Evolution of Cosmic Structure · 2004-02-13 · 7 Evolution of Cosmic Structure The goal of...

FROM BIG BANG TO BLACK

HOLES

Evolution of Cosmic Structure

Max Camenzind

Landessternwarte Konigstuhl HeidelbergFebruary 13, 2004

Contents

7 Evolution of Cosmic Structure 2117.1 Evolution of Relativistic Perturbations . . . . . . . . . . . . . . . . . . . . . 211

7.1.1 Homogeneous Universe in Conformal Time . . . . . . . . . . . . . . 2117.1.2 Perturbations in the metric and in the energy-momentum tensor . . . 2137.1.3 Perturbed Einstein’s Equations . . . . . . . . . . . . . . . . . . . . . 2167.1.4 Gauge Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 2187.1.5 Einstein’s Equations in Different Gauges . . . . . . . . . . . . . . . 220

7.2 Fluid equations in the Newtonian Gauge . . . . . . . . . . . . . . . . . . . . 2237.2.1 Fluid equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2237.2.2 Gravitational Equations . . . . . . . . . . . . . . . . . . . . . . . . 2267.2.3 Solutions inside horizon: matter domination . . . . . . . . . . . . . . 2267.2.4 Solutions inside horizon: radiation domination . . . . . . . . . . . . 2277.2.5 Solutions outside horizon . . . . . . . . . . . . . . . . . . . . . . . . 2287.2.6 Transfer functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2297.2.7 On Baryons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2337.2.8 Role of Scalar Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 234

7.3 Nonlinear Evolution of Cosmic Structure . . . . . . . . . . . . . . . . . . . 2357.3.1 Cosmic Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2367.3.2 Collisionless Components . . . . . . . . . . . . . . . . . . . . . . . 2387.3.3 Baryons and Gasdynamics . . . . . . . . . . . . . . . . . . . . . . . 2387.3.4 The Spectrum of Primordial Fluctuations . . . . . . . . . . . . . . . 241

7.4 Cosmological Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 2417.4.1 Particle Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2427.4.2 Progress of Cosmological Simulations . . . . . . . . . . . . . . . . . 2447.4.3 The Fractal Structure of Dark Matter Distribution . . . . . . . . . . . 2447.4.4 Statistical Measures for Cosmic Density Fields . . . . . . . . . . . . 2517.4.5 2000’s: Baryons in Simulations . . . . . . . . . . . . . . . . . . . . 258

7.5 Dark Baryons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2597.5.1 The Lymanα Forest . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

7.6 Galaxy Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2597.7 The Cosmic Microwave Background Probes Linear Perturbations . . . . . . . 262

7.7.1 Anisotropy Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . 2637.7.2 The Resulting CMB Power Spectrum . . . . . . . . . . . . . . . . . 2677.7.3 More on Acoustic Oscillations . . . . . . . . . . . . . . . . . . . . . 269

7.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

Bibliography 277

4 Contents

7.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

A Perturbation Calculations 281A.1 Perturbations of the Einstein Tensor in Newtonian Gauge . . . . . . . . . . . 281

A.1.1 3+1 Slice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281A.1.2 Perturbation of Extrinsic Curvature . . . . . . . . . . . . . . . . . . 281A.1.3 Perturbation of the Spatial Ricci Tensor . . . . . . . . . . . . . . . . 281A.1.4 Perturbed Einstein Tensor . . . . . . . . . . . . . . . . . . . . . . . 281A.1.5 Curvature Perturbations . . . . . . . . . . . . . . . . . . . . . . . . 283

7 Evolution of Cosmic Structure

The goal of studying cosmological perturbations is to understand the evolution of the structurein the universe, from small ripples generated sometime after the initial quantum state all theway to galaxies, clusters and large–scale structure we see today. Relativistic perturbation the-ory describes this evolution in a general relativistic context. In cosmology such an approach isrequired, because we want to describe perturbations not just on small scales, where Newtonianlaws suffice, but also on scales comparable or larger to the Hubble radius, which can be usedas a typical scale of observable universe. This is nowadays a very active field of internationalresearch and I will concentrate on a presentation of only the most important aspects.

7.1 Evolution of Relativistic Perturbations

Before we introduce the perturbations we must first recapitulate the unperturbed (homoge-neous) evolution of the universe.

7.1.1 Homogeneous Universe in Conformal Time

The fundamental assumption of cosmology is that the universe is homogeneous and isotropicon average. To be more precise, the paradigm is that we live in a weakly perturbed Robertson-Walker universe, in which the metric perturbations are small, so the averaging procedure iswell defined and the backreaction of the metric fluctuations on the homogeneous equationsis negligible. Note that we only require metric perturbations to be small and there is norequirement on the density perturbations, which we know can be large on small scales.

One can write the line element in a homogeneous and isotropic universe usingconformaltime τ and comoving coordinatesxi as

ds2 = γµνdxµdxν = a2(τ)−dτ2 + γijdxidxj

. (7.1)

This is the Robertson-Walker metric. We will often use greek indices to denote 4-tensors andlatin to denote spatial 3-tensors. Herea(τ) is the scale factor expressed in terms of conformaltime τ , which is related to the proper timet via dt = a dτ . Similarly, proper coordinatesri

are related to the comoving coordinatesxi via ri = axi. We adopt units such thatc = 1. Thespace part of the background metric can be written as

γijdxidxj = dχ2 + r2(dθ2 + sin2 θdφ2),

r = sinK χ ≡

K−1/2 sin K1/2χ, K > 0χ, K = 0(−K)−1/2 sinh(−K)1/2χ, K < 0

(7.2)

212 7 Evolution of Cosmic Structure

whereK is the curvature term which can be expressed using the density parameter in allcomponentsΩ and the Hubble parameterH asK = (Ω − 1)H2. The present day values weusually denote with an additional subscript 0, egH0 andΩ0. The density parameterΩ canhave contributions from nonrelativistic matter (Ωm), such as baryons, cold dark matter (CDM)or massive neutrinos (mixed dark matter MDM, warm dark matter WDM etc.), relativisticmatter (Ωr), such as photons and massless neutrinos and from cosmological constant (ΩΛ)or some more general form of dark energy (or quintessence,Ωφ). The advantage of usingthe conformal timeτ is that the metric becomes conformally Euclidean (K = 0), 3-sphere(K > 0) or 3-hyperboloid (K < 0) and leads to a simple geometrical description of lightpropagation and other processes.

The usual starting point in general relativity is the Einstein’s equation

Gνµ = 8πGT ν

µ , (7.3)

whereGνµ is the Einstein tensor andTµ

ν is the stress-energy tensor. Einstein’s tensor is relatedto the spacetime Ricci tensorRµν by

Gµν ≡ Rµν − R

2gµν , R ≡ Rµ

µ , Rµν ≡ Rκµκν . (7.4)

An alternative starting point is the Einstein–Hilbert action

S =∫

dx4√−g

(1

16πGR + L

), (7.5)

whereL is the matter Lagrangian density. The action principleδS = 0 leads to equations ofmotion, which applied to equation (7.5) gives rise to Einstein’s equation (7.3) with

Tµν = −2∂L

∂gµν+ gµνL. (7.6)

This form will be useful when we introduce scalar field degrees of freedom.The Einstein field equations (7.3) show that the stress-energy tensor provides the source

for the metric variables. The stress-energy tensor takes the well–known form

Tµν = (ρ + p)uµuν + pgµν + pΠµν , (7.7)

whereρ andp are the energy density and pressure,uµ = dxµ/dλ (wheredλ2 ≡ −ds2)is the fluid 4-velocity andpΠµν is the shear stress absent for a perfect fluid. In locally flatcoordinates in the fluid frame,T 00 = ρ, T 0i = 0, andT ij = pδij for a perfect fluid.

Einstein’s equations applied to the background metric gives the evolution of the expansionfactora(τ),

(a

a

)2

≡ H2 =8π3

Gρa2 −K , (7.8)

H =1a

d2a

dt2= −4π

3Ga2(ρ + 3p) . (7.9)

Overdots denote derivatives with respect to the conformal timeτ . For convenience weintroduced comoving Hubble parameterH = a/a, which will appear often in the equations

7.1 Evolution of Relativistic Perturbations 213

below. Its value today isH0. These are the Friedmann equations applied to the Robertson-Walker metric. The density todayρ0 is related to the density parameterΩ0 via 8πGρ0/3 =H2

0Ω0. Remember that forw = p/ρ < −1/3 the universe is accelerating.The mean density of the universeρ (and similarly the mean pressurep) can be written as

a sum of matter, radiation, cosmological constant or any other dark energy contributions,

ρ = ρm a−3 + ρr a−4 + ρΛ + ρφ a−3(1+w), (7.10)

wherew is the equation of state of the dark energy.The energy-momentum tensor is required to obey local conservation lawTµν

;ν = 0, whichgives

ρ + 3H(ρ + p) = 0 . (7.11)

One can also derive the evolution equation for a homogeneous scalar fieldφ evolving in apotentialV (φ). Its Lagrangian is

L =√−g

[12gµν∂µ∂νφ + V (φ)

], (7.12)

wheregµν is the metric. Stress energy tensor is given by

Tµν = ∂µφ∂νφ + gµνL, (7.13)

which gives

ρφ =φ2

2a2+ V (φ) pφ =

φ2

2a2− V (φ). (7.14)

Equation of statew = p/ρ is in general a function of time. Continuity equation (7.11) gives

φ + 2Hφ + a2V ′ = 0 . (7.15)

The scalar field source has to be added to the Friedmann equations above and modifies the ex-pansion of the universe. It obeys the same energy-momentum conservation as the other fluidsand can be easily integrated to find energy density as a function of time (or expansion factoras in equation 7.10). In the limit where kinetic term is negligible compared to the potentialscalar field reduces to the cosmological constant withw = −1. This case is relevant bothfor inflation and for the possible late time contribution from the dark energy (Quintessencemodels).

7.1.2 Perturbations in the metric and in the energy-momentum tensor

Our universe is not homogeneous: we see inhomogeneities caused by gravity present on allscales, from planets to clusters, superclusters and beyond. We want to describe the deviationsfrom the isotropy and homogeneity of the universe using general relativity. Small perturba-tionshµν around the Robertson-Walker metric are

gµν = a2(γµν + hµν). (7.16)


In the most general form one can write the perturbed line element using conformal timeτand comoving coordinatesxi as

ds2 = a2(τ)−(1 + 2A) dτ2 − 2Bid τ dxi + [(1 + 2HL)γij + 2hij ] dxi dxj

. (7.17)

The perturbations have been decomposed into

• time–time component−2A (scalar potential),

• time–space component−2Bi (shift vector),

• trace of the space–space component2HL (spatial curvature perturbation)

• and traceless space–space component2hij (traceless spatial metric perturbation).

In total we have introduced 10 metric perturbations. Vector fieldBi can be further decomposedinto a scalar component, which arises from a gradient of a scalar field, and a pure vectorcomponent, which is the remainder of what is left. Similarly we can decompose tensorhij

into a scalar, vector and tensor components. As we show below these perturbations can bedecomposed into

• 4 scalar (m = 0, compressional),

• 4 vector (m = ±1, vortical) and

• 2 tensor (m = ±2, gravitational wave) eigenmode components,

which differ in their transformation properties under spatial rotations. The advantage of thisdecomposition is that thelinearized equations decouple into separate scalar, vector andtensor components, with no cross–coupling between them.

Eigenmodes of the Laplacian

In linear theory, each eigenmode of the Laplacian for the perturbation evolves independently,and so it is useful to decompose the perturbations via the eigentensorQ(m), where

∇2Q(m) ≡ γijQ(m)|ij = −k2Q(m), (7.18)

with “ |” representing covariant differentiation with respect to the 3–metricγij . Note thatto the lowest order one can raise and lower indices on 3-tensors using the three metricγij .The eigentensorQ(m) has |m| indices (suppressed in the above). To obtain a pure vectorcomponent, it has to be obtained from a scalar field. In real space this is a gradient of a scalarfield. This mean that vector modes satisfy the auxiliary condition

Q(±1)i

|i = 0 , (7.19)

which represents the divergence–free condition forvorticity waves.Similarly, to obtain a pure tensor mode we must subtract out components that can be

formed from a scalar and vector field. The auxiliary conditions are

γijQ(±2)ij = 0 = Q

(±2)ij

|i , (7.20)


which represent the transverse–traceles and divergence–free conditions respectively, as appro-priate forgravity waves.

We will often focus on perturbations in flat space, both because they lead to simplifiedexpressions and because they seem to be observationally favored. In this case the eigenmodesare particularly simple. If we assume the direction of the wavevector~k is e3 then

Q(±m)i1...im

∝ (e1± ie2)i1 ⊗ . . .⊗ (e1± ie2)imexp(i~k ·~x) , (K = 0,m ≥ 0) , (7.21)

where the presence ofei, which forms a local orthonormal basis withe3 = k, ensures thedivergenceless and transverse-traceless conditions. One can see now the transformation prop-erties of eigenmodes under rotation in the plane perpendicular toe3 is such that

Q(±m) = Q(±m)e∓imψ, (7.22)

whereψ is the rotation angle. The eigenmodes are essentially plane waves

Q(0) = exp(i~k · ~x) (7.23)

Q(±1)i =

−i√2

(e1 ± ie2)i exp(i~k · ~x) (7.24)

Q(±2)ij = −

√38

(e1 ± ie2)i ⊗ (e1 ± ie2)j exp(i~k · ~x) . (7.25)

The vectors~e1 and~e2 span the plane tranverse to~k.It is also useful to construct (auxiliary) vector and tensor objects out of the fundamental

scalar and vector modes through covariant differentiation

Q(0)i = −k−1Q

(0)|i , (7.26)

Q(0)ij = k−2Q

(0)|ij +

13γijQ

(0) , (7.27)

Q(±1)ij = −(2k)−1(Q(±1)

i|j + Q(±1)j|i ). (7.28)

ForK = 0 this becomes

Q(0)i = −i(ki/k)Q(0) , (7.29)

Q(0)ij = −(kikj/k2)Q(0) +

13γijQ

(0) , (7.30)

Q(±1)ij = −i(kjQ

(±1)i + kiQ

(±1)j )/(2k) . (7.31)

The eigenmodes form a complete set, so that any perturbation can be expanded in terms ofthese.

The metric perturbations can be broken up into the normal modes of scalar (m = 0),vector (m = ±1) and tensor (m = ±2) type,

A = A(0)Q(0) , (7.32)

HL = H(0)L Q(0) , (7.33)

Bi = −1∑

m=−1

B(m)Q(m)i , (7.34)

hij =2∑

m=−2

H(m)T Q

(m)ij . (7.35)


Note that this gives 4 scalar, 4 vector and 2 tensor components.

The stress energy tensor can likewise be broken up into scalar, vector, and tensor contri-butions. The fluctuations can be decomposed into the normal modes as

δT 00 = −δρ(0) Q(0),

δT 0i =

∑1m=−1[(ρ + p)(v(m) −B(m))] Q(m)

i ,

δT i0 = −∑1

m=−1(ρ + p)v(m) Q(m)i,

δT ij = δp(0)δi

jQ(0) +

∑2m=−2 pΠ(m)Q(m)i

j .

(7.36)

Note that the two mixed time–space components are not equal. We introduced above thedensity perturbationδρ(0), pressure perturbationδp(0), velocity perturbationv(m), which hasa scalar (m = 0) and vector (±1) component and anistropic stress perturbationpΠ(m), whichcan be scalar (m = 0), vector (m = ±1) or tensor (m = ±2) type. These are in general a sumfrom all of the species present,

δTµν =∑

I

δTµνI , (7.37)

where the indexI stands for baryons, CDM, photons, neutrinos (massive and massless), darkenergy etc.

A minimally coupled scalar fieldϕ also has perturbations,ϕ = φ + δφ. They can berelated to the fluid quantities by expanding equation (7.13) to the next order

δρ(0)φ = a−2(φδφ−A(0)φ2) + V ′δφ ,

δp(0)φ = a−2(φδφ−A(0)φ2)− V ′δφ ,

(ρφ + pφ)(v(0)φ −B(0)) = a−2kφδφ ,

pφΠ(0)φ = 0 , (7.38)

whereV ′ denotes derivative with respect toφ and only the lowest order terms have beenkept. This shows that there are no vector or tensor modes associated with the scalar field, asexpected in the linear order. Applying energy-momentum conservation equation (7.11) onefinds

δφ(0)+2Hδφ(0)+(k2+a2V ′′)δφ(0) = (A(0)−3HL(0)−kB(0))φ−2a2V ′A(0). (7.39)

7.1.3 Perturbed Einstein’s Equations

Einstein’s equations are also decomposed into various modes (see Appendix and Exercise).


Scalar Equations

Einstein’s equations for the scalar modes become (suppressing 0 superscripts)

(k2 − 3K)

[HL +

13HT +H

(B

k− HT

k2

)]= 4πGa2

[δρ + 3H(ρ + P )

v −B

k

]

k2(A + HL +13HT ) + (∂τ + 2H)(kB − HT ) = −8πGa2 PΠ (7.40)

HA− HL − 13HT − K

k2(kB − HT ) = 4πGa2(ρ + P )

v −B

k[2a

a− 2H2 +H∂τ − k2

3

]A− (∂τ +H)

(HL +

kB

3

)= 4πGa2

(δP +

13δρ

).

The equations of motion provide two equations

(∂τ + 3H)δρ + 3HδP = −(ρ + P )(kv + 3HL) (7.41)

(∂τ + 4H)(ρ + P )v −B

k= δP − 2

3(1− 3K/k2)PΠ + (ρ + P )A . (7.42)

These are in total 8 variables (4 metric and 4 matter) and 6 equations, leaving two gaugefreedoms.

Vector Equations

The vector modes are determined by two equations

(1− 2K/k2)(kB(±1) − H(±1)T ) = 16πGa2(ρ + P )

v(±1) −B(±1)

k(7.43)

[∂τ + 2H

](kB(±1) − H

(±1)T ) = −8πGa2PΠ(±1) , (7.44)

and one equation from the Euler equations

[∂τ + 4H

](ρ + P )

v(±1) −B(±1)

k= −1

2(1− 2K/k2)PΠ(±1) . (7.45)

Tensor Equations

The Einstein equation for the tensor mode is[∂2

τ + 2H∂τ + (k2 + 2K)]H

(±2)T = 8πGa2PΠ(±2) . (7.46)

In the absence of anisotropic stresses andK = 0, the tensor equation becomes a source–freegravitational wave equation which has solutions in terms of Hankel functions

H(±2)T (kτ) = C1H1(kτ) + C2H2(kτ) , (7.47)

where

H1(x) ∝ x−mjm(x) , H2(x) ∝ x−mnm(x) . (7.48)

Therebym = (1 − 3w)/(1 + 3w). If w > −1/3, then the gravitational wave amplitude isconstant above horizon,x ¿ 1, and then oscillates and damps.


7.1.4 Gauge Transformations

So far we have described the perturbations in a given coordinate system, but we did not saymuch about the coordinate system itself. In the absence of perturbations there is a preferredcoordinate system, which corresponds to comoving frame, in which the observers see themomentum density to be zero at that point and the comoving observers are free falling. Thespatial slices (defined as slices of constant time) are orthogonal to the time threading (definedas a constant coordinate~x) and forK = 0 they are spatially flat. ForK < 0 they correspondto a 3-hyperboloid and forK > 0 to a 3-sphere. This defines the background SpaceTime.

The choice of coordinates becomes nontrivial once we discuss the perturbations and thereis no unique choice. One can, for example, choose the coordinates in which the observers arefree–falling (corresponding to thesynchronous gauge), but this is no longer the same choiceas if the observers are comoving so that the momentum density vanishes (corresponding tothe comoving gauge). One can instead choose a spatially flat gauge so that spatial compo-nents of the metric vanish, or a gauge with zero metric shear with threading and slicing beingorthogonal (corresponding toconformal Newtonian or longitudinal gauge).

