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arXiv:astro-ph/0202215v2 8 May 2002

Dynamics of Cosmological Perturbations

in Position Space

Sergei Bashinsky

Department of Physics

Princeton University

Princeton, New Jersey 08544

Edmund Bertschinger

Department of Physics

Massachusetts Institute of Technology

Cambridge, Massachusetts 02139

(May 8, 2002)

We show that the linear dynamics of cosmological perturbations can be described by coupled wave

equations, allowing their efficient numerical and, in certain limits, analytical integration directly in

position space. The linear evolution of any perturbation can then be analyzed with the Green’s

function method. Prior to hydrogen recombination, assuming tight coupling between photons and

baryons, neglecting neutrino perturbations, and taking isentropic (adiabatic) initial conditions, the

obtained Green’s functions for all metric, density, and velocity perturbations vanish beyond the

acoustic horizon. A localized primordial cosmological perturbation expands as an acoustic wave

of photon-baryon density perturbation with narrow spikes at its acoustic wavefronts. These spikes

provide one of the main contributions to the cosmic microwave background radiation anisotropy

on all experimentally accessible scales.The gravitational interaction between cold dark matter

and baryons causes a dip in the observed temperature of the radiation at the center of the initial

perturbation. We first model the radiation by a perfect fluid and then extend our analysis to account

for finite photon mean free path. The resulting diffusive corrections smear the sharp features in the

photon and baryon density Green’s functions over the scale of Silk damping.

I.INTRODUCTION

The nearly perfect black body spectrum and isotropy

of the cosmic microwave background radiation indicates

that the universe at large redshift was highly uniform

and in thermal equilibrium, with fluctuations in temper-

ature of only about 1 part in 105on observable scales.

Under these conditions, the dynamics of matter, radia-

tion, and gravity is described accurately by linearizing

the governing equations about their spatially homoge-

neous solutions representing an unperturbed expanding

background. In place of the strongly nonlinear fluid and

Einstein equations, we have a system of coupled linear

partial differential equations. The linear approximation

continues to apply much later on large scales even af-

ter nonlinear structures such as stars and galaxies have

formed on smaller scales.

The cosmological perturbations can be described by a

set of classical fields, e.g. the gravitational potential φ.

After the perturbations are created in the early universe,

each field may be expanded over a convenient set of basis

functions

φ(r,τi) =

?

k

φkfk(r) ,(1)

where the φk are expansion coefficients at some initial

time τi. Under linear evolution at a later time τ

φ(r,τ) =

?

k

φkFk(r,τ) ,(2)

where Fk(r,τ) is the solution to the linearized equation

for φ that satisfies the initial conditions Fk(r,τi) = fk(r).

The range of possible basis functions fk(r) is enor-

mous. Since the pioneering work by Lifshitz, Ref. [1],

nearly all the cosmological perturbation theory calcula-

tions, e.g. Refs. [2, 3, 4, 5, 6, 7], have used harmonic

plane waves or their generalizations in curved spaces.

There is a good reason for this: because the dynami-

cal equations are translationally invariant, each harmonic

mode evolves unchanged aside from a time-dependent

multiplicative factor Tk(τ) called the transfer function:

Fk(r,τ) = Tk(τ)fk(r). This separation of variables al-

lows the partial differential equations to be reduced to a

set of ordinary differential equations that are straightfor-

ward to solve numerically.

However attractive this reduction appears, it by no

means implies that cosmological perturbation theory re-

duces to the simplest form in Fourier space. The plane

wave expansion may be thought of as a localized (Dirac

delta) basis in Fourier space. From this perspective, it

is not unreasonable to consider a localized basis in real

space. In this case, the function Fk(r,τ) in eq. (2) no

longer factors into a simple product. The lack of separa-

tion of variables would seem to imply that perturbation

theory is more difficult in position space. In fact, we will

show that it is not only simple in the linear regime, but

the dynamics is clearer and more intuitive in real space.

Suppose, for example, that we consider the evolution

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of a perturbation originating from a point disturbance

φ(r,τi→ 0) = fk(r) = δ(3)

where δ(3)

D(r − r0) = δD(x − x0)δD(y − y0)δD(z − z0) is

the product of Dirac delta functions. What will such an

initial perturbation evolve to by time τ? The correspond-

ing function Fk(r,τ) = G(r − r0,τ) is called a Green’s

function. To see how the Green’s function language can

be simpler than the transfer function approach, we no-

tice that, as shown in Ref. [8], prior to recombination

the primordial isentropic (adiabatic) perturbation satis-

fying the initial conditions (3) expands in the photon-

baryon plasma as a spherical acoustic wave with sound

speed cs. During the radiation era, cs= c/√3 and the

gravitational potential is simply given by the properly

normalized Heaviside step function:

D(r − r0) ,(3)

φ(r,τ) =

3

4π(csτ)3

?1 , r ≤ csτ ,

0 , r > csτ .

(4)

Here, r is the comoving distance from the initial point

perturbation, and τ is conformal time related to the

proper time t by dτ = dt/a(t), where a(t) is the cosmolog-

ical scale factor. Later we will also use Green’s functions

produced by initial disturbances on two-dimensional

planes in space.

The position space representation is formally equiva-

lent to the Fourier space representation. It can be used

to describe the dynamics of cosmological perturbations

regardless of their origin and statistical properties. The

Green’s function method can be applied just as well to

the evolution of nearly scale-invariant perturbations gen-

erated by inflation, Refs. [8, 9], as it can be applied

to local perturbations produced by topological defects,

Refs. [10, 11].

So what new can be learned from this approach?

Green’s functions contain the same information as trans-

fer functions, but that information is packaged differ-

ently. The Green’s function approach reveals new sides

to cosmological dynamics that are of both phenomeno-

logical and theoretical interest.

Phenomenologically, Green’s functions are often char-

acterized by localized features such as the acoustic wave-

front.Through the uncertainty relation ∆x∆k>

localization in position space results in features being

spread over a broad range of wavenumbers in Fourier

space. An example of this is the acoustic peaks in the

cosmic microwave background (CMB) power spectrum

Cl. In Ref. [8] we have shown that the position-space

analogue of Cl, the angular correlation function C(θ), has

localized features instead of acoustic oscillations. These

features offer an alternative signature for experimental-

ists to measure.

Theoretically, acoustic and transfer processes are made

explicit in position space. This offers new methods for

solving the evolution equations or leads to substantially

simpler equations and solutions in certain cases. An ex-

ample is the spherical wave solution for the radiation era

∼1,

given by eq. (4).

this suggests that the position space view may provide

a more direct understanding of the dependence of CMB

anisotropy patterns on the underlying cosmological pa-

rameters.

Equations of perturbation dynamics can in principle

be numerically solved faster in position space than the

equivalent equations in Fourier space for same desired ac-

curacy. This is because the Green’s functions are mono-

tonic and limited in their spatial extent by the acoustic

horizon for perfect fluids, while the Fourier transfer func-

tions oscillate in both the wavenumber and time coordi-

nates requiring a larger number of sample points for their

accurate representation.

This paper describes cosmological linear perturbation

theory in position space using the Green’s function ap-

proach. The discussion is primarily focused on the pe-

riod of cosmological evolution prior to hydrogen recom-

bination and radiation decoupling from baryonic matter.

In the current paper we derive evolution equations that

are convenient for a position space analysis and consider

their Green’s function solutions. In a later paper we will

show how these results may be used to describe CMB

anisotropy. Sec. II presents the dynamical equations de-

scribing coupled perturbations in the metric and in radi-

ation and matter modeled by locally isotropic fluids. In

Sec. III we specify the initial conditions for the Green’s

functions and give explicit Green’s function solutions in

the radiation epoch. In Sec. IV we discuss general prop-

erties of the Green’s functions and their numerical inte-

gration in the fluid model, and we analyze the results of

numerical integration up to the time of hydrogen recom-

bination. Sec. V shows how the position-space descrip-

tion of perturbations can be implemented when the fluid

equations of Sec. II are replaced by the full equations of

Boltzmann phase space dynamics, which are summarized

in the Appendix. Then in Sec. VI we find the leading

corrections to the fluid approximation results. A brief

summary is given in the Conclusion.

We end this introduction with a summary of our con-

ventions and notations. Throughout the paper, Greek

indices range from 0 to 3 and label components of space-

time vectors.Components of spatial 3-vectors carry

Latin indices ranging from 1 to 3; if the indices are omit-

ted, 3-vectors are typed in bold. The speed of light is

c = 1. The 2π factors in the Fourier transforms always

appear in the denominator of the momentum integral, as

in the equation

The acoustic wave is manifest, and

φ(r) =

?

d3k

(2π)3eik·rφ(k) .(5)

We consider only the scalar perturbation mode, the one

involving radiation and matter density perturbations,

and work in the conformal Newtonian gauge for the per-

turbed Robertson-Walker metric, Refs. [4, 6].

gauge

In this

ds2= a2(τ)?−(1 + 2φ)dτ2+ (1 − 2ψ)dr2?

