First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Tests of Gaussianity

TL;DR

This paper tests the Gaussianity of primordial fluctuations in the WMAP 1-year CMB data by constraining the non-Gaussian coupling $f_{\rm NL}$ using two independent methods: the angular bispectrum and Minkowski functionals. The results show no evidence for significant non-Gaussianity, with $f_{\rm NL}=38\pm 48$ (68%) from the bispectrum and $f_{\rm NL}=22\pm 81$ (68%) from Minkowski functionals, corresponding to a 95% CL range roughly $-58<f_{\rm NL}<134$ and $f_{\rm NL}<139$, respectively. The findings are consistent with simple slow-roll inflation predictions ($|f_{\rm NL}|\sim 10^{-2}$–$10^{-1}$) and imply modest modifications to high-redshift cluster abundances; the SZ power spectrum remains largely unaffected by any allowed non-Gaussianity. The authors also quantify residual point-source contributions via the reduced bispectrum $b_{\rm src}$ and the related power spectrum $c_{\rm src}$, finding results in agreement with source-count models and independent measurements. Together, these analyses validate that the CMB power spectrum captures the relevant statistics for Gaussianity in the WMAP data and set the stage for tighter constraints with future data.

Abstract

We present limits to the amplitude of non-Gaussian primordial fluctuations in the WMAP 1-year cosmic microwave background sky maps. A non-linear coupling parameter, f_NL, characterizes the amplitude of a quadratic term in the primordial potential. We use two statistics: one is a cubic statistic which measures phase correlations of temperature fluctuations after combining all configurations of the angular bispectrum. The other uses the Minkowski functionals to measure the morphology of the sky maps. Both methods find the WMAP data consistent with Gaussian primordial fluctuations and establish limits, -58<f_NL<134, at 95% confidence. There is no significant frequency or scale dependence of f_NL. The WMAP limit is 30 times better than COBE, and validates that the power spectrum can fully characterize statistical properties of CMB anisotropy in the WMAP data to high degree of accuracy. Our results also validate the use of a Gaussian theory for predicting the abundance of clusters in the local universe. We detect a point-source contribution to the bispectrum at 41 GHz, b_src = (9.5+-4.4) X 1e-5 uK^3 sr^2, which gives a power spectrum from point sources of c_src = (15+-6) X 1e-3 uK^2 sr in thermodynamic temperature units. This value agrees well with independent estimates of source number counts and the power spectrum at 41 GHz, indicating that b_src directly measures residual source contributions.

Figures (9)

• Figure 1: The non-linear coupling parameter $f_{\rm NL}$ as a function of the maximum multipole $l_{\rm max}$, measured from the Q$+$V$+$W coadded map using the cubic (bispectrum) estimator [Eq. (\ref{['eq:f_NL']})]. The best constraint is obtained from $l_{\rm max}=265$. The distribution is cumulative, so that the error bars at each $l_{\rm max}$ are not independent.
• Figure 2: The left panels show the Minkowski functionals for WMAP data (filled circles) at $nside=128$ (28$\arcmin$ pixels). The gray band shows the 68% confidence interval for the Gaussian Monte Carlo simulations. The right panels show the residuals between the mean of the Gaussian simulations and the WMAP data. The WMAP data are in excellent agreement with the Gaussian simulations.
• Figure 3: Limits to $f_{\rm NL}$ from $\chi^2$ fit of the WMAP data to the non-Gaussian models [Eq. (\ref{['eq:phi']})]. The fit is a joint analysis of the three Minkowski functionals at $28\arcmin$ pixel resolution. There are 44 degrees of freedom.
• Figure 4: The limits to the effect of the primordial non-Gaussianity on the dark-matter halo mass function $dn/dM$ as a function of $z$. The shaded area represents the 95% constraint on the ratio of the non-Gaussian $dn/dM$ to the Gaussian one.
• Figure 5: The same as figure \ref{['fig:dndm']} but for the dark-matter halo number counts $dN/dz$ as a function of the limiting mass $M_{\rm lim}$ of a survey.