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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 γ Photons

subscript bBaryons

subscript r Radiation fluid (coupled photons and baryons)

subscript dDifference in photon and baryon perturbations

subscript ν Massless neutrinos

subscript c Cold dark matter (CDM)

subscript rel Background of relativistic species (γ plus ν)

subscript m Background 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

σ Entropy perturbation potential

fPhase space distribution

F Energy averaged perturbation of f

fl

Potentials for angular moments of F

βBaryonic 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

Meaning Defining 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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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