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To predict the shape of the temperature fluctuations ∆T /T observed in the CMB, one has to apply the Boltzmann equation describing the evolution of the photon distribution function in a photon-baryon gas with Compton scattering. The linear density perturbations will now be described using the gauge invariant potentials Ψ and Φ described in (Bardeen 1980). These are related to the total density perturbationsδ and the anisotropic stress Π by

k2Φ = 4πGρ0 _a a0 2 δk, (1.64) Φ + Ψ = −8πGp k2 _a a0 2 Π. (1.65)

Here (and in the rest of the section), δk, Φ, Ψ and Π denote the Fourier trans-

forms of the quantitiesδ, Φ, Ψ and Π respectively. The scale factor is as beforea, the pressure isp. In a completely matter dominated universe where the pressure pis unimportant, one sees that the two potentials are related by Ψ =−Φ.

The temperature anisotropy at position x in space at conformal time η =

R

dt(a0/a) in the direction n will now be denoted ∆(x,n, η), and its Fourier

transform ∆k(µ, η) where k = |k| and µ = k·n/k. The Legendre expansion is

given by,

∆k(µ, η) =

X

`

(−i)`∆k`(η)P`(µ), (1.66)

where P`(µ) is the Legendre Polynomial of order `. Using these definitions, the

Boltzmann equation for the temperature anisotropies becomes ˙

∆k(µ, η)+ikµ(∆k(µ, η)+Ψ) = −˙Φ+ ˙τ(η)[∆k0(η)−∆k(µ, η)−

1

10∆k2(η)P2(µ)−iµVk(η)]. (1.67) Here a dot represents the derivative with respect to conformal time, Vk is the

Fourier transform of the baryon velocity and ˙τ = xeneσTa/a0 is the differential

optical depth to Thompson scattering. The ionization fraction is given byxe, the

electron density by ne and the Thompson cross section by σT.

The evolution of the linear density perturbations in baryons δb and photons

δγ is given again by the continuity and Euler equations

( ˙δb)k = −k(Vk−∆k1) + 3 4( ˙δγ)k (1.68) ˙ Vk = − ˙ a aVk+kΨ + ˙ τ(∆k1−Vk) R . (1.69)

Here R = 3ρb/(4ργ). The universe is now supposed to consist of 3 species, the

photons, the baryons and the dark matter. The density fluctuations in the cou- pled baryon-photon fluid is described by the Boltzmann equation (1.67) and the

1.4 The Recombination Era and the CMB 30

Euler equations (1.68, 1.69). The dark matter is not coupled to any of these species and only shows up through its influence on the gravitational potentials Ψ and Φ.

Before recombination the photons were continuously scattered on the elec- trons. In such a case the photon distribution is isotropic in the electron rest frame, so the dipole ∆k1is only caused by the movement of the baryons, ∆k1 =Vk.

Inserting this into equation (1.68) one gets ( ˙δb)k = 3/4( ˙δγ)k, which is the equa-

tion for for adiabatic evolution of the density fluctuations. This also gives that ∆k` = 0 for `≥2. This is called the tight coupling limit.

Close to the tight coupling limit one can use thethe tight coupling approxima- tion. In this approximation equations (1.67), (1.68) and (1.69) can be expanded to first oder in the Compton scattering time ˙τ−1_{. This gives the resulting differ-}

ential equation ¨ ∆k0+ ˙ a a R 1 +R∆˙k0 +k 2_c2 s∆k0 =F(η), (1.70)

whereF(η) is a term containing the gravitational potentials and cs is the sound

speed cs= 1 3 1 1 +R. (1.71)

This is the equation of a forced damped oscillator. The gravitational attraction of the over density regions pull the matter into this region, but the photon pressure resists the contraction, giving acoustic oscillations in the photon-baryon fluid.

The solution to equation (1.70), can be written as ˆ

∆k0(η) =acoskrs(η) +bsinkrs(η) +c(F(η)), (1.72)

where the hat denotes that this is the solution in the thight coupling approxi- mation. The dipole goes as k∆ˆk1 =−3( ˙∆k0+ ˙Φ) which means that it oscillates

with an angleπ/2 out of phase with the monopole. Herers is the sound horizon,

given as

rs(η) =

Z η

0 csdη

0_{. (1.73)}

Clearly, the amplitude of the dipole will be damped by a factor ˙rs ∝(1 +R)−1/2

with respect to the amplitude of the monopole.

