Primordial Non-Gaussianity and Analytical Formula for Minkowski Functionals of the Cosmic Microwave Background and Large-scale Structure

TL;DR

This work derives analytical, perturbative formulas for Minkowski Functionals (MFs) of the CMB and large-scale structure (LSS) to quantify primordial non-Gaussianity via a constant $f_{NL}$. By expressing MF corrections through skewness parameters and the bispectrum, the authors connect real-space morphology to Fourier-space statistics, enabling fast Fisher-matrix forecasts without non-Gaussian simulations. The CMB MFs yield competitive constraints on $|f_{NL}|$ (e.g., $\sim$40 for WMAP and $\sim$20 for Planck), while LSS MFs require large-volume surveys to be competitive due to late-time non-Gaussianity from gravity and bias. The framework confirms the complementarity of MFs and the bispectrum, is extendable to scale-dependent $f_{NL}$, and offers a practical route to assess real-space data with masks, noise, and beams.

Abstract

We derive analytical formulae for the Minkowski Functions of the cosmic microwave background (CMB) and large-scale structure (LSS) from primordial non-Gaussianity. These formulae enable us to estimate a non-linear coupling parameter, f_NL, directly from the CMB and LSS data without relying on numerical simulations of non-Gaussian primordial fluctuations. One can use these formulae to estimate statistical errors on f_NL from Gaussian realizations, which are much faster to generate than non-Gaussian ones, fully taking into account the cosmic/sampling variance, beam smearing, survey mask, etc. We show that the CMB data from the Wilkinson Microwave Anisotropy Probe should be sensitive to |f_NL|\simeq 40 at the 68% confidence level. The Planck data should be sensitive to |f_NL|\simeq 20. As for the LSS data, the late-time non-Gaussianity arising from gravitational instability and galaxy biasing makes it more challenging to detect primordial non-Gaussianity at low redshifts. The late-time effects obscure the primordial signals at small spatial scales. High-redshift galaxy surveys at z>2 covering \sim 10Gpc^3 volume would be required for the LSS data to detect |f_NL|\simeq 100. Minkowski Functionals are nicely complementary to the bispectrum because the Minkowski Functionals are defined in real space and the bispectrum is defined in Fourier space. This property makes the Minksowski Functionals a useful tool in the presence of real-world issues such as anisotropic noise, foreground and survey masks. Our formalism can be extended to scale-dependent f_NL easily.

Figures (5)

• Figure 1: ( Left) Skewness parameters, $S^{(a)}$, for $a=0$ (solid), 1 (dotted), and 2 (dashed). ( Right) Skewness parameters multiplied by variance, $S^{(a)}\sigma_0$, for $a=0$ (solid), 1 (dotted), and 2 (dashed). Both have been divided by $f_{\rm NL}$, and are plotted as a function of a Gaussian smoothing width, $\theta_s$. Note that $S^{(2)}$ changes its sign at $\theta_s\sim 40$ arcmin.
• Figure 2: Analytical predictions for the Minkowski Functionals of CMB temperature anisotropy with primordial non-Gaussianity characterized by $f_{\rm NL}=-100$, $-50$ (dotted), 0, 50, and 100 (solid). Each MF, $V_{k}$ ($k=0$, 1, and 2), is plotted in the top panels. The other panels show the difference between non-Gaussian and Gaussian MFs, $V_k^G$, divided by the maximum amplitude of $V_k^G$. From the top to bottom, $\theta_s=10$, 20, 40, 70, and 100 arcmin are shown.
• Figure 3: ( Left) Skewness parameters, $S^{(a)}(z)$, for $a=0$ (solid), 1 (dotted), and 2 (dashed), at $z=0$, 2, 5. ( Right) Skewness parameters multiplied variance, $S^{(a)}(z)\sigma_0$, for $a=0$ (solid), 1 (dotted), and 2 (dashed), at $z=0$, 2, 5. In both figures, the thick lines show the skewness parameters from non-linear gravitational clustering, $S_{\rm grav}^{(a)}$ (Eq. [\ref{['eq:sgrav0']}--\ref{['eq:sgrav2']}]), while the thin lines show the primordial skewness parameters, $S_{\rm prim}^{(a)}$, with $f_{\rm NL}=100$ (and $10$ in the right panel). The symbols show the skewness parameters measured from numerical simulations of non-Gaussian matter density fluctuations. Note that $S_{\rm grav}^{(a)}$ and $S_{\rm prim}^{(a)}\sigma_0$ are independent of $z$.
• Figure 4: Analytical predictions for the Minkowski Functionals of LSS with primordial non-Gaussianity characterized by $f_{\rm NL}=100$, 50, 0 (solid), $-50$, and $-100$ (dotted), for a Gaussian smoothing length of $R=100~h^{-1}$ Mpc. The left panels show the MFs $V_k$ ($k=0$, 1, 2, and 3 from top to bottom), while the right panels show the difference between non-Gaussian and Gaussian MFs, $V_k^{G}$, divided by the maximum amplitude of $V_k^{G}$. The symbols show the average and error of the MFs calculated from 2000 realizations of simulated primordial non-Gaussian density fluctuations with $f_{\rm NL}=100$.
• Figure 5: Comparison between the MFs calculated from the analytical perturbation predictions (solid lines) and the numerical simulations (symbols) in the Sachs-Wolfe limit. We have used $\theta_s=100$ arcmin and $f_{\rm NL}=100$.