Sachs-Wolfe at second order: the CMB bispectrum on large angular scales

TL;DR

This work extends the Sachs-Wolfe result to second order to compute the CMB bispectrum on large angular scales under exact matter domination, using the generalized Poisson gauge and the flat-sky approximation. It includes intrinsic contributions at last scattering and a suite of integrated effects—Rees-Sciama, vector and tensor perturbations, and gravitational lensing—to produce a complete second-order bispectrum. The results show a characteristic $l^{-4}$ scaling with lensing dominating the total signal and a squeezed-limit form $f_{\rm NL}^{\rm local} = -1/6 - \cos(2\theta)$, reflecting the angular dependence from lensing, while the equilateral amplitude is around $f_{\rm NL}^{\rm equil} \approx 3.13$ for the total signal. Overall, the large-angle Sachs-Wolfe bispectrum in this limit is small and largely orthogonal to standard local or equilateral templates, implying negligible contamination to primordial non-Gaussianity searches; the authors also outline directions to incorporate late-time acceleration, ISW-lensing correlations, and a full-sky treatment for a more realistic comparison to data.

Abstract

We calculate the Cosmic Microwave Background anisotropy bispectrum on large angular scales in the absence of primordial non-Gaussianities, assuming exact matter dominance and extending at second order the classic Sachs-Wolfe result δT/T=Φ/3. The calculation is done in Poisson gauge. Besides intrinsic contributions calculated at last scattering, one must consider integrated effects. These are associated to lensing, and to the time dependence of the potentials (Rees-Sciama) and of the vector and tensor components of the metric generated at second order. The bispectrum is explicitly computed in the flat-sky approximation. It scales as l^(-4) in the scale invariant limit and the shape dependence of its various contributions is represented in 3d plots. Although all the contributions to the bispectrum are parametrically of the same order, the full bispectrum is dominated by lensing. In the squeezed limit it corresponds to f_NL^local = -1/6 - cos(2 θ), where θis the angle between the short and the long modes; the angle dependent contribution comes from lensing. In the equilateral limit it corresponds to f_NL^equil ~ 3.13.

Figures (9)

• Figure 1: The CMB bispectrum on large angular scales induced by primordial non-Gaussianities of the local form for $f_{\rm NL}^{\rm local} = 1$. According to its definition, the bispectrum is negative for positive $f_{\rm NL}^{\rm local}$; thus, we have plotted it with an overall minus sign.
• Figure 2: The CMB bispectrum on large angular scales induced by primordial non-Gaussianities of the equilateral form for $f_{\rm NL}^{\rm equil} = 1$. According to its definition, the bispectrum is negative for positive $f_{\rm NL}^{\rm equil}$; thus, we have plotted it with an overall minus sign.
• Figure 3: The CMB bispectrum induced by the momentum dependent intrinsic contribution in eq. (\ref{['IntrinsicdT']}).
• Figure 4: The CMB bispectrum induced by the Rees-Sciama effect, eq. (\ref{['CMBRS']}).
• Figure 5: The CMB bispectrum induced by the vector contribution, eq. (\ref{['CMBV']}).