an "early" effect, due the changeover from a radiation dominated

universe to a matter dominated universe, and a "late" effect, due to the changeover

from a matter dominated universe to a A dominated one. (Thought to happening

around now.) The origin of the early ISW lies in the fact that when radiation

is controlling the expansion of the universe, potentials on the scale of the sound

horizon are damped, giving photons crossing them an overall boost in energy. The

early effect provides a contribution to the angular power spectrum in the region of

the first acoustic peak, at low e's. This effect has been seen by comparing data

from WMAP to data from the Sloan Digital Sky Survey (SDSS).32 The late effect is

expected to be strongly damped at medium and higher 's as well, as the photons

pass through a great many regions of overdensities and underdensities. It is hoped

that detections of the late ISW effect at low 's will provide more information about








the dark energy content of the universe, as it is sensitive to the time of switch-over

from matter domination to A domination.

4.5 Thermal and Non-Thermal Sunyaev-Zeldovich Effects

At small angular scales, corresponding to 's over about 1200, the thermal and

kinetic SZ effects should be dominant. The SZ effect is the result of the interaction

of the CMB photons with regions of charged particles between the observer and the

surface of last scattering. The thermal SZ effect is an inverse Compton scattering of

the CMB off thermal electron populations. The result is a temperature fluctuation,

which, at low frequencies, like those of most of the CMB photons, is

AT f kBTe
A JT= -2 eTneaTdl. (4.9)

Recent analysis of the data obtained by the WMAP satellite confirms that the

SZ effect is swamped by the primary anisotropies for all angular scales available to

it, corresponding to e's up to roughly 900.33 Though there is not yet a large amount

of angular power spectrum data taken at high e's (see figure [4-3]), at least one

mission, the Cosmic Background Imager, has measured power up to e 3500 which

seems to be consistent with the power expected from the thermal SZ effect.34'35 More

accurate measures of this power may be useful in constraining the values of some

cosmological parameters.36

There is an additional effect due to scattering off non-thermal electron popu-

lations, such as would be found in cluster radio halo sources. Computation of this

effect is made difficult by the highly relativistic nature of the electrons, but the

low density of such electrons, and the relative rarity of such populations, lead to

insignificant contributions to the angular power spectrum. (Though the search for

the non-thermal SZ effect in specific clusters can prove useful.)








4.6 Kinetic Sunyaev-Zeldovich Effect

There is an additional temperature fluctuation induced in the CMB if the fore-

ground material is in bulk motion. This effect, called the kinetic SZ effect, is simply

due to the doppler shift induced by the motion. The temperature fluctuation along

a line-of-sight direction, j, is given by

= dle- rne v, (4.10)
T c

where T is the optical depth to scattering through the electron cloud, ne is the

number density of electrons in the cloud, and v is the peculiar velocity of the cloud.

Measurement of this effect for particular clusters can be used to calculate the ra-

dial component of its peculiar velocity, which is of significant interest, as the method

does not require an independent distance measurement.37 (It does, however, require

an accurate estimation of optical depth and electron density.) In addition, one must

disentangle it from the thermal effect, as well as other small-scale anisotropies. This

is typically done by making use of the fact that the kinetic SZ effect has a different

frequency dependence than the thermal effect.38














CHAPTER 5
KINETIC SZ ANGULAR POWER SPECTRUM INGREDIENTS

5.1 Introduction

In this chapter we lay out the ingredients for calculating the angular power

spectrum of CMB anisotropies due to the kinetic Sunyaev-Zeldovich effect. The

angular power spectrum can be written in terms of a projection of the actual three-

dimensional power spectrum, which we will explore in the first few sections of the

chapter. We will compare several of the popular models used to build these power

spectra. The final sections will include calculations of the angular spectrum in both

the perturbative regime and the nonlinear regime, without inclusion of the rota-

tional modes. This gives us the necessary background for the calculation including

rotational modes in the next chapter.

5.2 Linear Power Spectrum

One of the main ingredients in all the calculations to follow is a linear order

density-density power spectrum. The most commonly used model is one originated

by Bardeen et al.39 The authors matched a fitting function to results from numer-

ical simulation to produce an analytic form for the spectrum. A slightly improved

version, used by Ma,40 and to be used as the input linear power spectrum in all that

follows is given by

P, Akn[D(a)/Do]2 [ln(1 + alq)/a1 q]2
S[1 + a2q + (a3q)2 + (a4q)3 + (a5q)4]1/2'






43

where

k
q h (5.2)
Fh
Omh
e(l+3/ ,) (5.3)
D(a) = ag(a) (5.4)

Do = D(a= 0) (5.5)
2.50m(a) 5
g(a).5 (a) (5.6)
g(a) (a)4/' QA(a) + [1 + Om(a)/2][1 + 2A(a)/70]
A =6(C/Ho)n+3
A 63 (5.7)
47r
SH = (1.94 x 10-5)QM 1-)2 'ln(")-)e-3(n-j-f4(n-1)2 (5.8)

m(a) = + (5.9)
Qm + a3QA
QA(a) 1- Qm(a). (5.10)

When the cosmological parameters Qm, QA, Qb appear without functional de-

pendence on a, they refer to the values today. Which values to use are determined

by the cosmology one wants to explore. The values used in the calculations that

follow are based on a ACDM cosmological model and values found by recent ob-

servations,4 and they can be found in Appendix C. The values of the numerical

constants are given in table [5-1].


Table 5-1. Linear Power Spectrum Parameters

Parameter Value
al 2.34
a2 3.89
a3 16.1
a4 5.46
a5 6.71
)p 0.785
32 0.05
/33 0.95
/34 0.169








The factor of k" reflects the assumption that the initial power spectrum, (at

decoupling) was a scale-free Guassian process and thus the initial spectrum was a

simple power law. We typically define a dimensionless version of the power spectrum,

given by, A = k3P(k)/27r2. Figure [5-1] shows a plot of the dimensionless linear

power spectrum given above.