In equation (1.72), the cos-term comes from adiabatic perturbations and the sin-term from isocurvature perturbations. Adiabatic fluctuations are perturba- tions for which the entropy is constant causing the relation between the baryon and photon perturbation toδb = 3/4δγ. These kind of fluctuations are predicted

by inflation. The isocurvature perturbations are perturbations in the entropy. For these perturbations which are expected by cosmic string models, δb = −δγ

1.4 The Recombination Era and the CMB 31

so that the total energy density is constant. Recent observations indicate that adiabatic perturbations are dominant so that b ≈ 0 in the equation. For this reason I will mainly discuss adiabatic perturbations in this thesis.

One sees that differentkmodes oscillate with a different frequency. At recom- binationη=ηrec, the monopole for some scalesk is at its maximum or minimum

of the oscillations and will show up as peaks in the CMB temperature anisotropy power spectrum. For some other scales the monopole will be zero, and the CMB temperature power spectrum has a trough, which is not zero due to the dipole (which has its maximum/minimum here due to theπ/2 phase shift). The peaks and troughs in the temperature power spectrum resulting from these oscillations will be described in more detail in section (1.4.2).

When recombination started, the mean free path of the photons became longer as there were less and less electrons present on which they could scatter. The photons were able to diffuse through the baryons and in this way the tempera- ture anisotropies at small scales were getting smaller. This is called Silk damping (Silk 1968). Another result of the increasing diffusion length, is that the CMB photons that one can observe did not last scatter at the same time. The last scat- tering surface has a finite width and the photons come from different depths in the surface where the oscillations in the monopole ∆k0 had different phases. This

has the similar effect of smearing out the temperature anisotropies on small scales. When scattering became less frequent the tight coupling limit breaks down and the equations (1.67), (1.68) and (1.69) have to be expanded to second oder in the Compton scattering time ˙τ−1_{. The solution to this set of equations can be}

written as

(∆k0+ Ψ) = ( ˆ∆k0+ Ψ)e−(k/kD(η))

2

, (1.74)

wherekD(η) is dependent on the diffusion length of the photons at time η. This

exponential damping of small scales is caused by the above mentioned effects of photon diffusion and finite width of the last scattering surface.

To find the CMB temperature anisotropies today, the multipoles `<sub>≥</sub>2 of ∆k`

have to be found. One can get these by solving the boltzmann equation (1.67), ignoring the quadrupole which disappears in the tight coupling limit and make a multipole expansion of the solution. The result is

∆k`(η0) ≈ (∆k0+ Ψ)(ηrec)(2`+ 1)j`(k∆ηrec) (1.75)

+∆k1(ηrec)[`j`−1(k∆ηrec)−(`+ 1)j`+1(k∆ηrec)]

+(2`+ 1)

Z η0

ηrec

( ˙Ψ− ˙Φ)j`(k∆η)dη,

1.4 The Recombination Era and the CMB 32

damping and the damping due to finite with of the last scattering surface are included by

(∆k0+ Ψ)(ηrec) = ( ˆ∆k0+ Ψ)(ηrec)D(k) ∆k1(ηrec) = ˆ∆k1(ηrec)D(k), (1.76)

where D(k) = Z η0 0 τ˙(η)e −τ(η,η0) e−(k/kD(η))2_{dη. (1.77)}

One can see from equation (1.75) that the anisotropies in the CMB have three main contributions. The first term in the equation is called the ordinary Sachs-Wolf effect(Sachs and Wolfe 1967). This is the sum of the monopole tem- perature differences on the last scattering surface and the gravitational potential Ψ. The gravitational potential accounts for the redshift of the photons as they climbed out of the potential wells of the density inhomogenieties at the last scat- tering surface. The second term in equation (1.75) is due to a Doppler shift. Because of the acoustic oscillations before and at recombination, the baryons were moving. When the photons were last scattered on these moving electrons, they experienced a Doppler shift. Finally, the last main contributor to the CMB anisotropy is theintegrated Sachs-Wolf effect. This effect aroseafterthe photons last scattered. This is due to a possible time change in the gravitational potential after the recombination epoch. This term becomes important if matter-radiation equality occurred close to recombination, or if the universe has got Λ-dominated after recombination.

Clearly, these three terms are dependent on cosmological parameters. If the amount of dark matter and baryons are changed, the size of the density pertur- bations and the gravitational potentials are also changed. This clearly results in changes of the CMB anisotropies. The speed of expansion of the universe mea- sured by the Hubble constant, is of course also important in these calculations. As are the initial perturbations power spectrum produced by (possibly) inflation. In the next section, I will define the CMB temperature power spectrum of fluc- tuations and outline how it is dependent on some of these parameters. Equation (1.75) shows how important the CMB power spectrum can be for measuring the cosmological parameters.