The choice of coordinate system has some freedom without affecting anything physical.One can change the coordinatesdxµ and the metric perturbationshµν in such a way that themetric distanceds2 (equation (7.17)) remains invariant.To represent the perturbations wemust thus make a gauge choice. A gauge transformation in GR is a change from onecoordinate choice to another. The most general form isxµ = xµ + δxµ or explicitly

τ = τ + T,xi = xi + Li.

T corresponds to a choice in time slicing andLi a choice of spatial coordinates. This can alsobe decomposed into Fourier modes

τ = τ + T (0)Q(0),

xi = xi +∑1

m=−1 L(m)Q(m)i .

Since gauge transformation only uses scalar (m = 0) and vector quantities (m = ±1), it isclear that tensor modes (m = ±2) will not change under gauge transformation. They arethus gauge-invariant. Even though the coordinates can change the metric distance,ds2 mustremain invariant, i.e.gµνdxµdxν = gµνdxµdxν . Sincegµν is a tensor and transforms in thesame way as other tensors we can derive the transformation property of a general tensorWµν

Wµν(xγ) =∂xα

∂xµ

∂xβ

∂xνWαβ(xγ − δxγ)

= Wµν −Wµβ∂νδxβ −Wαν∂µδxα − δxα∂αWµν . (7.49)

A gauge transformation equals a coordinate transformation of the perturbed first–order quan-tities generated by

xµ → xµ + Xµ . (7.50)

A symmetric 2–tensor transforms under such a gauge transformation as

δW → δW + LXW . (7.51)


For the metric this means

δgµν → δgµν + Xσgµν,σ + gσµXσ,ν + gσνXσ

,µ . (7.52)

This transformation law applied to the metrichµν in equation (7.17) gives explicitly

A(0) = A(0) − T (0) −HT (0) ,

B(m) = B(m) + L(m) + kT (m) ,

H(0)L = H

(0)L − k

3L(0) −HT (0) ,

H(m)T = H

(m)T + kL(m) , (7.53)

wherem = 0,±1.The stress-energy perturbationTµν in different gauges is similarly related by the gauge

transformation

δρ(0) = δρ(0) − ˙ρT (0) ,

δp(0) = δp(0) − ˙pT (0) ,

v(m) = v(m) + L(m) ,

Π(m) = Π(m) , (7.54)

wherem = 0,±1 in the velocity equation andm = 0,±1,±2 in anisotropic stress equation.The anisotropic stress is gauge-invariant.

Finally, a scalar function transforms as

f(τ , xi) = f(τ, xi)− ∂f

∂τT − ∂f

∂xiLi. (7.55)

The spatial gradient of the scalar function is of first order, so the last term in the expressionabove is of second order and can be dropped. The scalar field thus transforms as

δφ(0) = δφ(0) − φT (0) . (7.56)

Gauge–Invariant Bardeen Potentials

The most generalscalar perturbation of the metric is of the form

δg = a2 [−2ψ dt2 − 2(∇iB) dxi dt + (2φγij −∇i∇jE) dxi dxj ] . (7.57)

One can then derive that the four potentials transform as follows

ψ → ψ + T + HT (7.58)

φ → φ−HT (7.59)

B → B + T − L (7.60)

E → E + L . (7.61)

With the four potentialsψ, φ, B andE and the two gauge freedomsT andL we can findtwo gauge–invariant quantities, which are not unique. The simplest two potentials have beenintroduced by Bardeen [1] and are defined as

Ψ = ψ − 1a

d

dτ

[a(E + B)

](7.62)

Φ = φ + H[E + B] . (7.63)


By using the gauge transformations one can show the invariance of these two potentials (seeexercises).

TheNewtonian (or longitudinal) gaugeis defined by the choice

L = −E , T = −E −B (7.64)

with the consequence that nowB = 0 and E = 0. Obviously, in this gaugeE and Bwill vanish after the gauge transformation, and we are left with the two invariant Bardeenpotentials.

7.1.5 Einstein’s Equations in Different Gauges

The choice of gauge can be governed by the simplicity of equations, numerical stability ofsolutions, Newtonian intuition or other considerations. As we discussed above there is nogauge ambiguity for tensor modes. For vector modes the choice can either beB(±1) = 0 orH

(±1)T = 0. The latter specifies the gauge completely, since it fixesL(±1), while the former

only fixesL(±1) and thus leads to unspecified integration constant inH(±1)T . This however

does not lead to any dynamical effects. Since vector modes are unlikely to be generated in theearly universe we will not discuss them further. Instead we will look at four popular gaugechoices for scalar modes that we will have the chance to use in the rest of the lectures. Wewill drop the superscripts(0) from now on.

Synchronous gauge

This gauge was very popular in the early development of perturbation theory. Its main advan-tage from today’s perspective is its numerical stability, which is why it is still the gauge ofchoice in CMBFAST numerical package [19]. It corresponds to settingA = B = 0, so thatonly the spatial metric is perturbed. This implies that slicing is orthogonal to the threadingand that a set of freely falling observers remains at a fixed coordinate position. To show thisone must show that the spatial part of 4-velocityuµ = dxµ/dλ, whereλ is affine parameterparametrizing the geodesic, vanishes. Geodesic equation is

Duµ

Dλ= uµ

;ν

dxν

dλ=

duµ

dλ+ Γµ

αβuαuβ = 0. (7.65)

SinceA = B(m) = 0 impliesΓi00 = 0 it follows thatui = 0 is a geodesic.

The property that the fundamental observers follow geodesics means that the coordinatesare Lagrangian and this gauge can only be used whileδρ/ρ ¿ 1. In the nonlinear regimewhere this condition is not satisfied one can have orbit crossings where two observers withdifferent Lagrangian coordinates find themselves at the same real (Eulerian space) position.This can only happen if the metric perturbations diverge and the linear perturbation theoryis no longer valid. While this limits the use of this gauge at late times it can still be usedsuccessfully in the early universe, as long as the density perturbations are small.

Another shortcoming of this gauge is that the gauge choice does not fully specify it. Onecan see this by using a gauge transformation from a general gauge. ImposingA = B = 0 tothe equations (7.53) we find

T = a−1

∫aAdτ + c1a

−1

L = −∫

(B + kT )dτ + c2, (7.66)


wherec1 and c2 are integration constants. These remain unspecified in this gauge. Theylead to unphysical gauge mode solutions for the density perturbations outside horizon. Whilehistorically this caused some confusion in their interpretation they are not really a problemsince these modes do not show up in any observable quantity and are at any rate decayingfaster than the physical modes.

The remaining two scalar variables in this gauge areHL andHT . Instead of these oneoften introducesη ≡ −HL − HT /3 andh ≡ 6HL, but we will not make this replacementhere. Einstein’s equations are

(k2 − 3K)(HL +13HT ) + 3HHL = 4πGa2δρ ,

HL +13(1− 3K/k2)HT = −4πGa2(ρ + p)v/k ,

HL +HHL = −4πGa2[13δρ + δp] ,

HT +HHT − k2(HL +13HT ) = −8πGa2pΠ . (7.67)

Two of these are redundant. The conservation equations are

(∂∂τ + 3H)

δρ + 3HδP = −(ρ + P )(kv + 3HL)(∂∂τ + 4H)

([ρ + P )v/k

]= δP − 2

3 (1− 3K/k2)PΠ.(7.68)

Newtonian gauge

In Newtonian gauge one setsB = HT = 0. The gauge is also called conformal Newtonian orlongitudinal. The remaining two scalar perturbations are renamed intoA ≡ Ψ andHL ≡ −Φ,

ds2 = a2(τ)−(1 + 2Ψ)dτ2 + (1− 2Φ)γijdxidxj

. (7.69)

This gauge is popular since it reduces to Newtonian gravity in the appropriate limit, is suitablefor analytical work because of algebraic relations between matter and metric perturbationsand does not have gauge ambiguities. It can however be numerically unstable, which is whyit is usually not used in numerical codes. Instead one can do numerical computations insynchronous gauge and use the gauge transformation into the Newtonian gauge. A generalgauge transformation into Newtonian gauge gives

HT = 0 → L = −HT /k

B = 0 → T = −B/k + HT /k2. (7.70)

One can see that there is no remaining gauge freedom, so the gauge is entirely fixed. The mainadvantage of this gauge is that there is a simple Newtonian correspondence and the equationsreduce to Newtonian laws in the limit of small scales. The Einstein’s equations are,

−(k2 − 3K)Φ− 3H(Φ +HΨ

)= 4πGa2δρ

Φ +HΨ = 4πGa2 [(ρ + p)v/k]Φ−KΦ +H(Ψ + 2Φ) + (2H+H2)Ψ + 1

3 k2(Φ−Ψ) = 4πGa2δp ,

k2(Ψ− Φ) = −8πGa2pΠ.


On small scales the second term on the left hand side of first equation above becomes negligi-ble compared to the first one. Similarly we can also neglectK relative tok2, since curvaturescale, if present, is of the order of the Hubble length. The result is a Poisson equation in an ex-panding universe. This means that we can identify metric perturbationΦ with the Newtonianpotential on small scales. From the last of equations above we findΦ = Ψ in the absence ofanisotropic stress. This is a good approximation in the matter era where ideal fluids dominatethe energy density. We know that for astrophysical sources gravitational potential is roughly|Φ| ∼ v2 (in units wherec = 1), wherev is a typical velocity of the object. This impliesΦ ∼ Ψ ¿ 1 almost everywhere in the universe, except near a black hole. So in this gaugethe linear perturbation theory is almost always valid and one can use these equations alsoto describe the late time nonlinear evolution in the density field, as long as the gravitationalpotential remains small.

The conservation equations are(

∂∂τ + 3H)

δρ + 3Hδp = −(ρ + p)(kv − 3Φ)(∂∂τ + 4H)

([ρ + p)v/k] = δp− 23 (1− 3K/k2)PΠ + (ρ + P )Ψ.

(7.71)

The Newtonian gauge is useful for analytic treatments, the equations are however numericallyunstable.

Comoving gauge

This gauge is convenient because it gives algebraic relations between matter and metric pertur-bations. It introduces the curvature perturbation, which is useful when one wants to describethe evolution of perturbations, generated for example by inflation, outside the horizon. It turnsout that this quantity is conserved for adiabatic perturbations (see more on this below) and soevolution is particularly simple in this case. On the other hand, this gauge is not particularlyintuitive inside the horizon, so one is better off to transform into the Newtonian gauge in thislimit.

The gauge is defined so that the momentum densityT 0i vanishes. From equation (7.36)

this impliesB = v. The second constraint can be set toHT = 0. The remaining two scalarperturbations are renamed asA ≡ ξ andHL ≡ ζ. Gauge transformation from a general gaugeinto comoving gauge gives

v − B = 0 → T = (v −B)/k

HT = 0 → L = −HT /k. (7.72)

The gauge is entirely fixed, since these are just algebraic relations between the quantities.Einstein’s equations are

(k2 − 3K) (ζ +Hv/k) = 4πGa2δρ

Hξ − ζ − Kk v = 0

(2 a

a − 2H2 + η ∂∂τ − 1

3k2)ξ − (

∂∂τ +

)(ζ + k

3v) = 4πGa2(δp + 13δρ)

k2(ξ + ζ) +([ ∂∂τ + 2H)

kv = 8πGa2pΠ .

(7.73)

Corresponding energy–momentum tensor conservation equations are(

∂∂τ + 3H)

δρ + 3Hδp = −(ρ + P )(kv + 3ζ)

(ρ + P )ξ = −δP + 23 (1− 3K/k2)PΠ.

(7.74)

7.2 Fluid equations in the Newtonian Gauge 223

Note that the gauge conditionB = v implies δφ = 0 from 3rd equation (7.38), so thescalar field perturbations vanish in this gauge.

Spatially flat gauge

While comoving gauge is a useful gauge to describe evolution of perturbations after they crossthe horizon, it has the shortcoming that the scalar field perturbations vanish in this gauge. Itthus cannot be used to calculate scalar fluctuations from inflation, for example. One way tosolve this is to calculate them in another gauge and use the gauge transformation to calculatecurvature perturbation in comoving gauge. The simplest gauge to choose is the spatially flatgauge, whereHL = HT = 0,

(1− 3K/k2)HkB = 4πGa2 [δρ + 3H(ρ + p)(v −B)/k]

HA− Kk B = 4πGa2(ρ + P )(v −B)/k

(2 a

a − 2H2 +H ∂∂τ − 1

3k2)A− (

∂∂τ +H)

k3B = 4πGa2(δP + 1

3δρ)

k2A +(

∂∂τ + 2H)

kB = 8πGa2PΠ.

(7.75)

Corresponding energy-momentum tensor conservation equations are

(∂∂τ + 3H)

δρ + 3HδP = −(ρ + P )kv(∂∂τ + 4H)

([ρ + P )(v −B)/k

]= δP − 2

3 (1− 3K/k2)PΠ + (ρ + P )A.

(7.76)

Scalar field equation is

δφ + 2H ˙δφ + (k2 + a2V ′′)δφ = (A− kB)φ− 2a2V ′A. (7.77)

7.2 Fluid equations in the Newtonian Gauge

For the analysis in this section we use the perturbed metric in the Newtonian gauge

ds2 = a2(τ)[(1 + 2Ψ) dτ2 − (1− 2Φ)γij dxi dxj

], (7.78)

whereΦ is a measure for the fractional perturbation of the scale–factor (curvature perturba-tion) andΨ is the effect of the Newtonian potential. As we have seen, when the Universe isfilled with an ideal fluid, thenΦ = Ψ. τ denotes the conformal time. We will restrict our-selves to the scalar fluctuations, given that only these can lead to growth of perturbations byself-gravity and thus to structure formation in the universe.

7.2.1 Fluid equations

Energy momentum tensor is determined by the matter content in the universe. This can bedivided into two classes. In the first class are matter components, which can be describedwithin the fluid approximation. This class includes cold dark matter, baryons and scalar fields.In this section we derive evolution equations for the fluid ingredients that contribute to the


energy-momentum tensor. Photons and neutrinos that cannot be described as fluids will bediscussed in the following Sect. Together with the Einstein’s equations these form a closedset of equations that can be evolved forward in time from some given initial conditions. Wewill focus ourselves to the Newtonian gauge, since the equations are easiest to interpret andthe solutions do not suffer from the gauge modes or breakdown in the nonlinear regime.

To specify the evolution in fluid approximation one only needs the equations for overden-sity δ = δρ/ρ and velocityv, δuµ = vµ/a, both of which can be obtained from the energy-momentum conservation equations. The perturbations of the energy–momentum tensor arethen given by (in this Newtonian gauge)

δT 00 = δρ (7.79)

δT 0i = (ρ + P )vi (7.80)

δT i0 = (ρ + P )vi (7.81)

δT ij = −δP δi

j . (7.82)

Matter is given by an equation of stateP = wρ (w = 0 for cold dark matter) and the adiabaticsound speedc2

S = ∂P/∂ρ. The equations of motion,Tµν;ν = 0, split into two sets of equations,

one equation for the density perturbation

δ + 3(c2S − w)Hδ + (1− w) ∂iv

i − 3(1 + w) Φ = 0 (7.83)

and the Euler equations

(1 + w) ∂τvi + (1 + w)(1− 3w)Hvi − c2S∂iδ − (1 + w)∂iΨ = 0 . (7.84)

The effect of the expansion is seen in the second terms of these equations – leading to a kindof damping. When we differentiate the first equation by time and take the divergence of thesecond equation and neglect the small termsH2 anda/a, we arrive at the following expression

δ + (1− 6w + 3c2S)Hδ + c2

S ∂i∂iδ + (1 + w)∂i∂iΨ + W = 0 , (7.85)

whereW is a pure relativistic term

W = 3(1 + w)[(3w − 1)HΦ− Φ

]. (7.86)

This is a hyperbolic equation for density perturbations (acoustic waves propagating with thesound speedcS), including some damping term and the gravitational field as a driving source.In the Newtonian limit this leads to the equation (for a direct derivation of this equation fromNewtonian hydrodynamics, see Sect. 7.3.3)

δ + 2Hδ − c2S∇2δ −∇2Φ = 0 . (7.87)

Thereby, the gravitational potential is given by the Poisson equation

∇2Φ = 4πGδρ = 4πGρ δ . (7.88)

By assumptioncold dark matter (CDM) is cold and its pressure and anisotropic stressare zero. In Fourier space (flat model), one obtains the following evolution equation,

δc +H δc = −k2Ψ + 3HΦ + 3Φ . (7.89)


If we further assume the anisotropic stress is negligible, which is almost always a valid ap-proximation except on very large scales in radiation era where neutrinos make a non-negligiblecontribution, we haveΨ = Φ, which takes the role of Newtonian potential. This equation canbe solved analytically for two cases of interest, matter domination and radiation domination.In the first case the souce of potential are CDM fluctuations themselves, while in the secondcase the souce of potential is coming from radiation and can be treated as external source. Wediscuss both cases below.

Forbaryonsone must also include pressure and Thomson scattering between photons andelectrons. Thomson scattering has a well specified angular dependence in the rest frame of theelectron and rapid scattering leads to isotropic photon distribution in this frame. The change inphoton velocity is proportional to the difference between photon and baryon velocities timesthe scattering probability and so leads to the exchange of momentum between the photons andbaryons (see the lecture on Boltzmann equation for more detail). Momentum conservationrequires an opposite term in the baryon momentum conservation equation,

δb = −kvb + 3Φ ,

vb = −Hvb + c2skδb +

4ργ

3ρbanexeσT (vγ − vb) + kΨ , (7.90)

wherene is the electron density,xe the ionization fraction andσT Thomson cross section.We also included the pressure term, relating it to density gradients via adiabatic sound speedc2s = (∂p/∂ρ)S , neglecting entropy gradients.

While equation (7.89) was derived for CDM it can also be used to describe baryons onlarge scales and after the recombination, where the pressure term from baryons and the cou-pling between baryons and photons can be neglected. We will denote withδm the matterperturbation when it applies both to CDM and baryons.

Oscillations in the Radiation Fluid

Let us consider now a radiation fluid (baryons and photons before recombination, whereThomson scattering leads to a rapid momentum exchange between photons and baryons),w = 1/3 andc2

S = 1/3, resulting in the equation

δ − 13∇2δ − 4

3∇2Ψ− 4Φ = 0 . (7.91)

This can be written for temperature perturbationsδ = 4(δT/T ) = 4Θ in Fourier space

Θk +k2

3Θk = −k2

3Ψk + Φk . (7.92)

This represents a forced acoustic oscillator driven by gravitational forces. When baryons areincluded, the sound speed is somewhat less thanc/

√3. Since pressure is mainly given by

radiation and the density is a unique function of temperature, we may write for the soundspeed

c2S =

(∂P/∂T )ad

(∂ρ/∂T )Rad + (∂ρ/∂T )B, (7.93)

which provides the expression

c2S =

13

4ρRad

4ρRad + 3ρB=

c2

31

1 +R , (7.94)


whereR = 3ρB/4ρRad denotes the baryon loading of the radiation fluid. With this we canwrite the equation for the temperature fluctuation as follows

Θk +13

k2c2

1 +RΘk = −k2

3Ψk + Φk . (7.95)

This is now the fundamnetal equation which governs the evolution of temperature fluctuationsbefore recombination.

7.2.2 Gravitational Equations

For the scalar modes we have only two equations (see Appendix A,K = 0)

3H(HΦ + Φ) + ∂i∂iΦ = −4πGa2δρ (7.96)

Φ + 3HΦ + (2H+H−K)Φ = 4πGa2 δP (7.97)

UsingδP = c2Sδρ, we can combine the two equations

Φ + 3H(1 + c2S)Φ + (2H+H2 + 3c2

SH2)Φ + c2S∂i∂iΦ = 0 . (7.98)

This is a kind of a Klein–Gordon equation with signal velocity given by the sound speed andincluding a damping term. Please note that in the Newtonian limit all the time–derivativesvanish and only the Poisson equation survives.

