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

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)similar to 200, and then weakens to 0.3 at l(max)similar to 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 similar to 0.4 at l(max)similar to 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(NL)(local) similar to 0.9 at l(max) similar to 200, and f(NL)(local) similar to 0.5 at l(max) similar to 2000.