5.3 Nonlinear Power Spectrum

Here we present a model for the nonlinear power spectrum first constructed

by Ma and Fry.41 Before moving on, we note that there are other persciptions for

describing the nonlinear power spectrum, the most notable due to Zhang et al.,42

but we follow the Ma and Fry description.

As we noted in chapter 2, the two main ingredients in this construction are the

halo mass function and the halo density profile. We use three different models for

the halo density profile. The first two are the NFW and Moore profiles mentioned

in chapter 2. Following the convention of Ma and Fry we write the profiles, given

by equations [2.38] and [2.39], as

(r)= au(r/rs), (5.11)
Po

where


UNFW(X) =
x(1 + z)2'
1
UM(X) = x.(1 + x) (5.12)

A commonly introduced parameter, which we can use to write 6a and r, in terms of

contained mass, is the concentration parameter, c r20oo/r, where r200 is the radius

within which the density is 200 times the mean. This radius can be written in terms

of mass by, M = 8007rporo/3. Thus we can write the scale radius as


rs = /(5.13)
c 800iPo















I11 I 7ll 11111 I IIIj I 111111 I I I IIII1 I 111111


10






1






0.1






0.01




0.01 0.1 1 10 100 1000
k(h/Mpc)


Figure 5-1. A plot of the dimensionless linear power spectrum, A, =
k3P(k)/27r2, for use in upcoming calculations. This represents
the linear power spectrum today, a = 1.








The characteristic density, Sa can also be written in terms of the concentration

parameter, and is given by

200c3
.,NFW = (5.14)
aN = 3[ln(1 + c) c/(l + c)]
100c3
kaM = ln(1 + c3/2) (5.15)

For the plots in this section we use a model for the concentration parameter based
on N-body simulation data of Jing and Suto43 given by

5a(M) : (M/M) < 1014 (5.16)
c = ^(5.16)
9a(M) : (M/M) > 1014

M = 1.989 x 1030 kg is one solar mass. The third halo profile we will use is
simply a combination of the previous two, as suggested by Jing and Suto.43 Their
simluations imply that the NFW profile is appropriate for cluster halos and the
Moore profile is appropriate for galactic halos and the halos of small groups of
galaxies. The cutoff between the mass of a cluster and that of a small group of
galaxies is about M/Me = 1014, so the halo profile we refer to as "mixed" is Moore's
for (M/M) < 1014 and NFW's for (M/MQ) > 1014.
The other important ingredient is the halo mass function. In this section, we
will look at both the Press-Schechter mass function, given in equation [2.36] and the
Sheth-Tormen mass function, given in equation [2.37]. The principle ingredient in
each of the mass functions, as we saw in chapter 2, is a, the standard deviation of
the density contrast, which is a function of the radius of the sphere within which we
are interested. Written in terms of the power spectrum, we have

R) j 4jr(k2dk.
2 (R)(2) P(k)W2 (kR), (5.17)
Jo (2r1)3








where W is the Fourier transform of the top-hat window function,

3(sin x x cos x)
W(x) = 3 (5.18)

The top hat window function has the value one within the radius R and zero every-
where else and is used to remove the need to worry about tiny fluctuations outside
the region of interest. This can also be expressed as a function of enclosed mass
using the relation, M = 47rpoR3/3. Figure [5-2] shows a plot of a vs. M for the
input linear power spectrum given in equation [5.1].
Finally, we must put this together to make a power spectrum. We first note
that we can write the spectrum as a sum of two distinct contributions, a one-halo
term, from particles within the same halo, and a halo-halo term, from particles in
two separate halos. From Fry and Ma,41 we can write these terms as

P (k) = dMd [r~afi(kr,)]2, (5.19)

P2h(k) = E dMdn r3Oui(krs)b(M) Pun(k), (5.20)

where b(M) is a bias factor given by Jing (1998),44 as

b(M)= + )( +) 2 (5.21)
& 2v4

where, as before, v =- c/a(M). This bias factor is introduced to account for the
fact that dark matter halos cluster differently than the general mass density field.
Clusters form at the peaks of the density fields, thus the bias accounts for the
difference between the statistics of the the peaks and those of the general field.
Figure [5-3] shows the behavior of b(M). Its value is near unity for masses of the
most common clusters. It becomes very large for very large mass clusters, but these
are exceedingly rare occurances. (Which is, of course, the very reason the bias is so
high.) The second factor in the bias parameter is a phenomenological addition to the
original formula based on N-body results and the Sheth-Tormen perscription, and







48








"I 1 '.1 I ["'" I 1111111'"I 1' I 1 '"I 111111111 1. ....1 I 1. I "I





1-








0.1

b






0.01








ml i I 1111 I i 1 l- n il l II 1111l 111n11l i 111111l i I ni iil l 1 1
1012 1013 1014 1016 101' 10 1018 1011 1020 1021
M/M0



Figure 5-2. The linear rms fluctuation of the density contrast as a function
of the mass within a top hat window. The mass is expressed in
multiples of one solar mass (1M = 1.989 x 103kg).








it is therefore neglected when the Press-Schechter formalism is used in the figures

below.

Figures [5-4], [5-5], and [5-6] show the nonlinear power spectra for the NFW,

Moore, and mixed halo profiles, respectively. In each plot the spectrum is plotted

twice, once for the Press-Schechter mass function and once for the Sheth-Tormen

mass function. Also, the linear power spectrum of figure [5-1] is included for com-

parison.

Figure [5-7] shows the nonlinear power spectrum, using the mixed halo profile

and ST mass function, as we will henceforth, broken down into its one-halo and halo-

halo contributions. Of course the halo-halo term dominates at small k, corresponding

to large distances, or large masses, and the one-halo term dominates at large k,

corresponding to small scales.