7.2.3 Solutions inside horizon: matter domination

The solutions for perturbations differ depending on whether the mode scalek−1 is larger orsmaller than the Hubble radiusc/H (we will loosely call this horizon scale, since for bothmatter and radiation epochs the Hubble radius and causal horizon distance are proportionalto each other with a prefactor of order unity). On scales smaller than the Hubble length wecan ignore the time derivatives of potential relative to the spatial derivatives. This only worksif the solution is not oscillating and must be justified aposteriori from the obtained solution.Under this assumption we dropΦ andΦ terms in equation (7.89). To relate potential to densityperturbation we use Poisson’s equation, ignoringHΨ andK terms in addition toΦ. We thusobtain a second order differential equation,

δm +Hδm − 4πGρma2δm = 0 . (7.99)

The general solution to equation (7.89) consists of a growing and a decaying solution. It isinstructive to look at the solutions in some limits. In the matter domination forΩm = 1 wehavea ∝ τ2 from Friedmann equation and so4πGρma2 = 6/τ2. Setting the solution asa power lawδ = τα one findsα = 2,−3. Sincea ∝ τ2 in matter era the growing modesolution grows as a scale factor,

δm(Ωm = 1, a) = aδm(Ωm = 1, a = 1). (7.100)

From Poisson’s equation one finds that gravitational potential remains constant on smallscales,Φ = const. It is customary to introduce the growth factorD(a) as a ratio of den-sity fluctuation at a given expansion factora relative to today. ForΩm = 1 one hasD(a) = a.


In a universe filled with a cosmological constant, curvature or dark energy the growthwill be slowed down relative toΩm = 1. The effect is only important at late times whenthe additional component can be dynamically important. No analytic solutions exist in thiscase, but for a cosmological constant model (w = −1) with or without curvature a very goodapproximation to the growth factor is [3]

D(a) ≈ 5Ωm(a)a2

[1− ΩΛ(a) + Ωm(a)4/7 + 1

2Ωm(a)] . (7.101)

Note that for a givenΩm the growth factor suppression is smaller in a flat universe (ΩΛ > 0)than in an open universe (ΩΛ = 0). This is because the effects of cosmological constantbecome significant later than those from curvature, as evident from the Friedmann’s equation.

7.2.4 Solutions inside horizon: radiation domination

In radiation era the photons are the dominant component to the energy density (we ignore theneutrinos, which contribute 40% of the total energy density). We show first that because ofthe pressure effects the photon perturbations do not grow. To show this one takes the pressurecomponent of Einstein’s equations (??) and subtracts it from one third of the density equation.Sinceδp = c2

Sδρ with c2S = c/3 this gives the following equation,

Φ + 4HΦ +[2H+ 2H2

]Φ = −c2

Sk2Φ . (7.102)

We have again ignored the curvature terms. In radiation epoch one hasa ∝ τ , from whichfollowsH = 1/τ and so the last term on the left hand side vanishes. Then the equation is

Φ +4τ

Φ = −c2Sk2Φ (7.103)

which has the growing mode solution

Φ = 3Φi

(sin z

z3− cos z

z2

), (7.104)

wherez = kcsτ and the solution was normalized to the initial potential valueΦi. As expectedthe photon pressure causes the potential to oscillate and decay away as(kcsτ)−2 inside thehorizon. While the inclusion of baryons prior to recombination complicates the equations andchanges the sound speed somewhat this conclusion does not change significantly. Gravita-tional potential therefore decays inside horizon in radiation era.

To solve for CDM we take equation (7.89) and evaluate it in radiation epoch,

δc +1τ

δc = −k2Ψ + 3ηΦ + 3Φ. (7.105)

The full solution consists of a homogeneous solution (ie solution to the above equation withoutthe source) and a particular solution, which can be written as an integral of Green’s functionover the source term. Equation (7.104) shows that the latter decays away on small scales andso we can ignore the particular solution. One is left with the homogeneous solution, whichhas the growing mode

δc = C + ln(τ), (7.106)


whereC is a constant. So theCDM density perturbation grows only logarithmically insidethe horizon in radiation era. This has a simple physical interpretation: Prior to horizoncrossing both radiation and matter evolve similarly and aquire a velocity term as they crossthe horizon. After that CDM decouples from radiation and its velocity decays asτ−1 dueto the Hubble drag (damping term). This gives rise to the logarithmic growth of densityperturbation.

7.2.5 Solutions outside horizon

Prior to horizon crossing (ck/H ¿ 1), the solutions depend on the adopted gauge. As abovewe will use equations in Newtonian gauge, but we emphasize that the solutions can differdrastically if a different gauge is adopted. For example, while in Newtonian gauge adoptedhere density perturbations do not grow outside horizon, they in fact do grow in synchronousand conformal gauge. If we ignore the anisotropic stress (makingΦ = Ψ), spatial derivativesand curvatureK then we again combine first and third of Einstein’s equations to obtain

Φ +[3H(1 + c2

s,mr)]Φ +

[2H+H2(1 + 3c2

s,mr)]Φ =

3H2

2ρ

[δp− c2

mrδρ], (7.107)

where we introduced adiabatic speedca

c2a =

13 [1 + 3y/4]

. (7.108)

Here

y =ρm

ρr≡ a

aeq= x2 + 2x, x =

(Ωm

4aeq

)1/2H0

cτ, (7.109)

andaeq is expansion factor when matter and radiation densities are equal,

(1 + zeq) = a−1eq = 2.5× 104Ωmh2 (7.110)

for CMB temperatureTγ = 2.73K (scaling asT−4γ ). This equation can easily be derived from

Friedmann’s equation (??) if only matter densityρm (CDM+baryons) and radiation densityρr (photons and neutrinos) are included. Conformal time at matter-radiation equality is givenby

τeq =2(√

2− 1)cH0

√aeq

Ωm' 16Mpc

h2Ωm. (7.111)

The right-hand side of equation (7.107) is proportional to the specific entropy fluctuationσ = 3

4δr − δm,

δp− c2aδρ = ρc2

aσ. (7.112)

If we restrict ourselves to the adiabatic initial conditions where entropy fluctuations vanishinitially they cannot be generated on scales outside the horizon. In that case we can solveequation (7.107) analytically obtaining the growing and decaying solution [17]

Φ+ = 1 +29y− 8

9y2+

16x

9y3, Φ− =

1 + x

y3. (7.113)


In the radiation era (y ¿ 1) the solution isΦ+ = 10/9(1− y/16), while in the matter eraΦ+ = 1. In general thus

Φ =910

Φ+Φi, (7.114)

whereΦi is the initial perturbation. One can see thaton scales outside horizon the gravita-tional potential Φ does not change both in radiation and in matter epochs, but between thetwo it changes the amplitude by 10% (this gives rise to so called early integrated Sachs-Wolfeeffect in CMB).

Density perturbations are related to potentialΦ through Poisson’s equation. On largescales one can ignore spatial derivatives and curvature terms to obtain

δ = −2(Φ + Φ/H), (7.115)

whereδ is the total density perturbation. In the matter and radiation domination limits thisgivesδ = −2Φ. Since

δ =δγ + yδm

1 + y(7.116)

one findsδm = −3Φ/2 in radiation era andδm = −2Φ in the matter era.Density perturba-tions are therefore also constant on large scales (figure 7.2). We stress again that this isa gauge dependent statement. Of course, for any quantity that is directly observable such asCMB anisotropies the predictions are independent of the gauge choice.

7.2.6 Transfer functions

Any initial power spectrum of density perturbations gets modified because of the differentgrowth of perturbations in different regimes. This can be conveniently expressed in terms ofthe transfer function, which is defined as how muchδX of a given mode grows (or decays)relative to initial value. For convenience this is divided byk2 and normalized relative to theamplitude of some lowk = ki mode at early times, when the mode is outside the horizon(kiτ0 ¿ 1),

TX(k) =k2

i δX(k, τ0)δX(k = ki, τi)k2δX(k, τi)δX(k = ki, τ0)

. (7.117)

The transfer function on large scales is unity,T (k ∼ ki) = 1. The transfer function canbe defined for any speciesX, such as CDM, baryons, photons, neutrinos and dark energy.Of particular relevance is the transfer function for the total density perturbationδ (equation7.116), defined as an average over all the species weighted by their mean density, which onsmall scales determines the gravitational potentialΦ. In this case the transfer function is alsoreflecting the transfer of Newtonian potential from early time until today,

T (k) =Φ(k, τ0)Φ(k = ki, τi)Φ(k, τi)Φ(k = ki, τ0)

. (7.118)

Todayδ is dominated by CDM,δ ∼ δc, but it also has contributions from baryons and, ifpresent, massive neutrinos. To obtain the processed power spectrum one multiplies the squareof the transfer function with the primordial power spectrumP (k) = Pi(k)T 2(k).


Outside the horizon the modes do not grow in both matter and radiation era andδc isconstant. When the modes enter horizon in radiation era the density perturbation only growslogarithmically until the matter era as discussed above. If the mode enters in the matter era itbegins to grow immediately asτ2. By definition the modes enter the horizon whenk ∼ τ−1,so the growth is proportional tok2 as long as the mode entered in the matter era. Sincein our definion of transfer function we divide byk2 the transfer function remains unity forall the modes entering horizon after the matter-radiation equality. The transfer function formodes entering prior to that will suffer suppression in the transfer function which scales as(kτeq)−2[ln(kτeq) + 1] on small scales, whereτeq is the conformal time at matter-radiationequality (equation 7.111).

The most important parameter that determines the transfer function shape is conformaltime at equalityτeq, which depends onΩmh when the power spectrum is expressed againstk/h (as opposed toΩmh2 whenk is measured inMpc−1, equation 7.111). For CDM modelsthe asymptotic slope of the transfer function is the same regardless of the value of cosmo-logical parameters. Below we discuss baryons which can also affect the transfer function.

The transfer function for models with the full above list of ingredients was first computedaccurately by Bond & Szalay (1983), and is today routinely available via public-domain codessuch as CMBFAST (Seljak & Zaldarriaga 1996). These calculations are a technical challengebecause we have a mixture of matter (both collisionless dark particles and baryonic plasma)and relativistic particles (collisionless neutrinos and collisional photons), which does not be-have as a simple fluid. Particular problems are caused by the change in the photon componentfrom being a fluid tightly coupled to the baryons by Thomson scattering, to being collisionlessafter recombination. Accurate results require a solution of the Boltzmann equation to followthe evolution in detail.

The different shapes of the functions can be understood intuitively in terms of a few speciallength scales, as follows (Fig. 7.1):

• Horizon length at matter–radiation equality : The main bend visible in all transferfunctions is due to the Meszaros effect, which arises because the universe is radiationdominated at early times. Fluctuations in the matter can only grow if dark matter and ra-diation fall together. This does not happen for perturbations of small wavelength, becausephotons and matter can separate. Growth only occurs for perturbations of wavelengthlarger than the horizon distance, where there has been no time for the matter and radia-tion to separate. The relative diminution in fluctuations at highk is the amount of growthmissed out on between horizon entry andzeq, which would beδ ∝ D2

H in the absenceof the Meszaros effect. Perturbations with largerk enter the horizon whenDH ' 1/k;they are then frozen untilzeq, at which point they can grow again. The missing growthfactor is just the square of the change inDH during this period, which is∝ k2. The ap-proximate limits of the CDM transfer function are thereforeTk ' 1 for kDH(zeq) ¿ 1and

Tk ' [kDH(zeq)]−2 , kDH(zeq) À 1 . (7.119)

This process continues, untilzeq = 23900 ΩMh2, where the Universe becomes mat-ter dominated. We therefore expect a characteristic ‘break’ in the fluctuation spectrumaround the comoving horizon length at this time. Using a distance-redshift relation that


Figure 7.1: A plot of transfer functions for various adiabatic models, in whichT (k) ' 1 atsmallk. A number of possible matter contents are illustrated: pure baryons; pure CDM; pureHDM. For dark–matter models, the characteristic wavenumber scales proportional toΩMh2,marking the break scale corresponding to the horizon length at matter–radiation equality. Thescaling for baryonic models does not obey this exactly; the plotted case corresponds toΩM = 1,h = 0.5.

ignores vacuum energy at high z, one obtains

DH(zeq) = (√

2− 1)2c

H0(ΩMzeq)−1/2 ' 16Mpc

ΩMh2. (7.120)

• Free-streaming length: This relatively gentle filtering away of the initial fluctuations isall that applies to a universe dominated by Cold Dark Matter, in which random velocitiesare negligible. A CDM universe thus contains fluctuations in the dark matter on all scales,and structure formation proceeds via hierarchical process in which nonlinear structuresgrow via mergers.

Examples of CDM would be thermal relic WIMPs with masses of order 100 GeV/c2.Relic particles that were never in equilibrium, such as axions, also come under this head-ing, as do more exotic possibilities such as primordial black holes. A more interestingcase arises when thermal relics have lower masses. For collisionless dark matter, pertur-bations can be erased simply by free streaming: random particle velocities cause blobs


to disperse. At early times (kT > mc2), the particles will travel atc, and so any pertur-bation that has entered the horizon will be damped. This process switches off when theparticles become non–relativistic, so that perturbations are erased up to proper length-scales of' ct (kT = mc2). This translates to a comoving horizon scale (2ct/a duringthe radiation era) atkT = mc2 of

LFree−streaming ' 112 Mpcm/eV

. (7.121)

A more interesting (and probably practically relevant) case is when the dark matter is amixture of hot and cold components. The free-streaming length for the hot componentcan therefore be very large, but within range of observations. The dispersal of HDMfluctuations reduces the CDM growth rate on all scales belowLfree−stream – or, relativeto small scales, there is an enhancement in large–scale power.

• Acoustic horizon length: The horizon at matter–radiation equality also enters in theproperties of the baryon component. Since the sound speed is of orderc, the largestscales that can undergo a single acoustic oscillation are of order the horizon. The transferfunction for a pure baryon universe shows large modulations, reflecting the number ofoscillations that have been completed before the universe becomes matter dominated andthe pressure support drops. The lack of such large modulations in real data is one ofthe most generic reasons for believing in collisionless dark matter. Acoustic oscillationspersist even when baryons are subdominant, however, and can be detectable as lower–level modulations in the transfer function.

• Silk damping length: Acoustic oscillations are also damped on small scales, wherethe process is called Silk damping: the mean free path of photons due to scattering bythe plasma is non-zero, and so radiation can diffuse out of a perturbation, convecting theplasma with it. The typical distance of a random walk in terms of the diffusion coefficientD is x ' D

√t, which gives a damping length of

λS '√

λDH , (7.122)

the geometric mean of the horizon size and the mean free path. Sinceλ = 1/(nσT ) =44.3(1 + z)−3 (Ωbh

2)−1 proper Gpc, we obtain a comoving damping length of

λS = 16.3Gpc (1 + z)−5/4 (Ω2bΩMh6)−1/4 . (7.123)

This becomes close to the horizon length by the time of last scattering,1 + z = 1100.

• Fitting formulae : It is invaluable in practice to have some accurate analytic formulaethat fit the numerical results for transfer functions. We give below results for some com-mon models of particular interest (illustrated in Fig. 7.1 along with other cases where afitting formula is impractical). For the models with collisionless dark matter,Ωb ¿ ΩM

is assumed, so that all lengths scale with the horizon size at matter–radiation equality,leading to the definition

q ≡ k

Ωh2 Mpc−1 (7.124)


Figure 7.2: Evolution ofδc (solid) andδb (dashed). The longer wavelength mode (k = 2×10−2h/Mpc)enters the horizon almost entirely in the matter domination and subsequently grows as the scale factor.Higherk mode enters the horizon well in the radiation era and grows slowly until the matter domination.Baryon growth is suppressed until recombination (log(a) ∼ −3), after which it catches up with CDM.

We consider the following cases: (1) Adiabatic CDM; (2) Adiabatic massive neutrinos(1 massive, 2 massless); (3) Isocurvature CDM; these expressions come from Bardeen etal. (1986; BBKS). Since the characteristic length–scale in the transfer function dependson the horizon size at matter–radiation equality, the temperature of the CMB enters. Inthe above formulae, it is assumed to be exactly 2.7 K; for other values, the characteristicwavenumbers scale∝ T−2. For these purposes massless neutrinos count as radiation,and three species of these contribute a total density that is 0.68 that of the photons.

TCDM (k) =log(1 + 2.34q)

2.34q

[1 + 3.89q + (16.1q)2

+(5.46q)3 + (6.71q)4]−1/4

(7.125)

Tν(k) = exp(−3.9q − 2.1q2) (7.126)

Tcurv(k) = (5.6q)2(

1 +[15.0q + (0.9q)3/2 + (5.6q)2

]1.24)−1/1.24

.(7.127)

These expressions are useful for work at a level of 10% precision, but increasingly it isnecessary to do better. In particular, these expressions do not include the weak oscillatoryfeatures that are expected if the universe has a significant baryon content. Eisenstein &Hu (1998) give an accurate (but long) fitting formula that describes these wiggles for theCDM transfer function. This was extended to cover MDM in Eisenstein & Hu (1999).

7.2.7 On Baryons

Prior to decoupling baryons are tightly coupled with photons and their effective sound speed isclose toc/

√3. In this case baryon perturbations do not grow for the modes inside the horizon.

Since decoupling occurs atz ≈ 1100, which is typically after matter-radiation equality, CDMmodes inside horizon will have grown relative to baryons during this epoch (Fig. 7.2). Afterdecoupling baryon sound speed drops to very low values and the Thomson scattering term inequation (7.90) can be dropped. In this case baryons obey the same equation (7.89) as CDM.If one subtracts baryons from CDM one finds,

δbc +Hδbc = 0 , (7.128)

whereδbc = δb−δc. This equation is valid on scales where baryons pressure can be neglected.The growing mode solution for this equation is a constant, so the difference between baryonand CDM density perturbation does not change in time. However, the CDM density pertur-bation grows asδc ∝ τ2 ∝ a in the matter domination, so the relative difference betweenbaryon and CDM density contrast decreases asa−1 ∝ τ−2. Thusδb catches up withδc afterdecoupling and then continues to grow at an equal rate (Fig. 7.2).


We have so far assumed that CDM grows following the solution to equation (7.99). Thisis valid if CDM is the dominant component to the matter density. If baryon density is not neg-ligible compared to CDM then baryons will also contribute to the gravitational potential fromPoisson’s equation. But baryons after decoupling still reflect acoustic oscillations from theepoch before decoupling, when they were tightly coupled with photons (equation 7.104). Asa result the total potential will reflect these oscillations and CDM evolution will be modifiedbecause of this. Sufficiently large baryon density leads to acoustic oscillations in the transferfunction (Fig. 7.1).

Another effect of baryons that are dynamically important is that they suppress the transferfunction on small scales. This is again expected, since baryons are damped prior to decouplingon small scales due to imperfect coupling with photons (this is the so-called Silk damping,which we will discuss in more detail in later sections). If they are dynamically important thisleads to a suppression of the gravitational potential. The latter is the source for the CDMdensity fluctuations, which are thus suppressed as well (Fig. 7.1).

7.2.8 Role of Scalar Fields

Scalar fields have many applications in cosmology. One is as an inflaton field, which wediscussed in some detail in the Sect. on Inflation. Another is as a candidate for the darkenergy (or quintessence), which is responsible for the observed late time acceleration in theuniverse. Here we analyze its late time dynamics by focusing on the equations of motion.

The evolution equation for the scalar field perturbations in Newtonian gauge follows fromequation (7.39)

Scalar field—

δφ + 2ηδφ + [k2 + a2V ′′]δφ = (Ψ + 3Φ)φ− 2a2V ′Ψ. (7.129)

In the limit wφ = −1 we have no gravitational source forδφ and there are no perturbations inthe scalar field assuming none existed initially. The easiest way to see this is by observing thesame evolution equation (7.39) in synchronous gauge, whereA(0) = 0. Forwφ = −1 one hasφ = 0, so all the sources on the right hand side of equation (7.39) vanish (in Newtonian gaugethe same conclusion holds, sinceV ′ = 0, as shown in next section). Cosmological constantand scalar field withwφ = −1 are thus indistinguishable. To solve for perturbations in generalwe must specifyφ, V ′ andV ′′. It is often more convenient to express these in terms of thepresent densityΩφ0 and the equation of statewφ as a function of time.