,(6)

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Symbol

a

y

τ

τe

r

superscript (3) Spherical (3D) Green’s function

superscript (1) Plane-parallel (1D) Green’s function

x Spatial coordinate of a plane-parallel Green’s function

superscript (T) Fourier space transfer function

k

Comoving wave vector

k|k|

φ,ψMetric perturbations

subscript γPhotons

subscript bBaryons

subscript rRadiation fluid (coupled photons and baryons)

subscript dDifference in photon and baryon perturbations

subscript νMassless neutrinos

subscript cCold dark matter (CDM)

subscript relBackground of relativistic species (γ plus ν)

subscript mBackground of non-relativistic matter (baryons plus CDM)

δEnergy density enhancement /

If applied to a variable, the perturbation of that variable

δD(x)Dirac delta function

uPeculiar velocity potential

vi

Peculiar 3-velocity

δpPressure perturbation

πShear stress potential

σEntropy perturbation potential

fPhase space distribution

F Energy averaged perturbation of f

fl

Potentials for angular moments of F

β Baryonic fraction of nonrelativistic matter (ρb/ρm)

cs

Speed of sound in the photon-baryon plasma

cw

“Isentropic sound speed” (∂p/∂ρ)1/2

A and B1/(3c2

S(τ)Radius of acoustic horizon

τc

Mean conformal time of a photon collisionless flight

MeaningDefining Equation

(6)

(15)

(6)

(16)

(6)

(38)

(39)

(39)

(41)

(5)

–

(6)

–

–

–

–

–

–

–

–

(10a)

–

–

(10b)

(10b)

(10c,A2)

(A2)

(21–22)

(A4)

(A8)

(A10)

(15)

(7,19)

(20)

(29)

(49)

(75)

Scale factor relative to the present

Scale factor relative to the radiation-matter density equality

Conformal time

Characteristic τ of the radiation-matter density equality

Comoving 3-space coordinate

adiab

w) and 1/(3c2

s) respectively

TABLE I: Frequently used notations.

where dr2is the three-metric of a Robertson-Walker

space. For perfect fluids, there is a single gravitational

potential φ = ψ, but in general two distinct potentials

are required. (Note that the perturbation φ of the met-

ric (6) is called ψ, and ψ is φ, in Refs. [6, 12]. Our present

choice agrees with Ref. [4].) Other frequently used vari-

ables and notations are summarized in Table 1. They

will be introduced systematically in what follows.

II. COSMOLOGICAL DYNAMICS IN THE

FLUID APPROXIMATION

A. The model

In this and the following two sections we study pertur-

bation dynamics adopting a simplified model where the

photon-baryon plasma and cold dark matter are approx-

imated by coupled fluids. The photon gas is assumed

to behave as a locally isotropic fluid characterized by

its density and velocity at every point in space and its

pressure equals one third of the photon energy density.

This is a good approximation before recombination when

photons intensively scatter against free electrons and, be-

cause of the Coulomb interaction between the electrons

and baryons, the velocity of baryons is locked to equal

the mean velocity of the photon fluid. We will arrive

at these results consistently from the Boltzmann equa-

tion and will consider the leading corrections to the fluid

approximation in Secs. V–VI. The effects of global cur-

vature and of the cosmological constant should be very

small on the scale of the acoustic dynamics before recom-

bination and we do not include them in the discussion.

Neutrinos contribute a large fraction (about 40%) of

the radiation energy and require specification of their

full phase space distribution even before recombination.

We do not include a full treatment of neutrino pertur-

bations in the present paper for the following reasons:

The fluid model, corrected for photon diffusion and for

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4

the neutrino contribution to the background energy den-

sity as shown below, can describe perturbations in the

early universe at least up to 5% accuracy, Ref. [9], when

compared in its predictions of CMB anisotropy with full

numerical calculations. The fluid description is adequate

for baryons, dark matter, and photons before recombina-

tion. Its simplicity and intuitive appeal make the fluid

approximation an attractive starting point for applying

perturbation theory in position space. The position space

approach can also be used for accurate description of neu-

trino dynamics and offers substantial advantages over the

traditional Fourier decomposition, but that requires new

constructions different from the spirit of the fluid descrip-

tion, Ref. [9]. We postpone the phase space analysis of

neutrino perturbations to a later paper.

To account for the substantial contribution of neutri-

nos to the background radiation density while excluding

them cleanly from the perturbation dynamics, we con-

sider a fictitious universe filled with photons, baryons,

and cold dark matter (CDM), but without neutrinos. In

this model universe the photon background energy den-

sity, ρ(model)

γ

, equals the actual (physical) energy density

of the combined relativistic backgrounds of photons and

neutrinos:

ρ(model)

γ

= ρ(phys)

rel

≡ ρ(phys)

γ

+ ρ(phys)

ν

= (1 + Rν)ρ(phys)

γ

where

Rν≡

ρν

ργ+ ρν

=

?

1 +

8

7Neff

ν

?11

4

?4/3?−1

,

is the fraction of the radiation energy density in neutri-

nos that equals Rν ≃ 0.408 for Neff

Ref. [13].

The sound speed in the photon-baryon fluid is deter-

mined by the ratio of baryon and photon energy densities

as

ν

≃ 3.04, e.g. from

c2

s≡

?

∂pγ

∂(ργ+ ρb)

?

adiab

=

1

3[1 + (3ρb)/(4ργ)]. (7)

The sound speed controls the dynamics of acoustic per-

turbations via the length scale?csdτ. To preserve this

crease the baryon density by a factor (1 + Rν) over its

physical value:

scale in our model, where ργ is replaced by ρrel, we in-

ρ(model)

b

= (1 + Rν)ρ(phys)

b

. (8)

Finally, the total mean density of non-relativistic mat-

ter ρm, including baryons and CDM, is taken to equal its

physical value:

ρ(model)

m

= ρ(phys)

m

.(9)

For eqs. (8–9) to hold, the mean density of CDM in

our model must be reduced slightly compared with its

physical value: ρ(model)

c

= ρ(phys)

c

− Rνρ(phys)

b

. With

these definitions, we arrive at a self-consistent two fluid

model that preserves the important time scale of the

radiation-matter density equality as well as the acoustic

length scale

?csdτ. From now on, we drop the super-

script “(model)”.

B.Dynamical equations

The dynamics of perturbations in the fluid model is

governed by the linearized Einstein and fluid equations.

We give these equations in the conformal Newtonian

gauge, Refs. [4, 6, 14], and then derive an equivalent but

a more intuitive and easier to solve system of equations.

Metric perturbations are induced by perturbations in

the energy-momentum tensor Tµν =?

(a = γ,b,c). For every species, Tµ

ized by an energy density enhancement δaand a velocity

potential ua, denoted by W in Ref. [14],

aTµ

aν where a

runs over all radiation or matter species in our model

aν can be parameter-

T0

T0

Ti

a0 = −(ρa+ δρa) ,

ai = (ρa+ pa)vai,

aj = δi

δρa= ρaδa,

vai= −∇iua,

(10a)

(10b)

(10c)

j(pa+ δpa) .

The stress Ti

sure perturbations equal δpγ=1

Tight coupling between photons and baryons implies the

equality of mean local velocities of the radiation and the

baryon fluids, Sec. VI. Assuming adiabatic initial condi-

tions,

ajis isotropic for perfect fluids and the pres-

3ργδγand δpc= δpb= 0.

δb=3

4δγ≡3

4δr

andub= uγ≡ ur

(11)

before recombination.

The linearized Einstein equations for metric perturba-

tions φ and ψ in eq. (6) are given by eqs. (A1) of the

Appendix. The last of eqs. (A1) states that for perfect

fluids, when anisotropic stress is negligible, ψ = φ. Then

the remaining equations are

∇2φ − 3˙ a

a

?

aφ = 4πGa2?

?˙ a

˙φ +˙ a

aφ

?

= 4πGa2?

(ρa+ pa)ua,

a

δρa,(12a)

˙φ +˙ a

a

(12b)

¨φ + 3˙ a

a

˙φ +

?

2¨ a

a−

a

?2?

φ = 4πGa2?

a

δpa,(12c)

where “∇” and “˙” denote partial derivatives with respect

to the comoving spatial coordinate r and the conformal

time τ.

The fluid equations for density and velocity evolution

in the conformal Newtonian gauge were derived in Ref. [6]

and follow from the general formulae of Sec. V. In the

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tight coupling limit and our notations they are

˙δc = ∇2uc+ 3˙φ ,

˙ uc = −˙ a

˙δr =

3∇2ur+ 4˙φ ,

(13a)

auc+ φ ,(13b)

4

(13c)

˙ ur =

1

1 + (3ρb)/(4ργ)

?1

4δr−3ρb

4ργ

˙ a

aur

?

+ φ . (13d)

The scalar gravitational potential φ does not present an

independent dynamical variable in addition to the densi-

ties and velocity potentials because on any given hyper-

surface of constant time the gravitational potential can

be determined from the energy and momentum density

perturbations on the same hypersurface via the general-

ized Poisson equation

∇2φ = 4πGa2?

following from eqs. (12a–12b).

We define

a

?

δρa+ 3˙ a

a(ρa+ pa)ua

?

,(14)

y(τ) ≡

a

aeq

=ρm

ρrel

,β ≡ρb

ρm

, (15)

where aeq≃ 1/(2.40×104Ωmh2) is the scale factor value

at the time of matter-radiation density equality and

ρrel = ργ in our model. The ratio β is time indepen-

dent. An important scale in our problem is a character-

istic conformal time of the transition from radiation to

matter domination

τe≡

?

aeq

0Ωm

H2

.(16)

For the following ΛCDM model set of cosmological pa-

rameters: Ωm = 0.35, ΩΛ = 0.65, Ωbh2= 0.02, and

h ≡ H0/(100kms−1Mpc−1) = 0.65, the numerical value

of τeis cτe≃ 130Mpc.

With the above notations, the factor 4πGa2on the

right hand side of the Einstein equations becomes:

4πGa2=

3

2τ2

e

1

yρm

.(17)

If the dark energy is neglected in the early epoch, the

Friedmann equation yields:

˙ y2=1 + y

τ2

e

,y(τ) =τ

τe

+1

4

?τ

τe

?2

.(18)

The equality of matter and radiation densities, at which

y(τeq)=1, occurs at τeq=2(√2−1)τe≃0.83τe.

Dynamics of perturbations in our model depends on

two characteristic speeds, given by

c2

s=

1

3?1 +3

4βy?

(19)

and

c2

w≡

?∂p

∂ρ

?

adiab

=

1

3?1 +3

4y? ,(20)

p =?

ρ(phys)

b

speed, cw, is not a true sound speed.

finitesimal pressure and density changes for perturba-

tions of constant radiation entropy per unit mass of non-

relativistic matter, η, defined by

apa and ρ =?

/ρ(phys)

γ

aρa, see note [15]. Eq. (19) fol-

lows from the sound speed definition of eq. (7) since

= ρ(model)

b

/ρ(model)

γ

= βy.The second

It relates in-

η ≡ δ

?

lnT3

ρm

r

?