5.4 Angular Power Spectrum Machinery

Now that we have three dimensional power spectra for both linear and nonlinear

regimes, we need to know how to use these to calculate the angular spectrum of

CMB temperature fluctuations due to the kinetic SZ effect. This machinery is due

to Ma and Fry.45 First we remember from chapter 4 that the fractional temperature

anisotropy due to the kSZ effect is

AT
-() = / dle-'neaT (v/c). (5.22)

We write the electron density as ne = 'ie(1 + 6) where the mean electron density

can be written, f~ = XebPc(l + z)3/mp. Here Xe is the ionization fraction, mp is

the mass of the proton, and Pc = 3H2/8irG is the critical density for closure of the

universe. We can then write the temperature fluctuation as


T () = J dle-r'eal (q/c), (5.23)



















I I """I I "" """ I I I I IIII I I llll I I I


106





105





104





9 1000
-o



100





10





1

liii I 11ln I i 111 li il i i l i i l i l i i l l llid I I i ll I I il I i l il I I Il i lim i
1012 1013 1014 10's 1016 101' 1018 1019 1020 1021

M/Mo



Figure 5-3. The bias parameter plotted as a function of halo mass. Most
halos fall into the mass range M/M = 1012 1015, where the
bias is near unity.


n a i s un i s i s l ss


.. .I ...I .I


I





























--*
.... I Ih f I 1111 .... I .. .. "









r/










I ...












r i r i I i i i iliI i i i i liii 1 11 1111i i i iili[11 i I i l i i-i i


0.01


1000


k(h/Mpc)


Figure 5-4.


Dimensionless nonlinear power spectra using the NFW halo pro-
file. The solid line uses the Press-Schechter mass function, the
dashed line the Sheth-Tormen mass function. The dotted line
shows the linear power spectrum for comparison.


1000




100




10




1




0.1




0.01
















rr1 I Ir 11111 I 111111r I I 1111111 I ~J-IlI IIl


104




1000




100




10


[III I I I 1111 I I I 1111 11 I I 1 11 I I 111111 I I I111111I


10 100


Figure 5-5.


Dimensionless nonlinear power spectra using the Moore halo pro-
file. The solid line uses the Press-Schechter mass function, the
dashed line the Sheth-Tormen mass function. The dotted line
shows the linear power spectrum for comparison.


/
/ .-


0.1




0.1


0.01


0.01


k(h/Mpc)


1000


- -







53






I IFF I F ,'F I- I fill



1000




100




10









0.1




0.01

] iIll I I I I If ill I I 1 I I I l I I I I -L ill I I I il l ll I I 1 1IFl
0.01 0.1 1 10 100 1000
k(h/Mpc)


Figure 5-6. Dimensionless nonlinear power spectra using the NFW profile for
large mass halos and the Moore profile for smaller mass halos.
The solid line uses the Press-Schechter mass function, the dashed
line uses the Sheth-Tormen function. The dotted line shows the
linear power spectrum for comparison.














1II"i I 1111111 I 111111 I I 111111 1111111 111111


II II I t I I 1 1111 1 I 111111 I 1 111111 1 1 1
0.01 0.1 1 10 100
k(h/Mpc)


Figure 5-7.


Dimensionless nonlinear power spectrum compared to linear ver-
sion. The solid line is the nonlinear spectrum, using the mixed
halo profile and the ST mass function. The dotted line repre-
sents the linear power spectrum. The short dashed line is the
one-halo contribution to the nonlinear spectrum and the long
dashed line is the contribution from the halo-halo term.


104



1000



100



10



1



0.1



0.01


1000
1000







where q = v(1 + 6) is the density weighted peculiar velocity, which has the Fourier

transform

q= i+ (2) i 3v(k')S(k k'). (5.24)

The angular two-point correlation function of the CMB temperature fluctua-

tions is defined by equation [4.2]. The power spectrum, Ce can be calculated as a
projection of the three-dimensional power spectrum. This result is due to Kaiser,46

and is simply the Fourier version of a projection originally due to Limber.47

For the density weighted velocity, Ma and Fry arrive at an expression for the

angular power spectrum for the kSZ effect,

d 4--2 2,
C H T x (1 +z)e (5.25)

where P.y is the power spectrum of q along the line of sight vector. The integration

variable, x, is simply the co-moving distance to an object of redshift z. In a model

with matter and cosmological constant, but no curvature, the relationship between

x and z is given by
dz
dx z (5.26)
Ho Qm(1 + Z)3 + (526
Figure [5-8] shows this distance plotted against (1 + z).

One very interesting aspect of a projection such as this is that modes parallel to

the wave-vector k do not contribute, as the crests and troughs cancel each other when

collapsed on top of one other.48 (In actuality this cancellation is only approximate

for long wavelengths, or large angular scales, but is very near total at the small

angular scales we are interested in.) This implies that what we really need to

find is Pq, 2Pq. The factor of two comes from the fact that there are two

independent directions perpendicular to k. It also implies that there is no linear
perturbative contribution to this effect, as in linear theory (without the rotational

mode) q = b(k) oc k. It is the second order contribution, called the Ostiker-Vishniac

(OV) effect, that we will look at in the next section.



















104





8000




S6000




4000




2000




0

0 5 10 15 20
(1+z)


Figure 5-8. The relationship between redshift and distance to an object in a
zero-curvature universe.








The result obtained by Ma and Fry for Pq_ is
d 3k' r
Pqi(k) = (2r)3 (1- pPa(lek k'I)P,(k')
(1 p2)k'
-k k'I P t( k k' )Pg.(k')
Sd3-k d3 k"

+ 1- (27r2)3 (2)3 cos(' -
xP66,,(k k', -k k", k', k"), (5.27)

where 1/' k k', p" k k", and the power spectra are defined by

(6(ka)S(kb)) = (2wr)3D(ka + kb)P66(ka) (5.28)

(i'(ka)ij(kb)) = (27r)36D(ka + kb) ka f P,,(ka) (5.29)

(6(ka)'i(kb)) = (2r)36D(ka + kb)kibPS (ka) (5.30)

(S(ka)S(kb)Vi(kc)i)j(kd))c = (27)36 (Eka)Ie) dP66,(ka, kb, ke, kd). (5.31)