Forw > −1 we must look at the stability analysis of evolution equation. In general gravitywill cause the perturbations to grow, but if there is a significant positive pressure to counteractit they will not grow on scales smaller than the Jeans scale, defined roughly as the sound speedtimes time. For ordinary matter one would expect the sound speed to be given bywφ = p/ρ,which forw < 0 would even accelerate the collapse of perturbations.

However, for the scalar fields the stability of perturbations is not determined by the equa-tion of state. On small scales one hasa2V ′′ ∼ H2 ¿ k2, so ignoring the sources the evolutionequation (7.39) is

δφ + 2Hδφ + k2δφ = 0 . (7.130)

The solution to this equation is a damped oscillator with effective speed of soundcS = c. Thesound speedc2

S = δPφ/δρφ equal to unity also follows from equations (7.38) in comoving

7.3 Nonlinear Evolution of Cosmic Structure 235

gauge, wherev = B implies δφ = 0. Note thatδρ in comoving gauge is the true densityperturbation on small scales, since it satisfies the Poisson’s equation (A.19).

The scalar field perturbations are thus unimportant except on horizon scale. This justifiesour treatment of matter perturbations inside horizon in the presence of dark energy (section7.2.3), where we assumed it only affects the background evolution. Just as in the case ofcosmological constant there are no exact analytical solutions available in the general case. Thesmooth component leads to a decay of the gravitational potential and the growth of structureis slowed down relative toΩm = 1 case.

For CMB on large scales the scalar field perturbations can make a small difference, sinceon large scales and at late times scalar field can have a significant contribution to the metricperturbation. These effects are difficult to observe because of the sampling variance. Morecomplicated and interesting are cases of the non-canonic kinetic term (k-essence) and non-minimally coupling of the scalar field (Brans-Dicke type of models), both of which can pro-duce in special casesc2

S ¿ c2. In this case the scalar field perturbations may be more impor-tant for the structure evolution even on small scales.

7.3 Nonlinear Evolution of Cosmic Structure

The equations of motion are nonlinear, and we have previously only solved them in the limit oflinear perturbations. We now discuss evolution beyond the linear regime, first for a few specialdensity models1, and then considering full numerical solution of the equations of motion.

In order to account for the observable Universe, any comprehensive theory or model ofcosmology must draw from many disciplines of physics, including gauge theories of strongand weak interactions, the hydrodynamics and microphysics of baryonic matter, electromag-netic fields, and spacetime curvature, for example. Although it is difficult to incorporate allthese physical elements into a single complete model of our Universe, advances in computingmethods and technologies have contributed significantly towards our understanding of cosmo-logical models, the Universe, and astrophysical processes within them. A sample of numericalcalculations addressing specific issues in cosmology are reviewed in the following: from theBig Bang singularity dynamics to the fundamental interactions of gravitational waves; fromthe quark-hadron phase transition to the large scale structure of the Universe. The emphasis,although not exclusively, is on those calculations designed to test different models of cosmol-ogy against the observed Universe.

In this Section we provide general notions about the formation and evolutions of cosmicstructures, like galaxies and clusters of galaxies, within the standard cosmological scenario,based on a spatially homogeneous and isotropic expanding Universe. For this we introducethe notation for describing the degree of inhomogeneity which characterizes the structure ofthe Universe on scales' 100 Mpc, thus much smaller than the cosmological event horizon(' 4300 Mpc). We explicitely show how the nature (either cold or hot) of the DM affects theformation and evolution of cosmic structures. We finally discuss how modern numerical com-putational techniques and observations can be joined together to improve our understandingof the nature of our Universe and its evolution.

The most successful theory of cosmological structure formation is inflationary cold darkmatter (ΛCDM) in which nearly scale–invariant adiabatic initial fluctuations are set up by a

1For the Zeldovich approximation and spherical models, see e.g. Padmanabhan III, pp. ...; for time constraints,these approximations are not discussed in these lectures


period of inflation in the early universe. Thus theinitial conditions are:

• the fluctuations in the gravitational potential (which can be related to density fluctuationthrough Poisson’s equation) are almost independent of scale and

• that the fluctuations in the pressure are proportional to those in the density (which keepsthe entropy constant, hence the term adiabatic).

A direct consequence of the adiabatic assumption is that a cosmic overdense region containsoverdensities of all particle species. The alternative mode, where overdensities of one speciescounterbalance underdensities in another, is known as isocurvature because the spatial curva-ture is unchanged. Models incorporating isocurvature initial conditions fare very badly whencompared to the observations.

Most (though not all) inflation models also predict that the spectrum of fluctuations isgaussian, with zero mean and variance given by the power spectrum. The preponderance ofobservational evidence suggests that the initial fluctuations which produced the large-scalestructure in the universe were gaussian, with non-linear gravitational clustering producing allof the non-gaussianity in the distribution observed today. All the front-running candidates formodels of structure formation use gaussian initial conditions.

Viable models of structure formation thus differ mostly in what is assumed for the back-ground cosmological parameters, specifically the density in CDM, spatial curvature or a cos-mological constant/dark energy component (and its evolution), baryonic component, expan-sion rate etc. Of the possible models, the only currently viable one is L(ambda)CDM withroughly 1/3 of the energy in the universe being cold dark matter, 2/3 in a cosmological con-stant or dark energy component, and a few percent being in the form of normal or ”baryonic”matter (most of which is also dark). The initial fluctuations have to be almost exactly Gaussianwith a close to scale-invariant spectrum which is nearly power-law in scale.

Once the initial spectrum and type of fluctuations is known, the linear evolution (at earlytimes) is determined by the background cosmology and the type of dark matter. Because hotdark matter moves rapidly, it is able to stream out of overdense regions, escaping from theenhanced gravitational potential. This tends to erase fluctuations in the matter distribution onscales smaller than the free-streaming scale (approximately the speed of the HDM particlestimes the age of the universe). In contrast to this, cold dark matter is able to support perturba-tions on all scales of cosmological interest. This behaviour is much closer to what is inferredfrom observations of clustering. The dependence on the background cosmology enters be-cause fluctuations only grow when the universe is matter dominated. Lowering the matterdensity thus decreases the length of time the perturbations can grow (both at early times, bydelaying the dominance of matter over radiation and at late times when the universe becomescurvature or cosmological constant dominated).

7.3.1 Cosmic Matter

Having indirectly probed the nature of matter in the Universe using the previous estimates, it isnow time to turn to direct constraints that have been derived in the past decade. Here, perhapsmore than any other area of observational cosmology, new observations have changed the waywe think about the Universe.

Perhaps the greatest change in cosmological prejudice in the past decade relates to theinferred total abundance of matter in the Universe. Because of the great intellectual attractionInflation as a mechanism to solve the so–called Horizon and Flatness problems in the Universe,


it is fair to say that most cosmologists, and essentially all particle theorists had implicitlyassumed that the Universe is flat, and thus that the density of dark matter around galaxiesand clusters of galaxies was sufficient to yieldΩ = 1. Over the past decade it became moreand more difficult to defend this viewpoint against an increasing number of observations thatsuggested this was not, in fact, the case in the Universe in which we live.

The earliest holes in this picture arose from measurements of galaxy clustering on largescales. The transition from a radiation to matter dominated universe at early times is depen-dent, of course, on the total abundance of matter. This transition produces a characteristicsignature in the spectrum of remnant density fluctuations observed on large scales. Makingthe assumption that dark matter dominates on large scales, and moreover that the dark mat-ter is cold (i.e. became non-relativistic when the temperature of the Universe was less thanabout a keV), fits to the two point correlation function of galaxies on large scales yieldedΩMh = 0.2 − 0.3. Unlessh was absurdly small, this would imply thatΩM is substantiallyless than 1.

For the moment, however, perhaps the best overall constraint on the total density of clus-tered matter in the universe comes from the combination of X-Ray measurements of clusterswith large hydrodynamic simulations. The idea is straightforward. A measurement of boththe temperature and luminosity of the X-Rays coming from hot gas which dominates the totalbaryon fraction in clusters can be inverted, under the assumption of hydrostatic equilibriumof the gas in clusters, to obtain the underlying gravitational potential of these systems. Inparticular the ratio of baryon to total mass of these systems can be derived. Employing theconstraint on the total baryon density of the Universe coming from BBN, and assuming thatgalaxy clusters provide a good mean estimate of the total clustered mass in the Universe, onecan then arrive at an allowed range for the total mass density in the Universe. Many of theinitial systematic uncertainties in this analysis having to do with cluster modelling have nowbeen dealt with by better observations, and better simulations, so that now a combination ofBBN and cluster measurements yieldsΩM = 0.35± 0.1.

The foremost HDM candidate is a particle known as the neutrino. This elusive particlewas predicted to exist as early as 1931 by physicist Wolfgang Pauli in order to account for theconservation of momentum and energy in beta decay. One of the most revealing criteria asto whether the Universe is dominated by CDM or HDM is the way that matter, in particulargalaxies, are distributed throughout the sky. HDM, as represented primarily by neutrinos,does not account for the pattern of galaxies observed in the Universe. Neutrinos would haveemerged from the Big Bang with such highly relativistic velocities (i.e. close to light speed)that they would tend to smooth out any fluctuations in matter density as they streamed outthrough the Universe. In the early Universe, the neutrino density was enormous, and so mostof the matter density could be accounted for by neutrinos. Given their great speeds, neutrinoswould tend to free stream out of any overdense regions–that is, regions with densities greaterthan the average density in the Universe. This process implies that density fluctuations couldappear only after the neutrinos slowed down considerably. (i.e. As the Universe expanded, itstemperature decreased, thereby resulting in neutrino cooling.)

With this we obtain the following class of models which are consistent with the idea ofinflation:

• Classical CDM (sCDM): it would require a low Hubble constant,H0 ' 50 km/s/Mpc,Λ = 0 undΩB + ΩCDM = 1. This was the standard model in the 90es.

• νCDM (MDM) : H0 ' 50 km/s/Mpc, withΛ = 0, ΩB = 0.1, ΩHDM = 0.2 and


ΩCDM = 0.7. Neutrinos can however not provide all the dark matter, since galaxieswould form to late in a such a Universe. Such models are now also ruled out.

• ΛCDM : This is by now the standard model which consists of about two third of darkenergy and one third of dark matter. This is often called theconcordance modelwhichis defined by the most important parameters following from WMAP.

The WMAP satellite has allowed the very precise determination of many cosmological pa-rameters

Parameter Value Quantity

H0 71± 5 km/s/Mpc Hubble constantt0 13.7± 0.2 Gyr Age of UniverseΩM 0.27± 0.04 Matter densityΩb 0.044± 0.004 Baryon densityΩΛ 0.73± 0.04 Dark energyΩtot 1.02± 0.02 Total density

Table 7.1: Parameters of the concordance model following from WMAP.

7.3.2 Collisionless Components

Photonen, Neutrinos und Dunkle Materie verhalten sich heute stoßfrei, d.h. sie entwickelnsich im Universum ohne gegenseitige Wechselwirkung. Diese Formen der Materie sind wiePhotonen zu beschreiben, d.h.uber ihre Phasenraumfunktionen – Photonen werden durchdie Planck–Funktion beschrieben, Neutrinosuber eine Fermi–Verteilung. Die DM kann auchim Teilchenbild beschrieben werden, d.h. wir konnen ihre Dynamik der Teilchentrajektorienberechnen.

7.3.3 Baryons and Gasdynamics

Im Unterschied zu all diesen Teilchen ist die baryonische Materie stoss–dominiert. Ihre En-twicklungsgleichungen folgen aus der Gasdynamik fur die DichteρB , die Geschwindigkeit~v,den DruckP , sowie das GravitationspotentialΦ

• Continuity equation (mass conservation)

∂ρB

∂t+∇ · (ρB~v) = 0 (7.131)

• Euler equation (momentum conservation)

∂~v

∂t+ (~v · ∇)~v = − 1

ρB∇P −∇Φ (7.132)


• Poisson equation (gravitation)

∇2Φ = 4πGρ0 (7.133)

Here we have to take into account that all form of matter generates gravity.

• Der Zusammenhang zwischen Druck und Dichte folgt aus einer Zustandsgleichung,P =P (ρB , sB), wennsB die spezifische Entropie der Baryonen beschreibt.

Zur Vereinfachung nehmen wir im folgenden an, dass nur baryonische Materie beteiligt istund lassen deshalb den IndexB entfallen. Wir betrachten nun kleine Storungen um eine Gle-ichgewichtsverteilung mit Dichteρ0, Geschwindigkeit~v0 und PotentialΦ0, die als homogenangenommen wird,∇ρ0 = 0 = ∇P0,

ρ = ρ0 + δρ , ~v = ~v0 + δ~v , P = P0 + δP , Φ = Φ0 + δΦ . (7.134)

Durch Subtraktion der Gleichgewichtsgleichungen von den vollen Gleichungen ergeben sichdamit folgende Beziehungen. Aus der Kontinuitatsgleichung erhalten wir

d

dt

(δρ

ρ0

)=

d∆dt

= −∇ · δ~v . (7.135)

Dabei bedeutet∆ ≡ δρ/ρ0 den Dichtekontrast, der sich aufgrund der Storung des Geschwindigkeits-feldes entwickelt.

Dann ergibt die Euler–Gleichung die Bewegungsgleichung fur die Storungen

∂δ~v

∂t+ (δ~v · ∇)~v0 = − 1

ρ0∇δP −∇δΦ , (7.136)

sowie die Poisson–Gleichung

∇2δΦ = 4πGδρM . (7.137)

Da das Universum expandiert, ist es gunstig, mitbewegte Koordinaten einzufuhren,~x =R(t)~r (s. Metrik). Damit gilt fur eine gestorte Position der Teilchen

δ~x = ~r δR(t) + R(t) δ~r . (7.138)

Die Geschwindigkeit setzt sich deshalb aus zwei Anteilen zusammen

~v =δ~x

δt= R(t)~r + R(t)

d~r

dt. (7.139)

Der erste Term beschreibt die Hubble–Expansion, der zweite Term die Abweichung von derHubble–Expansion, die wir alsR(t) ~u schreiben. Damit kann man die Euler–Gleichung wiefolgt darstellen

d

dt(R~u) + (R~u · ∇)R ~r0 = − 1

ρ0∇cδP −∇cδΦ . (7.140)

Wir betrachten jetzt Ableitungen in Raumrichtungen als Ableitung bez. mitbewegter Koor-dinaten,d/dx = (1/R) d/dr, ∇x = ∇c/R, und damit(R~u · ∇)R ~r = ~u R, womit dieEuler–Gleichung lautet

d~u

dt+ 2

R

R~u = − 1

ρ0R2∇cδP − 1

R2∇δΦ . (7.141)


Im folgenden betrachten wir sog.adiabatische Storungen, bei denenδP = c2S δρ, d.h.

Anderungen des Drucks beruhen nur aufAnderungen der Dichte, und nicht der Entropie.c2S ≡ (∂P/∂ρ)s bezeichnet die Schallgeschwindigkeit des Mediums. Dazu bilden wir die

Divergenz der gest”orten Euler–Gleichung

∇c · ~u + 2R

R(∇c · ~u) = − 1

ρ0∇2

cδP −∇2cδΦ . (7.142)

Die gestorte Kontinuitatsgleichung leiten wir nach der Zeit ab

d2∆dt2

= −∇c · ~u . (7.143)

Durch Erstzen der Euler–Gleichung folgt daraus

d2∆dt2

+ 2R

R

d∆dt

=c2S

ρ0R2∇2

cδρ + 4πG δρM . (7.144)

Dies ist eine lineare Wellengleichung in mitbewegten Koordinaten mit einem Quell–Term undeinem Dampfungsterm, der durch die kosmische Expansion zustande kommt. Wir machendeshalb den Ansatz ebener Wellen,∆(t, ~r) = ∆(t) exp i(~kc ·~r), was zu folgender fundamen-taler Gleichung fur die Amplituden∆(t) fuhrt

d2∆dt2

+ 2R

R

d∆dt

+ ∆(~k2cc2

S − 4πGρB) = 4πGρDM ∆DM . (7.145)

The proper wave–vector is~k = R(t)~kc. This is the equation of a forced damped oscillator.In the absence of dark matter, the nature of the solution is determined by the sign of the thirdterm. If k2 > k2

J , where

k2J =

4πGρB

c2S

(7.146)

the frequency of the oscillator is real - the perturbation in the baryonic component oscillatesas an acoustic vibration. Ifk2 < k2

J , then the frequency is imaginary and we get growing anddecaying amplitudes. In such a case, perturbations can grow.

The term proportional to the Hubble parameter represents the damping that is due to ex-pansion, and the dark matter on the right hand side provides the driving force. When bothdark matter and baryons are present, the dark matter source usually outnumbers the baryonicterm – the growth in dark matter can induce a corresponding growth in baryons at large scales.Modes withλc < λJ do not grow, whereas long–wavelength modesλc > λJ can grow, where

λJ =

√πc2

S

Gρ(7.147)

is theJeans length. The correspondingJeans massis the mass contained within a sphere ofradiusλJ/2

MJ =4π

3ρB

(λJ

2

)3

(7.148)

7.4 Cosmological Simulations 241

Putting in numbers, the Jeans mass before recombination is huge

M<J ' 1016 M¯

ΩB

ΩM

√1

ΩMh2. (7.149)

The Jeans mass after decoupling is small

M>J ' 104 M¯

ΩB

ΩM

√1

ΩMh2. (7.150)

7.3.4 The Spectrum of Primordial Fluctuations

The transfer function for models with the full above list of ingredients was first computed ac-curately by Bond & Szalay (1983), and is today routinely available via public-domain codessuch as CMBFAST (Seljak & Zaldarriaga 1996). These calculations are a technical challengebecause we have a mixture of matter (both collisionless dark particles and baryonic plasma)and relativistic particles (collisionless neutrinos and collisional photons), which does not be-have as a simple fluid. Particular problems are caused by the change in the photon componentfrom being a fluid tightly coupled to the baryons by Thomson scattering, to being collisionlessafter recombination. Accurate results require a solution of the Boltzmann equation to followthe evolution in detail.

The gravitational potential shown in Fig. 7.3 is a statistical fractal at the end of inflation(gaussian random field). During the radiation era, acoustic damping smooths the potentialon small scales. Because the density fluctuations are related to the potential by two spatialderivatives, the corresponding linear density field is dominated by high spatial frequency com-ponents. The nonlinear evolution of the density field causes the mass to cluster strongly intowispy filaments and dense clumps – most of the volume has low density. This intermittencyof structure is a sign of strong deviations from gaussian statistics.

7.4 Cosmological Simulations

The structures observed in the present Universe cannot be obtained by applying a simplelinear perturbation theory. As soon asδ ≥ 1, structures will collapse and follow their owngraity, thereby deviating away from the common Hubble–flow. Such investigations can onlybe obtained by means of large–scale computer simulations.

The Universe at Redhsiftz = 30 − 50: In einem CDM–Modell werden kleine Skalenschon sehr fruh nichtlinear, da das Fluktuationsspektrum zu kleinen Skalen langsam ansteigt.Massenskalen von(106−1011)M¯ klumpen bereits im Rotverschiebungsbereichz ' 10−30.Dies entspricht einer Zeitskala von einigen 100 Mio. Jahren. Die dunkle Materie des Uni-versums zerfallt in einezellenartige Struktur , die eine Massenverteilung bis zu1011 M¯aufweist. Das Zentrum jeder Zelle ist ein Massenzentrum dunkler Materie, das von einemHalo umgeben ist. Diese Strukturen bilden die Keimzellen der Spharoide (Bulges) der Galax-ien.


