The shape of the CMB lensing bispectrum

TL;DR

This work characterizes the CMB lensing bispectrum beyond leading order, revealing a non-perturbative, high-precision framework that accounts for both temperature and polarization signals. It shows that the ISW–lensing coupling produces a robust temperature bispectrum and a sizable polarization counterpart via the $E$–ψ correlation, with the non-perturbative short-leg approach capturing ~10% corrections relative to the first-order result. By deriving accurate estimators that include signal variance, the authors quantify detection significances (Planck ~5σ; cosmic-variance limit ~9σ) and quantify biases on local non-Gaussianity from lensing, while providing public software (CAMB/LensPix) to compute these effects. The results have important implications for unbiased primordial non-Gaussianity constraints and for joint analyses with large-scale structure, offering a practical path to isolate and marginalize the lensing contribution in high-precision CMB data.

Abstract

Lensing of the CMB generates a significant bispectrum, which should be detected by the Planck satellite at the 5-sigma level and is potentially a non-negligible source of bias for f_NL estimators of local non-Gaussianity. We extend current understanding of the lensing bispectrum in several directions: (1) we perform a non-perturbative calculation of the lensing bispectrum which is ~10% more accurate than previous, first-order calculations; (2) we demonstrate how to incorporate the signal variance of the lensing bispectrum into estimates of its amplitude, providing a good analytical explanation for previous Monte-Carlo results; and (3) we discover the existence of a significant lensing bispectrum in polarization, due to a previously-unnoticed correlation between the lensing potential and E-polarization as large as 30% at low multipoles. We use this improved understanding of the lensing bispectra to re-evaluate Fisher-matrix predictions, both for Planck and cosmic variance limited data. We confirm that the non-negligible lensing-induced bias for estimation of local non-Gaussianity should be robustly treatable, and will only inflate f_NL error bars by a few percent over predictions where lensing effects are completely ignored (but note that lensing must still be accounted for to obtain unbiased constraints). We also show that the detection significance for the lensing bispectrum itself is ultimately limited to 9 sigma by cosmic variance. The tools that we develop for non-perturbative calculation of the lensing bispectrum are directly relevant to other calculations, and we give an explicit construction of a simple non-perturbative quadratic estimator for the lensing potential and relate its cross-correlation power spectrum to the bispectrum. Our numerical codes are publicly available as part of CAMB and LensPix.

Figures (11)

• Figure 1: The geometry of the polarization signal generated by scattering at reionization: the figure is roughly to scale in comoving distance, with reionization (shading) starting at $z\sim 11$, about 2/3 of the way to the last-scattering surface (dark red outer shell). After the universe has reionized, the probability of scattering falls off as the universe expands, so most scattering occurs between $\sim 1/2$ and $\sim 2/3$ of the distance to recombination. An electron at reionization sees its own last scattering surface as indicated by the red shell, and Thomson scattering of the quadrupolar component of the distribution of photons originating from this surface generates $E$-polarization. For an electron at the start of reionization its last scattering shell extends from our last-scattering surface down to a redshift of about $z\sim 2$. Perturbations that generate a large-scale polarization signal will be correlated on large-scales, and therefore be correlated to perturbations at $z\alt 2$ (see Fig. \ref{['waves']}).
• Figure 2: An illustration of the correlation between the quadrupole component of the polarization from reionization and the local density quadrupole, here generated by a superposition of two orthogonal plane waves.
• Figure 3: The power spectrum (left) and dimensionless correlation coefficient (right) for the correlation of the CMB lensing potential with the CMB temperature anisotropy (top) and $E$-mode polarization (bottom) for a standard $\Lambda$CDM cosmology with an optical depth to reionization $\tau=0.09$ (solid lines) and $\tau=0$ (dashed lines). Note that without reionization the $E$-polarization power spectrum on large scales is very small, so although the correlation is still significant the actual size of the signal is very tiny. The $T$--$\psi$ correlation is due mainly to the ISW effect; the $E$--$\psi$ correlation to the reionization signal.
• Figure 4: The fractional difference between the flat-sky lensed temperature power spectra and the unlensed power spectra for $\tilde{C}^{TT}_l/C^{TT}_l-1$ (top; dashed), $\tilde{C}^{T\nabla T}_l/C^{TT}_l-1$ (top; solid) and analogously for the polarization in the lower two plots (see Appendix \ref{['Cgards_correlation']}). The lensing bispectrum depends on $\tilde{C}^{T\nabla T}_l$, with $\tilde{C}^{T\nabla T}_l \approx \tilde{C}^{TT}_l$ to within about one percent.
• Figure 5: The CMB temperature lensing reduced bispectrum $b_{4,l,l+4}$ using the squeezed-limit approximation of Eq. \ref{['diff_approx']} with the unlensed (thin blue dot-dashed) and lensed (thin blue solid) small-scale power spectrum, compared to the full spherical result of Eq. \ref{['fullspherbispec']} using the lensed small-scale spectrum (thick red). For comparison the thick black line shows the result for a local-model primordial bispectrum with $f_{\rm{NL}}=10$ (which has not itself been lensed here; see Ref. Hanson:2009kg); note the difference in phase and scale-dependence.