Non-Gaussianities from Perturbing Recombination

TL;DR

The paper analyzes how second-order perturbations to the recombination era, dominated by fluctuations in the free-electron density δ_e, imprint a non-Gaussian bispectrum on the CMB. Using a line-of-sight formalism and analytic approximations for diffusion damping, the authors decompose the second-order source into diffusion-related and visibility-perturbation contributions, then perform a full second-order calculation within the horizon to obtain the bispectrum. They find a non-scale-invariant signal that peaks in squeezed configurations with an effective local f_NL of about -3.5, suggesting Planck could marginally detect it if polarization is included. The results reveal three main physical effects—time-shift of recombination, diffusion-scale perturbations, and recombination-area changes—each contributing at order unity and partially cancelling, underscoring the need for comprehensive second-order analyses for accurate CMB non-Gaussianity constraints.

Abstract

We approximately compute the bispectrum induced on the CMB temperature by fluctuations in the standard recombination epoch. Of all the second order sources that can induce non-Gaussianity during recombination, we concentrate on those proportional to the perturbation in the free electron density, which is about a factor of 5 larger than the other first order perturbations. This term induces some non-Gaussianity by delaying the time of recombination and by changing the photon diffusion scale. We find that the signal is not scale invariant, peaked on squeezed triangles with the smaller multipole around the scale of the first acoustic peak, and that its size corresponds to an effective f_NL ~ -3.5, which could be marginally detected by Planck if both temperature and polarization are measured.

Figures (13)

• Figure 1: Schematic representation of the effect of a perturbation $\tilde{\delta g}$ to the shape of the visibility function for the case high-$k_e$ and low-$k_s$. Notice that the location of the peak is unchanged, so that $\delta\eta_r=0$. However, $g$ is much more peaked around the central value than in the homogenous case. This means that more CMB photons originate from the region near the peak, where the power of the mode is not yet damped (in red we plot the diffusion damping). This means that the resulting CMB anisotropy is larger. The shaded black region of $g^{(0)}$ represent how much the 'Effective Area' in the homogeneous case differs from one and it is a measure of how many CMB photons originate from the region where the mode is already beginning to be damped. We see that $g^{(0)}+\tilde{\delta g}$ has a larger 'Effective Area', implying a positive $\delta$Area and the possibility for a larger CMB anisotropy.
• Figure 2: The first-order perturbations responsible for the bispectrum generated from recombination. The computation was done in synchronous gauge, and the normalization of the perturbations is the one in CMBFAST cmbfast-- on superhorizon scales $\zeta=1$. $\delta_{k_D}$ represents the perturbation to the diffusion scale at the peak of the visibility function ($\eta\simeq 288\,$Mpc) due to $\delta n_e$; $\delta_{k_g}$ is the perturbation to $k_D$ due to $\delta\eta_r$ -- the shift in the position of the last scattering surface. $\delta\mathrm{Area}$ approximately represents the perturbation to the probability that a photon originates from the last scattering surface before the perturbation decays due to diffusion damping. The timeshift $\delta\eta_r$ also gives rise to a change in the phase at which we evaluate the first order source, which is schematically given by $(-c_s k_s \delta \eta_r/3)$. The plot is for $k_s\eta_0=3000$. From the plot we can see that the largest contribution to the second order temperature anisotropies for the given $k_s$ is from $k_e\eta_0\sim 200$, i.e. from around the first acoustic peak. Note that the two largest contributions to $\Theta_l^{(2)}$ -- the perturbations of $k_D$ coming from $\delta_e$ and the change in the position of the last scattering surface, partially cancel each other.
• Figure 3: The same as in Fig. \ref{['fig:kDL']} but for $k_s\eta_0=200$. As one can see, all effects for high $k_e$ suffer from some kind of suppression as described in the text.
• Figure 4: When computing the bispectrum, we take the expectation value of three $\Theta$ modes, and one of them has to be taken at second order in order not to have a null result. Each of the two first order perturbations contained in the second order mode, approximately $\delta_e$ and $\Theta^{(1)}$, need to be matched with one of the two first order $\Theta$ modes, forcing $\vec{k}_e=-\vec{k}_2$ and $\vec{k}_s=-\vec{k}_3$, where $\vec{k}_1=\vec{k}_e+\vec{k}_s$. The sum of $\vec{k}_1$, $\vec{k}_2$ and $\vec{k}_3$ must be equal to zero, so that the three wave vectors form a closed triangle. The same is done for all the symmetric combinations.
• Figure 5: Approximate contributions to the signal-to-noise of the three point function per triangle in the squeezed limit as obtained from (\ref{['blllsecond']}) for $l_1=200$ as a function of $l_2=l_3$. The contribution to the bispectrum from $\delta_{k_D}$ comes from the perturbation due to $\delta n_e$. The different contributions to the bispectrum from $\delta_g$ are shown in Figures \ref{['fig:bL']} and \ref{['fig:bH']}.