Figure 7.3: Potential and density for a mixed cold+hot dark matter model,Ων = 0.2

(Bertschinger 1995). A slice is shown 50h−1 Mpc wide. Upper left: potential at the endof inflation; upper right: potential at the end of recombination; lower left: density at the end ofrecombination; lower right: density at redshift zero.

7.4.1 Particle Models

Um diese Situation auf dem Computer zu simulieren, wird die Bewegung dieser Massen-zellen unter ihrer gegenseitigen Gravitationswirkung untersucht. Im allgemeinen beschranktman sich aufN Massenlemente zur selben Masse (sog. Testmassen). Die Bewegung solcherTestmassen wird am besten in mitbewegten Koordinaten analysiert,i = 1, . . . , N , N = 109

(!)

~x′i = ~xi/a(t) , ~x′(0) = ~x(0) (7.151)

~v′i =1a

(~vi −H~xi) . (7.152)


Die wichtige Geschwindigkeit ist nur die Bewegung relativ zur RaumexpansionH~x. Dieurspr”unglichen Bewegungsgleichungen lauten

~v = −∇Φ (7.153)

~x = ~v (7.154)

∆Φ = 4πGρ(t, ~x) . (7.155)

Dies ergibt folgende Gleichungen in mitbewegten Koordinaten

~v′ + 2H(t)~v′ = −1a∇Φ− a

a~x′ . (7.156)

Der letzte Term entspricht der Gezeitenkraft. Durch die Transformation

Φ′ = a(t)Φ +a2a

2x′2 (7.157)

erh”alt man die Bewegungsgleichung

~v′ + 2H(t)~v′ = − 1a3∇′Φ′ . (7.158)

Die Poisson–Gleichung in mitbewegten Koordinaten lautet dann

∆′Φ′ = 4πGa3 ρ + 3a2a , ρ′ = ρa3 (7.159)

Da in der materiedominierten Epochea = −(4/3)4πGρa(t), bedeutet dies

∆′Φ′ = 4πG(ρ′(t, ~x′)− ρ(0)) . (7.160)

Nur die Differenzρ′−ρ(0) tritt als Quelle der Gravitation auf. N Massenelemente werden nunin eine kubische Box verteilt. Die totale Masse verschwindet. Haufen weisen einen positivenDichteuberschuß auf,ρ′ − ρ > 0, Voids einen negativen. Dies fuhrt dazu, dass sich Haufenund Voids gegenseitig abstoßen.

In total, the following system of differential equations has to be integrated

~v′i = ~x′i (i = 1, ..., N) (7.161)

~v′i + 2H(t)~v′i = ~Fi (7.162)

~Fi = − 1a3∇′Φ′|xi . (7.163)

∆Φ′ = 4πG(ρ′(t, ~x′)− ρ(0)) (7.164)

a = H0

√Ω0

(1

a(t)− 1

)+ ΩΛ(a2 − 1) + 1 , a(0) = 1 (7.165)

H(t) =a

a(7.166)

ρ(t) = ρ(0)/a3 . (7.167)

Mit Hilfe von N–Korper Methoden k”onnen die Bewegungsgleichungen von einzelnenMassenteilchen auf dem expandierenden Hintergrund integriert werden. Die Poisson–Gleichungwird auf einem Gitter gelost. Heute konnen Simulationen auf10243–Gittern gefahren wer-den mit typisch mindestens10003 einzelnen Massen (Abb 7.4). Bei1283 Teilchen und


einer Boxlange von 720 Mpc entspricht dies einer Teilchenmasse von1.2 × 1013 M¯, bei144 Mpc einer typischen Galaxienmasse' 1011 M¯ (1283/Np). Im ersten Falle ergibt dieseine raumliche Auflosung von 5.6 Mpc, im zweiten Falle von 1.1 Mpc. Diese Rechnungenber”ucksichtigen nur die dunkle Materie, in neueren Simulationen kann auch die baryonischeMaterie mitgerechnet werden (Hydrodynamik). Die Anfangsbedingungen beia = 1 werdennach dem linearen St”orungsspektrum generiert mit einer Amplitude= 1/16, normiert aufder Skala von 16 Mpc. Die Simulationen werden zeitlich entwickelt, bis das Powerspektrumdie COBE–Werte erreicht. Dies entspricht einem Skalenfaktora0 = R0/Rin ' 20.

7.4.2 Progress of Cosmological Simulations

Cosmological N–boby simulations started in late 1970’s, and since then have played an im-portant part in describing and understanding the nonlinear clustering in the Universe. SuchN–body calculations in a comoving periodic cube were done for the first time by Miyoshi andKihara (1975) usingN = 400 particles. Suto [22] has recently pointed out that the progressis well fitted by a trend of the form

N = 400× 100.215(yr−1975) , (7.168)

where the amplitude is normalized to the first publication by Miyoshi and Kihara. Just forcomparison, the total number of CDM particles of massmCDM in a box of the Universe ofone sideL is given by

N =ΩCDML3

mCDM' 1083 ΩCDM

0.23

(hL

Gpc

)3 keVmCDM

h

0.7. (7.169)

If we extrapolate this equation, then the number of particles that one can simulate in a (1Gpc/h)3 box will reach the real numbers of CDM particles in the box in the year 2348 formCDM = 1 keV and in the year 2386 formCDM = 10−5 eV (Fig. 7.4) !

After the first simulations, the primary goal of the simulations in 1980’s was to predictobservable galaxy distribution from dark matter clustering. So one had to distinguish betweengalaxies and simulation particles which represent dark matter, i.e. to introduce the notion ofgalaxy biasing. Davis, Efstathiou, Frenk and White (1985) was one of the most influentialand seminal paper in cosmological N–body simulations. The most important message theywere able to show is the fact that large–scale simulations can provide numerous realistic andtestable predictions of dark matter scenarios against the observational data. At that time, theyonly usedN = 323 particles.

Simulations in the 1990’s revealed astonishing progress in understanding the large–scalestructure. One of the breakthrough was the amazing scaling property in the two–point corre-lation functions. It was found that it can be well approximated by a universal fitting formulawhich empirically interpolates the linear regime and the non–linear stable solution (Peacock& Dodds [?]; Smith et al. [?]). The results for all particles agree well with the theoreticalpredictions. Only at small scales, where the force resolution is no longer sufficient, deviationsoccur.

7.4.3 The Fractal Structure of Dark Matter Distribution

Fig. 7.5 illustrates the spatial distribution of dark matter at the present day, in a series of sim-ulations covering a large range of scales. Each panel is a thin slice of the cubical simulation


Figure 7.4: Evolution of the number of particles in cosmological N–body simulations. [Source:Suto 2003]

volume and shows the slightly smoothed density field defined by the dark matter particles. Inall cases, the simulations pertain to the ”CDM” cosmology, a flat cold dark matter model inwhich ΩM = 0.3, ΩΛ = 0.7 andh = 0.7. The top-left panel illustrates the Hubble volumesimulation: on these large scales, the distribution is very smooth. To reveal more interestingstructure, the top right panel displays the dark matter distribution in a slice from a volumeapproximately 2000 times smaller. At this resolution, the characteristic filamentary appear-ance of the dark matter distribution is clearly visible. In the bottom-right panel, we zoomagain, this time by a factor of 5.7 in volume. We can now see individual galactic-size haloswhich preferentially occur along the filaments, at the intersection of which large halos formthat will host galaxy clusters. Finally, the bottom-left panel zooms into an individual galactic-size halo. This shows a large number of small substructures that survive the collapse of thehalo and make up about 10% of the total mass (Klypin et al. 1999, Moore et al. 1999). Forsimulations like the ones illustrated in Figure 7.5, it is possible to characterize the statisticalproperties of the dark matter distribution with very high accuracy. The statistical error bars inthis estimate are actually smaller than the thickness of the line. Similarly, higher order cluster-ing statistics, topological measures, the mass function and clustering of dark matter halos andthe time evolution of these quantities can all be determined very precisely from these simula-tions (e.g. Jenkins et al. 2001, Evrard et al. 2002). In a sense, the problem of the distributionof dark matter in the CDM model can be regarded as largely solved [22].


Figure 7.5: Slices through 4 different simulations of the dark matter in the ”CDM” cosmology.Denoting the number of particles in each simulation byN , the length of the simulation cubeby L, the thickness of the slice byt, and the particle mass bymp, the characteristics of eachpanel are as follows. Top-left (the Hubble volume simulation, Evrard et al. 2002):N = 109,L = 3000 h−1 Mpc, t = 30 h−1 Mpc, mp = 2.2 × 1012 h−1 M¯. Top-right (Jenkins et al.1998): N = 16.8 × 106, L = 250 h−1 Mpc, t = 25 h−1 Mpc, mp = 6.9 × 1010 h−1 M¯.Bottom-right (Jenkins et al. 1998):N = 16.8 × 106, L = 140 h−1 Mpc, t = 14 h−1 Mpc,mp = 1.4 × 1010 h−1 M¯. Bottom-left (Navarro et al. 2002):N = 7 × 106, L = 0.5 h−1

Mpc, t = 1 h−1 Mpc, mp = 6.5× 105 h−1 M¯.

Biasing of Dark Matter Halos

The second progress where N–body simulations played a major role in the 1990’s is relatedto the statistics of dark halos, their mass function and spatial biasing. The standard picture ofstructure formation predicts that luminous objects form in the gravitational potential of dark


Figure 7.6: Dark matter distribution at redshiftz = 6 (top) and in the present Universe (bottom).Parameters:ΩM = 0.30, Ωb = 0.035, σ8 = 0.90, L = 25 Mpc/h, Ncell = 7683. [Source:Cen Princeton]


matter halos. Both the Press–Schechter theory and high resolution N–body simulations havemade significant contributions in constructing a semi–analytical framework for halo cluster-ing.

Density Profiles for Dark Halos and Cluster Gas

One of the most useful result from N–body simulations is the discovery of the universal densityprofile of dark halos. Navarro, Frenk and White (1995, 1997) found that all simulated densityprofiles can be well fitted by the following simple model (socalled NFW–profile)

ρDM (r) =δcρc(z)

(r/rS)(1 + r/rS)2(7.170)

by an appropriate choice of the scaling radiusrS = rS(M) as a function of halo massM .ρc(z) = 3H2(z)/8πG is the critical density of the Universe at redshiftz, rS is the scaleradius, andδc is a characteristic dimensionless density, which takes the form when expressedin terms of the concentration parameterc

δc =2003

c3

ln(1 + c)− c/(1 + c). (7.171)

Hence, there are two free parameters for the NFW density profile:rS andc. c is related to thevirial radius rvir, defined to be the radius where the enclosed average mass density equals200ρc(z), i.e. rvir = crS . The mass enclosed within radiusr is found to be

M(< r) = 4πρc(z) δcr3S

[ln

(rS + r

rS

)− r

r + rS

]. (7.172)

The scaling radius found in clusters is typically 100 – 200 kpc. Forr ¿ rS we findM(<r) ∝ r2.

Many others have indicated that the inner slope of the density profile is steeper than theNFW–value, and current census among N–body workers is given by

ρDM ∝ 1(r/rS)α(1 + r/rS)3−α

. (7.173)

with α ' 1.2 − 1.5, rather than the NFW–valueα = 1 for r > 0.01 Rvir (for recent simula-tions of halo density profiles see [5]).

In the meanwhile, many clusters have been investigated with Chandra and XMM. The gasdensity is then typically found to obey aβ–law

ρG(r) = ρG0 (1 + r2/r2c )−3β/2 (7.174)

with the core radiusrc ' 30− 50 kpc andβ ' 0.3− 0.5. The cluster A2589 e.g. (at redshiftz = 0.042) has a core radiusrc = 40 ± 12 kpc andβ = 0.39 ± 0.04 with a central densityρG0 = 2 × 10−26 g/cm3 (Buote and Lewis 2003 [2]). This cluster has roughly a constantX–ray temperature of 3.16 keV between 15 and 150 kpc.


Figure 7.7: Total gravitational mass and gas mass as a function of radius for Abell 2589 [2].Also plotted is the mass range expected from the stars (dot–dahsed curves): The lower curveassumesM∗/LV = 1 M¯/L¯ and the upper curveM∗/LV = 12 M¯/L¯. The upper axisshows the radius in units of the virial radius,Rvir = 1.72 Mpc.

Cluster Temperature

The baryons confined by the gravity of the dark matter in clusters are heated up to virial tem-perature which is in the range of X–ray temperatures. We determine the virial temperatureTvir

of the system of massM by equating the gravitational potential(3/5)GM2/r of a dark haloto (3M/µmp)kBTvir. UsingM = (4π/3)ρhr3, µ = 0.59 2 andρ0 = 2.78×1011 Ωmh2 M¯Mpc−3, we find the virial temperature at redshiftzh where the halo has been formed

Tvir = 1.26× 107 K

(M

1015 M¯

)2/3

(fhΩmh2)1/3 (1 + zh) , (7.176)

2If the He fraction isY = 025 by weight and the gas is fully ionised

µ =mHnH + mHenHe

2nH + 3nHe=

mH

2

1 + Y

1 + 0.375Y' 0.57 mH . (7.175)


whereρh = fhρ0(1 + zh)3 denotes the mass density in the dark halos andfh ' 180/Ω0.7m .

The corresponding virial radius can be written as

Rvir = 2.58Mpc(

M

1015 M¯

)1/3

(1 + zh)−1 h−2/350 (7.177)

and the virial velocity (velocity of dark sub–halos and baryons in the cluster) as

Vvir = 1000 km s−1

(M

1015 M¯

)1/3

(1 + zh)1/2 h1/350 . (7.178)

This hot cluster gas cools by emission of Bremsstrahlung photons which can be detected bymeans of X–ray telescopes (Einstein, ROSAT, ASCA, Chandra and XMM–Newton). Thecooling time is typically longer than the Hubble time, except in the very center of massiveclusters and galaxies. Simple scaling relations (form the above we haveMB ∝ M ∝ T

3/2X

andLX ∝ MBT1/2X ∝ T 2

X ) predict that the X–ray luminosity scales withT 2X . It has been

known for many years that the actual relation is steeper (see Fig. 7.8).

Figure 7.8: X–ray luminosity vs. X-ray temperature derived from ASCA observations of 270galaxy clusters. The best–fit isLX ∝ T 3

X . [Source: Mushotzky 2003]


7.4.4 Statistical Measures for Cosmic Density Fields

A key question is how to treat the statistical analysis for these data – this is how observationsand theory are confronted with each other. The concept of an ensemble is used every timewe apply probablity theory to such events. The density at a given point in space will havedifferent values in each member of the ensemble, with some overall variance< δ2 > betweenmembers of the ensemble. Statistical homogeneity of theδ–field means that this variancemust be independent of space–points. The actual field found in a given simulation is just onerealisation of the statistical process.

There are two problems with this line of argument: (i) we have no evidence that the ensem-ble exists, (ii) in any case we only get to observe one realisation. In order to solve the secondquestion, we just have to look at widely separated parts of space, since theδ–fields should bethere uncorrelated. This means we have to mesasure the variance< δ2 > by averaging overlarge volumes. Fields that satisfyvolume average = ensemble averageare called ergodic –which is a kind of common–sense axiom in Cosmology.

A Gaussian Random Field

We consider the density contrastδ(~x) = ρ/ρ − 1 at positions in comoving coordinates. Thisdensity field is considered as a stochastic variable, i.e. a random field. The conventionalassumption is that the initial perturbations are Gaussian, i.e. i.e. itsm–point joint probabilitydistribution obeys a multi–variate Gaussian

P (δ1, ..., δm) dδ1...dδm =1√

(2π)m detMexp

1

2

m∑

i,j=1

δi(M−1)ijδj

dδ1...dδm

(7.179)

Here,Mij =< δiδj > is the covariance matrix andM−1 its inverse. SinceMij = ξ(~xi, ~xj)is specified by the two–point correlation function, this implies that the statistical nature of aGaussian density filed is completely given by the correlation function.

The Gaussian nature of the density field preserves in the linear stage of evolution, but notin the non–linear stage. Here, it can be approximated by one–point log–normal (Fig. 7.9)

P (1)(δ) =1√2πσ2

1

exp(− [ln(1 + δ) + σ2

1/2]2

2σ21

)1

1 + δ. (7.180)

σ1 depends on the smoothing scaleR

σ21(R) = ln(1 + σ2

NL(R)) , (7.181)

where the variance of the non–linear clustering is computed from the non–linear power spec-trum

σ2NL(R) =

12π2

∫ ∞

0

W 2(kR) k2PNL(k) dk . (7.182)

W (x) is the window function, e.g. a GaussianW (x) = exp(−x2/2) or a top–hatW (x) =


Figure 7.9: Probability distribution function (PDF) in the non–linear regime. Smoothing win-dows:2/h Mpc (cyan),6/h Mpc (red), and18/h Mpc (green).

3(sinx/x− cos x)/x2. The smoothed density field is then given by

δρ(~x,R) =∫

d3y W (|~x− ~y|; R) ρ(~y)

=∫

d3k

(2π)3W (kR) ρ(~k) exp(−~k · ~x) . (7.183)

In the case of a top-hat window function we find

W (kR) =∫ |~x|<R

exp(i~k · ~x) d3x

=3

(kR)2

(sin(kR)(kR)

− cos(kR))

' 1 , kR ¿ 1' 0 , kR À 1 . (7.184)


The functionW (kR) effectively truncates the sum so that waves withλ ¿ 2R will notcontribute.

Actually in a stochastic sample, we cannot measure the density at a single point~x. It ismore relevant to consider the total massMR(~x) in a sphere of radiusR centered on the point~x

MR(~x) =∫ |~x−~x′|<R

ρ(~x′) d3x′

= ρVR

(1 +

1V

∑

k

δkW (kR) exp(i~x · ~k)

). (7.185)

To calculate the fluctuations inMR we need

[MR(~x)]2 = ρ2V 2R

[1 +

2√V

∑

k

δkW (kR) exp(i~k · ~x)

+1V

∑

k

∑

k′δkδk′W (kR)W (k′R) exp(i(~k + ~k′) · ~x)

]. (7.186)

Taking the average over the volumeV , the oscillating terms integrate to zero leaving

< M2R >= ρ2V 2

R

[1 +

1V

∑

k

|δk|2W 2(kR)

]. (7.187)

With this we obtain thevarianceof MR

(∆MR

MR

)2

=< M2

R > − < MR >2

< MR >2=

1V

∑

k

|W (kR)|2|δk|2 . (7.188)

SinceVR ¿ V , we can replace the sum by an integral

(∆MR

MR

)2

=1

(2π)3

∫d3k |δk|2 |W (kR)|2 . (7.189)

This often written in terms of the variance∆2k = k3|δk|2 of the power spectrum as (for

isotropic fluctuations)

(∆MR

MR

)2

=1

2π2

∫dk

k|∆k|2 |W (kR)|2 . (7.190)

The window function cuts off the integral fork > 1/R. Since∆k is an increasing function ofk (see Fig. 7.10), a good approximation forR ' 8/h Mpc is

∆MR

MR' ∆k|kR'1.38 . (7.191)


Fourier Transformations

If a field were periodic in some large box of sideL, then we could just sum over all modes

F (~x) =∑

Fk exp(−i~k · ~x) (7.192)

The periodicity restricts then the wave numbers tokx = 2πn/L. If we let the box becomearbitrarily large, the sum will go to an integral

F (x) =(

L

2π

)n ∫Fk exp(−ikx) dnk (7.193)

Fk =(

L

2π

)n ∫F (x) exp(ikx) dnx . (7.194)

Correlation Function and Power Spectra

Of particular importance is the autocorrelation function

ξ(~r) =< δ(~x)δ(~x + ~r) > (7.195)

simply referred as the correlation function. Now we use thatδ is real

ξ(~r) =<∑

k

∑

k′δ∗kδk′ exp(i(~k′ − ~k) · ~x) exp(−i~k · ~r) > (7.196)

By periodic boundary conditions, all the cross–terms with~k′ 6= ~k average to zero, and theremaining sum can be written as an integral

ξ(~r) =V

(2π)3

∫|δk|2 exp(−i~k · ~r) d3k . (7.197)

In short, thecorrelation function is the Fourier transform of the power spectrum. InCosmology, the alternative notation is often used

P (k) ≡< δ2k > (7.198)

for the ensemble–average power.In an isotropic Universe, the density perturbation spectrum cannot contain a preferred

direction, and so we must have an isotropic power spectrum,< |δk(~k)|2 >= |δk|2. Theangular part of thek–space integral can immediately be done: introduce spherical polars withthe polar axis along~k and use the reality ofξ so thatexp(−i~k · ~x) → cos(kr cos θ). In 3dimensions this yields then

ξ(r) =V

(2π)3

∫P (k)

sin(kr)kr

4π2 k2 dk . (7.199)

It is now quite common to express the power spectrum in dimensionless form as the varianceper logarithmic decade:

∆2(k) ≡ d ln(< δ2 >)d ln k

=V

(2π)34πk3P (k) =

2π

k3

∫ξ(r)

sin(kr)kr

r2 dr . (7.200)


The function∆ is dimensionless and∆2(k) = 1 means that there are order–unity fluctuationsfrom modes in a logarithmic bin aroundk (Fig. 7.10). Density fluctuations of order unityare achieved on scales of about8/h Mpc (the scale of large clusters). Large volume redshiftsurveys (2dFGRS and SDSS) allow to determine the cosmic variance even beyond this scale.

