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

Understand

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.