Cosmological Perturbations at Second Order and Recombination Perturbed

TL;DR

This paper develops the full second-order perturbation theory for cosmological evolution including the first-order electron-density perturbations $\delta_e$, and analyzes how recombination perturbations modify the ionization history and Ly$\alpha$ photon escape. It shows that a perturbed version of the Peebles effective 3-level atom suffices for $k<1\,\mathrm{Mpc}^{-1}$ and derives the perturbations to the Ly$\alpha$ escape probability, which are governed by the local baryon velocity divergence. Crucially, $\delta_e$ is amplified by about a factor of $\sim 5$ for modes shorter than the photon diffusion scale, suggesting a potentially detectable non-Gaussian signature in the CMB, which is explored in a companion paper via the CMB bispectrum. The work also provides a rigorous second-order framework in the Poisson gauge, including the full energy-momentum tensor and Einstein equations, and lays the groundwork for precise predictions of recombination-era non-Gaussianity.

Abstract

We derive the full set of second-order equations governing the evolution of cosmological perturbations, including the effects of the first-order electron number density perturbations, δ_e. We provide a detailed analysis of the perturbations to the recombination history of the universe and show that a perturbed version of the Peebles effective 3-level atom is sufficient for obtaining the evolution of δ_e for comoving wavenumbers smaller than 1Mpc^{-1}. We calculate rigorously the perturbations to the Lyαescape probability and show that to a good approximation it is governed by the local baryon velocity divergence. For modes shorter than the photon diffusion scale, we find that δ_e is enhanced during recombination by a factor of roughly 5 relative to other first-order quantities sourcing the CMB anisotropies at second order. Using these results, in a companion paper we calculate the CMB bispectrum generated during recombination.

Figures (6)

• Figure 1: We summarize all important timescales related to recombination in this figure and in Fig. \ref{['fig:recomb']}. Here we concentrate on the timescales that control the 3-level atom approximation. The peak of the CMB visiblity function is at $z=1090$. The expansion timescale is defined as $(3\mathcal{H})^{-1}$; the Ly$\alpha$ relaxation timescale is given by eq. (29) of rybdell; the collisions timescale corresponds to atomic collisions, and is irrelevant at early times; the Kompaneets timescale corresponds to the timescale of Compton heating. Of the timescales given in this figure, the Ly$\alpha$ relaxation imposes an upper limit of $k\approx1\,$Mpc$^{-1}$ on the validity of the perturbed Peebles equation.
• Figure 2: The comoving rates per HII ion governing Hydrogen recombination. The recombination rate is defined as $|\dot n_e/n_e|$. Here we concentrate on the rates that affect directly the recombination rate.
• Figure 3: The amplitude of the perturbations to the electron density in the Newtonian gauge compared to other first order perturbations for a mode well outside the horizon. The peak of the CMB visibility function is at $\eta=288\,$Mpc. The normalization of the perturbations is the one in CMBFAST -- superhorizon $\zeta=1$. The perturbations in the Newtonian gauge for superhorizon modes are equivalent to time shifted zeroth order evolution. This results in two regions of enhancement of $\delta_e$ corresponding to HeIII$\to$HeII, and the combined HeII$\to$HeI and HII$\to$HI recombination. Similarly, $\delta_{T_M}$ coincides with $\delta_{T_R}$ while Compton scattering is effective in keeping the photons and electrons in thermal contant. As $T_M$ decouples, $\delta_{T_M}$ converges to $2\delta_{T_R}$.
• Figure 4: The amplitude of the perturbations to the electron density in the Newtonian gauge compared to other first order perturbations for a mode corresponding approximately to the second acoustic peak. The enhancement of $\delta_e$ is still well approximated by eq. (\ref{['dedb']}). This mode is well inside the horizon, and therefore, this enhancement is not a gauge artefact, and will leave an imprint on the CMB bispectrum.
• Figure 5: A plot of some first order quantities in the Newtonian gauge at the peak of the visibility function showing the enhancement of $\delta_e$. The first order $\delta_b$ shows the typical magnitude of the variables sourcing the CMB anisotropies at second order, and one can see that $\delta_e$ dominates. We also plot the expected electron density perturbation coming from eq. (\ref{['dedb']}) (dotted line) which is a good approximation to the true $\delta_e$ down to the recombination scale ($k\sim0.1\,$Mpc$^{-1}$). The enhancement for superhorizon modes is gauge dependent, but persists for modes inside the horizon, where the effect is observable, approximately down to the photon diffusion scale ($k_D\approx0.15\,$Mpc$^{-1}$).