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}~ 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.
