The CMB bispectrum in the squeezed limit

Abstract

The CMB bispectrum generated by second-order effects at recombination can be calculated analytically when one of the three modes has a wavelength much longer than the other two and is outside the horizon at recombination. This was pointed out in \cite{Creminelli:2004pv} and here we correct their results. We derive a simple formula for the bispectrum, $f_{NL}^{loc} = - (1/6+ \cos 2 \theta) \cdot (1- 1/2 \cdot d \ln (l_S^2 C_{S})/d \ln l_S)$, where $C_S$ is the short scale spectrum and $\theta$ the relative orientation between the long and the short modes. This formula is exact and takes into account all effects at recombination, including recombination-lensing, but neglects all late-time effects such as ISW-lensing. The induced bispectrum in the squeezed limit is small and will negligibly contaminate the Planck search for a local primordial signal: this will be biased only by $f_{NL}^{loc}\approx-0.4$. The above analytic formula includes the primordial non-Gaussianity of any single-field model. It also represents a consistency check for second-order Boltzmann codes: we find substantial agreement with the CMBquick code.