Cosmic microwave background bispectrum on small angular scales

TL;DR

The paper addresses how non-linear evolution after inflation induces non-Gaussianity in the CMB on sub-Hubble scales by focusing on second-order perturbations. It combines analytical insight with numerical integration of the CDM–radiation–baryon system to show that, on small angular scales, the second-order temperature fluctuations are driven by the gravitational potential dominated by CDM, with the dominant term given by $\Theta^{(2)}\approx -R\Phi^{(2)}$ and Silk damping suppressing oscillations. The resulting CMB bispectrum for equilateral configurations corresponds to an effective primordial non-Gaussianity of order $f_{\rm NL}\sim 25$ in the range $1000\lesssim l\lesssim 3000$, highlighting a sizable, intrinsic non-Gaussian signal that must be accounted for when interpreting measurements of primordial $f_{\rm NL}$. The study provides a physically transparent picture of how post-inflationary non-linearities imprint on the small-scale CMB, with implications for cosmological parameter estimation and the search for primordial non-Gaussianity in current and future experiments.

Abstract

This article investigates the non-linear evolution of cosmological perturbations on sub-Hubble scales in order to evaluate the unavoidable deviations from Gaussianity that arise from the non-linear dynamics. It shows that the dominant contribution to modes coupling in the cosmic microwave background temperature anisotropies on small angular scales is driven by the sub-Hubble non-linear evolution of the dark matter component. The perturbation equations, involving in particular the first moments of the Boltzmann equation for the photons, are integrated up to second order in perturbations. An analytical analysis of the solutions gives a physical understanding of the result as well as an estimation of its order of magnitude. This allows to quantify the expected deviation from Gaussianity of the cosmic microwave background temperature anisotropy and, in particular, to compute its bispectrum on small angular scales. Restricting to equilateral configurations, we show that the non-linear evolution accounts for a contribution that would be equivalent to a constant primordial non-Gaussianity of order fNL~25 on scales ranging approximately from l~1000 to l~3000.

Figures (9)

• Figure 1: Evolution of the two second order gravitational potentials, $\phi^{(2)}(\textbf{k}_{1},\textbf{k}_{2})$ and $\psi^{(2)}(\textbf{k}_{1},\textbf{k}_{2})$ for $k_1=k_2= 10k_\mathrm{eq}$ (top panel) or $k_1=k_2=20k_\mathrm{eq}$ (bottom panel) and $\textbf{k}_{1}\cdot\textbf{k}_{2}=0$.
• Figure 2: Top: Comparison of the baryons and photons velocity perturbation at order 2 for $k_1=k_2=10k_\mathrm{eq}$ and $\textbf{k}_{1}\cdot\textbf{k}_{2}=0$. It shows that $v_{\rm r}^{(2)}=v_{\rm b}^{(2)}$ with a good approximation until decoupling. Bottom: l.h.s and r.h.s of Eq. (\ref{['CI2abiab']}) for the adiabaticity condition at order 2. It can be seen that this adiabaticity condition holds until recombination, hence justifying the approximation of § \ref{['sec3a']}.
• Figure 3: The time at which the contribution of cold dark matter and of radiation are comparable in the Poisson equation as a function of $k/k_\mathrm{eq}$. $k_\mathrm{eq}$ is the wave-number of the mode that becomes sub-Hubble at the time of quality. For most of the scales of interest, i.e. for $k\gg k_{\rm eq.}$, the contribution of CDM in the Poisson equation is dominant before equality, i.e. $y_\star<1$).
• Figure 4: (solid line) The second order potential computed in the tight coupling limit as a function of time $y$ and of $\theta$, angle between the wave vectors. It is compared to its expected late time behaviour (\ref{['Phi2MD']}). It is to be noted that the convergence toward this solution is extremely rapid and takes place as soon as equality is reached, e.g. $y=1$. The results correspond to $k_{1}=6k_\mathrm{eq}$ and $k_{2}=12k_\mathrm{eq}$. The difference in the amplitude of the function is due to the fact that the baryons component has been neglected in the derivation of Eq. (\ref{['Phi2MD']})
• Figure 5: Behavior of $\Theta_{SW}^{(2)}$ (black) and $k v_{\rm r}^{(2)}/\sqrt{3}$ (red). The numerical integration is depicted with solid lines while the analytical estimates is plotted in dashed lines. From top to bottom, we have $k_{1}=k_{2}=30\,k_{\rm eq.}$ and $k_{1}=k_{2}=40\,k_{\rm eq.}$. The vertical grey zone represents the "surface" of last scattering.