The subscript c in the last equation refers to the connected, or irreducible, moment.
It is important to note that equations [5.29], [5.30], and [5.31] all depend on the
assumption of v oc k, a condition we will later modify.
5.5 Ostriker-Vishniac Angular Power Spectrum
Calucluation of the angular power of CMB fluctuations in the quasi-linear
regime, using second-order perturbation theory, produces the Ostiker-Vishniac spec-
trum.
In this regime, we use first order peculiar velocities and density perturbations
to get the second order power spectra. Taking the Fourier transform of equation
[2.22], we have

"(k) = 6 Sk, (5.32)
ki








so that


pi ) 2- p (5.33)







) 2 3k' ) k(k 2k'')(1 )
P f(k) = i-j (k (5.35)
S(27r)3 k' (k2 + k'2 2kk') (53)

Using the linear power spectrum given in section [5.2] and inserting this in equation
[5.25] produces the spectrum plotted in figure [5-9]. In that figure we have integrated
equation [5.25] from the present, z = 0, to several different possible redshifts of
reionization, zr. The OV spectrum, and the more refined nonlinear spectrum of the

next section, are sensitive to zr and thus observations of this effect will put good
constraints on the ionization history of the universe.42

5.6 Nonlinear Angular Power Spectrum
In the nonlinear regime Ma and Fry suggest an approximate form for Pqg which
is valid for high k, the region of interest in a nonlinear analysis. They use

2 d3k' 2
Pq l(k) = 3 ~-~k(P,,' ( v(k')= 2) P6,(k). (5.36)

In figure [5-10] we plot the angular spectrum obtained from this form, using the
one-halo power spectrum given in section [5.3]. We use a value of (v2)1/2 = 200km/s
for the velocity dispersion. The key feature is that the OV spectrum underestimates
the power at high C's. In figure [5-11] we show the nonlinear kSZ spectrum as it
compares to the expected spectrum from the thermal SZ effect. Note that for very
high t the kSZ effect is dominant.
One factor that is not taken into account in this model is the bias on small
scales between baryonic matter and dark matter. The SZ effect is due to scattering

















10-9


Figure 5-9.


The second-order, perturbative Ostriker-Vishniac angular power
spectrum. The solid line is the OV spectrum, calculated using
a redshift of reionization of 20, the dashed line is the same with
reionization redshift of 6, and, for reference, the dotted line is
the primary anisotropy spectrum.


.-.






























10 10 1000 1 105 10




10 100 1000 104 105 106


10-1o


10-11






+ 10-12






10-13






10-14








off free electrons which may (in fact, most certainly do) cluster differently than the

surrounding dark matter halo. Ma and Fry do address this by also plotting the

spectrum using a somewhat different profile function based on the modeling of hot

gas in galaxies and clusters. This generally causes the power to be lower, but still

greater than in the linear model.

Before moving on to the next chapter and the effect of rotational velocities on

these power spectra, we note that we have not considered the possibility of non-

uniform reionization in these calculations.49,50 Santos et al. calculate that such
"patchy" reionization would lead to significantly higher values for the temperature

power spectrum, and in fact would likely swamp the effect of density perturbations.

Since little is known about the actual inhomogeneity of reionization, we have chosen

to assume it uniform for all that follows.


















I. I


iN

iN

//N




//





11 1111111 I I 1111111 Itl S, 111 111


10 100 1000 104 106 106


Figure 5-10.


The nonlinear angular power spectrum, based on the model of
Ma and Fry. (solid line) The dashed line is the OV spectrum
and the dotted line is the primary anisotropy spectrum. Both
the nonlinear and OV spectra are evaluated to z, = 20.


10-9


10-10 PL


10-"


C2


+ 10-12






10-13






10-14


HI11 1 Tllrllrl 1 1 1111111 1 1 1111711 .r


'"


I I I I I I I I


















10-9





10-10





10-11





10-12





10-13





10-14





10-15


100


1000


Figure 5-11.


The solid line represents the nonlinear kSZ effect spectrum,
while the dotted line represents the thermal SZ spectrum. (Plot
for this spectrum was obtained from C.-P. Ma.) The dashed line
is the primary spectrum.


-1 I IIIII" .111111 I I 111.11. I I 111'111 I 11711 "1



/ T \







I



-- I




















i i i. ." I I l i I i i s i i l l i i i i ii- i i













CHAPTER 6
SZ POWER SPECTRUM INCLUDING ROTATIONAL MODES

6.1 Introduction
In this chapter we calculate the angular power spectrum of temperature an-
isotropies in the CMB due to the kinetic SZ effect with the inclusion of rotational
velocity modes. Before beginning our discussion, we note that attempts have been
made to include rotation in a calculation of the kinetic SZ effect. Cooray and Chen
(2002)51 calculate the angular power spectrum of the rotational kSZ effect using a
perscription for the halo rotations based on work by Bullock and collaborators.52
Importantly, however, this is angular momentum gained through events such as
halo interactions and is different from the rotational modes that would be present
due to conservation of initial angular momentum. Cooray and Chen find a power
spectrum contribution several orders of magnitude smaller than the contributions
seen in the previous chapter, and in fact they anticipate this result given the accuracy
with which Ma and Fry's spectrum matches the N-body data. Of course these N-
body simulations do not include the initial rotational modes, so they would not be
expected to include the contribution we calculate here.

6.2 First-Order Results
We now include the effect of rotational velocities in the SZ angular spectrum by
following the procedure, due to Ma and Fry, laid out in the last chapter. Remem-
bering that we are looking for the power spectrum of the density weighted peculiar
velocity, we had written the Fourier transform of this quantity as

q(k) = (k) + (2)3(k')6(k k'). (6.1)
B~)- B~k~f (27r~)3d3~








We have previously ignored the contribution from the first term above, because for
the growing mode v oc k. The first order rotational mode, however, is perpendicular
to k so we have the simple relationship, Pq = (ai/a)PIv, where P,i is the

power spectrum of the rotational velocity at decoupling and ai is the scale factor at
decoupling.
We expect the rotational velocity dispersion at decoupling to be approximately
equal to the longitudinal dispersion, which is approximately 20 km/s. This means

(2Jd3k'_ (k = ((v)2) w. 400(km/s)2. (6.2)

We can determine the functional dependence of the initial spectrum on the
wavenumber by considering the mass contained within the horizon at a time t.

