The full CMB temperature bispectrum from single-field inflation

Abstract

We compute the full cosmic microwave background temperature bispectrum generated by nonlinearities after single-field inflation. By integrating the photon temperature at second order along a perturbed geodesic in Newtonian gauge, we derive an expression for the observed temperature fluctuations that, for the first time, clarifies the separation of the gravitational lensing and time-delay effects from the purely second-order contributions. We then use the second-order Boltzmann code CosmoLib$2^{\rm nd}$ to calculate these contributions and their bispectrum. Including the perturbations in the photon path, the numerically computed bispectrum exactly matches the expected squeezed limit. Moreover, the analytic squeezed-limit formula reproduces well the signal-to-noise and shape of the full bispectrum, potentially facilitating the subtraction of the bias induced by second-order effects. For a cosmic-variance limited experiment with $l_{\rm max} = 2000$, the bias on a local signal is $f_{\rm NL}^{\rm loc} =0.73$ negligible for equilateral and orthogonal signals. The signal-to-noise ratio is unity at $l_{\rm max} \sim 3000$, suggesting that second-order effects may hopefully be measured in the future.