The cosmic microwave background bispectrum from the non-linear evolution of the cosmological perturbations

TL;DR

<p>We present the first complete computation of the CMB temperature bispectrum arising from the non-linear evolution of all cosmological fluids up to second order, including neutrinos, by solving the full second-order Boltzmann and Einstein equations and employing a line-of-sight, flat-sky formalism. The work systematically separates primary (early) and secondary (late) effects and expresses the result as an equivalent primordial non-Gaussianity amplitude $f_{_{ m NL}}$ for local and equilateral shapes, finding ${\widehat{f}_{_{ m NL}}}\approx 5$ for $\ell$ up to $2000$ in both cases. Analytically, the authors validate limiting-case behavior on large scales and provide small-scale intuition, while numerically they quantify the spectrum and bispectrum, assess detectability via signal-to-noise, and compare to ISW-lensing as a secondary source. The study demonstrates that the evolution-induced bispectrum can mimic primordial non-Gaussianity at a few units in $f_{_{ m NL}}$, highlighting the need to account for these effects in Planck-era and future analyses, and it provides publicly available tools (CMBquick) for broader investigations.</p>

Abstract

This article presents the first computation of the complete bispectrum of the cosmic microwave background temperature anisotropies arising from the evolution of all cosmic fluids up to second order, including neutrinos. Gravitational couplings, electron density fluctuations and the second order Boltzmann equation are fully taken into account. Comparison to limiting cases that appeared previously in the literature are provided. These are regimes for which analytical insights can be given. The final results are expressed in terms of equivalent fNL for different configurations. It is found that for moments up to lmax=2000, the signal generated by non-linear effects is equivalent to fNL~5 for both local-type and equilateral-type primordial non-Gaussianity.

Figures (8)

• Figure 1: First order scalar emitting sources $S_\ell^0$ with $\ell=0,1,2$ in respectively solid, dashed and dotted lines. The plot on the left corresponds to a mode which is well inside the Hubble radius around recombination time, whereas the plot on the right corresponds to a mode which is still super-Hubble around recombination time. In both cases the contributions are sharply peaked and this defines the last-scattering surface and validates the flat-sky approximation.
• Figure 2: Top left: $\Phi^{(2)}$ and $\Psi^{(2)}$ are in plotted in solid and dotted lines respectively. Top right: Vector perturbation $B^{(2)}$ (solid line); Bottom left: Tensor perturbation $H^{(2)}$ (solid line). For all these panels, the asymptotic transfer functions in the limit $a/a_{\rm eq} \gg 1$ are depicted in dashed line. Bottom right: Second order energy density contrasts for the radiation, $\delta_\gamma$ (solid), and baryons; $\delta_b$ (long dashed). The tight coupling approximation which holds as long as $\tau'/k \gg 1$, is drawn in dotted line.
• Figure 3: Comparison of the bispectrum induced by the non-linear dynamics (thick solid line) to a primordial bispectrum with $f_{_{\rm NL}}^{\Phi}=5$ (thick dashed line for local type, thick dotted line for the equilateral type). We also plot the approximation of Ref. PUB2008 that is considering only purely second order scalar sources in thin solid line, and the contribution from sources for $m=0,1,2$ in respectively thin dashed, dotted, and dot-dashed lines. We considered three different configurations in $\ell$ space, from top to bottom: an equilateral configuration with $\ell_1=\ell_2=\ell_3$ and then two squeezed configurations with $10 \ell_1=\ell_2=\ell_3$ and $\ell_1=20$ with $\ell_2=\ell_3$.
• Figure 4: Comparison of $\widehat{f}_{_{\rm NL}}$ (solid line) induced by the non-linear dynamics to $\widehat{f}_{_{\rm NL}}$ obtained by including only the scalar ($m=0$) modes (dashed line) of the purely second order sources, which corresponds to the approximation of our previous work PUB2008. We also plot $\widehat{f}_{_{\rm NL}}$ (dot-dashed lined) induced by lensing-ISW secondary effect. Right panel corresponds to local type couplings and the left panel to equilateral type couplings and the $\pm \sigma$ detection limit of $f_{_{\rm NL}}^\Phi$ is depicted in dotted lines for the local type couplings case.
• Figure 5: The signal to noise ratio as a function of the maximum multipole $\ell_{\rm max}$ for an ideal experiment. The signal to noise ratio of the total bispectrum generated by non-linear effects (solid line), of the local type primordial bispectrum (dashed line) and of the equilateral type (dotted line) when $f_{_{\rm NL}}^{\Phi}=1$. We also plot the signal to noise ratio of the bispectrum due to the ISW-lensing correlation in dot-dashed line.