Euclid preparation. Non-Gaussianity of 2-point statistics likelihood: Precise analysis of the matter power spectrum distribution

TL;DR

This study shows that the distribution of matter power-spectrum multipoles under realistic nonlinear evolution is distinctly non-Gaussian and that this non-Gaussianity is tied to higher-order density-field statistics, notably the trispectrum and pentaspectrum. Using 100,000 COVMOS mock realizations in real and redshift space, the authors derive analytical expressions linking the third cumulant of the estimator to $P$, $B$, $T$, and $Q$, and quantify how redshift-space distortions, survey geometry, Fourier binning, shot noise, and the integral constraint affect the skewness. They demonstrate that the dominant source of non-Gaussianity on intermediate scales is the pentaspectrum, while the bispectrum contribution is suppressed by geometric considerations; per-mode statistics remain exponential, indicating inter-mode correlations drive shell-averaged non-Gaussianity. Validation against N-body simulations confirms COVMOS reliably captures these features, with implications for Euclid-like likelihood analyses, including a companion work showing minimal bias in parameter inference under non-Gaussian likelihoods.

Abstract

We investigate the non-Gaussian features in the distribution of the matter power spectrum multipoles. Using the COVMOS method, we generate 100\,000 mock realisations of dark matter density fields in both real and redshift space across multiple redshifts and cosmological models. We derive an analytical framework linking the non-Gaussianity of the power spectrum distribution to higher-order statistics of the density field, including the trispectrum and pentaspectrum. We explore the effect of redshift-space distortions, the geometry of the survey, the Fourier binning, the integral constraint, and the shot noise on the skewness of the distribution of the power spectrum measurements. Our results demonstrate that the likelihood of the estimated matter power spectrum deviates significantly from a Gaussian assumption on nonlinear scales, particularly at low redshift. This departure is primarily driven by the pentaspectrum contribution, which dominates over the trispectrum at intermediate scales. We also examine the impact of the finiteness of the survey geometry in the context of the Euclid mission and find that both the shape of the survey and the integral constraint amplify the skewness.

Figures (13)

• Figure 1: Estimated skewness $S_3^{(\ell)}$ (left panels) and kurtosis $S_4^{(\ell)}$ (right panels) of the distribution of power spectrum multipoles $P^{(\ell)}(k)$ in real space for the reference $\Lambda$CDM cosmology and for the 5 redshifts considered in this work (from top to bottom panels $\ell = 0,2,4$). In each panel the black line shows the prediction for a Gaussian field.
• Figure 2: Same as Fig.\ref{['fig:S3S4multipolesreal']} but now in redshift space.
• Figure 3: The number of triplets $\vec{k}_1$, $\vec{k}_2$, $\vec{k}_3$ per $k$-shell depending on the considered configuration $N_1$, $N_2$, $N_3$ and $N_4$. The purple diamonds are showing the total number of triplets and the black line is showing the corresponding expected number.
• Figure 4: Measurement of $r_3$ from the $\Lambda$CDM simulation at z=0 and its different contributions. The solid line is showing the total relative difference $r_3$ while the dashed and dot-dashed lines are showing the trispectrum and bispectrum contribution, $3r_2$, and $b$ respectively. The black dotted line shows a rough estimation of $b$ based on Eq. \ref{['aproxr3']}.
• Figure 5: Relative excess skewness $r_3$ which is made of two contributions $3r_2$ and $r_3-3r_2$ respectively in dashed and solid lines. The colours correspond to the different redshifts. For clarity of the figure we do not show $r_3 - 3r_2$ for $z=2$ as it is compatible with 0. There are two horizontal black dotted lines showing the levels $1$ and $3$.