= (1 − β)

=

?δpa

c2

wρm

−

?δρa

ρm

= (21)

?3

4δr− δc

?

.

We describe η by a dimensionless entropy potential σ

such that

∇2σ ≡

3

2τ2

e

η . (22)

Using φ,˙φ, σ, ˙ σ for independent dynamical variables,

¨φ +?3 + 3c2

¨ σ +?1 + 3c2

The first equation follows from the substitution of the

left hand side of eqs. (12a,12c) into the first line of

eq. (21) and using eqs. (22,18). The second is derived

from eqs. (22,21,13,12a,12b).

Next, we replace the potentials φ and σ by a pair

of their linear combinations φr ∝ φ + σ/y and φc ∝

?c2

¨φr+?

w

?˙ y

y

˙φ +

3c2

4τ2

w

eyφ = c2

w∇2?

s− c2

φ +σ

y

?

,

w− 3c2

s

?˙ y

y˙ σ = y?c2

w

?∇2?

φ +σ

y

?

.

(23)

s/c2

second derivative terms in the system (23):

w− 1?φ−σ/y, which are chosen to diagonalize the

i=r,c(ari˙φi+ briφi) = c2

i=r,c(aci˙φi+ bciφi) = 0 .

s∇2φr,

¨φc+?

(24)

The new variables φrand φcare uniquely defined if they

are normalized so that

φ = φr+ φc.(25)

Then

σ = y

??c2

s

c2

w

− 1

?

φr− φc

?

(26)

and

φr =

c2

c2

?

w

s

1 −c2

?

φ +σ

y

?

?

,(27a)

φc =

w

c2

s

φ −

?c2

w

c2

s

?σ

y.

(27b)

The matrices aijand bijare obtained by straightforward

and somewhat tedious substitution of eqs. (25–26) in the

system of eqs. (23). The result is

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6

¨φr+ 3?1 + c2

¨φc− 3?c2

s

? ˙ y

? ˙ y

y

˙φr+ 3c2

w

˙ y

y

˙φc+3?c2

˙φc−

4τ2

w+ ϕ?

eyφr+(2/c2

4τ2

ey

φr−(2/c2

s) − 4 + 3c2

3τ2

w

ey2

φc= c2

s∇2φr,(28a)

s− c2

w

y

˙φr+ 3˙ y

y

3ϕ

s) − 3

3τ2

ey2

φc= 0(28b)

where ϕ ≡ (1−β)?2 + 2c2

evolution in the model of two perfect fluids. We would

like to stress that these are causal hyperbolic (wave)

equations in contrast to “action at a distance” type el-

liptic equation (14). This difference from the original

Einstein equations enables straightforward numerical in-

tegration of eqs. (28) directly in real space.

Eq. (28a) shows that perturbations in the radiation-

baryon plasma, described by φr, propagate as acoustic

waves with the sound speed cs. As the radiation wave

passes by, its gravity perturbs locally the cold dark mat-

ter as described by˙φcand φcterms in eq. (28b), and the

CDM potential φcstarts evolving.

s+ 12c4

s+ 3c2

sc2

w(2 − 3c2

s− 3c2

w)?.

Eqs. (28) are our principal equations of perturbation

C. Density and velocity fields

In order to analyze the CMB temperature anisotropy

or matter distribution in the universe, the potentials φr

and φcshould be related to perturbations in density and

velocity fields. The corresponding equations are derived

in this subsection.

It is convenient to replace c2

variables linear in y:

wand c2

sby the following

A ≡

1

3c2

w

= 1 +3

4y ,B ≡

1

3c2

s

= 1 +3

4βy .(29)

For the quantities

δ ≡

?δρa

ρm

,u ≡

?(ρa+ pa)ua

ρm

,(30)

the entropy perturbation η of eq. (21), and for an addi-

tional variable θ defined below, from eqs. (12,22,17) we

find

δ =B

yδr+ (1 − β)δc=

u =4B

3yur+ (1 − β)uc=

η = (1 − β)?3

θ ≡ (1 − β)?ur− uc

Since by the last two equations ˙ η + ∇ · jη= 0 with jη≡

−∇θ, physically, θ is a potential for the entropy current

density.

The above equations become particularly symmetric

in (yφ) and σ if the perturbation of energy density is

2τ2

3

e

?

e

∇2(yφ) − 3˙ y

3(yφ)˙,

2τ2

e

3∇2σ ,

2τ2

e

3

˙ σ .

y(yφ)˙

?

,

2τ2

4δr− δc

?=

?=

(31)

described in terms of

ǫ ≡ δ + 3˙ y

yu .

(32)

Then by eqs. (31)

ǫ =

2τ2

3∇2(yφ) ,

e

(33)

and ǫ satisfies the conservation equation ˙ ǫ + ∇ · jǫ= 0

with jǫ= −∇u.

The linear combination on the right hand side of

eq. (32) remains invariant under redefinition of constant-

time hypersurfaces, ˜ τ = τ + α(r,τ), and spatial coordi-

nates, ˜ r = r + ∇β(r,τ), when in the linear order

˜δa= δa−˙ ρa

ρaα ,˜ ua= ua− α .(34)

(For any α?=0 or β ?=0, the metric ds2will no longer be

conformally isotropic in space as it is in eq. (6).) Using

the equation of energy conservation,

˙ ρa= −3˙ a

a(ρa+ pa) ,

we can see that

˜ ǫ = ǫ ,˜ η = η ,

˜θ = θ .

Hence the invariant variable ǫ of eq. (32) equals the en-

ergy density perturbation˜δ in the gauge where ˜ u is iden-

tically zero, i.e.

the combined momentum density of

both fluids vanishes. Such a gauge is a generalization of

the comoving gauge to our multicomponent system, and

eq. (32) generalizes Bardeen’s gauge invariant variable ǫ

introduced in Ref. [2].

Using eqs. (31–32) and the definitions of φrand φcin

eqs. (25–26), it is straightforward to find:

δr= ǫr−3 ˙ y

Au ,δc= ǫc−3

4

3 ˙ y

Au

(35a)

where

ǫr≡

ǫc≡

y

A(ǫ + η) =

3y

4Aǫ −

2τ2

3

e

y

A∇2?A

3

Byφr

1

1−β∇2(yφc) ,

?

,

B

A(1−β)η =

2τ2

e

(35b)

and

ur=

uc=

3y

4A(u + θ) =

3y

4Au −

2τ2

3

e

3y

4A

?A

3

Byφr

2τ2

e

1−β

?˙,

B

A(1−β)θ =

1

?

(yφc)˙−(A/B)˙

(A/B)yφr

?

.(35c)

Page 7

7

In addition to determining the density and velocity

perturbations, eqs. (35) also help to establish the physical

meaning of the potentials φr and φc. From eqs. (35b),

remembering eq. (17),

∇2φr = 4πGa2

∇2φc = 4πGa2ρcǫc.

Thus, the potentials φr and φc are the metric pertur-

bations generated individually by the (comoving gauge)

disturbances in the radiation-baryon plasma and CDM.

The following conformal Newtonian gauge quantities

contribute to the primary CMB temperature anisotropy

in the tight coupling regime, Refs. [16, 17]:

ur, and˙φ.Of these, ur is related to φr by the first

of eqs. (35c) and φ is given by eq. (25). We can compute

1

4δr+φ from urand φ as

?

ργǫr+3

4ρbǫr

?

,

1

4δr+φ,

1

4δr+ φ = ˙ ur+3

4β

?

(yur)˙− yφ

?

.(37)

This equation follows from eq. (13d).

III.GREEN’S FUNCTIONS AND INITIAL

CONDITIONS

A.Defining equations

The evolution of cosmological perturbations in the lin-

ear regime may be studied in position space by super-

imposing their individual Green’s functions.

Green’s functions one can take any complete set of so-

lutions of eqs. (28) satisfying suitable initial conditions.

We consider two types of initial conditions: An initially

point-like perturbation at the origin, e.g.

For the

φ(3)(r,τ) → δ(3)

D(r)as τ → 0(38)

where δ(3)

functions δD(x)δD(y)δD(z), and a sheet-like perturbation

that is produced on the whole (0,y,z) plane and is inde-

pendent of the y and z coordinates, Ref. [8], e.g.

D(r) is the product of spatial coordinate delta

φ(1)(x,τ) → δD(x)as τ → 0 . (39)

The initial conditions in the form (39) will prove to be

the most attractive for describing perturbation evolution.

All the fields φr, φc, δr, etc. are functions of three

spatial coordinates x, y, and z, in addition to τ.

however, a configuration of fields has no initial depen-

dence on y and z, as in eq. (39), the fields will remain

independent of y and z for all time. For such a configu-

ration one needs to solve the evolution equations (28) in

the x and τ coordinates only. Thus a Green’s function

for the evolution equations with the plane-parallel initial

conditions, eq. (39), is also the Green’s function for the

If,

same equations considered in one spatial dimension. We

denote these Green’s functions by the superscript “(1)”,

and use the superscript “(3)” to distinguish the perturba-

tions originating from a point disturbance, as in eq. (38).

When setting the initial conditions, one should also

specify the ratio of perturbation magnitude for differ-

ent species, in our case the radiation and the dark mat-

ter. In this paper we consider only adiabatic initial con-

ditions, which are favored by the present experimental

data, Refs. [18, 19, 20, 21, 22], and by the simplest infla-

tionary models, Refs. [23]:

σ

yφ→ 0asτ → 0 .(40)

(A Green’s function approach to isocurvature initial con-

ditions, arising naturally in topological defect models,

Ref. [24], has been considered in Refs. [10, 11].)

