Fast Estimator of Primordial Non-Gaussianity from Temperature and Polarization Anisotropies in the Cosmic Microwave Background II: Partial Sky Coverage and Inhomogeneous Noise

TL;DR

This work extends a fast cubic bispectrum estimator for the local-type primordial non-Gaussianity amplitude $f_{NL}$ to realistically noisy and incomplete-sky CMB data by incorporating a Monte Carlo–derived linear correction term. The generalized estimator maintains computational efficiency, scaling as $O(N_{pix}^{3/2})$, and achieves near-Fisher-bound variance in simulations that mimic Planck-like noise and sky cuts. It also demonstrates unbiased recovery of $f_{NL}$ and shows substantial variance reductions compared to prior temperature-plus-polarization analyses. The methodology enables robust constraints on primordial non-Gaussianity from Planck and ground-based experiments, while outlining future work on polarization foreground non-Gaussian signals.

Abstract

In our recent paper (Yadav et al. 2007) we described a fast cubic (bispectrum) estimator of the amplitude of primordial non-Gaussianity of local type, f_{NL}, from a combined analysis of the Cosmic Microwave Background (CMB) temperature and E-polarization observations. In this paper we generalize the estimator to deal with a partial sky coverage as well as inhomogeneous noise. Our generalized estimator is still computationally efficient, scaling as O(N^3/2) compared to the O(N^5/2) scaling of the brute force bispectrum calculation for sky maps with N pixels. Upcoming CMB experiments are expected to yield high-sensitivity temperature and E-polarization data. Our generalized estimator will allow us to optimally utilize the combined CMB temperature and E-polarization information from these realistic experiments, and to constrain primordial non-Gaussianity.

Figures (4)

• Figure 1: Testing normalization of the linear term in the estimator of $f_{NL}$. The symbols show the standard deviation of $f_{NL}$ derived from the the Monte Carlo simulations using the estimator for a given normalization, $x$. The horizontal line shows the Fisher matrix prediction. Our formula gives $x=1$, while Creminelli_wmap1 give $x=0.5$ (their Eq. (30)). We have used simulated polarized Gaussian CMB maps with the Planck inhomogeneous noise as well as the WMAP Kp0 and P06 mask for temperature and polarization, respectively.
• Figure 2: Optimality of the generalized estimator. The solid lines show the Fisher matrix prediction for the standard deviation of $f_{NL}$, the triangles show the standard deviation derived from the Monte Carlo simulations using the estimator without the linear term, and the stars show the standard deviation derived from the Monte Carlo simulations using the generalized estimator (i.e., with the linear term). Left panel: The uncertainty vs the maximum multipole that is used in the analysis, $\ell_{max}$. The simulations contain the Gaussian CMB signal, inhomogeneous noise (which simulates the Planck satellite), WMAP Kp0 and P06 masks. Right panel: The uncertainty vs a fraction of the sky observed, $f_{sky}$, for $\ell_{max}=500$. The simulations include the Gaussian CMB signal, and flat sky-cut (which is azimuthally symmetric in the Galactic coordinates), while they do not include instrumental noise. This figure therefore shows that the sky cut contributes significantly to the linear term of polarization.
• Figure 3: The $\langle A_{sim}(\hat{n},r)B_{sim}(\hat{n},r)\rangle_{MC}$ and $\langle B^2_{sim}(\hat{n},r)\rangle_{MC}$ maps in dimension-less units for a slice near the surface of last scattering. These maps are calculated from Monte Carlo simulations with the Gaussian signal, Planck inhomogeneous noise, and WMAP Kp0 and P06 masks.
• Figure 4: The top panels show $\langle A_{sim}(\hat{n},r)B_{sim}(\hat{n},r)\rangle_{MC}$ and $\langle B_{sim}^2(\hat{n},r)\rangle_{MC}$ maps for the noise only analysis (i.e. no CMB signal or mask). The maps are in dimension-less units and are shown for a slice near the surface of last scattering. The Lower map shows the number of observations per pixel ($N_{obs}$) at the resolution of $N_{pix}=12582912$.