Primordial Bispectrum Information from CMB Polarization

TL;DR

This work shows that including CMB polarization in the optimal cubic bispectrum estimator significantly improves sensitivity to primordial non-Gaussianity, quantified by $f_{NL}$, with roughly a twofold gain over temperature-only analyses. It derives the full polarization-inclusive estimator, expresses the reduced bispectrum via line-of-sight integrals of transfer functions, and demonstrates substantial improvements for Planck and ideal experiments while quantifying the modest impact of gravitational lensing on the estimator's variance. The authors provide scaling arguments indicating $({S/N})^2$ grows with the number of observed pixels and remains robust to Silk damping through collapsed-triangle configurations, highlighting the importance of polarization measurements at small scales. Together, these results advocate strong emphasis on high-fidelity CMB polarization data to probe the physics of the primordial seeds and potential deviations from simple slow-roll inflation.

Abstract

After the precise observations of the Cosmic Microwave Background (CMB) anisotropy power spectrum, attention is now being focused on the higher order statistics of the CMB anisotropies. Since linear evolution preserves the statistical properties of the initial conditions, observed non-Gaussianity of the CMB will mirror primordial non-Gaussianity. Single field slow-roll inflation robustly predicts negligible non-Gaussianity so an indication of non-Gaussianity will suggest alternative scenarios need to be considered. In this paper we calculate the information on primordial non-Gaussianity encoded in the polarization of the CMB. After deriving the optimal weights for a cubic estimator we evaluate the Signal-to-Noise ratio of the estimator for WMAP, Planck and an ideal cosmic variance limited experiment. We find that when the experiment can observe CMB polarization with good sensitivity, the sensitivity to primordial non-Gaussianity increases by roughly a factor of two. We also test the weakly non-Gaussian assumption used to derive the optimal weight factor by calculating the degradation factor produced by the gravitational lensing induced connected four-point function. The physical scales in the radiative transfer functions are largely irrelevant for the constraints on the primordial non-Gaussianity. We show that the total (S/N)^2 is simply proportional to the number of observed pixels on the sky.

Figures (6)

• Figure 1: The lensed (dashed red) and unlensed (solid black) $C_l$ for the concordance $\Lambda CDM$ cosmology; the temperature power spectra are on the left and the $E$ polarization power spectra are on the right.
• Figure 2: Upper Left: A plot of $\beta^T_l(r)$ vs. $l$ for values of $r = \tau_0 - \tau_R$ (black, solid line), $r = \tau_0 - 0.6\tau_R$ (red, dashed line) and $r = \tau_0 - 1.4\tau_R$ (blue, dotted line); Upper Right: same but for $\beta^E_l(r)$; Lower Left: same, but for $\alpha^T_l(r)$; Lower Right: same, but for $\alpha^E_l(r)$.
• Figure 3: All figures are $(S/N)f^{-1}_{NL}$ vs. $l_{max}$ excluding the effects of gravitational lensing for TTT (dot dashed green), EEE (dashed red), TTT+TTE (dashed light blue), TTT,TTE+TEE (dotted blue) and all bispectra (solid black). Left: WMAP; Right: Planck.
• Figure 4: Plotted is $(S/N)f^{-1}_{NL}$ vs. $l_{max}$ for an ideal experiment (no instrument noise and infinitesimal beam size) without the effects of gravitational lensing for TTT (dot dashed green), EEE (dashed red), TTT+TTE (dashed light blue), TTT,TTE+TEE (dotted blue) and all bispectra (solid black).
• Figure 5: The ratio of the $(S/N)_{GL}$ including the various forms of gravitational lensing to the $(S/N)_0$ excluding the gravitational lensing is shown above. The curves indicate the inclusion in the estimator variance of the four-point function, but not the two-point function (dotted red), both the four and two-point functions (solid black) and two-point function, but not the four-point function (dashed blue).