CMB anisotropies at second order III: bispectrum from products of the first-order perturbations

TL;DR

This paper calculates the second-order bispectrum of CMB temperature anisotropies from products of first-order perturbations, showing it has a small impact on primordial non-Gaussianity detection in future experiments.

Contribution

It provides a detailed numerical calculation of the second-order bispectrum from first-order products, highlighting its shape differences and negligible contamination of primordial signals.

Findings

Bispectrum peaks in squeezed configurations similar to local primordial bispectrum.

Shape differences lead to a low cross-correlation coefficient (~0.5 at l=200).

Contamination of primordial f_NL is estimated to be below 1 for future experiments.

Abstract

We calculate the bispectrum of the Cosmic Microwave Background (CMB) temperature anisotropies induced by the second-order fluctuations in the Boltzmann equation. In this paper, which is one of a series of papers on the numerical calculation of the bispectrum from the second-order fluctuations, we consider the terms that are products of the first-order perturbations, and leave intrinsically second-order terms and perturbations in the recombination history to the subsequent papers. We show that the bispectrum has the maximum signal in the squeezed triangles, similar to the local-type primordial bispectrum, as both types generate non-linearities via products of the first-order terms in position space. However, detailed calculations show that their shapes are sufficiently different: the cross-correlation coefficient reaches 0.5 at the maximum multipole of l_{max}~ 200, and then weakens to 0.3 at l_{max}~ 2000. The differences in shape arise from (i) the way the acoustic oscillations affect the bispectrum, and (ii) the second-order effects not being scale-invariant. This implies that the contamination of the primordial bispectrum due to the second-order effects (from the products of the first-order terms) is small. The expected signal-to-noise ratio of the products of the first-order terms is ~ 0.4 at l_{max}~ 2000 for a full-sky, cosmic variance limited experiment. We therefore conclude that the products of the first-order terms may be safely ignored in the analysis of the future CMB experiments. The expected contamination of the local-form f_{NL} is f^{local}_{NL}~ 0.9 at l_{max}~ 200, and f^{local}_{NL}~ 0.5 at l_{max}~ 2000.