Limits on Primordial Non-Gaussianity from Minkowski Functionals of the WMAP Temperature Anisotropies

TL;DR

This work uses perturbative Minkowski Functionals (MFs) to constrain primordial non-Gaussianity, parameterized by $f_{ m NL}$, from the WMAP three-year temperature maps. The MF framework expresses the MFs as $V_k( u)=A_k v_k( u)$, with a Gaussian part and a non-Gaussian correction $igl| abla v_k( u,f_{ m NL})igr|$ driven by skewness parameters $S^{(0,1,2)}$, enabling direct comparison with data via a full covariance-based likelihood. The authors validate the analytical MF predictions against non-Gaussian simulations including full radiative transfer and observational systematics, finding excellent agreement and demonstrating robustness to beam, pixel window, noise, and masking. Applying a maximum-likelihood analysis to the WMAP data yields a 95% C.L. constraint of $-70<f_{ m NL}<91$ (for the Q+V+W map across multiple smoothing scales), consistent with prior bispectrum-based limits while highlighting mild discrepancies that warrant foreground treatment in future work. Overall, MFs provide a robust, complementary probe of primordial non-Gaussianity with strong potential for Planck-era constraints.

Abstract

We present an analysis of the Minkowski Functionals (MFs) describing the WMAP three-year temperature maps to place limits on possible levels of primordial non-Gaussianity. In particular, we apply perturbative formulae for the MFs to give constraints on the usual non-linear coupling constant fNL. The theoretical predictions are found to agree with the MFs of simulated CMB maps including the full effects of radiative transfer. The agreement is also very good even when the simulation maps include various observational artifacts, including the pixel window function, beam smearing, inhomogeneous noise and the survey mask. We find accordingly that these analytical formulae can be applied directly to observational measurements of fNL without relying on non-Gaussian simulations. Considering the bin-to-bin covariance of the MFs in WMAP in a chi-square analysis, we find that the primordial non-Gaussianity parameter is constrained to lie in the range -70<fNL<91 at 95% C.L. using the Q+V+W co-added maps.

Figures (4)

• Figure 1: Comparison between the analytical predictions ( lines) and the numerical estimations averaged over 200 realizations of non-Gaussian simulation maps ( symbols) for $f_{\rm NL}=100$; Upper-left: variances $\sigma_0$ and $\sigma_1$ (eq. [\ref{['eq:var']}]) Upper-right: skewness parameters $S^{(a)}$ ($a=0, 1,$ and $2$) in the equation (\ref{['eq:delmf_pb']}), Middle: MFs for non-Gaussian fields, $V_k$ (eq. [\ref{['eq:mf']}]), Lower: the difference ratio of MFs $\Delta v_k$ (eq. [\ref{['eq:delmf_pb']}]). CMB maps are smoothed with a Gaussian kernel $W_l=\exp[-l(l+1)\theta_s^2/2]$ where $\theta_s$ denotes the smoothing scale. The fully radiative transfer function is considered for both the theoretical predictions and the simulations. The simulations also include the various observational effects for WMAP three-year Q+V+W coadded map; pixel window function, beam smearing, inhomogeneous noise pattern, and Kp0 cut. The error-bars represent the errors for the averaged simulation results over 200 realizations (the sample variance divided by the square-root-of 200).
• Figure 2: The distribution function of the best-fit value of $f_{\rm NL}$ using WMAP three-year mock simulation maps (histogram). We use $200$ realizations of Gaussian simulations (solid) and non-Gaussian simulations with $f_{\rm NL}^{\rm (true)}=100$ (dotted) respectively. The best-fit values are obtained by fitting the analytical formulae (eq. [\ref{['eq:delmf_pb']}]) to all of the MFs for the simulations at $\theta_s=10'$ and $20'$ combined. For comparison, we plot the likelihood function of $f_{\rm NL}$ at $f_{\rm NL}^{\rm (true)}=0$ and 100 with $\sigma_{f_{\rm NL}}=44$ (eq. [\ref{['eq:likefunc_fnl']}]), which is the expected uncertainty of $f_{\rm NL}$ from all of the MFs for CMB maps at $\theta_s=10'$ and $20'$ combined.
• Figure 3: Comparison between MFs for WMAP three-year temperature maps (symbols) and the analytical formulae with the best-fit value of $f_{\rm NL}$ for each MF (lines). The MFs are calculated from the Q+V+W co-added map ( Left) and the V+W map ( Right). Top panels show the MFs $V_k$ ($k$=0,1, and 2) at $\theta_s$=20 arcmin, and other panels illustrate $\Delta v_k$ (eq. [\ref{['eq:delmf_pb']}]) at $\theta_s$=10, 20 and 40 arcmin respectively. Error-bars denote the standard deviation of MFs at each bin of $\nu$ computed from 1000 Gaussian realizations including the WMAP three-year noise distribution, Kp0 mask and pixel and beam window function. The systematics due to the pixelization effect is estimated from the Gaussian realizations and is subtracted from the observed MFs (see eq. [\ref{['eq:mfsub']}]).
• Figure 4: Same as Fig. \ref{['fig:mf3year_GA']} but for different $N_{\rm side}=256, 128$ and $64$ without Gaussian smoothing.