In CDM models, the shape of this spectrum only depends on the parameterΩmh. These

Figure 7.10: Cosmic variance as a function of wavenumber, as observed from galaxy clusteringin 2dFGRS. At small scales,∆ approaches a constant value for a primordial spectrumP (k) ∝k. The solid points with error bars show the power estimate. The window function correlatesthe results at differentk values, and also distorts the large-scale shape of the power spectrum.The solid and dashed lines show various CDM models, all assumingn = 1. For the case withnon–negligible baryon content, a big–bang nucleosynthesis value ofΩBh2 = 0.02 is assumed,together withh = 0.7. A good fit is clearly obtained forΩmh = 0.2 (in CDM models, thecosmic variance is only a function ofΩmh). The observed power at largek will be boosted bynonlinear effects. [Source: Peacock]

observations therefore clearly rule out models withΩm ' 1. It is also customary tonormalizethe spectrum in terms ofσ8 ≡ ∆MR/MR at the scale of8h−1 Mpc. This is one of theessential parameters appearing in cosmological simulations.

Using the halo model, it is then possible to calculate the correlations of the nonlinear den-sity field, neglecting only the large–scale correlations in halo positions. The power spectrumdetermined in this way is shown in Fig. 7.11, and turns out to agree very well with the ex-act nonlinear result on small and intermediate scales. The lesson here is that a good deal ofthe nonlinear correlations of the dark matter field can be understood as a distribution of ran-dom clumps, provided these are given the correct distribution of masses and mass-dependentdensity profiles.


Figure 7.11: The decomposition of the mass power spectrum according to the halo model. Thedashed line shows linear theory, and the open circles show the predicted 1–halo contribution.The full lines show the contribution of different mass ranges to the 1-halo term: bins of width afactor 10 in width, starting at1010−1011 h−1 M¯ and ending at1015−1016 h−1 M¯. [Source:Peacock]

By counting galaxies, one is not necessarily counting the distribution of the total mass.This introduces a kind of bias

∆nG

nG= b

∆ρ

ρ. (7.201)

b is called the bias parameter. Bias independent information can be obtained from counts ofgalaxy clusters. The largest clusters have masses of the order of the mean mass containedin spheres of8/h Mpc. Recent measurements giveσ8Ω0.56

m ' 0.5. So forΩm = 0.3 thiscorresponds to a value ofσ8 ' 0.7.

Topological Measures

Turning next to statistics for continuous density fields, we note first that N-point correlationfunctions and power spectra are naturally defined and very useful in this case much as theyare for point sets. A family of new statistics, reviewed by Melott (1990), is based on thetwo-dimensional surfaces of constant density (isodensity surfaces). The best known of theseis the genus per unit volumeg(ν) as a function of the standardized density contour levelν = δ/σ (Gott et al 1986). The total genus G of a surface is a topological invariant measuringthe number of ”holes” (as in a doughnut) minus the number of isolated regions. One ofits attractions is the fact that an exact prediction exists for the shape ofG(ν) for Gaussianrandom fields (Doroshkevich 1970, Bardeen et al 1986, Hamilton et al 1986), enabling a test


of Gaussianity on large scales where the matter distribution is expected to still approximate thelinearly growing initial conditions. When the smoothing scale is varied, the amplitude of thegenus curve varies in a spectrum-dependent way, providing another useful clustering statistic.Computer programs for computing the genus are given by Melott (1990); an alternative andsimpler algorithm is provided by Coles et al (1996).

Figure 7.12: Minkowski functionals as a function ofνf for Rg = 5/h Mpc in two regions ofSDSS. The results favour the LCDM model with initial Gaussian perturbations.

Minkowski functionals (Mecke et al 1994) have recently been introduced in cosmologyas a very powerful descriptor of the topology of isodensity surfaces. In three dimensions,there are four Minkowski functionals (V0, V1, V2, V3); two of them are the genus (actually,its relative the Euler characteristic) and surface area statistics discussed above, and the othertwo are the covered volume (related to the void probability function) and integral mean cur-vature. Analytical results for Gaussian random fields have been provided by Schmalzing &Buchert (1997), who also have made available a computer program for computing these statis-tics from a point process. The insight and unification provided by these recent results suggests


a promising future for Minkowski functionals as a statistic for both cosmological simulationsand redshift surveys.

7.4.5 2000’s: Baryons in Simulations

Understanding galaxy formation is a much more difficult problem than understanding the evo-lution of the dark matter distribution. In the CDM theory, galaxies form when gas, initiallywell mixed with the dark matter, cools and condenses into emerging dark matter halos. In ad-dition to gravity, a non-exhaustive list of the processes that now need to be taken into accountincludes: the shock heating and cooling of gas into dark halos, the formation of stars from coldgas and the evolution of the resulting stellar population, the feedback processes generated bythe ejection of mass and energy from evolving stars, the production and mixing of heavy ele-ments, the extinction and reradiation of stellar light by dust particles, the formation of blackholes at the centres of galaxies and the influence of the associated quasar emission. Theseprocesses span an enormous range of length and mass scales. For example, the parsec scalerelevant to star formation is a factor of108 smaller than the scale of a galaxy supercluster.

The best that can be done with current computing techniques is to model the evolution ofdark matter and gas in a cosmological volume with resolution comparable to a single galaxy.Subgalactic scales must then be regarded as ”subgrid” scales and followed by means of phe-nomenological models based either on our current physical understanding or on observations.In the approach known as ”semi-analytic” modelling (White & Frenk 1991), even the gas dy-namics is treated phenomenologically using a simple, spherically symmetric model to describethe accretion and cooling of gas into dark matter halos. It turns out that this simple modelworks suprisingly well as judged by the good agreement with results of full N-body/gas-dynamical simulations (Benson et al. 2001b, Helly et al. 2002, Yoshida et al. 2002).

The main difficulty encountered in cosmological gas dynamical simulations arises fromthe need to suppress a cooling instability present in hierarchical clustering models like CDM.The building blocks of galaxies are small clumps that condense at early times. The gas thatcools within them has very high density, reflecting the mean density of the Universe at thatepoch. Since the cooling rate is proportional to the square of the gas density, in the absence ofheat sources, most of the gas would cool in the highest levels of the mass hierarchy leaving nogas to power star formation today or even to provide the hot, X-ray emitting plasma detectedin galaxy clusters. Known heat sources are photoionisation by early generations of stars andquasars and the injection of energy from supernovae and active galactic nuclei. These pro-cesses, which undoubtedly happened in our Universe, belong to the realm of subgrid physicswhich cosmological simulations cannot resolve. Different treatments of this ”feedback” resultin different amounts of cool gas and can lead to very different predictions for the propertiesof the galaxy population. This is a fundamental problem that afflicts cosmological simula-tions even when they are complemented by the inclusion of semi-analytic techniques. In thiscase, the resolution of the calculation can be extended to arbitrarily small mass halos, perhapsallowing a more realistic treatment of feedback. Although they are less general than full gas-dynamical simulations, simulations in which the evolution of gas is treated semi-analyticallymake experimentation with different prescriptions relatively simple and efficient (Kauffmann,White & Guiderdoni 1993, Somerville & Primack 1999, Cole et al. 2000)

Serious attempts to create galaxies from hydrodynamical simulations have been initiatedin early 1990’s (Cen & Ostriker 1992; Katz, Hernquist & Weinberg 1992, Navarro et al. 2002),those simulated galaxies are still far from realistic.

7.5 Dark Baryons 259

7.5 Dark Baryons

7.5.1 The Lymanα Forest

The Lyα forest represents the optically thin (at the Lyman edge) component of Quasar Ab-

Figure 7.13: All quasars at high redshift exhibit huge numbers of narrow Absorption Systems.DLA: damped Lymanα lines, LLS: Lyman limit system.

sorption Systems (QAS), a collection of absorption features in QSO spectra extending back tohigh redshifts (Fig. 7.13). QAS are effective probes of the matter distribution and the phys-ical state of the Universe at early epochs when structures such as galaxies are still formingand evolving. Although stringent observational constraints have been placed on competingcosmological models at large scales by the COBE satellite and over the smaller scales of ourlocal Universe by observations of galaxies and clusters, there remains sufficient flexibility inthe cosmological parameters that no single model has been established conclusively. The rel-ative lack of constraining observational data at the intermediate to high redshifts (0 < z < 6),where differences between competing cosmological models are more pronounced, suggeststhat QAS can potentially yield valuable and discriminating observational data.

7.6 Galaxy Clusters

Clusters of galaxies are the largest gravitationally bound systems known to be in quasi–equilibrium. This allows for reliable estimates to be made of their mass as well as theirdynamical and thermal attributes. The richest clusters, arising from 3σ density fluctuations,can be as massive as1015 solar masses, and the environment in these structures is composedof shock heated gas with temperatures of order107−108 degrees Kelvin which emits thermalbremsstrahlung and line radiation at X–ray energies. Also, because of their spatial size' 1/hMpc and separations of order50/h Mpc, they provide a measure of nonlinearity on scalesclose to the perturbation normalization scale8/h Mpc. Observations of the substructure, dis-tribution, luminosity, and evolution of galaxy clusters are therefore likely to provide signaturesof the underlying cosmology of our Universe, and can be used as cosmological probes in theeasily observable redshift range0 > z > 2.


Figure 7.14: All quasars at high redshift exhibit huge numbers of narrow absorption lines start-ing at the wavelength of the quasar’s own Lyman alpha emission line and extending blueward.These are Lyman alpha absorption from foreground structures, in which the quasar light probesan otherwise invisible component of cosmic gas. This component evolves strongly with cosmictime, since we see dramatically more absorbers (be they clouds, filaments, or even crowding invelocity rather than in space) toward higher redshifts.

Internal Structure

Thomas et al. have investigated the internal structure of galaxy clusters formed in high resolu-tion N–body simulations of four different cosmological models, including standard, open, andflat but low density universes. They find that the structure of relaxed clusters is similar in thecritical and low density universes, although the critical density models contain relatively moredisordered clusters due to the freeze-out of fluctuations in open universes at late times. Theprofiles of relaxed clusters are very similar in the different simulations, since most clusters arein a quasi-equilibrium state inside the virial radius and follow the universal density profile ofNavarro et al. [1997]. There does not appear to be a strong cosmological dependence in the

7.6 Galaxy Clusters 261

Figure 7.15: Distribution of the gas density at redshiftz = 3 from a numerical hydrodynamicssimulation of the Lyα forest. The simulation adopted a CDM spectrum of primordial densityfluctuations, normalized to the second year COBE observations, a Hubble parameter ofh = 0.5,a comoving box size of 9.6 Mpc, and baryonic density of composed of 76% hydrogen and 24%helium. The region shown is 2.4 Mpc (proper) on a side. The isosurfaces represent baryons atten times the mean cosmic density (characteristic of typical filamentary structures) and are colorcoded to the gas temperature (red =3×104 K, yellow = 3×105 K). The higher density contourstrace out isolated spherical structures typically found at the intersections of the filaments. Asingle random slice through the cube is also shown, with the baryonic overdensity representedby a rainbow–like color map changing from black (minimum) to red (maximum). TheHe+

mass fraction is shown with a wire mesh in this same slice. Notice that there is fine structureeverywhere. To emphasize fine structure in the minivoids, the mass fraction in the overdenseregions has been rescaled by the gas overdensity wherever it exceeds unity. [Source: PeterAninos]


profiles as suggested by previous studies of clusters formed from pure power law initial densityfluctuations. However, because more young and dynamically evolving clusters are found incritical density universes, Thomas et al. suggest that it may be possible to discriminate amongthe density parameters by looking for multiple cores in the substructure of the dynamic clusterpopulation. They note that a statistical population of 20 clusters can distinguish between openand critically closed universes.

Number Density Evolution

The evolution of the number density of rich clusters of galaxies can be used to computeandΩM andσ8 (the power spectrum normalization on scales of8/h Mpc) when numericalsimulation results are combined with the constraintσ8

√ΩM ' 0.5, derived from observed

present-day abundances of rich clusters. Bahcall et al. computed the evolution of the clus-ter mass function in five different cosmological model simulations and find that the numberof high mass (Coma-like) clusters in flat, low models decreases dramatically by a factor ofapproximately103 from z = 0 to z = 0.5. For low ΩM , high σ8 models, the data resultsin a much slower decrease in the number density of clusters over the same redshift interval.Comparing these results to observations of rich clusters in the real Universe, which indicateonly a slight evolution of cluster abundances to redshiftsz ' 1, they conclude that criti-cally closed standard CDM and Mixed Dark Matter (MDM) models are not consistent withthe observed data. The models which best fit the data are the open models with low bias(ΩM = 0.3 ± 0.1 andσ8 = 0.85 ± 0.5), and flat low density models with a cosmologicalconstant (ΩM = 0.34± 0.13).

X–Ray Luminosity Function

The evolution of the X-ray luminosity function, and the size and temperature distributionof rich clusters of galaxies are all potentially important discriminants of cosmological mod-els. Bryan et al. investigated these properties in a high resolution numerical simulation of astandard CDM model normalized to COBE. Although the results are highly sensitive to gridresolution, their primary conclusion, that the standard CDM model predicts too many brightX-ray emitting clusters and too much integrated X-ray intensity, is robust since an increase inresolution will only exaggerate these problems.

Evrard et al. (2002) ..............

7.7 The Cosmic Microwave Background Probes LinearPerturbations

The cosmic microwave background (CMB) is the afterglow radiation left over from the hotBig Bang. Its temperature is extremely uniform all over the sky. However, tiny temperaturevariations or fluctuations (at the part per million level) can offer great insight into the origin,evolution, and content of the universe. Perturbations at that time were still in the linear regime- a fact which considerably simplifies the analysis.3

Even in the early Universe, all material was not distributed entirely uniformly. Observa-tions of the CMB suggest that slightly more dense regions of ionized Hydrogen gas and pho-

3A rigorous treatment of the anisotropies of the CMB is given in [11], [12], [7] and [8].

7.7 The Cosmic Microwave Background Probes Linear Perturbations 263

tons existed, which gradually collapsed because of their own gravitation. At about 300,000years of age, the Universe had cooled sufficiently so that the material became un-ionized, al-lowing electrons and protons to combine and form ordinary Hydrogen gas (a process referredto as recombination). This allowed the Universe to become transparent and photons wereable to travel freely. It is at this point when condensed regions of gas and photons appear asbright spots when viewed through today’s microwave telescopes. Measuring the size of thesespots with Viper or CBI, for example, can reveal their distance from Earth. After releasing thephotons, the cloud of un–ionized Hydrogen gas continued to collapse to form galaxy clusters.

7.7.1 Anisotropy Mechanisms

The evolution of first-order perturbations in the various energy density components and themetric are described with the following sets of equations:

• The photons and neutrinos are described by distribution functionsf(t, ~x, ~p). A funda-mental simplifying assumption is that the energy dependence of both is given by theblackbody distribution. The space dependence is generally Fourier transformed, so thedistribution functions can be written asΘ(t,~k, ~n), where the function has been normal-ized to the temperature of the blackbody distribution and~n represents the direction inwhich the radiation propagates. The time evolution of each is given by the Boltzmannequation. For neutrinos, collisions are unimportant so the Boltzmann collision term onthe right side is zero; for photons, Thomson scattering off electrons must be included.

• The dark matter and baryons are in principle described by Boltzmann equations as well,but a fluid description incorporating only the lowest two velocity moments of the distribu-tion functions is adequate. Thus each is described by the Euler and continuity equationsfor their densities and velocities. The baryon Euler equation must include the couplingto photons via Thomson scattering.

• Metric perturbation evolution and the connection of the metric perturbations to the matterperturbations are both contained in the Einstein equations. This is where the subtletiesarise. A general metric perturbation has 10 degrees of freedom, but four of these areunphysical gauge modes. The physical perturbations include two degrees of freedomconstructed from scalar functions (the Bardeen potenitalsΦ andΨ), two from a vec-tor, and two remaining tensor perturbations (see previous Sect.). Physically, the scalarperturbations correspond to gravitational potential and anisotropic stress perturbations;the vector perturbations correspond to vorticity and shear perturbations; and the tensorperturbations are two polarizations of gravitational radiation. Tensor and vector pertur-bations do not couple to matter evolving only under gravitation; in the absence of a “stiffsource” of stress energy, like cosmic defects or magnetic fields, the tensor and vectorperturbations decouple from the linear perturbations in the matter.

A variety of different variable choices and methods for eliminating the gauge freedom havebeen developed. The subject can be fairly complicated. A detailed discussion and comparisonbetween the Newtonian and synchronous gauges, along with a complete set of equations, hasbeen given previously (also see Hu et al. [13]). An elegant and physically appealing formalismbased on an entirelycovariant and gauge–invariant descriptionof all physical quantities hasbeen developed for the microwave background by Challinor and Lasenby (1999) and Gebbieet al. (2000), based on earlier work by Ehlers (1993) and Ellis and Bruni (1989).


The Boltzmann equations are partial differential equations,4 which can be converted tohierarchies of ordinary differential equations by expanding their directional dependence inLegendre polynomials. The result is a large set of coupled, first–order linear ordinary dif-ferential equations which form a well-posed initial value problem. Initial conditions must bespecified. Generally they are taken to be so-called adiabatic perturbations: initial curvatureperturbations with equal fractional perturbations in each matter species. Such perturbationsarise naturally from the simplest inflationary scenarios. Alternatively, isocurvature perturba-tions can also be considered: these initial conditions have fractional density perturbations intwo or more matter species whose total spatial curvature perturbation cancels.

Figure 7.16: Illustrating the physical mechanisms that cause CMB anisotropies. The shadedarc on the right represents the last–scattering shell; an inhomogeneity on this shell affects theCMB through its potential, adiabatic and Doppler perturbations. Further perturbations are addedalong the line of sight by time-varying potentials (ISW) and by electron scattering from hot gas(Sunyaev-Zeldovich effect). The density field at last scattering can be Fourier analysed intomodes of wavevectork. These spatial perturbation modes have a contribution that is in generaldamped by averaging over the shell of last scattering. Short–wavelength modes are more heavilyaffected (i) because more of them fit inside the scattering shell, and (ii) because their wavevectorspoint more nearly radially for a given projected wavelength. [Illustration by Peacock]

4The Boltzmann equation for a phase space distribution is of the form,~p = ε~n for photons,

d

dτf(τ, ~x, ~p) = ∂τ f + (~n · ∇)f − Γi

αβpαpβ ∂f

∂pi= C[f ] , (7.202)

whereC[f ] is the collision integral (given by Thomson scattering for photons). By using the geodesics equations,this can also be written in the form

∂τ f + (~n · ∇)f + ε∂f

∂ε+ ni ∂f

∂ni= C1 − C2 . (7.203)

C1 is the amount of photons scattered into the beam of radiation travelling in direction~n andC2 is the amount ofphotons scattered out of the beam.