Given the initial ratio of perturbations in various

species, e.g. as implied by eq. (40), the subsequent evo-

lution is completely described by a single set of Green’s

functions, either one-dimensional or three-dimensional,

for the relevant variables, in our case φr and φc. To il-

lustrate the use of Green’s functions, consider the total

gravitational potential φ. The time dependence of its

Fourier modes is given by

φ(k,τ) = φ(T)(k,τ)A(k) .(41)

Here, A(k) is a primordial potential perturbation created

by inflation, Ref. [25], or another mechanism of perturba-

tion generation, e.g. Refs. [24, 26], and φ(T)(k,τ) is the

Fourier space transfer function

φ(T)(k,τ) =

?+∞

−∞

dxe−ikxφ(1)(x,τ)(42)

or

φ(T)(k,τ) =

?

d3xe−ik·rφ(3)(r,τ) .(43)

Comparing eq. (43) and eq. (42), one can relate the two

types of Green’s functions as

φ(1)(x,τ) =

?+∞

|x|

2π rdrφ(3)(r,τ) .(44)

Eq. (44) says that in order to find the magnitude of the

perturbation at a distance x from the plane of the initial

excitation (39), one should add up the perturbations orig-

inating at all the points on the plane which are separated

from the considered point by r =

versely, a three-dimensional Green’s function can always

be obtained from a plane-parallel one by differentiation:

?x2+ y2+ z2. Con-

φ(3)(r,τ) = −

1

2πr

∂φ(1)(r,τ)

∂r

.(45)

Page 8

8

B.Radiation era solutions

When radiation dominates the background density,

y ≃τ

τe

≪ 1 ,

c2

s≃ 1/3 and c2

For the adiabatic initial conditions (40), eqs. (27) give

φc∼yφrduring the radiation era. Retaining in eqs. (28)

the leading in y terms only,

s− c2

w≃ (1/4)(1 − β)y.

¨φr+4

τ

1

τ2φc = 3(1 − β)y

˙φr =

1

3∇2φr,

¨φc+3

τ

˙φc+

?1

4τ

˙φr+

1

τ2φr

?

.

Their non-singular one-dimensional Green’s function so-

lutions satisfying initial conditions (39) and (40) are

φ(1)

r(x,τ) =

3

4

9(1−β)y

4

(csτ)2− x2

(csτ)3

θ(csτ − |x|) ,

?3(csτ)2− 2csτ|x| − x2

|x|

|x|

(47a)

φ(1)

c(x,τ) =

4(csτ)3

?

−

−

(csτ)2lncsτ

θ(csτ − |x|) , (47b)

where θ is the Heaviside step function: θ(x′) = 1 for x′>

0 and θ(x′) = 0 otherwise. The Fourier transforms of

these Green’s functions are the “growing mode” transfer

functions

φ(T)

r

(k,τ) = 3

?sinz

9(1−β)y

2

z3

−cosz

?sin(z) − z

z2

?

,

z ≡ kcsτ ,

+lnz + C − ci(z)

(48)

φ(T)

c

(k,τ) =

2z3

z2

?

where C =0.5772... is the Euler constant and

ci(z) ≡ −

?∞

z

cosz′

z′

dz′= lnz + C +

?z

0

cosz′− 1

z′

dz′

is the cosine integral.

IV. PROPAGATION OF PERTURBATIONS

The one dimensional Green’s functions for eqs. (28)

at a later time can be obtained by numerical integration

of these equations starting from the radiation era solu-

tions (47) at some τinit ≪ τe. The applied numerical

methods are described and compared with Fourier space

calculations in subsection IVC. The results of the inte-

gration up to the time of hydrogen recombination at red-

shift z ≃ 1.07×103are presented in Fig. 1. The original

delta function perturbation has separated into left-going

and right-going waves, whose evolution spreads φ(1)

space and diminishes the magnitude of the radiation dis-

turbance. The gravitationalpotential hill of the radiation

r

over

−0.5−0.250 0.250.5

0

0.5

1

1.5

2

φr(1)(x) and φc

(1)(x)

x / cτrec

FIG. 1:

φc (dashed) at the time of recombination τrec ≃ 2.15τe with

Ωm = 0.35, ΩΛ = 0.65, Ωbh2= 0.02, and h = 0.65.

Green’s functions for the potentials φr (solid) and

00.10.30.5

0

1

5

10

15

0.1 τe

φr

(1)

00.10.30.5

0

3

1

2

3

φc

(1)

00.10.30.5

0

0.5

τe

00.10.30.5

0

3

1

2

0 0.10.30.5

0

0.5

τrec

x / cτrec

00.1 0.30.5

0

1

2

x / cτrec

FIG. 2:

describing the gravitational potentials for radiation (left) and

CDM (right). The plots, from top to bottom, correspond to

the conformal time values 0.1τe, τe, and τrec ≃ 2.15τe.

Time evolution of the Green’s functions of Fig. 1

causes outward-directed gravitational forces which expel

the CDM away from x = 0. Three snapshots of the time

evolution of φ(1)

r

and φ(1)

c

are shown in Fig. 2. Only the

range x > 0 needs to be calculated since the potential

Green’s functions are even functions of x.

The Green’s functions are identically zero beyond the

acoustic horizon, |x| > S(τ), where

?τ

(S(τ) ≃ τ/√3 when τ ≪ τe). In a more careful treat-

ment one will find weak precursors extending beyond the

Green’s functions acoustic fronts up to the particle hori-

zon x = ±cτ. The precursors arise because of partial

photon, and in a full treatment neutrino, free streaming

S(τ) ≡

0

cs(τ′)dτ′=(49)

=

4τe

3√βln

?

1 +3

4yβ + τe˙ y

?

?

3

4β

1 +

3

4β

Page 9

9

−0.500.5

−10

0

10

δr(1)(x)

−0.500.5

−20

0

20

δc

(1)(x)

x / cτrec

FIG. 3: Green’s functions of the radiation density enhance-

ment δr(top) and the CDM density enhancement δc(bottom)

at the time of recombination. The vertical spikes represent

the delta-function singularities of δr(x) at its wavefronts and

of δc(x) at the origin.

with the speed of light. Postponing detailed discussion of

the photon diffusion until Sec. VI and of the neutrino free

streaming until a separate paper, we continue the analy-

sis of perturbation dynamics in the perfect fluid model.

The density and velocity perturbations of the photon-

baryon and CDM fluids are determined from the φrand

φc potentials by eqs. (35). The corresponding Green’s

functions are plotted in Figs. 3 and 4. Characteristic

features in the density and velocity Green’s functions

and their connection to cosmological parameters are dis-

cussed in subsection IVD. The delta function spikes at

the acoustic wavefronts of δ(1)

r

are fully calculable analytically and are considered in sub-

section IVB.

and v(1)

r

in the fluid model

A.Sum rules

Sum rules are simple, easy to apply integral relations

that offer a powerful tool for checking analytical formu-

lae or numerical calculations. The sum rules are more

general than the fluid approximation: they hold in the

linear regime regardless of how complicated the internal

dynamics of perturbations may be. They offer a simple

method for estimating the accuracy of numerical calcu-

lations of the Green’s functions. In addition, the sum

rules give insight into the connection between the posi-

tion and Fourier space descriptions. They become in-

dispensable for setting the initial conditions of Green’s

functions when one considers perturbations to the full

phase space distributions, Ref. [9].

The idea of the sum rules is to use eq. (42) with k = 0:

φ(T)(k =0,τ) =

?+∞

−∞

dxφ(1)(x,τ) .(50)

The time evolution of the k = 0 Fourier modes, the mag-

nitude of which we denote by upper case letters, e.g.

φ(T)(k =0,τ) ≡ Φ(τ), satisfies easily solvable ordinary

differential equations. The k-space equations at k =0 are

especially simple because, given the adiabatic initial con-

ditions, all the parameters characterizing relative motion

of different matter and radiation components are unper-

turbed at all time on infinitely large scales. For example,

in our case of the radiation-CDM model,

Σ(τ) ≡

?

dxσ(1)(x,τ) = 0. (51)

Trivially integrating first of eqs. (23) over x from −∞

to ∞, we obtain the equation for the gravitational po-

tential in the k = 0 mode:

˙Φ +3c2

¨Φ + 3?1 + c2

w

? ˙ y

y

w

eyΦ = 0 .

4τ2

(52)

With cw given by eq. (20), the linear independent solu-

tions of this equation are

f1 = 1 +

2

9y−

8

9y2−16

9y3,

1

16−f2 =

√1 + y

y3

≃ O(y) +

1

8y+

1

2y2+1

y3.

Taking the linear combination of f1and f2satisfying the

initial condition Φ(τ → 0) = 1, we obtain the sum rule

for the gravitational potential:

Φ(τ) ≡

?

dxφ(1)(x,τ) =

9

10f1+16

10f2. (53)

The sum rules for φr and φc then follow immediately

from eqs.(27,53,51):

Φr≡

?

?

dxφ(1)

r(x,τ) =

c2

c2

?

w

s

1 −c2

Φ ,(54a)

Φc≡ dxφ(1)

c (x,τ) =

w

c2

s

?

Φ .(54b)

The sum rules for the radiation velocity potential ur

and the combination1

4δr+φ are relevant for CMB tem-

perature anisotropy. The first sum rule simply follows

from the first of eqs. (35c) and from eq. (54a):

Ur≡

?

dxu(1)

r(x,τ) =

3c2

wτ2

2

ey

(yΦ)˙.(55)

The second can be derived starting from eq. (37) and

applying formulas of this subsection, including eq. (52),

1

4∆r+ Φ ≡

?

˙Ur= −3c2

dx

?1

4δ(1)

r

+ φ(1)

?

=

=

wτ2

2

ey2

?˙ y

yΦ

?˙. (56)

Page 10

10

−0.50 0.5

−5

0

5

vr(1)(x) , vc

(1)(x)

x / cτrec

−0.500.5

0

1

2

3

ur (1)(x) , uc

(1)(x)

x / cτrec

FIG. 4: Green’s functions of the velocities vr,c=−∇ur,c (left) and of the velocity potentials ur,c (right) at the time of

recombination. Solid and dashed lines are for the radiation and CDM respectively.

Finally, we give the sum rules for the radiation and

CDM energy density enhancements. They follow from

eqs. (35a–35b) and the second of eqs. (31):

∆r=4

3∆c= −2τ2

e˙ y

A

(yΦ)˙,(57)

with ∆r≡?dxδ(1)

r

and ∆c≡?dxδ(1)

c .

