CMB Constraints on Primordial non-Gaussianity from the Bispectrum (f_{NL}) and Trispectrum (g_{NL} and τ_{NL}) and a New Consistency Test of Single-Field Inflation

TL;DR

The paper develops and applies skewness and kurtosis power-spectrum estimators to CMB data to constrain local-type primordial non-Gaussianity, focusing on $f_{\rm NL}$ from the bispectrum and $\tau_{\rm NL}$, $g_{\rm NL}$ from the trispectrum. It introduces a window-function approach to correct for masks without relying on computationally heavy linear terms, and proposes the consistency test $A_{\rm NL} = \tau_{\rm NL}/(6 f_{\rm NL}/5)^2$ to differentiate single-field from multi-field inflation. Through theoretical Fisher forecasts and application to WMAP5 data, the study quantifies expected improvements for Planck and EPIC and reports current constraints with $g_{\rm NL}$ and $\tau_{\rm NL}$ consistent with zero within 95% C.L. The results underscore the trispectrum’s potential to reveal multi-field dynamics or self-interactions, and highlight the pivotal role of accurate mask handling in non-Gaussianity analyses.

Abstract

We outline the expected constraints on non-Gaussianity from the cosmic microwave background (CMB) with current and future experiments, focusing on both the third (f_{NL}) and fourth-order (g_{NL} and τ_{NL}) amplitudes of the local configuration or non-Gaussianity. The experimental focus is the skewness (two-to-one) and kurtosis (two-to-two and three-to-one) power spectra from weighted maps. In adition to a measurement of τ_{NL} and g_{NL} with WMAP 5-year data, our study provides the first forecasts for future constraints on g_{NL}. We describe how these statistics can be corrected for the mask and cut-sky through a window function, bypassing the need to compute linear terms that were introduced for the previous-generation non-Gaussianity statistics, such as the skewness estimator. We discus the ratio A_{NL} = τ_{NL}/(6f_{NL}/5)^2 as an additional test of single-field inflationary models and discuss the physical significance of each statistic. Using these estimators with WMAP 5-Year V+W-band data out to l_{max}=600 we constrain the cubic order non-Gaussianity parameters τ_{NL}, and g_{NL} and find -7.4 < g_{NL}/10^5 < 8.2 and -0.6 < τ_{NL}/10^4 < 3.3 improving the previous COBE-based limit on τ_{NL} < 10^8 nearly four orders of magnitude with WMAP.

Figures (17)

• Figure 1: Four point correlation function for the $\phi^3$ theory. The correlation functions breaks up into interaction-less unconnected diagrams and connected diagrams containing information about the interactions.
• Figure 2: Plot of the shape functions $S^{\rm local}(1, k_2, k_3)$ and $S^{\rm equil}(1, k_2, k_3)$ normalized such that $S(1, 1, 1) = 1$. In these plots only values satisfying the triangle inequality $k_2 + k_3 \geq k_1 = 1$ as well as the requirement $k_2 \leq k_3$ to prevent showing equivalent configurations are non-zero. The plot on top verified $S^{local}$ is maximized when $k_1 \sim k_3 \gg k_2$ whereas the bottom plot verifies $S^{equal}$ is maximised when $k_1 \sim k_2 \sim k_3$.
• Figure 3: The top plot compares various $\beta(r)$ for different $\tau_{*}$ and the bottom is the same for $\alpha(r)$.
• Figure 4: Beam transfer functions. The frequency band used for each experiment is in brackets.
• Figure 5: Noise and $b_l$ relation, $N_l/b_l^2$, for each experiment plotted against $C_l$ taken from WMAP 7-Year best fit parameters. The frequency band used for each experiment is in brackets.