Tight Coupling

Two basic time scales enter into the evolution of the microwave background. The first is thephoton scattering time scale

tT = 1/(σT nec) = 6.14× 107 s (T/eV )−3 (xeΩBh2)−1 , (7.204)

the mean time between Thomson scatterings. The other is the expansion time scale of theuniverse,1/H, whereH is the Hubble parameter. At temperatures significantly greater than0.5 eV, hydrogen and helium are completely ionized andtT ¿ 1/H. The Thomson scatteringswhich couple the electrons and photons occur much more rapidly than the expansion of theuniverse; as a result, the baryons and photons behave as a singletightly coupled fluid. Duringthis period, the fluctuations in the photons mirror the fluctuations in the baryons. (Rememberthat recombination occurs at around 0.5 eV rather than 13.6 eV because of the huge photon-baryon ratio; the universe contains somewhere around109 photons for each baryon, as weknow from primordial nucleosynthesis.)

The photon distribution function for scalar perturbations can be written asΘ(τ,~k, µ),whereµ = ~k · ~n, and the scalar character of the fluctuations guarantees the distribution can-not have any azimuthal directional dependence. (The azimuthal dependence for vector andtensor perturbations can also be included in a similar decomposition). The moments of thedistribution are defined as

Θ(τ,~k, µ) =∞∑

l=0

(−i)l Θl(τ,~k)Pl(µ) . (7.205)

Tight coupling implies thatΘl = 0 for l > 1. Physically, thel = 0 moment corresponds tothe photon energy density perturbation, whilel = 1 corresponds to the bulk velocity. Duringtight coupling, these two moments must match the baryon density and velocity perturbations.Any higher moments rapidly decay due to the isotropizing effect of Thomson scattering; thisfollows immediately from the photon Boltzmann equation with a source from Thomson scat-tering [13], [12]

Θ + iµ(Θ + Ψ) = Φ + tT [Θ0 −Θ− 110

Θ2P2(µ)− iµVB ] . (7.206)

tT = σT xene(a/a0) is the differential optical depth to Thomson scattering, and the scalefactora/a0 = 1/(1+ z). The appearance of the photon quadrupoleΘ2 represents the angulardependence of the Thomson scattering.5

The gravitational potentialsΦ andΨ have two effects on the temperature fluctuations, bothintroduced already in the paper by Sachs and Wolfe in 1967. The gradient of the Newtonian

5For a derivation using the perturbed Boltzmann equation, see e.g. Durrer [7]. The equation can be derivedfrom the following arguments: The intensity distribution is expected to be a Planckian with redshifted temperatureT = (T0/a)(1 + Θ), whereΘ denotes the fractional perturbation in temperature of the formΘ(τ, ~x, ~n). Thedistribution function for the photons isf = fP (aε/(1 + Θ)). The total variation is given by

df

dτ=

aε

1 + Θf ′Phd ln(aε)

dτ− dΘ

dτ

i=

df

dτ

c

. (7.207)

The time derivative ofΘ follows from

dΘ

dτ= ∂τΘ +

d~xi

dτ· ∇Θ +

∂Θ

∂εε +

d~ni

dτ· ∇nΘ , (7.208)


potential induces a gravitational redshift on the photons as they travel through the potentialwell. Since the potential difference merely induces a fractional temperature shift, the com-binationΘ + Ψ is the resultant temperature perturbation after the photon climbs out of thepotential with negative value forΨ. This is called the ordinarySachs–Wolfe effect. Thetime–dependence of the metric termΦ causes its own time–dilation effect known as theinte-grated Sachs–Wolfe (ISW) effect.

The baryons evolve under the continuity and Euler equations

∆B = −k(VB −Θ1) +34∆γ (7.212)

VB = −HVB + kΨ + tT (Θ1 − VB)/R . (7.213)

The Newtonian potential acts as a source for the velocity evolution through infall and givesrise to the adiabatic growth of perturbations.

Free–Streaming

In the other regime, for temperatures significantly lower than 0.5 eV,tT À 1/H and photonson average never scatter again until the present time. This is known as thefree streamingepoch. Since the radiation is no longer tightly coupled to the electrons, all higher moments inthe radiation field develop as the photons propagate. In a flat background spacetime, the exactsolution is simple to derive [11]

(Θ + Ψ)(τ0, µ) =∫ τ0

τ∗

[Θ + Ψ− iµVB ]tT + Φ + Ψ

exp[−τT (τ, τ0)] exp[ikµ(τ − τ0)] dτ .

(7.214)

The optical depth is measured from the momentτ to the present epochτ0, τT (τ, τ0) =∫ τ0

τtT dτ . We have dropped the quadrupole term, since it vanishes in the tight coupling

limit. The combinationtT exp(−τ) is called the conformal time visibility function, it is theprobability that a photon scattered withindτ of τ . It has a sharp peak at the last scatteringepoch.

After scattering ceases, the photons evolve according to the Liouville equation

Θ + ikµΘ = 0 (7.215)

where the last two terms vanish. For a metric given by Bardeen potentials, the geodesic equations give

d ln(aε)

dτ= −∇Ψ + Φ . (7.209)

Usin gthis expression in the Boltzmann equation, we get the evolution equation for temperature fluctuations

∂τΘ + (~n · ∇)Θ = −∇Ψ + Φ +

dΘ

dτ

c

. (7.210)

The collisional term can be expressed in terms of the Thomson scattering rate as

dΘ

dτ

c

= aneσT

h−Θ +

1

4∆Rad + ~n · ~Ve

i. (7.211)

The first two terms represent absorption and reemission of radiation. The third term is due to the Doppler effect fromthe velocity fields of the electrons.


with the trivial solution6

Θ(τ,~k, µ) = exp(−ikµ(τ − τ∗) Θ(τ∗,~k, µ) . (7.218)

τ∗ is the conformal time when free streaming begins. Taking moments on both sides

Θl(τ,~k) = (2l + 1)[Θ0(τ∗,~k)jl(kτ − kτ∗) + Θ1(τ∗,~k)j′l(kτ − kτ∗)

], (7.219)

with jl as the spherical Bessel functions.The process of free streaming essentially mapsspatial variations in the photon distribution at the last scattering surface (wavenumberk) into angular variations on the sky today (momentl). A more accurate expression followsfrom the solution (7.214) [13],l ≥ 2,

Θl(τ0, k) ' [Θ0 + Ψ](τ∗) (2l + 1)jl(k(τ0 − τ∗))+ Θ1(τ∗) [ljl−1(k(τ0 − τ∗))− (l + 1)jl+1(k(τ0 − τ∗))]

+ (2l + 1)∫ τ0

τ∗[Ψ + Φ]jl(k(τ0 − τ)) dτ . (7.220)

This shows how the integrated Sachs–Wolfe effect contributes to the temperature fluctuations.

Diffusion Damping

In the intermediate regime during recombination,tT ' 1/H, photons propagate a character-istic distanceLD during this time. Since some scattering is still occurring, baryons experiencea drag from the photons as long as the ionization fraction is appreciable. A second–order per-turbation analysis shows that the result is damping of baryon fluctuations on scales belowLD,known asSilk damping or diffusion damping. This effect can be modelled by the replacement

Θ0(τ∗,~k) → Θ0(τ∗,~k) exp[−(k/kD)2] . (7.221)

7.7.2 The Resulting CMB Power Spectrum

The fluctuations in the universe are assumed to arise from some random statistical process.We are not interested in the exact pattern of fluctuations we see from our vantage point, sincethis is only a single realization of the process. Rather, a theory of cosmology predicts anunderlying distribution, of which our visible sky is a single statistical realization. The mostbasic statistic describing fluctuations is their power spectrum. A temperature map on the skyT (~n) is conventionally expanded in spherical harmonics,

T (~n)T0

− 1 =∞∑

l=1

almY lm(~n) (7.222)

6Remember from quantum mechanics the identity

exp(iµτ) =∞X

l=0

(2l + 1)il jl(kτ) Pl(µ) (7.216)

with

(2l + 1)j′l = ljl−1 − (l + 1)jl+1 . (7.217)


where

alm =1T0

∫dΩT (~n)Y ∗

lm(~n) (7.223)

are the temperature multipole coefficients andT0 is the mean CMB temperature. Thel = 1term is indistinguishable from the kinematic dipole and is normally ignored. The temperatureangular power spectrumCl is then given by

< a∗lmal′m′ >= Cl δll′δmm′ . (7.224)

where the angled brackets represent an average over statistical realizations of the underlyingdistribution. The parametersCl determine the correlation function for temperature fluctua-tions

<∆T

T(~n)

∆T

T(~n′) > =

∑

ll′mm′< alma∗l′m′ > Ylm(~n) Y ∗

l′m′(~n′)

=∑

l

Cl

l∑

m=−l

Ylm(~n)Y ∗lm(~n′) =

14π

∑

l

(2l + 1)ClPl(µ = ~n · ~n′) . (7.225)

Here we have used the addition theorem for spherical harmonics.Since we have only a single sky to observe, an unbiased estimator ofCl is constructed as

Cl =1

2l + 1

m=l∑

m=−l

a∗lmalm . (7.226)

The statistical uncertainty in estimatingCl by a sum of2l+1 terms is known as “cosmic vari-ance”. The constraintsl = l′ andm = m′ follow from the assumption of statistical isotropy:Cl must be independent of the orientation of the coordinate system used for the harmonicexpansion. These conditions can be verified via an explicit rotation of the coordinate system.The measurements of the parametersCl is one of the most important issue in observationalcosmology. The results prior to WMAP are shown in Fig. 7.17. The data obtained withWMAP in the first year (Fig. 7.21) demonstrate the superb power of this instrument.

Integrating over allk–modes in the perturbation we get the following equality

2l + 14π

Cl =V

2π2

∫dk

k

k3|Θl(τ0, k)|22l + 1

. (7.227)

A given cosmological theory will predictCl as a function ofl, which can be obtained fromevolving the temperature distribution function as described above. This prediction can thenbe compared with data from measured temperature differences on the sky. Figure 7.20 showsa typical temperature power spectrum from the inflationary class of models. The distinc-tive sequence of peaks arise from coherent acoustic oscillations in the fluid during the tightcoupling epoch and are of great importance in precision tests of cosmological models. Theeffect of diffusion damping is clearly visible in the decreasing power abovel = 1000. Whenviewing angular power spectrum plots in multipole space, keep in mind thatl = 200 corre-sponds approximately to fluctuations on angular scales of one degree, and the angular scaleis inversely proportional tol. The vertical axis is conventionally plotted asl(l + 1)Cl, be-cause the Sachs–Wolfe temperature fluctuations from a scale–invariant spectrum of densityperturbations appears as a horizontal line on such a plot,l(l + 1)CSW

l ' const (see Exercise4).


Figure 7.17: CMB power spectrum prior to WMAP. [Source: Tegmark]

7.7.3 More on Acoustic Oscillations

Before decoupling, the matter in the universe has significant pressure because it is tightlycoupled to radiation. This pressure counteracts any tendency for matter to collapse gravita-tionally. Formally, the Jeans mass is greater than the mass within a horizon volume for timesearlier than decoupling. During this epoch, density perturbations will set up standing acous-tic waves in the plasma. Under certain conditions, these waves leave a distinctive imprint onthe power spectrum of the microwave background, which in turn provides the basis for preci-sion constraints on cosmological parameters. This section reviews the basics of the acousticoscillations.


The Oscillator Equation

In their classic 1996 paper, Hu and Sugiyama transformed the basic equations describing theevolution of perturbations into an oscillator equation (see also discussion in Sect. /.2.1). Com-bining the zeroth moment of the photon Boltzmann equation with the baryon Euler equationfor a givenk-mode in the tight–coupling approximation (mean baryon velocity equals meanradiation velocity) gives (in Newtonian gauge upto first order intT )

Θ0 +R

1 +R Θ0 + kc2SΘ0 = Φ +

R1 + 3R Φ− 1

3k2Ψ ≡ F (τ) , (7.228)

whereΘ0 is the zeroth moment of the temperature distribution function (proportional to the

Figure 7.18: Evolution of temperature fluctations before recombination. The tight couplinglimit is compared to a full numerical solution for the monopole and dipole component.

photon density perturbation),R = 3ρB/4ρ is proportional to the scale factora, H = a/a is


the conformal Hubble parameter, and the sound speed is given byc2S = 1/(3 + 3R). (All

overdots are derivatives with respect to conformal time.)Ψ andΦ are the scalar metric pertur-bations in the Newtonian gauge; if we neglect the anisotropic stress, which is generally smallin conventional cosmological scenarios, thenΦ = Ψ. But the details are not very important.The equation represents damped, driven oscillations of the radiation density, and the variousphysical effects are easily identified. The second term on the left side is the damping of os-cillations due to the expansion of the universe. The third term on the left side is the restoringforce due to the pressure, sincec2

S = dP/dρ. On the right side, the first two terms depend onthe time variation of the gravitational potentials, so these two are the source of theIntegratedSachs-Wolfe effect. The final term on the right side is the driving term due to the gravitationalpotential perturbations. As Hu and Sugiyama emphasized, these damped, driven acoustic os-cillations account for all of the structure in the microwave background power spectrum.

This is simply the equation of a forced damped oscillator. The homogeneous equationcan be solved by the WKB method in the limit where the frequency is slowly varying. Thesolutions are oscillatory functions with phaseskrS . The particular solution is then found by

Figure 7.19: Temperature fluctuation spectrum at recombination. Fluctuations on the last scat-tering surface free stream to the observer creating thereby anisotropies. The phase relationbetween the monopole and the dipole determine the Doppler peak structure. The dipole is sig-nificantly smaller than the monopole. At large scales, the Sachs–Wolf effect dominates. Atsmall scales, the oscillations are damped by the Silk damping.


the Green’s method[1 +R(τ)

]1/4

Θ0(τ) =

Θ0(0) cos[krS(τ)] +√

3k

[Θ0(0) +

14RΘ0(0)

]sin[krS(τ)]

+√

3k

∫ τ

0

[1 +R(τ ′)]3/4 sin[krS(τ) − krS(τ ′)] F (τ ′) dτ ′ , (7.229)

whererS(τ) is the sound horizon

rS(τ) =∫ τ

0

cS(τ ′) dτ ′ =

=23

1keq

√6Req

log

[√1 +R+

√R+Req

1 +√Req

]. (7.230)

Req is the value ofR at zeq ' 2.4 × 104ΩMh2 andkeq ' (14Mpc)−1ΩMh2. The dipolesolution can be obtained from the monopole by means ofkΘ1 = −3(Θ0 − Φ). The dipoleoscillatesπ/2 out of phase with the monopole.

The cos(krS) solution is the dominant adiabatic contribution. Therefore, the peaks inthe power spectrum will be located at the scalekm which satisfieskmrS(τ∗) = mπ wherem is an integer> 1. This locates the first Doppler peak roughly at the sound horizon,which is close to, but conceptually distinct from the Jeans scale (causal horizon).The lastsignificant feature in the power spectrum is the diffusion cutoff.

The current data [21] are contrasted with some CDM models in Fig. 7.21. The key featurethat is picked out is the peak atl ' 220, together with harmonics of this scale at higherl.Beyondl ' 1000, the spectrum is clearly damped, in a manner consistent with the expectedeffects of photons diffusing away from baryons (Silk damping), plus smearing of modes withwavelength comparable to the thickness of the last–scattering shell. This last effect arisesbecause recombination is not instantaneous, so the redshift of last scattering shows a scatteraround the mean, with a thickness corresponding to approximatelyσr = 7(Ωmh2)−1/2 Mpc.On scales larger than this, we see essentially an instantaneous imprint of the pattern of poten-tial perturbations and the acoustic baryon/photon oscillations. The significance of the mainacoustic peak scale is that it picks out the (sound) horizon at last scattering. The redshift oflast scattering is almost independent of cosmological parameters atzLS ' 1100. If we assumethat the universe is matter dominated at last scattering, the horizon size is

DLSH = 184/

√Ωmh2 Mpc . (7.231)

The angle this subtends is discussed in Sect. 5.4. The above Figure shows that heavily openuniverses yield a main CMB peak at scales much smaller than the observedl ' 220, and thesecan be ruled out. Indeed, open models were disfavoured for this reason long before any usefuldata existed near the peak, simply because of strict upper limits atl ' 1500. However, oncea non-zero vacuum energy is allowed, the story becomes more complicated, and it turns outthat large degrees of spatial curvature cannot be excluded using the CMB alone.

Evolution of CMB Data

The pace of progress in CMB experiments has maintained an astonishing rate for a decade.Following the 1992 COBE detection of fluctuations, 5 years of effort yielded the unclear


Figure 7.20: CMB spectrum. At largest scales (l < 30), the monopole from the ordinarySachs–Wolf effect dominates. The monopole is clearly important for the overall peak struc-ture. Diffusion damping significantly reduces fluctuations beyond the first peak and cuts theanisotropies beyondl > 1000.

picture of the first panel in Fig. 7.22, in which of order 10 experiments gave only vagueevidence for a peak inl2Cl. By the year 2000, this had been transformed to a clear picture ofa peak atl ' 200, although there was no model–independent evidence for higher harmonics.The present situation is much more satisfactory, with 3 peaks established in a way that doesnot require any knowledge of the CDM model.

The WMAP results (Spergel et al. 2003) measure the power spectrum about as well aspossible (i.e. hitting the limit of cosmic variance from a finite sky) up to the second peak. Atsmaller scales, however, there is still much scope for improvement, and the rate of advanceis unlikely to drop in the future (http://background.uchicago.edu/ whu/cmbex.html lists 14ongoing experiments).

Polarisation in CMB

Thomson scattering can produce polarisation provided the incident radiation is not isotropic,induced e.g. by velocity gradients in the baryon–photon fluid. Before recombination, succes-sive scatterings destroy the build up of any polarisation. One therefore expects a small degreeof polarisation created a recombination, partially correlated with the temperature anisotropies.Conveniently, the polarisation field is decomposed into two scalar fields denoted byE andB(in analogy to electromagnetic fields). The power spectrum of theE–part is expected to beabout ten times smaller than for the temperature fieldΘ, theB-part is only generated by ten-


Figure 7.21: CMB power spectrum for two different world models compared to the first yearWMAP data. In an open world model the first acoustic peak would occur at smaller angularseparations.

sor fields. The WMAP team did not yet release a measurement of theEE power spectrum,but did a measurement of theΘ − E cross power spectrum. This quantifies the expectedcorrelation of the temperature andE field. TheseΘ − E spectra turned out to be unexpectlylarge at lowl–values. This gives strong evidence that the optical depth to the last scatteringsurface is rather important,τes ' 0.17. Based on these observations, it has been suggestedthat reionisation happened rather early, at around redshift 17.

Future: WMAP vs. PLANCK

The WMAP and PLANCK experiments have the following similarities and differences:

• Both map the full sky, from an orbit around the Lagrangian point L2 of the Sun–Earthsystem, to minimise secondary radiation from the Earth. Both are based on the useof off–axis Gregorian telescopes in the 1.5 m class. Both aim at making polarisationmeasurements.

• The American WMAP has been designed for rapid implementation and is based on fullyexplored techniques. Its observational strategy uses differential scheme. Two telescopes


Figure 7.22: Dramatic change took place in CMB power spectrum measurements around theturn of the 21st century. Although some rise from the COBE level was arguably known even by1997, a clear peak aroundl ' 200 only became established in 2000, whereas by 2003 definitivemeasurements of the spectrum atl ≤ 800, limited mainly by cosmic variance, had been madeby the WMAP satellite.


are mounted back to back and feed different radiometers, based on High Mobility Tran-sistors for direct amplification of the radio signal. The angular resolution is not betterthan 10 arcmin.

