Fluctuations in the Cosmic Microwave Background II: Cℓ at Large and Small ℓ

General asymptotic formulas are given for the coefficient Cl of the term of multipole number l 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 l ≫ 1 and for ld/dA ≪ 1 (where dA 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 Cl receives its main contribution from wave numbers of order l/dA is found to be less accurate for the contribution of the Doppler effect than for the Sachs–Wolfe effect and intrinsic temperature fluctuations. For ld/dA ≪ 1, the growth of Cl with l is shown to be affected by acoustic oscillation wave numbers of all scales. The asymptotic formulas are applied to a simple model of acoustic oscillations before the time of last scattering. Electronic address: [email protected]