3 = PoH 0CH 3 = 2GfH) C3 tH
MH = porH = po(ctH) = (67rGpotH)( ). (6.3)

We write it in this manner because the solution to equation [2.9], for zero curvature,
is 67rGpot2 = 1. This means that tH oc MH oc k-3. Finally, we note that

A(k) = kP(k) oc (aH2 4/3 c (k-3)4/3 = k-4, (6.4)

which means that PO1o2(k) = Ak- We see that we need a low k cutoff for the
integral above to converge. We choose a cutoff of k = 10-5(h/Mpc). This gives us
the value A = 3 x 10-16h(Mpc)-4(km/s)2. Inserting this spectrum into equation
[5.25] gives us the angular spectrum of CMB temperature fluctuations. The result
is given in figure [6-1]. As expected, the contribution is many orders of magnitude
below other contributions because, to linear order, the rotational mode decays with
the expansion of the universe.
6.3 Non-Linear Results
Seeing that the first order rotational contribution is negligible, we look to the
second term in equation [6.1].







65














10-22

10-23

10-24

10-25

10-26

10-27

10-28

c2 10-29

10-30
+ 10-3 -

10-32

10-33

10-34

10-35 -

10-36

10-37

10-38 r

10-39 r

10-40
10 100 1000 104 105 108





Figure 6-1. Angular power spectrum of temperature variation on the CMB
due to the first order rotational velocities.








Before we do so we need a way to describe the rotational velocity field. We find
this by imposing angular momentum conservation on a collapsing object. As we
saw previously, collapsed objects have density profiles as given in equations [5.11]
and [5.12]. We consider the mass contained within an initial co-moving radius x,
in a region with a nearly homogeneous density. (The "nearly" indicating the slight
overdensity which will lead the region to collapse.) We denote the final co-moving
radial coordinate y and write the mass as

47r r [47 Y 126a
M = -o,iaix = poa3 y3 + dy'4ry 2u(y/ys) (6.5)
3 L J3

which is just expressing the fact that we are superimposing the halos onto a uniform
background. This will allow us to define the mapping from x to y for a given u(y/ys).
For the Moore profile (u(x) = 1/[x5(1 + x1-5)]) we find

Po,ia,3x =poa3 y3 +Ys -- In [1 + (y/y)] (6.6)

but since po oc a-3, po,iaa = poa3 and we have

x3 = y3 + 26ay In [1 + (y/ys)3/2 (6.7)

As our interest is in the small y regime, we find the asymptotic behavior at low y
to be
3 = 2a(yy)3/2. (6.8)

Figure [6-2] shows this behavior.
For the NFW profile (u(x) = l/[x(l + x)2]) we find

3 = + 36a) + n[1 + (y/ys)] (6.9)

which asymptotes to
3 = 36ay3 (6.10)


for low y. (See figure [6-3].)
















106


105


104


1000


100


10


1


0.1


0.01


Figure 6-2.


Mapping of the initial co-moving radius, x, to final co-moving
radius, y for a Moore profile cluster. The values 65 = 1/2 and
y, = 2 have been used. The dotted line represents the limiting
behavior at low y.


II' I I I I 1 l I I .I ''"1 i I I I I. '1






























0.01 0.1 1 10 100
-/





E.

.o

.




oo .-"


0.01 0.11 10 10







68









106





105





104 -


CT,


1000





100





------------------------ .-- .----- -- -- -- -- ---- ---
10

0.01 0.1 1 10 100
y


Figure 6-3. Mapping of the initial co-moving radius, x, to final co-moving
radius, y for a NFW profile cluster. The values 6a = 1/2 and
Ys = 2 have been used. The dotted line represents the limiting
behavior at low y.








We can express conservation of angular momentum as

aix ai
vt(y) = vo, = -vo,tA(y),
ay a


(6.11)


where


AM(y) = (1 + 2a(ys/y)31n [1+ (y/y)3/2])1/3, (6.12)
( 1 (y/ys) 1/3
ANW(Y) = 1 + 3a(ys/)3 ( ) +ln[1 + (y/y)] (6.13)
I I + (y/ys) I )
Now, returning to the definition, q = (1 + 6)v, we see that in the nonlinear
regime, where the second term above will be dominant, we have, for the rotational
velocity mode, q = Svt = (ai/a)6Avo,t. We then make the definition 5' = (ai/a)SA
and write q = 6'vo,t. This may, at first, seem like an odd thing to do, but we will
find that we just absorb the collapse factor into the halo density profile to produce
a new profile, making the calculation very similar to that in the last chapter.
The second moment of the components of q obey


= (2w)3D(kl + k2) [(bi1)

+ (2')3 (2 1)t (~k')(ki k')iO(k")6(k2 ki"))
= (27r)3D(k1i + k2) V(2i)


+ 3(2)3' (2d)3 [(i~(k')3(kL k')) i (k")S(k2 k"))
+(i'(k')S(k1 k"))(iV(k")S(k2 k'))

+(v'(k')Y(k"))((ki k')8(k2 k")) + (


6.14)


In the nonlinear regime behavior is dominated by the terms of higher order, so we
will neglect the first term above, which is the lowest order term. The second term
above is zero by our assumption of isotropy. The last term is the irreducible fourth
order term. If we had kept the collapse factor with the rotational velocity, we would
expect this irreducible term to be the term of interest, as then the velocity would