B.Wavefront singularities

We can obtain exact analytic formulae describing

Green’s functions at the acoustic wavefronts |x| = S(τ)

in the fluid model.As one will see below, the wave-

front spikes in the density and velocity Green’s functions

play a significant role in the acoustic dynamics. For the

wavefront analysis, it is convenient to factor the radiation

potential as

φr(x,τ) = C(τ)d(S(τ) − |x|,τ)

where for the perfect fluids, with x′≡ S(τ) − |x|,

(58)

d(x′≤ 0,τ) ≡ 0 ,

i.e. d(x′,τ) ≃ x′θ(x′) as x′→ 0.

equations in eq. (28a), setting |x| = S(τ), and taking

into account that at the wave fronts

∂

∂x′d(+0,τ) ≡ 1 ,(59)

Substituting these

φ(1)

r

= φ(1)

c

= 0 ,

˙φ(1)

c

= 0 ,(60)

we find the following simple relation for C(τ):

2˙C +

?

4˙ y

y+ 3˙ cs

cs

?

C = 0 .(61)

Here we have applied the formula

3c2

s

˙ y

y=˙ y

y+ 2˙ cs

cs

for the sound speed given by eq. (19). Integrating eq. (61)

and normalizing the result to agree with the radiation era

solution (47a) in the y → 0 limit, we obtain:

C(τ) =

9

2τ2

ey2(3c2

s)3/4. (62)

In particular, at x = ±S(τ),

˙φ(1)

r

= ∓csφ

′(1)

r

=

3√3

2τ2

ey2(3c2

s)1/4.(63)

The values of velocity potentials at the wave fronts

follow from eqs. (35c):

=3√3

4

u(1)

r

?3c2

s

?3/4,u(1)

c

= 0 .(64)

Because φrand urare identically zero when |x| > S(τ),

the velocity vr = −∇ur and the radiation density δr,

given by eqs. (35a–35b), both contain a delta function

singularity at the acoustic horizon:

δ(1)

r, sing=3√3

r, sing= ǫ(1)

r, sing= 3?3c2

4

s

?1/4δD(|x| − S) ,

?3/4sign(x)δD(|x| − S) .

(65a)

v(1)

?3c2

s

(65b)

The dynamics of the shock-like singularities in the

Green’s functions is linear, even if nonlinear effects would

take place in the propagation of a real shock wave in

the photon-baryon plasma. Indeed, the actual cosmo-

logical perturbations are the convolutions of the Green’s

functions with the smooth primordial potential field. As

long as the overall perturbations are small, they remain

as regular as the primordial perturbation field and their

dynamics is well in the linear regime. (On very small

scales, there may exist physically relevant nonlinear ef-

fects, Ref. [27], unrelated to the above Green’s function

singularities. These effects are absent when photons and

baryons are tightly coupled.)

C.Numerical integration

From the hyperbolic (wave) nature of eqs. (28), the

values of the potentials φrand φcat a point (x,τ + ∆τ)

are uniquely determined by their past values within the

interval [x − cs∆τ,x + cs∆τ] at time τ. Cusps at the

potential wavefronts will not be distorted by numerical

errors during the evolution if discretization intervals in

space and time are related so that cs∆τ is a multiple of

Page 11

11

∆x. In particular, for cs∆τ = ∆x, the second derivative

terms in eqs. (28) may be approximated by the second

order scheme

¨φr(x,τ) − c2

≃

s∇2φr(x,τ) ≃

1

(∆τ)2[φr(x,τ + ∆τ) + φr(x,τ − ∆τ)

− φr(x + cs∆τ,τ) − φr(x − cs∆τ,τ)] , (66)

¨φc(τ) ≃

≃

1

(∆τ)2[φc(τ + ∆τ) + φc(τ − ∆τ) − 2φc(τ)] .

In practice, we use the horizon radius S(τ) for the time

evolution coordinate and take ∆S = ∆x.

The first time derivatives of φrand φccan be approx-

imated by an implicit second order scheme

˙φ(τ) ≃

1

∆τ[φ(τ + ∆τ) − φ(τ)] −∆τ

2

¨φ(τ) .(67)

An explicit scheme should be avoided as numerically un-

stable. With the substitutions (66–67), eqs. (28) become

finite difference equations that can be straightforwardly

solved for the values of φr and φc at (x,τ + ∆τ). The

resulting finite difference scheme is of the second order

and it preserves the form of cusps or discontinuities in

propagating waves.

We start the integration from the radiation era solu-

tions (47b) at some small initial time, e.g. τinit≃ 10−4τe.

To follow the evolution over a large dynamic range, the

spacings of the space and time grids are increased by a

factor of 2 every time the extent of the acoustic hori-

zon S doubles. Then the number of discretization points

N = S/∆x used to represent the Green’s functions at

all later times remains in the range Nmin≤ N ≤ 2Nmin

where Nminis the initial number of points within the hori-

zon. The total number of φrand φcevaluations needed

to evolve the potentials to a final time τfinis of the order

of N2ln(τfin/τinit).

Whenever a Fourier space transfer function is needed,

it can be obtained from the corresponding Green’s func-

tion using the Fast Fourier Transform (FFT) algorithm.

The FFT can be efficiently applied to Fourier integrals

as described in Ref. [28] with its CPU time scaling as

N log2N. Due to the finite extent of the Green’s func-

tions, the related transfer functions oscillate in k and τ

requiring in general a larger number of evaluations for

their direct Fourier space calculation with the same ac-

curacy.

The sum rules introduced in Subsection IVA prove to

be a highly efficient debugging tool available for calcula-

tions in position space. The sum rule relations can also

be used to estimate the numerical error of computations.

Calculations of the Green’s functions in Figs. 1 and 2

proceeded from τinit ≃ 10−4τe to τrec ≃ 2.15τe. The

corresponding CPU time for our Pentium IV PC is 0.04s

with the minimal number of points in the spatial grid

Nmin= 64 and 0.1s with Nmin= 128. The accuracy of

these calculations as estimated by the sum rules is 0.19%

and 0.05% respectively.

D. Discussion of Green’s function features

The plane parallel Green’s functions for the potentials

φrand φc, plotted in Figs. 1 and 2, are continuous func-

tions of x and τ. In the fluid approximation, the radiation

potential is characterized by discontinuous first deriva-

tives (kinks) at its acoustic wavefronts as described by

eq. (63). The CDM potential has a central cusp reflect-

ing the initial repulsive singularity in the gravitational

potential φ(1)

r ; this cusp is preserved because the CDM

particles have no thermal motion. In fact, once the pho-

ton energy density becomes negligible in the matter dom-

inated era, the potential φc(x,τ) stops evolving, as may

be seen from eqs. (28) in the limit y ≫ 1.

The discontinuities in the potential derivatives give

rise to delta function singularities in the corresponding

density and velocity transfer functions, following from

eqs. (35) and plotted for the time of recombination in

Figs. 3 and 4. The singularities are visualized in the fig-

ures by the vertical spikes at x = ±S for δrand vrand

at x = 0 for δc.

The wavefront singularities in δ(1)

described analytically by eqs. (65). Their significance in

perturbation dynamics may be characterized by the delta

function contributions to the sum rule integrals for the

corresponding Green’s functions. For the radiation en-

ergy density, we find that in our ΛCDM model at recom-

bination

?dxδ(1)

turbation dynamics even on the largest scales. Their rel-

ative contribution to the CMB anisotropy even increases

on smaller scales because the oscillation amplitude of the

delta function Fourier transform does not fall off with k,

as opposed to the oscillations in the non-singular term

Fourier transforms. We have shown in Ref. [8] that in

real space the singularities give rise to a localized feature

in the CMB angular correlation function C(θ).

The delta function singularities acquire a finite width

when photon diffusion, or Silk damping, Ref. [29], is in-

cluded into the analysis. When the photon mean free

path is small compared with the scales considered, Silk

damping can be approximated as

r (x) and v(1)

r (x) are

r

sing/?dxδ(1)

r

≃ 2.7/(−2.3). Thus the

delta function singularities play a major role in the per-

δ(T)

r

(k) ≃ e−k2x2

Sδ(T)

r fluid(k) ,(68)

Refs. [30, 31, 32]. As found in Ref. [32],

x2

S(τ) ≃1

6

?τ

0

?

1 −14

15

?3c2

s

?+?3c2

s

?2?

cdτ

aneσT

(69)

where neis the proper number density of electrons and

σT is Thomson scattering cross section. Silk damping

broadens a delta function singularity in a position space

Page 12

12

−1−0.500.51

−2

0

2

∆(1)

eff = 1/4 δ(1)

γ + φ(1)

x / cτrec

Ωb= 0.02 h−2

Ωb= 0

|||||

FIG. 5: The position space transfer function for the intrinsic

(1

temperature anisotropy ∆T/T at recombination assuming

tight coupling between photons and baryons. The values of

cosmological parameters are taken as Ωm = 0.35, ΩΛ = 0.65,

and h = 0.65. Baryon drag effects are evident from comparing

the solid, Ωbh2= 0.02, and the dashed, Ωb= 0, lines.

4δγ) plus gravitational redshift contributions to the CMB

Green’s function to a Gaussian of finite width:

δD(|x| − S) →

1

2xS√π

exp

?

−(|x| − S)2

4x2

S

?

,(70)

as obtained by applying the filtering (68) to the con-

stant Fourier transform of a delta function.

erive eqs. (68–69) within our position space formalism

in Sec. VI.

The CDM density perturbations, with the Green’s

function shown in the bottom panel of Fig. 3, eventu-

ally seed the formation of galaxies. The central spike in

the figure arises from ∇2φc in the second of eqs. (35b).

It is negative because of the repulsive sign of the initial

gravitational potential peak, as shown in the top row of

Fig. 2. The spike is surrounded by positive tails because

the CDM pushed out from x = 0 piles up into the region

between x = 0 and the acoustic wavefront.

