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 $\ell \ge 2$, 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.

16 pages, one inessential figure. Improved treatment of damping in companion paper used here in calculations, reference added, minor corrections and changes made. Accepted for publication in Phys. Rev. D

References in corpus (1)

• Fluctuations in the Cosmic Microwave Background I: Form Factors and their Calculation in Synchronous Gauge

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