Hunting for Primordial Non-Gaussianity in the Cosmic Microwave Background

TL;DR

The paper surveys how the CMB can reveal primordial non-Gaussianity through higher-order statistics, focusing on the bispectrum and trispectrum as probes of inflationary physics. It presents the formalism for optimal f_NL estimators for local, equilateral, and orthogonal shapes, analyzes secondary and second-order contaminants, and highlights the trispectrum as a decisive test for multi-field models via $ au_{ m NL}$ and $g_{ m NL}$. Current results from WMAP7 provide constraints on the bispectrum with no definitive detection, while outlining the dominant contaminants (lensing-ISW, foregrounds) and the need for careful templates. Planck-era expectations are emphasized, with the potential to strongly constrain or rule out large classes of inflationary models, and the authors discuss complementary approaches and future directions in this rapidly evolving field.

Abstract

Since the first limit on the (local) primordial non-Gaussianity parameter, fNL, was obtained from COBE data in 2002, observations of the CMB have been playing a central role in constraining the amplitudes of various forms of non-Gaussianity in primordial fluctuations. The current 68% limit from the 7-year WMAP data is fNL=32+/-21, and the Planck satellite is expected to reduce the uncertainty by a factor of four in a few years from now. If fNL>>1 is found by Planck with high statistical significance, all single-field models of inflation would be ruled out. Moreover, if the Planck satellite finds fNL=30, then it would be able to test a broad class of multi-field models using the four-point function (trispectrum) test of tauNL>=(6fNL/5)^2. In this article, we review the methods (optimal estimator), results (WMAP 7-year), and challenges (secondary anisotropy, second-order effect, and foreground) of measuring primordial non-Gaussianity from the CMB data, present a science case for the trispectrum, and conclude with future prospects.

Figures (4)

• Figure 1: The angular power spectrum of the CMB temperature anisotropy, $C_l$, measured from the WMAP 7-year data larson/etal:prep, along with the temperature power spectra from the ACBAR reichardt/etal:2009 and QUaD brown/etal:2009 experiments. The solid line shows the best-fitting 6-parameter flat $\Lambda$CDM model to the WMAP data alone. The angular power spectrum contains all the information on fluctuations in the CMB, if fluctuations are Gaussian. If fluctuations are non-Gaussian, one must use the higher-order correlation functions (such as three- and four-point functions) to fully exploit the cosmological information contained in the CMB. This figure is adopted from komatsu/etal:prep.
• Figure 2: Visual representations of triangles forming the bispectrum, $B_\Phi(k_1,k_2,k_3)$, with various combinations of wavenumbers satisfying $k_3\le k_2\le k_1$. This figure is adopted from jeong/komatsu:2009.
• Figure 3: Shapes of the primordial bispectra. Each panel shows the normalized amplitude of $F(k_1,k_2,k_3)(k_2/k_1)^2(k_3/k_1)^2$ as a function of $k_2/k_1$ and $k_3/k_1$ for a given $k_1$, with a condition that $k_3\le k_2\le k_1$ is satisfied. As the primordial bispectra shown here are (nearly) scale invariant, the shapes look similar regardless of the values of $k_1$. The amplitude is normalized such that it is unity at the point where $F(k_1,k_2,k_3)(k_2/k_1)^2(k_3/k_1)^2$ takes on the maximum value. (Top Left) The local form given in equation \ref{['eq:Flocal']}, which peaks at the squeezed configuration. Note that the most squeezed configuration shown here has $k_1=k_2=100k_3$. (Top Right) The orthogonal form given in equation \ref{['eq:Forthog']}, which has a positive peak at the equilateral configuration, and a negative valley along the elongated configurations. (Bottom Left) The equilateral form given in equation \ref{['eq:Fequil']}, which peaks at the equilateral configuration. Note that all of these shapes are nearly orthogonal to each other.
• Figure 4: Shapes of the second-order bispectrum due to the second-order curvature perturbations in the Newtonian limit given in equation \ref{['eq:2nd']}. Each panel shows the normalized amplitude of $F_{\rm 2nd}(k_1,k_2,k_3)(k_2/k_1)^2(k_3/k_1)^2$ as a function of $k_2/k_1$ and $k_3/k_1$ for a given $k_1$, with a condition that $k_3\le k_2\le k_1$ is satisfied. The amplitude is normalized such that it is unity at the point where $F_{\rm 2nd}(k_1,k_2,k_3)(k_2/k_1)^2(k_3/k_1)^2$ takes on the maximum value. (Top Left) $k_1=10^{-3}~h~{\rm Mpc}^{-1}$. (Top Right) $k_1=10^{-2}~h~{\rm Mpc}^{-1}$. (Bottom Left) $k_1=10^{-1}~h~{\rm Mpc}^{-1}$. (Bottom Right) $k_1=1~h~{\rm Mpc}^{-1}$. The CMB data are sensitive to $k_1<k_{\rm max}\sim 0.2~h~{\rm Mpc}^{-1}(l_{\rm max}/2000)$, where the second-order bispectrum peaks at the equilateral configuration on large scales, and peaks along the elongated configurations on a smaller scale ($k_1\sim 0.1~h~{\rm Mpc}^{-1}$). On a very small scale ($k_1\sim 1~h~{\rm Mpc}^{-1}$), it peaks at the squeezed configuration. Note that the most squeezed configuration shown here has $k_1=k_2=100k_3$.