To understand the physics of the dip in δ(1)

in Fig. 3, let us examine the intrinsic radiation temper-

ature perturbation δr/4 corrected for the temperature

redshift by the local metric perturbation at x:

We red-

r (x) at x = 0

∆eff≡1

4δr+ φ . (71)

The plane-parallel Green’s function for ∆eff at recombi-

nation is shown in Fig. 5. The smoothness of ∆(1)

x = 0 in the absence of baryons (dashed line in Fig. 5)

can be understood as the condition of thermodynamic,

Ref. [33], and hydrostatic photon equilibrium near x = 0.

(Obviously the equilibrium does not apply at the acoustic

wavefronts, but it does hold as x → 0 where the elapsed

time is many sound-crossing times.) Linearizing the rel-

ativistic equation of hydrostatic equilibrium,

effat

∇ip + (ρ + p)∇iφ = 0 ,(72)

for the case of the radiation fluid containing no baryons

we obtain

∇i

?1

3δγ+4

3φ

?

=4

3∇i∆eff= 0 .(73)

Thus, if ρb = 0, ∆eff has zero gradient in equilibrium.

If Ωb ?= 0, on the other hand, the hydrostatic eq. (72)

applied to tightly coupled radiation-baryon fluid gives

∇i∆eff= −(3/4)βy∇iφ. The positive cusp of the CDM

gravitational potential becomes a dip of CMB tempera-

ture. The dip appears because the heavy pressureless

baryons are repelled by the cusp in the CDM gravi-

tational potential φc from the x = 0 region and they

drag the coupled photon fluid away from the origin. In

eq. (37), rewritten as

∆eff= ˙ ur+3

4β (y ˙ ur+ ˙ yur− yφr) −3

4βyφc,(74)

only the last, φc, term on the right hand side has a

cusp at x = 0. The contribution of this term and thus

the magnitude of the dip in ∆eff are roughly propor-

tional to βy(τrec) ∝ Ωbh2. Another effect of the baryonic

constituent on the radiation perturbations evident from

Fig. 5 is the decrease of the sound speed and thereby the

reduction of the acoustic distance compared with a pure

photon gas.

V.PERTURBATION DYNAMICS BEYOND

THE FLUID MODEL

Now we will distinguish between photon and baryon

perturbations and will not assume the fluid approxima-

tion for photons. We will continue to exclude neutrino

perturbations from our model as described in Sec. IIA.

The linear evolution equations for the photon and

baryon velocity potentials follow from linearized Boltz-

mann equation and are given in the Appendix as the

second of eqs. (A12) and the second of eqs. (A15). In

these equations, the terms proportional to the inverse of

the mean conformal time of a photon free flight,

τ−1

c

≡ aneσT , (75)

are due to Thomson scattering between photons and

baryons. The Thomson scattering damps the velocity

potential difference, for which we define

ud≡ ub− uγ. (76)

However, the scattering does not affect the overall mo-

mentum density of the plasma. Defining a momentum-

averaged velocity potential of photons and baryons

ur≡(ργ+ pγ)uγ+ ρbub

(ργ+ pγ) + ρb

=1

B

?

uγ+3βy

4

ub

?

, (77)

where B is given by eq. (29), from the velocity equations

in eqs. (A12,A15) we find:

˙ ur=

˙ ud= −˙ a

1

B

?

a

1

4δγ+ ∇2πγ−3β ˙ y

?ur+1

4ur

4δγ− ∇2πγ−

?

+ φ ,

Bud

?−11

τc

4B

3βyud.

(78)

Page 13

13

Substituting uγ= ur−3βy

the density evolution equations in eqs. (A12,A15), and

defining

4Budand ub=

1

Bud+ urin

δd≡ δb−3

4δγ,(79)

we obtain

˙δγ=4

˙δd= ∇2ud.

3∇2?

ur−3βy

4Bud

?

+ 4˙ψ ,

(80)

The cold dark matter density and velocity perturba-

tions evolve as

˙δc= ∇2uc+ 3˙ψ ,

˙ uc= −˙ a

auc+ φ .

(81)

The conformal Newtonian gauge potentials φ and ψ, de-

fined by eq. (6), can be determined non-dynamically from

the first, second, and fourth relations of eqs. (A1):

∇2ψ =

3

2τ2

ey

6

τ2

?B

yδγ+ (1 − β)δc+ βδd+ 3˙ y

yu

?

, (82a)

ψ − φ =

ey2πγ,(82b)

where formula (17) was applied.

eq. (82a) is defined as previously by eq. (30).

Two independent variables are needed to describe the

species number density perturbations3

ative to each other. For one such variable we use a poten-

tial generating the baryon density perturbation relatively

The variable u in

4δγ, δb, and δcrel-

to photons:

∇2σd≡

3

2τ2

e

δ

?

lnρb

T3

γ

?

=

3

2τ2

e

δd.(83)

The other can be the entropy potential σ from

eqs. (22,21) that now equals

∇2σ ≡

3

2τ2

e

δ

?

lnT3

ρm

γ

?

?3

=

=

3

2τ2

e

?

(1 − β)

4δγ− δc

?

− βδd

?

. (84)

The time derivatives of ∇2σdor ∇2σ equal a full ∇2

of a certain linear combination of velocity potentials ua.

Differentiating both sides of eq. (83) and eq. (84) with

respect to τ, remembering eqs. (80–81), and lifting ∇2,

one obtains:

˙ σ =

˙ σd=

3

2τ2

e

3

?

(1 − β)(ur− uc) −βA

eud.

Bud

?

,

2τ2

(85)

Taking another time derivative of both sides of

eqs. (85), using the dynamical

eqs. (78,81), and expressing all the resulting density

and velocity perturbations in terms of ∇2ψ, as given by

eq. (82a), ∇2σa, eqs. (83,84), and ˙ σa, eqs. (85),

˙ ua equations from

¨ σ +βc2

s

c2

w

¨ σd+?1 + 3c2

¨ σd+

w− 3c2

?

s

? ˙ y

+˙ y

y˙ σ +βc2

?

s

c2

w

˙ y

y˙ σd= y?c2

˙ σd+9˙ yc2

4

s− c2

w

?∇2

∇2

?

?

ψ +σ

y+

2πγ

c2

2πγ

c2

wτ2

ey2

?

?

,(86a)

4

9βyc2

sτc

y

w

˙ σ = −3c2

wy2

4

ψ +σ

y+

wτ2

ey2

. (86b)

Similarly to the fluid case, these equations are supplemented by the formula following from the first line of eq. (21),

eqs. (22,18), and the Einstein equations (A1):

¨ψ +?3c2

w+ 2? ˙ y

y

˙ψ +˙ y

y

˙φ +3c2

w

eyφ = c2

4τ2

w∇2

?

ψ +σ

y+

2πγ

c2

wτ2

ey2

?

.(86c)

Here φ is expressed via ψ and πγ using eq. (82b). The

set of eqs. (86,82b) is not closed as long as πγ remains

an independent variable.

We split the total potential ψ into potentials ψagen-

erated by the individual species:

ψ = ψγ+ ψc+ ψd.(87)

The dynamics of the potentials ψamay be described by

coupled wave equations provided the decomposition (87)

is performed according to the two following requirements:

First, on small scales, when the velocity term in eq. (82a)

is negligible, every ∇2ψa is proportional to the corre-

sponding δa and is independent of the density of the

other species. Second, the potentials ψa are given by

linear combinations of ψ and the entropy potentials σ

and σd. These linear combinations are unique and are

Page 14

14

easily found to be

ψγ =

c2

c2

?

βσd

y

w

s

1 −c2

?

ψ +σ

y

?

,(88a)

ψc =

w

c2

s

?

ψ −

?c2

w

c2

s

?σ

y−βσd

y

, (88b)

ψd =

. (88c)

The corresponding Laplacians are

∇2ψγ =

3

2τ2

ey

3

?B

?

?

yδγ+B

A3˙ y

yu

?

?

, (89a)

∇2ψc =

2τ2

ey

3

(1 − β)δc+

?

1 −B

A

?

3˙ y

yu

?

, (89b)

∇2ψd =

2τ2

ey

βδd

.(89c)

The evolution equations for the potentials ψaare ob-

tained by substituting eq. (87) and

σ = y

??c2

s

c2

w

− 1

?

ψγ− ψc− ψd

?

, σd= y1

βψd (90)

in eqs. (86,82b). Upon the substitution, one finds a sys-

tem of coupled second order wave equations generalizing

eqs. (28). For simplicity, here we write these equations

omitting the terms with no or only first derivatives of

the potentials ψaand πγunless a term contains the large

damping parameter τ−1

c . The neglected terms contribute

only to the dynamics on large scales λ>

fluid eqs. (28) can be used. With the remaining terms,

one obtains that on scales λ ≪ τ

?

¨ψd+

9βyc2

sτc

4

¨ψc ≈ 0 .

In general, eqs. (91) should be completed by the evolution

equations for the higher multipoles of the radiation phase

space density fγ l≥2and polarization multipoles gγ lthat

are given in the Appendix. However, in the following

section we find that in the next to the leading order in

the photon-baryon coupling τc, πγ≡1

mined by ψγand˙ψγ. Then, neglecting special features of

neutrino dynamics, the above equations provide a closed

system.

∼τ, when the

¨ψγ+¨ψd ≈ c2

4

˙ψd ≈ −3βyc2

s∇2

ψγ+

2πγ

c2

?

sτ2

ey2

?

, (91a)

s

∇2

ψγ+

2πγ

c2

sτ2

ey2

?

,(91b)

(91c)

2fγ 2is fully deter-

VI.TIGHT COUPLING: NEXT TO THE

LEADING ORDER

Now we can consider systematically how the general

formulae of the previous section reduce to the fluid equa-

tions in the limit of tight photon-baryon coupling and

find the O(τc) leading corrections to the fluid approxi-

mation.

