Analytic Spectra of CMB Anisotropies and Polarization Generated by Scalar Perturbations in Synchronous Gauge

TL;DR

This paper derives explicit analytical formulas for CMB temperature and polarization anisotropies caused by scalar perturbations, providing insights into how recombination parameters affect observed spectra.

Contribution

It introduces a detailed analytical approach to compute CMB anisotropies in synchronous gauge, including effects of recombination and scalar perturbations, extending previous numerical methods.

Findings

Analytic expressions match numerical results on large scales.

Longer recombination width increases polarization amplitude on large scales.

Recombination timing shifts the peaks of the spectra to larger angular scales.

Abstract

We present a detailed analytical calculation of CMB temperature anisotropies \alpha_k and polarization \beta_k generated by scalar metric perturbations in synchronous gauge, parallel to our previous work with RGW as a generating source. This is realized primarily by an analytic time-integration of Boltzmann's equation, yielding the closed forms of \alpha_k and \beta_k. Approximations, such as the tight-coupling approximation for photons a prior to the recombination and the long wavelength limit for scalar perturbations are used. The residual gauge modes in scalar perturbations are analyzed and a proper joining condition of scalar perturbations at the radiation-matter equality is chosen, ensuring the continuity of energy perturbation. The resulting analytic expressions of the multipole moments of polarization a^E_l, and of temperature anisotropies a^T_l are explicit functions of the scalar perturbations, recombination time, recombination width, photon free streaming damping factor, baryon fraction, initial amplitude, primordial scalar spectral index, and the running index. These results show that a longer recombination width yields higher amplitudes of polarization on large scales and more damping on small scales, and that a late recombination time shifts the peaks of C^{XX'}_l to larger angular scales. The analytic spectra C^{XX'}_l agree with the numerical ones and with those observed by WMAP on large scales (l \lesssim 500), but deviate considerably from the numerical results on smaller scales, showing the limitations of our approximate analytic calculations. Several possible improvements are pointed out for further studies.