(qi(k1)q (k2))








couple strongly to the density. However, as we have associated the collapse with the
density, we now expect the important term to be similar to the important term for
the longitudinal mode. Thus the irreducible fouth moment will vanish, just as in
Ma and Fry, and we drop it.
We now write the transformed velocity as


i(k) = if 1(k) + v(k) = 1vll(k) + (vi(k),


(6.15)


where e represents a unit vector perpendicular to k. This term could be written,

= (6.16)
v


Using this velocity we have the general definitions,


(6(kl)6(k2))



(f(k1)ji(k2))

(S(k1)V'(k2))


= (27r)3 D(kl + k2)P6s(kl) (6.17)

= ([i (ki) + i)l(kl)][Yj (k2) + V1L(k2)])
= (V1(kC1i)~(k2)) + (f)i(k)i(k2))

= (27r)36D(ki + kI2)[ flk2P, ll u(ki) + i^iP 1(kl)] (6.18)
= (6(kl)fl (k2)) + 6( (k)(l) k2))

= (27r)35d(kl + k2)[k2 P611(ki) + e2& Ps(k,)], (6.19)


The cross terms in the second expression vanish because the components are inde-
pendent. The last term in equation [6.19] is zero by isotropy.
Now, remembering that wherever we have a 6 associated with a rotational
velocity, we will replace it with a 6', and we will replace the velocity with vo,t, we







have

P(-k) d3k' d k"
P(k J (27r)3(27)3
x [(27r) 6D(k k' k")[k' k' P,,l,,1(k') P6(k") + E P ,,,_L P ")
+ k P, (k')P6, (k") + P,,, (k') P,, (k")

+ Ik" P6A,,,o(k')Pvl (k") + ~ie' P&,,,,o(k')PV,,,,o(k")]]. (6.20)

Note that ', e" refer to unit vectors perpendicular to k and k respectively. We
now calculate the quantity of interest using

Pql(k) = 2z'Pq '(k). (6.21)

The inner products of interest are

^,ijkk'j = (1 -/12)V'2 (6.22)
-y'k" = (P~(1 M2)1/2)v2 (6.23)
.ije'3 = t,/2v2 (6.24)

ijk'k" = (1 '2)/2v',-.I (6.25)
i," jk'ifI = (1 ,2)1/2i ." (6.26)
,iEi'=" = v'. I -k" (6.27)

ii'Velj = l'v"-. je", (6.28)

where 1' is defined as before and v' cos(0'). In equations [6.25] and [6.27] we have
the expression, -~ k which we evaluate by noting that in the next step we will
integrate over the delta function in equation [6.20], and thus replace k" with k k.
Since k" -+ (k k')/Ik k' and *k = 0, we end up with

(1 )k' (6.29)

S- k'(6.30)







Additionally, in equations [6.26] and [6.28], e" becomes i -'" where i'" is a unit
vector perpendicular to k k'.
Putting these in, we have
d3k',
Pqi(k) = ( [(1 i'2)v 2P,1I, (k')P5(Ik k'j)

+'2 v"/P ('k')P6,6(Ik k'|)

+(1 p2) k' P, (k')P,5, (Ik k'k)
Ik k'I
+(1 __'2)1/21I,' eP6v,, (k')P.Ly,0o(lk k'l)
/'(1 p2)1/2I2k'
+ A k ,- k' P|Ivo o(k')P6,,l ( k k'I)

+'v- i '"P&'v, 0(k')P,,,(,o( k k'|)]. (6.31)

Now, noting, as per Ma and Fry, that we are interested in high k, beyond the k' peak
of the above integrals, we drop terms of order k'/k and perform the integration over
pt' and v'. First, we notice that


7Y- e. = 1,


(6.32)


To perform the angular integration, we note that

-1
cos2()f (',,) = 7r dpk' of dkk'2f(', k')

= /(d3k'f (k'),
2 ,.__ .


Sd3k' cos( f')f (p', k')

d3k'(1- p2)f(k')



Sd3k''2 f (k')



d3k'p'(1 2)/2f(k')


= 0,

= 27rJ dk'k'2f(k')
= d d3k'f(k'),

S27r2 f dk'k2f (k')

=- Jdk'f (k'),
3J


= 0.


in this limit.


(6.33)

(6.34)



(6.35)



(6.36)

(6.37)








Putting this together gives us our result,

/ f d3k' )1
Pqi(k)= J(2)3 k')P (k) + P (k')Ps(k) (6.38)

The first term is, of course, the term introduced last chapter, due to Ma and
Fry. It is the second term we calculate here.
First, we must calculate Ps,'(k). As we hinted at earlier, the collapse factor,
A, simply multiplies the halo density profile, giving us a new profile, w(y/ys) =
Au(y/ys). Since we are ultimately interested in behaviors at low y, or equivalently,
high k, we will use the low y asymptotic form of A. For the Moore profile,

w(y/ys) = (26 a)1/3 1 (6.39)
(y/y ')2(1+ (y/)32) (39)

It is the Fourier transform of this function that we will need. Figure [6-4] shows
this transform as compared to the transform of the full w. The discrepancy is is the
relatively unimportant small k region.
Figure [6-5] shows the transform of w and an analytical fit, which we will use
to increase computational efficiency. This fit is given by

( ( 1/347r [ln(e + ) + 0.25 n(ln(e + ))
w(1 + 0.6k'.3)0-77 ( )

For the NFW profile, we have

w(y/ys) = (36a)1/3 (6.41)
(y/ys)2(1 + (y/y,))2"
This transform and the transform of using the full version of A are shown in figure
[6-6].
The analytical fit of the transform of this profile is

-(q) = (3) 1/3 4 (6.42)
1 + 0.6q


Figure [6-7] shows this fit compared to the transform.

















I I III I I I I I l I I I 1 1111 1 1 i 1i i I 1I 1 I I 1 1 11i l 1 1 1 11 li


. .. I I I I. 1.1111.


I,,, 1111 11111 1111,1,,,,1


0.001 0.01 0.1 1 10 100 1000

q


Figure 6-4.


104 105


The Fourier transform of the Moore w(y). The solid line rep-
resents the use of the full version of A, while the dotted line
represents the use of the high k asymptotic form of A. In both
versions we set 6, = 1/2.