First of all, ˙ udequation in eqs. (78) and all the equa-

tions for˙fγ l≥2and ˙ gγ lin the Appendix have the generic

form

˙fl= a −

1

rτc

(fl− b) ,(92)

where r is a positive number and a and b are some

linear combinations of variables other than fl. In the

tight coupling regime, τc ≪ τ, the evolution given by

eq. (92) will quickly, over a time of order τc, drive flto

fl≃ b+rτc(a−˙b), as is evident from the explicit solution

of eq. (92)

fl = b +?τdτ′(a −˙b) exp

?

−?τ

τ′

dτ′′

rτc

?

=

= b + rτc(a −˙b) + O(τ2

Thus at tight coupling the dynamical equations of the

form (92) reduce to algebraic equations fl≃ b up to O(τ0

or, if b in eq. (92) is absent, fl≃ rτca up to O(τc).

Applying this observation to the evolution of the pho-

ton polarization averaged multipoles fγ l in eqs. (A12)

and to the polarization difference multipoles gγ l in

eqs. (A14), we see that each l ≥ 3 multipole is suppressed

relative to the corresponding lower multipole by an extra

power of τc. Since fγ 1=4

fγl∼ τl−1

for l ≥ 2. The equations for˙fγ 2,˙˜ gγ 0and ˙ gγ 2reduce to

the following algebraic relations:

c) .

(93)

c)

3uγ∼ τ0

c, we then find that

c, and gγl∼ τl−1

c

for l ≥ 1, ˜ gγ0∼ τc, ˜ gγ1∼ τ2

c

fγ 2≃

˜ gγ 0≃1

1

10(fγ 2+ ˜ gγ 0+ gγ 2) + τc2

5fγ 1,

2(fγ 2+ ˜ gγ 0+ gγ 2) ,

1

10(fγ 2+ ˜ gγ 0+ gγ 2)

(94)

gγ 2≃

where the omitted terms are O(τ2

second of eqs. (78),

c). Similarly, from the

ud≃ −τc3βy

4B

?1

4δγ+˙ y

yur

?

. (95)

Resolving the linear algebraic system (94) in terms of fγ 1

and then remembering eq. (95) we find:

πγ≡1

2fγ 2≃ τc

4

15fγ 1≡ τc16

45uγ≃ τc16

45ur.(96)

Substitution of results (95,96) into the right hand side

of˙δγand ˙ urequations in eqs. (80,78) gives

˙δγ≃4

˙ ur≃

3

?

1 + τc

?

3βy

4B

?2

˙ y

y

?

∇2ur+1

16

45∇2ur

3τc

?

?

+ φ .

3βy

4B

?2

∇2δγ+ 4˙ψ ,

1

B

?

1

4δγ−3β ˙ y

4ur+ τc

Comparing these equations with their perfect fluid ana-

logues, eqs. (13c–13d), one can identify two qualitatively

Page 15

15

new terms appearing in the O(τc) order. The ∇2δγterm

in the first equation corresponds phenomenologically to

heat conduction and ∇2urin the second to bulk viscos-

ity of adiabatic scalar perturbations in the photon-baryon

plasma.

If Thomson scattering did not partially polarize the

scattered radiation, in place of eqs. (94) we would find

fγ 2 ≃ τc

and would wrongly yield a 25% smaller value of the ra-

diation bulk viscosity. The relevance of fluctuations in

photon polarization for dissipation of the perturbations

in photon-baryon fluid was pointed out in Ref. [32].

The remaining undetermined quantity describing the

relative motion of photons and baryons is their num-

ber density difference δd, which enters the Poisson equa-

tion (82a). We will calculate the related potential σdde-

fined by eq. (83). Within the O(τc) accuracy considered

here one can substitute the perfect fluid expressions (35)

into the right hand side of eq. (95). Then replacing φr

and φcby φ and σ and using the second of eqs. (23) to

eliminate ∇2(φ + σ/y), we find

ud≃ −τc2τ2

3

2

5fγ 1. This is 25% less than the result (96)

e

3β

4(1 − β)(y˙ σ)˙.

e)udfrom eq. (85) can then be

(97)

The relation ˙ σd= 3/(2τ2

integrated from τ = 0, with σd|τ=0= 0 for adiabatic

perturbations, to a given τ:

σd≃ −τc

3β

4(1 − β)y˙ σ . (98)

Thus to first order in τc one can continue to de-

scribe the perturbation dynamics by only two indepen-

dent fields φr≡ ψγand φc≡ ψc. By eqs. (88,82b), these

fields reduce to our earlier definitions (27) in τc → 0

limit. In the O(τc) order, the potential ψdis related to

φrand φcvia eq. (88c) and eq. (98), in which σ may be

substituted by the O(τ0

c) order expression (26).

Since πγin eq. (96) is an O(τc) quantity, uron the right

hand side of eq. (96) may also be expressed in terms of φr

using the leading order relations (35c). When the corre-

sponding expression for πγand eq. (98) are used to close

the dynamical equations of the previous subsection, one

finds two coupled differential equations for φrand φcthat

contain all the terms of eqs. (28) and additional terms

proportional to τc. (Note that the third time derivative

of σ arising from ¨ σd may be reduced within our O(τc)

accuracy by relations (23) to terms containing a lesser

number of time derivatives.) When τc ≪ τ, the O(τc)

terms containing three derivatives of the potential may

still contribute appreciably to small scale variations of

the potentials when their higher τ or spatial derivatives

become increasingly large.

derivative terms with O(τ0

c) coefficients and all the third

derivative terms, from eqs. (91a,91c) we arrive at the

following system, valid on small scales λ ≪ τ:

¨φr ≃ c2

¨φc ≃ 0

Retaining only the second

s∇2φr+ 2τcg∇2˙φr, (99a)

(99b)

where

g(τ) ≡1

6

?

1 −

14

15B+

1

B2

?

.(100)

The last term in eq. (99a) and so the expression (100)

receive contributions from both¨ψd and πγ terms in

eq. (91a).

The ∇2˙φrterm in eq. (99a) describes the famous Silk

damping of perturbations in the photon-baryon plasma

on small scales, Ref. [29]. In momentum space, the dis-

persion relation imposed by eq. (99a) on a plane wave

φr= Arexp(ik · r − iωτ) is

ω2+ 2iτcgk2ω − k2c2

The solutions to first order in τcare ω = ±kcs−iγ with

the damping rate

s= 0 .(101)

γ ≃ τcgk2.(102)

This rate coincides with the result of Ref. [32], where

derivations were done with a different approach that also

took into account the polarizing property of Thomson

scattering.

One can quantitatively describe photon diffusion at the

sharp wavefront of φ(1)

r (x,τ), as it evolves from τinit→ 0

to a certain finite τ, by considering the full eq. (28a) with

the third derivative term 2τcg∇2˙φr added to its right

hand side. We look for a solution in the wavefront region

|S(τ) − |x|| ≪ τ using the ansatz (58):

φ(1)

r

= C(τ)d(x′,τ) ,

where C(τ) is given by eq. (62) and initially d(x′,0) ≈

x′θ(x′). Assuming that

x′≡ S(τ) − |x|

|∂d/∂x′| ≫ |d/τ| and |∂d/∂x′| ≫ |∂d/∂τ|

and remembering eq. (61) as the condition of cancellation

of the ∂d/∂x′terms, we find:

(103)

∂2d

∂x′∂τ≃ τcg∂3d

∂x′3.

Integrating over dx′from −∞ to a given x′, we arrive at

the classical diffusion equation

∂d

∂τ≃ τcg∂2d

∂x′2.(104)

The solution of this equation satisfies the second condi-

tion in eq. (103) provided τcis much less than the charac-

teristic scale over which d(x′,τ) varies in x′. Individual

Fourier modes of any function satisfying eq. (104) are

damped as

d(k,τ) ≃ e−k2x2

S(τ)d(k,0)(105)

where

x2

S(τ) =

?τ

0

g(τ′)τc(τ′)dτ′, (106)

in agreement with eqs. (68–69).

By eq. (99b), small scale CDM dynamics is not affected

by Silk damping.

Page 16

16

VII.CONCLUSIONS

We have shown how to reduce the linearized cosmo-

logical dynamics of gravitationally interacting species in

the conformal Newtonian gauge to a system of coupled

wave equations. These equations indicate that the linear

evolution of not only the density fluctuations but also of

the corresponding gravitational potentials (metric per-

turbations) proceeds causally and locally, as opposed to

the instantaneous action at a distance seemingly implied

by the Poisson equation. In the fluid approximation,

a disturbance in the gravitational potential propagates

through the photon-baryon fluid at the speed of sound.

The locality of the linear dynamics permits the efficient

analysis of the perturbation evolution using the Green’s

function method. The Green’s functions are simply the

Fourier transforms of the familiar transfer functions, but

the causal nature of the Green’s functions provides in-

sights that were previously unknown.

We find that the Green’s functions of primordial

isentropic perturbations prior to photon decoupling are

sharp-edged acoustic waves expanding with the speed of

sound in the photon-baryon plasma. As shown in Fig. 3,

much of the integral weight of the photon density pertur-

bation is localized at the acoustic wavefront of the cor-

responding Green’s function. When the finite mean free

path to Thomson scattering is accounted for, the photon

density perturbation is broadened to the width of the Silk

damping length. The photon dynamics in the presence of

Thomson scattering is described by the Boltzmann equa-

tion, which we have analyzed to first order in the photon

mean free path. The result is a diffusive damping cor-

rections to the radiation wave equation. We show how

photon polarization affects Silk damping and provide an

alternative derivation of the results of Kaiser, Ref. [32].

Another insight obtained with the Green’s function

method concerns the effects of baryons. The baryonic

component of the plasma is responsible for a distinctive

central dip in the Green’s function for the gravitation-

ally redshifted CMB photon temperature perturbation

1

4δr+ φ as shown in Fig.

Ref. [9], to the known effect of odd peak enhancement and

even peak suppression, Ref. [7], in the CMB power spec-

trum. It is readily understood in our analysis through

the gravitational effect of the cold dark matter acting on

the baryons.

