CMB at 2x2 order: the dissipation of primordial acoustic waves and the observable part of the associated energy release

Silk damping of primordial small-scale perturbations in the photon-baryon fluid due to diffusion of photons inevitably creates spectral distortions in the CMB. With the proposed CMB experiment PIXIE it might become possible to measure these distortions and thereby constrain the primordial power spectrum at comoving wavenumbers 50 Mpc^{-1} < k < 10^4 Mpc^{-1}. Since primordial fluctuations in the CMB on these scales are completely erased by Silk damping, these distortions may provide the only way to shed light on otherwise unobservable aspects of inflationary physics. A consistent treatment of the primordial dissipation problem requires going to second order in perturbation theory, while thermalization of these distortions necessitates consideration of second order in Compton scattering energy transfer. Here we give a full 2x2 treatment for the creation and evolution of spectral distortions due to the acoustic dissipation process, consistently including the effect of polarization and photon mixing in the free streaming regime. We show that 1/3 of the total energy (9/4 larger than previous estimates) stored in small-scale temperature perturbations imprints observable spectral distortions, while the remaining 2/3 only raises the average CMB temperature, an effect that is unobservable. At high redshift dissipation is mainly mediated through the quadrupole anisotropies, while after recombination peculiar motions are most important. During recombination the damping of the higher multipoles is also significant. We compute the average distortion for several examples using CosmoTherm, analyzing their dependence on parameters of the primordial power spectrum. For one of the best fit WMAP7 cosmologies, with n_S=1.027 and n_run=-0.034, the cooling of baryonic matter practically compensates the heating from acoustic dissipation in the mu-era. (abridged)