100





10





1





0.1





0.01





0.001


I 1 .111


I I,,,,,,I


I I1 I I .. I


I 1 111





















10





1





0.1





0.01





0.001




,,,,1 .. I 1 I I ,, ,I i,,I .... I I 1 I I ii,,,i ,,I ,I
0.001 0.01 0.1 1 10 100 1000 104 105
q


Figure 6-5. The Fourier transform of w(y) for the Moore profile using the
asymptotic form of A. The solid line represents the actual trans-
form and the dotted line represents the fit given in equation
[6.40]


:1_1_1 1 111111 1 T 11-rll -1 .1 1 111 1 II. .. -- I _


""1I


""'I """I """I'
















11111I ...... '" I 1 1111111 1 '""1 1 111111 I "111 1 11111'" 1


I,, I ,,.


0.001 0.01 0.1 1 10
q


100 1000


104 106


Figure 6-6.


The Fourier transform of the NFW w(y). The solid line rep-
resents the use of the full version of A, while the dotted line
represents the use of the high k asymptotic form of A. In both
versions we set 6a = 1/3.


100




10




1




0.1




0.01




0.001


...... ...... ...... E I I l llil I I III ll I I UI [I I I I111


,,,,,I















































III I im 1 111111 1 111 1 III 11111 111111 1 1 11 111111 11111 I 11111


0.001 0.01 0.1 1 10


100 1000


104 105


q


Figure 6-7. The Fourier transform of w(y) for the NFW profile using the
asymptotic form of A. The solid line represents the actual trans-
form and the dotted line represents the fit given in equation
[6.42]


10





1





0.1





0.01





0.001


...11 I .. 1.111 1 1 1.. ..11.1 1 1. 1-








We now simply calculate P6,,'(k) just as we did before, in equation [5.19], with

the substitution of w for u. Figure [6-8] shows P6,&,(k), with the factor of (a,/a)2
left out, as compared to PS6(k). Both the overall magnitude and the shape are, of
course, markedly different.

Figure [6-9] shows the difference in Pa, (k) when the different halo profiles are

used. Notice that use of the NFW profile, rather than the Moore profile, leads to a
difference of about a factor of ten in the spectrum for high k.

We now have

Pq = (v,t)P&'5'(k). (6.43)

We will use a value of (v ,t)1/2 = 20km/s, as we did in the first section of this chapter.

Putting this into equation [5.25], we find the result given in figures [6-10] and
[6-11].
Figure [6-12] shows how the nonlinear rotational velocity contribution compares
to the linear contribution. This shows, in stark fashion, that this study was well

motivated.







79










1011

101o


109

108

10

106

105


100

1000

100
10



0.1

0.01

0.001

0.0001
0.01 0.1 1 10 100 1000 104 105
k


Figure 6-8. Plots of P6s6,(k), the solid line, and Pss(k), the dashed line.



















1011

1010

109

108

107

106

105

104

1000

100

10

1

0.1

0.01


0.01


10 100


1000


k



Figure 6-9. Pay (k) plotted using the NFW, dashed line, Moore, dotted line,
and mixed, solid line, halo profiles.


- "~








-
r T I









rJ o
r '

o
/ .."

/ -."
,,,ni n m i ,,,,ni u n m i a ,1 o'i a n i a



















10-15 1' 1 -"11 1 1 "1 1 1"1 1 1 1 1 f i"1
F -



10-16




10-17




10-18




10-19




10-20




10-21




10-22

10 100 1000 104 105 106 107





Figure 6-10. Angular power spectrum of temperature fluctuations in the
CMB due to nonlinear rotational velocity contribution.

















10-9

10-10

10-11

10-12

10-13

10-14

10-15

10-16

10-17

10-e1

10-19

10-20

10-21


E"'' '' ''"I, -: '.7'


I i 11 1 Ii i I II I I 1 111 i


Figure 6-11.


The solid line shows the angular spectrum due to the rotational
velocity mode. The nonlinear spectrum due to the longitudi-
nal mode, dotted line, and the primary anisotropy spectrum,
dashed line, are shown for comparison.


I


10-22


10-23

10-24

10-25


I I I I i"


I I ,,,,


100


1000


~


-- --- .
-- \

\
\
\
\



















10-15
10-1 r
10-17
10-18
10-1-
10-20
10-21
10-22
10-23
10-24
10-25 r
10-2


S10-28 r

10-29
10-30
10-31

10-32
10-33
10-34r
10- r
10-30 r

10-3 r


10-33
10-39 r



1 0-39
10-40
10 100 1000 104 105 106 107
I



Figure 6-12. The angular power spectrum of CMB temperature fluctuations
due rotational velocity modes. The top line is the nonlinear
contribution, the bottom line the linear contribution.














CHAPTER 7
CONCLUSION
Our goal in this dissertation has been to include the effects of an initial ro-

tational velocity mode in large-scale structure calculations. We have retained this

mode in perturbation theory and we found that, although it decays with time in

linear theory, at higher orders it couples to the longitudinal mode and plays a non-

trivial role in quasi-linear theory.

We expected that the rotational velocity mode might play an even more signifi-

cant role in the nonlinear regime. Specifically, we believed that the inclusion of this

mode in a calculation of the kinetic Sunyaev-Zeldovich effect might prove significant.

The results of last chapter indicate, however, that the inclusion of such modes is not

significant on any angular scales available to observation now or in the near future.

This fact provides evidence for the validity of the assumption typically made

in previous calculations that the rotational mode is largely unimportant in most

cosmological situations. It is important, however, to note that the inclusion of

rotational modes in the nonlinear calculation leads to temperature fluctuations many

orders of magnitude higher than those in the linear regime. It seems that the success

of the a priori exclusion of initial rotational modes was fortuitous.

