Measuring primordial non-Gaussianity in the cosmic microwave background

TL;DR

This paper introduces a fast, near-optimal cubic statistic for measuring primordial non-Gaussianity in nearly full-sky CMB maps, dramatically reducing computational cost compared with the full bispectrum while preserving sensitivity. It builds a Wiener-filtered reconstruction of the underlying primordial fluctuations and defines A and B maps to form a cubic statistic S_prim that directly constrains f_NL with $N^{1.5}$ scaling. The method extends to mixed adiabatic/isocurvature fluctuations and provides a fast estimator for foreground point-source non-Gaussianity, plus a simple treatment for incomplete sky coverage. The approach enables extensive simulations and robust separation of primordial and foreground signals, with broad applicability to inflationary scenarios and early-universe physics.

Abstract

We derive a fast way for measuring primordial non-Gaussianity in a nearly full-sky map of the cosmic microwave background. We find a cubic combination of sky maps combining bispectrum configurations to capture a quadratic term in primordial fluctuations. Our method takes only N^1.5 operations rather than N^2.5 of the bispectrum analysis (1000 times faster for l=512), retaining the same sensitivity. A key component is a map of underlying primordial fluctuations, which can be more sensitive to the primordial non-Gaussianity than a temperature map. We also derive a fast and accurate statistic for measuring non-Gaussian signals from foreground point sources. The statistic is 10^6 times faster than the full bispectrum analysis, and can be used to estimate contamination from the sources. Our algorithm has been successfully applied to the Wilkinson Microwave Anisotropy Probe sky maps by Komatsu et al. (2003).

Figures (2)

• Figure 1: Wiener filters for the primordial fluctuations applied to a CMB sky map, $O_l(r)=\beta_l(r)/C_l$ [Eq. (\ref{['eq:Ol']})]. We plot (a) $O_l$ for an adiabatic SCDM ($\Omega_m=1$, $\Omega_\Lambda=0$, $\Omega_b=0.05$, $h=0.5$), (b) for an adiabatic $\Lambda$CDM ($\Omega_m=0.3$, $\Omega_\Lambda=0.7$, $\Omega_b=0.04$, $h=0.7$), (c) for an isocurvature SCDM, and (d) for an isocurvature $\Lambda$CDM. The filters are plotted at five conformal distances $r=c(\tau_0-\tau)$ as explained in the bottom-right panel. Here $\tau$ is the conformal time ($\tau_0$ at the present). The SCDM models have $c\tau_0=11.84$ Gpc and $c\tau_{dec}=0.235$ Gpc, while the $\Lambda$CDM models $c\tau_0=13.89$ Gpc and $c\tau_{dec}=0.277$ Gpc, where $\tau_{dec}$ is the photon decoupling epoch.
• Figure 2: Performance of cubic statistics. The left panels and right panels show errors of $f_{NL}$ and $b_{src}$, respectively, which are obtained from 300 Gaussian simulations. Each point has been computed for a given straight sky cut with $|b|_{cut}$. From the right to left, $|b|_{cut}=0$, 10, 20, 30, 40, 50, 60, 70, and $80^\circ$. The solid lines show the minimum variance which would be obtained by the full bispectrum analysis. The top, middle, and bottom panels show simulations of the CMB signal only, CMB plus homogeneous noise, and CMB plus inhomogeneous noise, respectively. Noise properties assume WMAP 1-year data in V band.