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

TL;DR

The paper develops a formal framework to compute the CMB bispectrum generated by second-order perturbations in the Boltzmann equation, focusing on terms that are products of first-order perturbations. It derives a general angular-averaged bispectrum and then explicitly evaluates four product-based contributions, $B^{(0,0,0)}$, $B^{(1,1,0)}$, $B^{(1,0,1)}$, and $B^{(1,1,2)}$, using line-of-sight transfer functions and detailed angular-momentum algebra. The results show that the second-order product bispectrum peaks in squeezed configurations much like local-type primordial non-Gaussianity but differs in detail due to acoustic oscillations and non-scale-invariance, yielding a cross-correlation with the local form that is modest (up to ~0.5 at low $l_{max}$) and a signal-to-noise around ~0.4 at $l_{max}=2000$. Consequently, the contamination of the local $f_{NL}$ parameter by these second-order products is small (about $f^{con}_{NL}\sim0.9$ at $l_{max}=200$ and ~0.5 at $l_{max}=2000$), supporting the neglect of these terms in future CMB analyses of $f_{NL}$ with current and upcoming data, though intrinsically second-order effects and perturbed recombination remain for future work.

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.

Figures (7)

• Figure 1: Shape dependence of the second-order bispectrum from products of the first-order terms (top) and that of the local primordial bispectrum (bottom). We show $l_1l_2\langle a_{l_1m_1}^{(1)}a_{l_2m_2}^{(1)}a_{l_3m_3}^{(2)}\rangle {({\cal{G}}_{l_1l_2l_3}^{m_1m_2m_3})}^{-1}/(2\pi)^2\times 10^{22}$ as a function of $l_1/l_3$ and $l_2/l_3$ where $l_3=200$. Both shapes have the largest signals in the squeezed triangles, $l_1\ll l_2\approx l_3$.
• Figure 2: Same as Fig. \ref{['fig:shape1']} for $l_3=1000$. The acoustic oscillations are clearly seen.
• Figure 3: Signal-to-noise ratios for the local primordial bispectrum for $f_{NL}=1$ (dashed), and the second-order bispectrum from the products of the first-order terms (solid), for an ideal full-sky and cosmic-variance-limited (noiseless) experiment.
• Figure 4: Absolute values of the contributions to the signal-to-noise ratio from each component, $(S/N)_{ab}$, as defined by Eq. (\ref{['eq:sncomp']}).
• Figure 5: The cross-correlation coefficient between the second-order bispectrum from the products of the first-order terms and the local primordial bispectrum.