182 Cosmic Microwave Background
high퓁must be included. Anisotropies on the largest angular scales corresponding to
quadrupoles are manifestations of truly primordial gravitational structures.
For the analysis of temperature perturbations over large angles, the Legendre poly-
nomial expansion in Equation (8.19) will not do; one has to use tensor spherical
harmonics. The temperature푇(n) in the directionncan be expanded in spherical
harmonics
푇(n)=푇 0 +∑∞
퓁= 1∑퓁
푚=−퓁푎T퓁푚푌퓁푚(n), (8.20)where푎T퓁푚are the powers ortemperature multipole components. These can be deter-
mined from the observed temperature푇(n)using the orthonormality properties of the
spherical harmonics,
푎T퓁푚=1
푇 0 ∫
dn푇(n)푌퓁∗푚(n). (8.21)Expressing the autocorrelation function퐶as a power spectrum in terms of the
multipole components, the average of all statistical realizations of the distribution
becomes
⟨푎퓁T푚∗푎T퓁′푚′⟩=퐶퓁T훿퓁퓁′훿푚푚′=퐶T퓁. (8.22)The last step follows from statistical isotropy which requires statistical independence
of eachlmmode, as manifested by the presence of the two Kronecker deltas and the
absence of anm-dependence.
Sources of Anisotropies. Let us now follow the fate of the scalar density perturba-
tions generated during inflation, which subsequently froze and disappeared outside
the (relatively slowly expanding) horizon. For wavelengths exceeding the horizon,
the distinction between curvature (adiabatic) and isocurvature (isothermal) pertur-
bations is important. Curvature perturbations are true energy density fluctuations or
fluctuations in the local value of the spatial curvature. These can be produced, for
example, by the quantum fluctuations that are blown up by inflation. By the equiva-
lence principle all components of the energy density (matter, radiation) are affected.
Isocurvature fluctuations, on the other hand, are not true fluctuations in the energy
density but are characterized by fluctuations in the form of the local equation of state,
for example, spatial fluctuations in the number of some particle species. These can be
produced, for example, by cosmic strings and other cosmic defects that perturb the
local equation of state. As long as an isocurvature mode is superhorizon, physical
processes cannot re-distribute the energy density.
When the Universe arrived at the radiation- and matter-dominated epochs, the
Hubble expansion of the horizon reveals these perturbations. Once inside the hori-
zon, the crests and troughs can again communicate, setting up a pattern of standing
acoustic waves in the baryon–photon fluid. The tight coupling between radiation and
matter density causes the adiabatic perturbations to oscillate in phase. After decou-
pling, the perturbations in the radiation field no longer oscillate, and the remaining
standing acoustic waves are visible today as perturbations to the mean CMB temper-
ature at degree angular scales.