The CMB bispectrum in the squeezed limit

TL;DR

The paper derives an exact analytic expression for the CMB bispectrum in the squeezed limit arising from second-order recombination effects, correcting earlier results. It frames the long-wavelength mode as a coordinate transformation, derives how it modulates short-scale CMB perturbations, and combines short-mode modulation with lensing to obtain a compact B_{l_L l_S l_S} formula. The result maps to a local f_NL^{loc} with explicit θ dependence, and shows the isotropic piece is small while Planck-level contamination is negligible. The work also provides a cross-check against second-order Boltzmann codes like CMBquick and outlines extensions to polarization and beyond-squeezed configurations.

Abstract

The CMB bispectrum generated by second-order effects at recombination can be calculated analytically when one of the three modes has a wavelength much longer than the other two and is outside the horizon at recombination. This was pointed out in \cite{Creminelli:2004pv} and here we correct their results. We derive a simple formula for the bispectrum, $f_{NL}^{loc} = - (1/6+ \cos 2 θ) \cdot (1- 1/2 \cdot d \ln (l_S^2 C_{S})/d \ln l_S)$, where $C_S$ is the short scale spectrum and $θ$ the relative orientation between the long and the short modes. This formula is exact and takes into account all effects at recombination, including recombination-lensing, but neglects all late-time effects such as ISW-lensing. The induced bispectrum in the squeezed limit is small and will negligibly contaminate the Planck search for a local primordial signal: this will be biased only by $f_{NL}^{loc}\approx-0.4$. The above analytic formula includes the primordial non-Gaussianity of any single-field model. It also represents a consistency check for second-order Boltzmann codes: we find substantial agreement with the CMBquick code.

Figures (4)

• Figure 1: Plot of $-1/6 \cdot \left(1- 1/2 \cdot d \ln (l_S^2 C_{l_S})/d \ln l_S\right)$, which acts as an effective $f^{\rm loc}_{\rm NL}$ in our final formula \ref{['final_formula']}.
• Figure 2: Comparison of the effect of the coordinate transformation \ref{['ct']} on the gravitational potential $\Phi$ and on the radiation energy density $\delta_{\rm rad} = 4 \Theta$ to the full second-order evolution in the squeezed limit. We consider the radiation-to-matter transition in the fluid approximation. On the left-hand panel we compare eq. \ref{['Phi_sec']} (solid line) to the numerical solution of Einstein's equation (dashed line). On the right-hand panel we do the same for eq. \ref{['delta_sec']}. We chose $k_L = 0.01 \eta^{-1}_{\rm eq}$, $k_S = 10 \eta^{-1}_{\rm eq}$ and $\cos \theta=0.5$.
• Figure 3: Comparison of eq. \ref{['final_conv']} with the bispectrum in the flat-sky approximation produced by CMBquick in the absence of lensing. Cosmological parameters are the best-fit ones of WMAP5 Komatsu:2008hk. Solid lines represent our equation while dashed lines the bispectrum computed by the code. In the top panels $l_1=20$ and $l_2=l_3=l$; in the bottom panels $l_1=l/10$ and $l_2=l_3=l$. For the top panels the squeezed limit corresponds to $l \ {\hbox{$\buildrel>\over\sim$}}\ 300$ and for the bottom panels to $l \ {\hbox{$\buildrel<\over\sim$}}\ 300$. On the right-hand panels we have removed neutrinos and set the photon temperature today to $T=3.08$ K to have the same radiation density.
• Figure 4: Signal-to-noise ratio for the bispectrum in eq. \ref{['final_formula']}, evaluated summing over squeezed configurations only, as a function of $l_{\rm max}$. This roughly oscillates around $\sim 10^{-7} l_{\rm max}^2$. For comparison, the dashed line shows $S/N$ for a primordial local signal with $f_{\rm NL}^{\rm loc}=1$.