It is important to note that there are many possible ways to improve upon

our results. One important factor is that the halo profiles used to build the halo

model are the result of N-body simulations which do not include the initial rotational

mode. It is certainly conceivable that the profile in the central regions of halos would

be considerably different if this mode were included in the simulations. Analytical

calculations which allow for the inclusion of angular momentum do indicate that a

shallower inner profile is likely.53 One would expect that if a numerical simulation






85

were performed where the initial rotational velocities were retained, such a profile

would be seen. Having said this, we note that, as suggested by the fairly small profile

differences seen in figure [6-9] even a somewhat different inner profile would not be

likely to produce an observable effect on the angular power spectrum of temperature

fluctuations.

Besides the aforementioned inclusion of rotational modes in N-body simulations,

there are many ways in which the work begun in this dissertation could continue.

First, the role of the rotational modes in quasi-linear theory should be further ex-

plored. We have shown here that it plays potentially important roles, but not fully

explored what those roles might be. Additionally, the inclusion of rotational veloc-

ities will also have an impact on the thermal Sunyaev-Zeldovich effect. An initial

rotational velocity dispersion should contribute to the overall velocity dispersion,

and thus effective temperature, of collapsed objects.












APPENDIX A
CONSERVATION OF ANGULAR MOMENTUM IN AN EXPANDING
UNIVERSE
We begin by defining the angular momentum about an arbitrary origin as

L = d3rpr x v, (A.1)

which we can express in comoving coordinates as

L = d3xpoa3(1 + 6)ax x v. (A.2)

We want to look at the time derivative of the angular momentum. Noting that the
quantity poa3 is constant in time (remembering that Po oc a-3) we have

dL 8 0
= poa3 d3xx x ([(1 + 6)av]. (A.3)
dt 5t

Using the fluid flow equations from chapter 2, equations [2.12] and [2.15], this be-
comes

dL = -poa3 d3x[xx {(1 + 6)[(v V)v + V] + v(V [(1 + )v])}] (A.4)

= -poa3 d3x[xx {([V.(1 + )v]v)+ (1 + )V}]. (A.5)

The cross product of x with the second term vanishes because 0 leads to a
central force. Writing the remainder in component notation, we have

dL= d3zijkxV[(1 6)vkvl], (A.6)

which we can write as

dL = d3Eijk [Vt (1 + 6)vkvI + (1 + 8)vkVlVXj]. (A.7)






87

The first term is a surface term and is thus zero, and the second term is just

dL=d d36ijk1VkVi(1 + 6) = 0, (A.8)


as it is just proportional to the cross product of v with itself. Thus dL/dt = 0,
and, as we would suspect, the total angular momentum in an expanding universe is
conserved.













APPENDIX B
SPHERICAL COLLAPSE AND THE DERIVATION OF bc

A reasonable value for the critical density contrast required for collapse of a

halo can be obtained through a simple analysis of spherical collapse model. We

begin by writing the Newtonian equation of motion for a mass shell surrounding a

mass, M.
d2r GM (
=-r- (B.1)
dt2 r2
This can be solved using the parametric approach,

r = A(1 cos0) (B.2)

t = B(O sin 0). (B.3)

Simple substitution will verify this solution and provide the relation

A3 = GMB2. (B.4)

Figure [B-1] shows the radius and time plotted as functions of the parameter

0.

Now, expanding these solutions for small 0, corresponding to small initial over-

densities, we have

r(0) A 20I4 '(B.5)


t(O) B 12 (B.6)
(6 120/







89











6








4




















0 2 4 6

0



Figure B-1. Plots of radius and time as functions of the parameter 0. t/A is
the solid line, r/B is dashed.
.v N
^ / N
^7N
/'N










the solid line, r/B is dashed.








Feeding the leading order solution for t into the expression for r we find

S A 6t 2/3 1[ (6t23
2 B 206 [ (B

210(2/3 1/3 1 f6t \2/3
S (6t)2/3(GM)/ 1- .(


Remembering that p = M/(4rr3/3) (this is actually the
sphere of radius r) and po = 1/67rt2G, we find


mean densit


(B.7)


(B.8)

y within a


p-Po 3 /6t\2/3
6 P= -PO 2/(B.9)
Po 20 B

Equation [B.2] tells us that total collapse occurs at 0 = 27r, where t = 2irB. Putting
this in we get the result, 6,c 1.686. The reader may object, as 0 = 2r is clearly
not predicts an overdensity of 1.686.
As a point of further interest, we can calculate the exact density run for
this spherical collapse. To start we consider the energy of a spherical shell at its
turnaround,


GM(x)
2A


(B.10)


as, at turnaround, all energy is potential and r = 2A. Here x refers to the co-moving
radius. We also note that E oc 1/x because we are perturbing an Einstein-De Sitter
universe, where E = 0, by an initial overdensity a co-moving distance x from the
shell. When combined with the fact that M(x) oc 23 we see that A oc x4 and
B oc x9/2. Using these scale relations we can write


r A(1-cos0) 4N 1-cos0
rta 2Ata xJta 2


X / B 7 F
'Xta} Bta ~0- sin0'


and


(B.11)


(B.12)








so that
r ( 8/9 1 coso
rta 0 sin 0 2(B13)
In each of the previous equations the subscript "ta" indicates a quantity for the shell

at turnaround at the chosen time t. We compute the mass density run using

dM 1
p(r) dr 41 (B.14)
dr 4wr2

and using the chain rule,
dM dMdx dr\-1
dr- dx dO I) (B.15)
and the dependence we found above. The result, when divided by the mean density,

becomes
p(r) 9 ( -s in 0)2 9sin(O- sin )-1
4- (B.16)
Po 2(1 -cos)3 [ 2 (1-cos0)2 J (B16)
We can also write down the mean density within the sphere, to compare with the
linear prediction,

S M 2 9 MGt2 9 (0 sin 0)2
6wGt2 = (B.17)
po rr 2 r3 2 (1 -cos )3 (B )

Figure [B-2] shows the density contrast as a function of the scaled radius, for the
mean density, as above, and for the linear prediction of equation [B.9]. Also shown

the the density run. None of these are expected to be accurate at small r, where

shell crossing causes virialization.