• The European PLANCK mission is more ambitious and a complex project (too ambitious?) and will be launched in 2007. Several channels of the High frequency Instrument(HFI) will reach the ultimate possible sensitivity per detector of 2.5µK/K at 100 GHz,limited by the photon noise of the CMB (operating at 100, 143, 217, 353 and 857 GHz).Bolometers cooled to 0.1 K will improve the angular resolution to 5 arcmin (only a factor2 better than WMAP, extending thel–range from 800 to 1600, where Silk damping isvery important). The Low Frequency Instrument (LFI) limited to frequencies less than100 GHz (operating at 30, 44 and 70 GHz) will use HEMT amplifiers cooled to 20 K toincrease the sensitivity. The PLANCK scan strategy is to measure the total power. TheLFI use 4K radiative loads for internal reference to achieve this goal. The combinationof these two instruments will allow to map the foregrounds in a very broad frequencyrange: from 30 GHz to 850 GHz.

7.8 Concluding Remarks

The understanding of the formation of structure in the Universe has gone through major stepsin the last ten years. It is by now clear that the large–scale structure evolved from tiny fluctua-tions in the early Universe. Observational access is possible at the recombination era when thephotons decouple from the rest of the cosmic matter. At that time, fluctuations on all scales arestill in the linear regime – they determine the initial conditions for the later non–linear growthof structures into clusters, filaments and galaxies. These structures can obviously only be un-derstood within cold dark matter models, baryons alone would grow into structures differentfrom those observed in the present Universe.

To determine the evolution of cosmic matter density, linear perturbation theory has to beextended with the theory of non–linear Newtonian gravity. All these methods are presentlybased on N–body techniques which are thought to simulate correctly the clustering of colli-sionless dark matter and massive neutrinos. The story is, however, much more complicated,since for the understanding of the evolution of baryonic structures one needs complicated hy-drodynamical processes including shock physics and cooling mechanisms. These are the mostchallenging projects in astrophysics.

Our world could be just the border of some higher dimensional SpaceTime. In such anenvironment, linear perturbations are expected to evolve in a different manner. This opens upa completely new vision on the origin of fluctuations – a subject for intense future research.

Bibliography

[1] Bardeen, J.M.: 1980,Gauge Invariant Cosmological perturbations, Phys. Rev. D 22,1882.

[2] Buote, D.A., Lewis, A.D.: 2003,The dark matter radial profile in the core of the relaxedcluster A2589, astro–ph/0312109.

[3] S. M. Carroll, W. H. Press, & E. L. Turner, ARAA 30, 499, 1992.

[4] Cen, : 2002, http: //www.princeton.edu/

[5] Diemand, J., Moore, Ben, Stadel, J.: 2004,Convergence and scatter of cluster densityprofiles, astro–ph/0402267

[6] Durrer, R.: 1994, Fund. Cosmic Phys. 15, 209.

[7] Durrer, R.: 2001,Theory of CMB Anisotropies, astro-ph/0109522.

[8] Durrer, R.: 2004,Cosmological Perturbation Theory, astro-ph/0402129

[9] Evrard, A.E. et al.: 2002, ApJ 573, 7 (Hubble simulation)

[10] Hu, W., Eisenstein, D. J., Tegmark, M.: 1998, PRL, 80 5255.

[11] Hu, W., Eisenstein, D. J.: 1999, Phys. Reports 215, 203.

[12] Hu, W.: 2004,Covariant Linear Perturbation Formalism, astro–ph/0402060

[13] Hu, W., Seljak, M. et al.: 1998,A Complete Treatment of CMB Anisotropies in FRWUniverse, Phys. Rev. D 57, 3290.

[14] Jenkins, A., Frenk, C.S., Pearce, F.R., Thomas, P.A., Colberg, J.M., White, S.D.M.,Couchman, H.M.P., Peacock, J.A., Efstathiou, G., Nelson, A.H.: 1998, ApJ 499, 20

[15] K. Scholberg,et al., hep-ph/9905016 (Super-Kamiokande collaboration).

[16] Kodama, H. & Sasaki, M.: 1984,Cosmological Perturbation Theory, Prog. Theor. Phys.Suppl. 78, 1.

[17] Kodama, H. & Sasaki, M.: 1986, Int. J. Mod. Phys. A1, 265.

[18] Miyoshi, K., Kihara, T.: 1975, Publ. Astron. Soc. Japan 27, 333

[19] Seljak, U. & Zaldarriaga, M.: 1996, ApJ 469, 437.

[20] Steinhardt, P.J., Turok, Neil: 2002,The Cyclic Universe: An Informal Introduction,astro–ph/0204479.

[21] Spergel, D.N. et al.: 2003,First Year Wilkinson Microwave Anisotropy Probe (WMAP)Observations: Determination of Cosmological Parameters, astro-ph/0302209

[22] Suto, : 2003,Simulations of Large–Scale Structure in the New Millenium, astro–ph/0311575

[23] M. Zaldarriaga, U. Seljak and E. Bertschinger, ApJ, 494, 491, 1998.

[24] W. Hu, U. Seljak, M. White & M. Zaldarriaga, Phys. Rev. D 57, 3290, 1998.

278 Bibliography

7.9 Exercises

1. Bardeen Potentials

Prove the gauge-invariance of the Bardeen potentialsΦ andΨ.

2. Density Perturbations

Derive from the conservation of the energy–monetum tensor of a perfect fluid the equations ofmotion for the density and velocity perturbations in the Newtonian gauge. Combine the twoequations in order to derive a second order equation for the density perturbation. Show thatthe Newtonian limit leads to the equation

δ + 2Hδ = δ

(4πGρ− c2

Sk2

a2

). (7.232)

This equation follows directly from the perturbations of the Newtonian hydrodynamical equa-tions.Derive from this equation the Jeans lengthλJ and the corresponding Jeans massMJ . Derivethe Jeans mass for the radiation–dominated era and its value atzeq. Discuss the behaviour ofthe Jeans mass after recombination.

3. Gunn–Peterson Trough in Quasar Spectra

Derive the optical depth for the Gunn–Peterson trough as a function of redshift and the abun-dancexHI of neutral hydrogen. The width of the Lyα line can be considered to have a profilethat is given by a delta function

σλ dλ =πe2

mec

fλ20

cδ(λ− λ0) dλ , (7.233)

whereλ0 = 121.6 nm is the wavelength andf = 0.416 the oscillator strength of the line. Theoptical depth

dτ(λ) = nHI(z)σ[λ/(1 + z)] c dt (7.234)

takes into account the cosmological redshift of the absorber. The integration provides us thenthe total optical depth at given wavelength

τ(λ) =∫ zem

0

nHI(z)σ[λ/(1 + z)] cdt

dzdz

=πe2

mec

fλ20

c

∫ zem

0

cnHI(z)H(z)(1 + z)

δ[λ/(1 + z)− λ0] dz (7.235)

which depends on the cosmological model. The integral can be written as

τ =πe2fλ0 nHI(z)

mecH(z)= 6.45× 105 xHI

ΩBh

0.03

√0.3Ωm

(1 + z

10

)3/2

. (7.236)

The last expression is valid in a flat Universe withΩΛ = 0. Observations tell us thatτ < 0.1for Quasars with redshift 2.6, butτ À 1 for z > 5.6, i.e. below redshift 6 the Universe hasbeen reionized.

7.9 Exercises 279

4. Temperature Fluctuations on Superhorizon Scales

The dominant contribution to temperature fluctuations on superhorizon scales is given by(SW)

Θ(τ0, ~x0, ~n) =13Ψ(τ∗, x∗) (7.237)

The Fourier transform gives

Θ(τ0,~k, ~n) =13Ψ(τ∗, k) exp(i~k · ~n[τ0 − τ∗]) . (7.238)

Derive from this an expression for the temperature correlation function on the sky (using theParseval theorem for Fourier transforms)

< Θ(τ0, ~x0, ~n)Θ(τ0, ~x0, ~n′) >

=1

9(2π)3

∫d3k < Ψ2 >

∞∑

ll′=0

(2l + 1)(2l′ + 1)il−l′

jl(k(τ0 − τ∗))jl′(k(τ0 − τ∗)) Pl(~k · ~n)Pl′ (~k · ~n′) . (7.239)

Using the addition theorem for spherical harmonics

Pl(~k · ~n) =4π

2l + 1

∑m

Y ∗lm(~k)Ylm(~n) (7.240)

we get with the notationµ = ~n · ~n′,

< Θ(τ0, ~x0, ~n)Θ(τ0, ~x0, ~n′) >=

29π

∑

l

(2l + 1)4π

Pl(µ)∫

dk

k< Ψ2 > k3j2

l [k(τ0−τ∗)] .

(7.241)

Comparing this with the general expression for the temperature correlation function, we obtainfor adiabatic perturbations on scales2 < l ¿ χ(τ0 − τ∗)/τ∗ ' 100

CSWl ' 2

9π

∫ ∞

0

dk

k< Ψ2 > k3 j2

l (k(τ0 − τ∗)) . (7.242)

For a pure power–law spectrum

< Ψ2 > k3 = A2 kn−1 τn−10 (7.243)

this integral can be perforemd analytically. In particular, for a scale–invariant (Harrison–Zeldovich) spectrum,n = 1, it leads to

l(l + 1) CSWl = const '< [∆T (θl)]2 > (7.244)

with θl = π/l. This behaviour of the spectrum was first observed by the DMR experiment onboard of the COBE satellite.

A Perturbation Calculations

A.1 Perturbations of the Einstein Tensor in NewtonianGauge

A.1.1 3+1 Slice

We perform the perturbations in thelongitudinal gauge, where perturbations of the metricare given by

δgtt = −2a2ψ (A.1)

δgti = 0 (A.2)

δgij = 2a2φγij = −2g(0)ij ψ . (A.3)

We have not to worry about gauges, since according to above the result will be gauge–invariant. The above 3+1 slice of the metric implies then the following decomposition ofthe Einstein tensor (see Appendix to Sect. 5)

Gtt = −1

2R(g) +

12

[Ki

jKji −Ki

iKjj

](A.4)

Gti =

1α

[∇jK

ji −∇iK

jj

](A.5)

Rij = Ri

j(g) + KssKi

j − 2KisK

sj −

1α

[∇i∇jα + gisKjs

](A.6)

R = R(g) + KijK

ji + Ki

iKjj −

2α

[∆α + Kss ] . (A.7)

We have now to calculate the perturbation of each of these terms.

A.1.2 Perturbation of Extrinsic Curvature

A.1.3 Perturbation of the Spatial Ricci Tensor

A.1.4 Perturbed Einstein Tensor

Collecting all terms, we find

Gtt = −1

2R(g) +

12

[Ki

jKji −Ki

iKjj

](A.8)

and for the perturbation

δGtt = −1

2δR +

12

δ(KijK

ji )− 1

2δ(Ki

iKjj )

= −12

4a2

(3k + ∆)ψ +12

... (A.9)

282 A Perturbation Calculations

The next equation is the energy flow

Gti =

1α

[∇jK

ji −∇iK

jj

](A.10)

with the perturbation

δGti = ...

= (A.11)

The spatial components ...Using the transformation

φ → Φ , ψ → Ψ (A.12)

with the Bardeen potentials, we find the gauge–invariant expressions for the perturbed Einsteintensor

δGtt =

2a2

[3H(HΦ + Ψ

]− (3k + ∆)Ψ

](A.13)

δGti = − 2

a2∇i(HΦ + Ψ) (A.14)

δGij = −2kΨ

a2δij − ... (A.15)

The general covariant form of linearized perturbed Einstein’s equations follows from thedefinitions above. The algebra is straight-forward (although lengthy) and leads to the follow-ing set of equations

Scalar perturbations

(k2 − 3K)[HL + 1

3HT + ηk−2(kB − HT )]

= 4πGa2 [δρ + 3η(ρ + p)(v −B)/k]

ηA− HL − 13HT − K

k2 (kB − HT ) = 4πGa2(ρ + p)(v −B)/k(2 a

a − 2η2 + η ∂∂τ − 1

3k2)A− (

∂∂τ + η

)(HL + k

3B) = 4πGa2(δp + 13δρ)

k2(A + HL + 13HT ) +

(∂∂τ + 2η

) (kB − HT

)= 8πGa2pΠ.

(A.16)

These are density (Poisson), momentum, pressure and anisotropic stress equations, respec-tively. The superscripts(0) have been dropped.

Corresponding energy-momentum tensor conservation equations are continuity and Eu-ler’s equations, which in the general covariant form are

(∂∂τ + 3η

)δρ + 3ηδp = −(ρ + p)(kv + 3HL)(

∂∂τ + 4η

)[ρ + p)(v −B)/k] = δp− 2

3 (1− 3 Kk2 )pΠ + (ρ + p)A.

(A.17)

Energy-momentum tensor conservation equations are not independent of Einstein’s equations,since they follow from Bianchi identities. They are nevertheless useful, since they involveonly first derivatives instead of second as in the case of Einstein’s equations. In total wehave 4 equations for 4 metric perturbations, which are sourced by 4 components of energy-momentum tensor. As we will see below the number of equations can be further reduced bythe gauge freedom.

A.1 Perturbations of the Einstein Tensor in Newtonian Gauge 283

A.1.5 Curvature Perturbations

Curvature perturbationζ defined in comoving gauge (section 7.1.5) is a useful quantity be-cause, as shown below, it is constant outside the horizon in the absence of entropy perturba-tions. It also relates simply to the Newtonian perturbationΦ, which allows one to useζ toevaluateΦ outside the horizon even if the underlying equation of state changes. To relate thetwo we first use the gauge transformation between the Newtonian and comoving density,

δρcom = δρN − ˙ρvN/k = δρN + 3η(ρ + p)vN/k, (A.18)

where the quantities on the left are in comoving gauge and those on the right in Newtoniangauge. Poisson’s equation in Newtonian gauge gives

−(k2 − 3K)Φ = 4πGa2δρcom. (A.19)

Inserting this into Poisson’s equation in comoving gauge leads to

ζ + ηvcom/k = −Φ. (A.20)

But sinceHT = 0 in Newtonian gaugevcom = vN , which follows from the gauge transfor-mation in equations (7.54) and (7.72). From the velocity equation (??) ignoring curvatureKone finds the curvature perturbation is thus related toΦ as

ζ = −Φ− 23

η−1Φ + Ψ1 + w

, (A.21)

wherew = p/ρ.Curvature perturbationζ obeys the evolution equation (7.73), which in the absence of

curvature gives

ζ = ηξ = −ηδp(1 + αΠ)(ρ + p)

, (A.22)

where the latter relation follows from Euler’s equation (7.74) in comoving gauge. We haveintroduced the anisotropic stress fractionαΠ ≡ −2pΠ/3δp. To show thatζ is constant outsidethe horizon one must showζ ¿ ηζ. In the absence of entropy perturbations we can relatepressure perturbations to density perturbations through the adiabatic speed of sound, which isindependent of position. Then

ζ = −ηδpcom(1 + αΠ)(ρ + p)

= −c2sηδρcom(1 + αΠ)

(ρ + p)

=c2sk

2ηΦ(1 + αΠ)4πGa2(ρ + p)

=23

(csk

η

)2ηΦ(1 + αΠ)

1 + w=

(csk

η

)2 2ηζ(1 + αΠ)(1 + αΦ)5 + 3w

,(A.23)

whereαΦ ≡ Φ/(ηΨ). This showsζ ¿ ηζ in the limit k/η → 0, as long asc2s, αΠ and

αΦ are all of order unity or smaller.ζ thus remains constant whenΦ ∼ ηΦ, such as duringmatter-radiation transition and at late times during matter-dark energy (or curvature) transi-tion. Similarly, it is constant even in the presence of a significant anisotropic stress outsidehorizon, as in the case of massless neutrinos during radiation domination.

The above shows that sinceζ is constant we can predict evolution ofΦ without solvingits equations of motion. As we have shown in previous subsection outside the horizonΦ does

284 A Perturbation Calculations

not change in adiabatic case in matter and radiation epochs. Moreover, in the absence ofanisotropic stress we haveΦ = Ψ. Then

Φ = −3 + 3w

5 + 3wζ, (A.24)

so thatΦ = Ψ = −2ζ/3 in radiation andΦ = Ψ = −3ζ/5 in matter era (bottom right panelin figure??). Thus, changing fromw = 1/3 to w = 0 from radiation to matter dominationproduces a 10% decrease inΦ, as already shown in previous subsection. Similarly, if at latetimes the universe is accelerating one hasw < 0 today. One can easily show from equation(A.21) that in this case one must haveΦ < 0, so the potential decays in time,Φ → 0. In adark energy dominated universe the structure ceases to grow.

In the presence of massless neutrinos one must include their anisotropic stress, which doesnot vanish even on large scales. During radiation era neutrinos are dynamically importantcontributingRν ∼ 0.4 to the total energy density. We will show later that in this case theanisotropic stress in equation??on large scales and in radiation era gives

Φi = Ψi(1 +25Rν) ∼ 1 + 0.166(Nν/3)Ψi, (A.25)

whereNν is the number of massless neutrinos. Sinceζ is conserved even in this case wecan relateΦ andΨ in radiation and matter epochs. In matter era neutrinos are dynamicallyunimportant and the solution isΦm = Ψm = −3ζ/5 by equation (A.24). Then in matter era

Ψm =35

(32

+25Rν

)Ψi

Φm =35

32 + 2

5Rν

1 + 25Rν

Φi. (A.26)

Note that for 3 neutrino familiesΨ remains practically unchanged from radiation to matter,while Φ changes by 0.166 (upper left panel in figure??).

We can now qualitatively explain the evolution of the potentials for any mode or time.Figure?? shows the evolution for several representative examples. The simplest is upper leftcase, which corresponds to a mode outside horizon throughout its evolution, in a model withno neutrinos and withΩm = 1. In this caseΦ = Ψ and equation (A.24) can be used to obtaintheir values in the two epochs, assuming initiallyζ = 1. Upper right shows the evolutionof the corresponding mode for the model with 3 neutrino families and with cosmologicalconstant dominating at late times. Equation (A.25) can be used to obtain the relation betweenthe potentials in this case. Lower left panel of figure?? shows a mode which enters horizonshortly before matter radiation equality, so all of the potentials are suppressed (in particular,ζis not conserved on small scales and solution in equation 7.104 applies to it as well). Once inmatter domination they remain constant until cosmological constant causesΦ andΨ to decayagain, whileζ remains unchanged. Finally, for a mode that enters the horizon very early inradiation era the potentials oscillate and decay as in equation (7.104). This is shown in thebottom right panel of figure??.

A.1 Perturbations of the Einstein Tensor in Newtonian Gauge 285

Vector perturbations

Since density and pressure are scalar quantities there are only two Einstein’s equations forvectors, momentum and anisotropic stress,

(1− 2K

k2

)(kB − HT ) = 16πGa2(ρ + p)(v −B)/k(

∂∂τ + 2η

)(kB − HT ) = −8πGa2pΠ.

(A.27)

Energy-momentum conservation consists of Euler’s equation only,

2(

∂

∂τ+ 4η

)[(ρ + p)(v −B)/k] =

(2K

k2− 1

)pΠ (A.28)

and we suppressed±1 superscript on all the variables. Here again we have 2 Einstein’s equa-tions for 2 metric variables, but fixing the gauge freedom reduces this to a single equation.

Tensor perturbations

There is only one equation in this case, corresponding to the tensor part of the anisotropicspatial component in Einstein’s equations,

HT + 2ηHT + (k2 + 2K)HT = 4πGa2pΠ, (A.29)

where again superscript±2 was suppressed. There is no gauge freedom in the tensor case, aswe discuss next. Equation above is a wave equation and describes gravity wave propagationand sourcing in an expanding universe.

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