In this paper we considered perturbations only in pho-

tons, baryons, and CDM. The Green’s function method

can be applied to the linearized phase space dynamics

of an arbitrary number of species with rather general in-

ternal dynamics and mutual interactions, including neu-

trinos, Ref. [34]. The advantage of the position space

approach over the traditional Fourier space expansion is

its simple and explicit treatment of acoustic and trans-

fer phenomena underlying the dynamics of all particle

species. In addition to providing an intuitive, compact

framework for the many effects shaping the fluctuations

of matter and radiation, this approach can give simpler

analytical and faster computational methods for the still

challenging aspects of cosmological perturbations, such

as those of relic neutrinos.

lar wave equations and the Green’s function approach

to other areas of astrophysics where gravity influences

acoustic or transfer phenomena might be fruitful.

5.This feature gives rise,

Application of the simi-

APPENDIX A: BOLTZMANN HIERARCHY

IN POSITION SPACE

Two independent physical potentials are needed to

specify scalar metric perturbations in the general case,

see e.g. any of Refs. [2, 3, 4, 5, 6]. In the conformal

Newtonian gauge, these are φ(r,τ) and ψ(r,τ) defined

by eq. (6). The linearized Einstein equations in a flat

universe reduce to the following equations, following cor-

respondingly from the 0-0, 0-i, summed i-i, and trace-

less i-j components of Gµν= 8πGTµν, Ref. [12]:

∇2ψ − 3˙ a

a

?

˙ψ +˙ a

˙ψ +˙ a

aφ

aφ = 4πGa2?

3(ψ − φ) = 4πGa2?

?

= 4πGa2?

aρaδa,

a(ρa+ pa)ua,

aδpa,

a(ρa+ pa)πa.

¨ψ +˙ a

a

?

2˙ψ +˙φ

?

+

?

2¨ a

a−?˙ a

a

?2?

φ −1

3∇2(ψ − φ) = 4πGa2?

1

(A1)

The energy density enhancement δaand the velocity po-

tential uaof each of the matter or radiation species “a”

are defined, as previously, in terms of the corresponding

energy-momentum tensor components by eq. (10). The

variables δpaand πa, giving the isotropic and anisotropic

components of stress perturbation, are defined by

Ti

j(pa+ δpa) +

+ (ρa+ pa)3

aj = δi

2

?

∇i∇j−1

3δi

j∇2

?

πa

(A2)

where ∇i= ∇i, assuming negligible 3-space curvature.

The anisotropic stress potentials πavanish for perfect flu-

Page 17

17

ids and we see from the last of eqs. (A1) that the gravi-

tational potentials φ and ψ are then equal.

Six variables specify the coordinates of a particle in

phase space at a given time. For them, we take the co-

moving coordinates of the particle riand the comoving

momenta

qi≡ api

(A3)

where pi are the proper momenta measured by a co-

moving observer, Refs. [6, 35]. The momentum coordi-

nates qiare not canonically conjugate to the variables ri;

the canonical momenta of a particle of mass ma are

Pi= madxi/√−ds2= (1 − ψ)qi.

The particle density in phase space is specified by the

canonical phase space distribution fa(ri,Pj,τ):

dNa= fa(ri,Pj,τ)d3rid3Pj

(A4)

for every species of particles and their states of polariza-

tion a. Time evolution of the phase space distributions

is given by Boltzmann equation

∂fa

∂τ

+ ˙ ri∂fa

∂ri+ ˙ q∂fa

∂q

+ ˙ ni∂fa

∂ni

=

?∂fa

∂τ

?

C

,(A5)

where fais considered as a function of the coordinates ri,

q ≡|qi|, ni ≡ qi/q, and τ. The energy-momentum ten-

sor of the species a is given in the conformal Newtonian

gauge by the following simple expression up to the first

order of cosmological perturbation theory, Refs. [6, 35]:

Tµ

aν=

?

d3pipµpν

p0

fa

(A6)

with p0≡ −p0≡

later omit the species label a if referring to any sort of

particles in general.

As in the tight-coupling case, an arbitrary perturba-

tion δf(r,q, ˆ n,τ) of the phase space distribution about

the unperturbed distribution f0(q,τ) can be expanded

over plane waves, in which the value of δf at a given q,

ˆ n, and τ remains constant along two spatial directions.

To describe the dynamics of such a plane wave, we set the

spatial coordinates y and z to vary along these invariant

directions, and the coordinate x to vary along the remain-

ing direction. The perturbation δf(x,q, ˆ n,τ) can be ex-

panded over Green’s functions satisfying the delta func-

tion initial conditions δf(1)(x,q, ˆ n,τ) → A(q, ˆ n)δD(x −

x0) as τ → 0 with various initial locations x0and func-

tional forms of A(q, ˆ n). In the translationally invariant

background, the time evolution of the Green’s functions

is independent of x0.

If ϕ is the angle in the y–z plane between the vec-

tor (ny,nz) and the axis y, we may expand an arbitrary

A(q, ˆ n) in the Fourier series

?(q/a)2+ m2

aand pi≡ pi= qi/a. We

A(q,nx,ny,nz) =

+∞

?

m=−∞

eimϕAm(q,nx) .(A7)

By definition, the scalar perturbations have initially zero

Amfor m ?= 0. For a homogeneous and isotropic back-

ground, subsequent evolution according to the Boltz-

mann equation (A5) in the linear regime can not ex-

cite m ?= 0 components of δf through scalar pertur-

bations alone.Therefore, we will consider only the

perturbations that are axially symmetric about x axis:

δf = δf(x,q,nx,τ).

The number of coordinates needed to describe δf can

be reduced further when the particle mass is zero and the

relevant scattering cross-sections are energy independent.

Then one can work with the energy averaged perturba-

tion of the distribution, Ref. [36],

F(x,nx,τ) ≡

?q2dq q δf(x,q,nx,τ)

∞

?

?q2dq q f0(q)

(2l + 1)Fl(x,τ)Pl(nx) ,

= (A8)

=

l=0

where Plis a Legendre polynomial. The energy density

enhancement, mean velocity, and anisotropic pressure of

the particles described by an energy averaged distribu-

tion F are given by

δ =

?1

4

3

4

−1

3

?1

dnx

2F(nx) = F0,

dnx

2

nxF(nx) =3

?1

∂

∂x, since we are considering plane-

parallel perturbations that are constant in the y and

z directions. Eqs. (A9) follow directly from the defini-

tions (10a,10b,A2) and the formula (A6).

Continuing our practice of reducing the number of gra-

dients in the evolution equations, we define for all l

−∇u =

∇2π =

−1

4F1,

−1

dnx

2

?n2

x−1

3

?F(nx) =1

2F2

(A9)

for every massless species a. In these and the follow-

ing equations ∇ ≡

Fl≡ (−1)l∇lfl.(A10)

The evolution equations for the variables in eqs. (A9–

A10) can be derived in position space from Boltzmann

equation (A5), Ref. [9], or written down by replacing

momenta k by spatial gradients in previously derived

Fourier space equations, e.g. Ref. [6]. We give the cor-

responding hierarchy equations in the conformal Newto-

nian gauge (6) for neutrinos, photons, baryons, and CDM

particles.

The evolution of neutrinos is given by

˙δν =

4

3∇2uν+ 4˙ψ ,

1

4δν+ ∇2πν+ φ ,

l

2l + 1fν(l−1)+

˙ uν =(A11)

˙fνl =

l + 1

2l + 1∇2fν(l+1), l ≥ 2 ,

with πν=1

The collision terms for Thomson scattering of photons

and baryons were derived in Refs. [37]. The resulting

2fν2and uν=3

4fν1.

Page 18

18

equations read

˙δγ =

4

3∇2uγ+ 4˙ψ ,

1

4δγ+ ∇2πγ+ φ −1

l

2l + 1fγ(l−1)+

−1

τc

˙ uγ =

τc(uγ− ub) ,

l + 1

2l + 1∇2fγ(l+1)−

(A12)

˙fγl =

?

fγl−δl2

10(fγ2+ ˜ gγ0+ gγ2)

?

, l ≥ 2 .

Again, πγ =

eq. (75). The quantities gγl(x,τ) and ˜ gγl(x,τ) are de-

fined as

1

2fγ2, uγ =

3

4fγ1, and τc is defined by

Gγ0= ∇2˜ gγ0,

Gγ1= −∇3˜ gγ1,

Gγl= (−1)l∇lgγl,l ≥ 2

(A13)

where G is the energy averaged distribution describing

the difference of two linear photon polarization compo-

nents. The equations of their evolution are

˙˜ gγ0 = ∇2˜ gγ1−1

1

3˜ gγ0+2

τc

3gγ2−1

?

˜ gγ0−1

2(fγ2+ ˜ gγ0+ gγ2)

?

,

˙˜ gγ1 =

τc

l + 1

2l + 1∇2gγ(l+1)−

˜ gγ1,(A14)

˙ gγl =

l

2l + 1gγ(l−1)+

−1

τc

?

gγl−δl2

10(fγ2+ ˜ gγ0+ gγ2)

?

, l ≥ 2 ,

where gγ1≡ ∇2˜ gγ1in the last equation at l = 2.

Applying momentum conservation to photon-baryon

interactions, one can easily modify the non-relativistic

fluid equations of baryon evolution to include Thomson

scattering:

˙δb = ∇2ub+ 3˙ψ ,

˙ ub = −˙ a

(A15)

aub+ φ −1

τc

4

3βy(ub− uγ) .

The parameters y and β are defined by eq. (15). Aside

from the distinction between φ and ψ metric perturba-

tions, the equations of CDM evolution are the same as

in the fluid model:

˙δc = ∇2uc+ 3˙ψ ,

˙ uc = −˙ a

(A16)

auc+ φ .

ACKNOWLEDGMENTS

This work has benefitted from discussions with

J. R. Bond, A. Loeb, L. Page, P. J. E. Peebles,

P. L. Schechter, and U. Seljak. We also thank A. Shi-

rokov for his help with numerical calculations.

port was provided in part by NSF grant ACI-9619019,

by Princeton University Dicke Fellowship, and by NASA

grant NAG5-8084.

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