Fluctuations in the Cosmic Microwave Background II: $C_\ell$ at Large and Small $\ell$

Abstract

General asymptotic formulas are given for the coefficient $C_\ell$ of the term of multipole number $\ell$ in the temperature correlation function of the cosmic microwave background, in terms of scalar and dipole form factors introduced in a companion paper. The formulas apply in two overlapping limits: for $\ell\gg 1$ and for $\ell d/d_A\ll 1$ (where $d_A$ is the angular diameter distance of the surface of last scattering, and $d$ is a length, of the order of the acoustic horizon at the time of last scattering, that characterizes acoustic oscillations before this time.) The frequently used approximation that $C_\ell$ receives its main contribution from wave numbers of order $\ell/d_A$ is found to be less accurate for the contribution of the Doppler effect than for the Sachs--Wolfe effect and intrinsic temperature fluctuations. For $\ell d/d_A\ll 1$ and $\ell\geq 2$, the growth of $C_\ell$ with $\ell$ is shown to be affected by acoustic oscillation wave numbers of all scales. The asymptotic formulas are applied to a model of acoustic oscillations before the time of last scattering, with results in reasonable agreement with more elaborate computer calculations.