Fluctuations in the cosmic microwave background. II.Clat large and smalll

Abstract

General asymptotic formulas are given for the coefficient ${C}_{l}$ 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 $l\ensuremath{\gg}1$ and for ${ld/d}_{A}\ensuremath{\ll}1$ (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 ${C}_{l}$ receives its main contribution from wave numbers of order ${l/d}_{A}$ is found to be less accurate for the contribution of the Doppler effect than for the Sachs-Wolfe effect and intrinsic temperature fluctuations. For ${ld/d}_{A}\ensuremath{\ll}1$ and $l>~2,$ the growth of ${C}_{l}$ with l 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.