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

TL;DR

This work develops a rigorous 2×2 perturbation framework to compute CMB spectral distortions produced by Silk damping of small-scale primordial acoustic waves. By separating distortions from the blackbody temperature evolution and including second-order Compton energy transfer, polarization, and photon mixing, the authors find that exactly a fraction $\frac{1}{3}$ of the dissipated energy appears as observable distortions while $\frac{2}{3}$ raises the average CMB temperature. The distortions are dominated by the $\mu$-era at high redshift and by the Doppler/masters at recombination, with the observable signal highly sensitive to the primordial power spectrum parameters $n_S$ and $n_{run}$. Using CosmoTherm and analytic refinements, they quantify $\mu$ and $y$ contributions for various models and assess PIXIE’s ability to constrain small-scale power, including balanced-energy-release scenarios where $\mu$ may vanish. This work thus provides a principled link between early-universe microphysics, energy redistribution in the radiation field, and a potential cosmological observable probing scales inaccessible to galaxy surveys.

Abstract

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)

Figures (17)

• Figure 1: Comparison of different source functions for spectral distortions according to Eq. \ref{['eq:sss_a']}. $\mathcal{G} \,\mathcal{B}$ arises because of the recoil effect and the first order Klein-Nishina correction, while $\mathcal{D}_x \mathcal{G}$ describes the effect of Doppler broadening and boosting on temperature perturbations.
• Figure 2: Comparison of different source functions for spectral distortions according to Eq. \ref{['eq:Final_EQ_elim_Te']}. To make things more comparable we rescaled $\mathcal{H}$ by a factor of $4$.
• Figure 3: Monopole and dipole transfer functions for different values of $k$, normalized to initial curvature perturbation in comoving gauge $\zeta\equiv 2C=1$. We compared all of the cases with CMBfast and found agreement at the level of $\lesssim 1\%$. In the free-streaming regime many multipoles needed to be included to avoid errors caused by truncation of the Boltzmann hierarchy.
• Figure 4: Same as Fig. \ref{['fig:modes_MD']} but for the quadrupole and octupole.
• Figure 5: Main source function, $\left<\mathcal{S}_{\rm ac}\right>$, in units of $A_\zeta H(z) / \dot{\tau}$ for $n_{\rm S}=0.96$. For the dotted/red line multipoles up to $l=80$ were included, while for all the other curves only the moments up to the quadrupole were accounted for. The result obtained with Eq. \ref{['eq:heat_SZ_appr']} (dashed/blue) is compared with the total result from the perturbation code (dash-dotted/black). The solid lines give partial contributions to the source function for different ranges in $k$ (in units of ${\rm Mpc}^{-1}$), starting from the right with $10^{-4}\leq k\leq 10^{-2}$, $10^{-2}\leq k\leq 10^{-1}$, $10^{-1}\leq k\leq 1$, $1\leq k\leq 10^{2}$, $10^{2}\leq k\leq 10^{4}$, and $10^{4}\leq k\leq 10^{6}$.