4.3 CMB Physics
4.3.1 The multipole expansion
The observed temperature anisotropy today in direction ˆn is conveniently expanded in terms of spherical harmonics: power spectrum is then defined as the average of the observed variances
Cˆ≡ 1 2+ 1
m
a∗mam. (4.34)
However, we do cannot predict any particularam, but rather the distribution from which they were drawn. The variance ofam is independent ofm. Theoretically, we can predict only this variance, but not its actual realization. The expected CMB spectrum is the expectation value of the theoretical spectra,
C≡<|am|2>= 1 2+ 1
m
< a∗mam> . (4.35) These equalities hold because the primordial fluctuations, and thus the CMB anisotropy, is assumed to be result of a statistically isotropic random process. The differentam are independent random variables, and thus we have that
< a∗mam>=Cδδmm. (4.36) The so called cosmic variance arises from the fact that we can measure only a finite number of realizations of these random variables. For the quadropole, we have 2×2+1 = 5 of them, which leaves significant uncertainty. One sees that the uncertainty due to cosmic variance is
2/(2+ 1) for each multipole. So when considering smaller scales, sayequal to few dozens, the cosmic variance is smaller than the uncertainty due to measurement errors.
We want to have theCspectrum determined by the spectrum of primordial curvature perturbationR,
PR(k)≡ k3
2π2 <R∗kRk>=AknS−1, (4.37) whereAis the amplitude andnS the (scalar) spectral index of the primordial fluctuation spectrum. We define the transfer functionTΘ(k, μ) so that in Fourier space Θ(τ0,k,n) = TΘ(k,kˆ·nˆ)Rk. Next we expand the transfer function in Legendre series and Fourier transform the temperature anisotropy to position space. Then we have that
Θ(τ0,x,n) = 1 By using equations (4.33)-(4.37), and some useful formulas for the exponent function and spherical harmonics, we arrive at the following expression for the angular power spectrum:
C= 4π dk
kTΘ2(k, )PR(k). (4.39)
It remains to find the transfer function.
We begin from the Boltzmann equation for photons [10], with source term from Thom-son scattering included:
Θ +˙ ikμΘ−κ˙TΘ = ˙Ψ−iμkΦ−κ˙T(1
4δγ+iμkvb−1
2P2(μ)Π). (4.40) The scattering term depends on the number density of free electronsne,
˙
κT ≡ −aneσT, (4.41)
whereσT is the Thomson cross section. Here again Θ is a function of conformal time, k andμ, the cosine of the angle betweenkand ˆn. The LHS of Eq.(4.40) can be written as
e−iμkτe+κT(τ) d
dτ[eiμkτe−κT(τ)Θ]. (4.42) Thus the derivative of the square brackets above is the RHS of equation (4.40) multiplied by the two exponents inside those brackets. So one can integrate that product to get the term inside the square brackets today. When that is again multiplied by the inverse of those exponents, evaluated today, the result is the present temperature anisotropy:
Θ(μ, τ0) = To get rid of theμ-prefactors, we integrate this by parts (twice, sinceP2(μ) involvesμ2).
The boundary terms can be dropped, because at τ = 0 they vanish, and atτ =τ0 they would contribute only to the monopole and the dipole. The result is conveniently expressed in terms of the visibility function,g≡ −κ˙TeκT(τ)−κT(τ0): in terms of jl and Pl. Remembering the definition of the transfer function, we are then ready to pick the coefficients in its Legendre expansion. They are:
RkTΘ(k, l) =
The prefactor Rappears in the left hand side since we want to relate the perturbations to their primordial values: transfer function gives the time evolution of the perturbations for each k-mode, but their amplitude is determined by the primordial spectrum5. The
5RHS of Eq.(4.45) can be considered as the right expression for the transfer function when the initial curvature perturbation is set to unity. Then the normalization and possible tilt in the primordial spectrum are taken into account in Eq.(4.39).
anisotropy sources in (4.45), are multiplied by a Bessel function. That is the geometrical part of the transfer function, which governs how the anisotropies contribute to different multipoles in the spherical expansion. In addition, the perturbations are weighted by the visibility factors,g,gandeκT(τ)−κT(τ0), so that the anisotropy is gathered in the integral from the relevant parts of the universe. Let us briefly discuss each of the source terms.
• 14δγ is the temperature anisotropy present at last scattering. Because in thermal equilibriumργ is proportional toT4, there is the factor 1/4 in front of the fractional density perturbation. That term is the primary cause of the acoustic peaks at the smaller angular scales.
• The Sachs-Wolfe effect stems from the Φ. The photons coming from overdense re-gions suffer a loss of energy due to their climbing out of the gravitational well induced by the overdensity. A proper temperature anisotropy taking into account the grav-itational shift is thenδγ/4 + Φ, and since it is weighted by theg which is sharply peakead at last scattering, the main contribution comes directly from there.
• Also the combined effect of termsgvb+gvb, which corresponds to Doppler shift of photons due to movement of baryons, is also important for the shape of the angular power spectrum6.
• Ψ + ˙˙ Φ is responsible for the integrated Sachs-Wolfe effect (ISW). The red- and blueshifts of photons travelling into and out from static gravitational wells along their path to us cancels out. This canceling is not exact if the gravitational poten-tials evolve. During matter domination, the gravitational potenpoten-tials are constant.
Therefore there are two distinct ISW contributions: the early one when radiation is not yet negligible and the late one when dark energy takes over matter. The early ISW comes from the scales corresponding roughly to the last scattering surface, and the late one comes mainly from large scales.
• The terms involving Π and its derivatives have smaller but still non-negligible ef-fects on the CMB spectrum. The higher multipoles of the photon distribution are suppressed by the tight coupling to baryons, but begin to evolve at the decoupling.
In solving the evolution of Π, one has to take into account that it is coupled to higher multipoles. But since they are suppressed, it is enough to consider multipoles up to about = 8 to get accurate results. In addition, Π is coupled to the polar-ization, which it in turn sources. Thus polarization is smaller than the temperature anisotropy, and cannot be detected with equal accuracy. Among many other impor-tant aspects of CMB, here we omitted a more detailed study of polarization. Exotic dark energy might couple non-minimally to photons[128], but usually we have only indirect effect from dark energy perturbations to the polarization. Schematically the coupling goes like δde ↔ Ψ ↔ vγ ↔ Πγ ↔ ΠP. Nevertheless, at least in future,
6Adiabatic initial conditions require that theδγ is negative (of the opposite sign thanR) at super-horizon scales. At decoupling, the monopole contribution to effective temperature,δγ/4 + Ψ, is still smaller than zero at large scales. The first acoustic peak however corresponds to compression of the photon-baryon fluid. Would the anisotropy atτLSbe due only to the photon density and the gravitational potential, the spectrum would have a minimum before the rise to the first acoustic peak. The reason that there is no such minimum is the Doppler (and to a smaller amount, the early ISW) effect.
precise constraints on dark energy should also take advantage of the information inferrable from the polarization spectra.