Cosmic Microwave Background Bispectrum from Recombination

We compute the cosmic microwave background temperature bispectrum generated by nonlinearities at recombination on all scales. We use CosmoLib2nd, a numerical Boltzmann code at second order to compute cosmic microwave background bispectra on the full sky. We consistently include all effects except gravitational lensing, which can be added to our result using standard methods. The bispectrum is peaked on squeezed triangles and agrees with the analytic approximation in the squeezed limit at the few percent level for all the scales where this is applicable. On smaller scales, we recover previous results on perturbed recombination. For cosmic-variance limited data to $l_{\max }=2000$, its signal-to-noise ratio is $S/N=0.47$, corresponding to $f_{\mathrm{NL}}^{\mathrm{eff}}=-2.79$, and will bias a local signal by $f_{\mathrm{NL}}^{\mathrm{loc}}\simeq 0.82$.

Figure 1

Reduced bispectrum of $\tilde{\Theta }$ for $l_1=6$ as a function of $l_2=l_3$ (black line), compared to the analytic approximation from the spatial coordinate rescaling equation (2) (dashed red line). The other lines are the different contributions to the bispectrum. In the squeezed limit, i.e., for $l_3\gg 6$, the agreement is at the few percent level.

Figure 2

Reduced bispectrum for $l_1=l_3/10$ as a function of $l_2=l_3$. At small $l$ the bispectrum satisfies Eq. (2); deviations around $l_3\sim 100$ are due to integrated effects such as vector and RS. For $l_1\gtrsim 50$ ($l_3\gtrsim 500$) the long mode is inside the Hubble radius at recombination and the analytic approximation fails.

Figure 3

The reduced bispectrum with two $l$ fixed, $l_2=l_3$ as a function of $l_1$. As expected, for $l_1\lesssim 100$ the difference between the computed bispectrum and the analytic approximation is of order $\sim (l_1/l_H{)}^2$.

Figure 4

Signal-to-noise density, $l_3^{3/2}{b}_{l_1{l}_2{l}_3}/(C_{l_1}{C}_{l_2}{C}_{l_3}{)}^{1/2}$, for $l_3=1720$ and $l_1\le l_2\le l_3$. The signal is peaked on squeezed triangles and the acoustic oscillations are clearly displayed. For $l_1\ge 100$ the amplitude and shape agree with that computed in Ref. 13.