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

1. Problem Statement

Cosmological parameter inference from large-scale structure (LSS) surveys like Euclid typically relies on the assumption that the likelihood of two-point statistics (specifically the matter power spectrum) follows a multivariate Gaussian distribution. However, this assumption is theoretically flawed:

• Non-Gaussian Fields: The non-linear clustering of matter generates non-vanishing higher-order statistics (bispectrum, trispectrum, etc.), which induce non-Gaussian features in the distribution of the power spectrum.
• Low Mode Counts: On large scales, the number of independent Fourier modes is low, causing the distribution to deviate from Gaussianity even for Gaussian fields (following a $\chi^2$ distribution).
• Risk: If the likelihood is incorrectly modeled as Gaussian when the data is non-Gaussian, it could bias cosmological parameter constraints.

The paper aims to quantify the skewness and kurtosis of the matter power spectrum distribution, identify the physical mechanisms driving these non-Gaussianities, and determine how survey settings (redshift space, binning, mask, shot noise) influence them.

2. Methodology

Analytical Framework

The authors derived an analytical framework linking the cumulants of the power spectrum estimator ($\hat{P}^{(\ell)}(k)$) to higher-order correlation functions of the density field:

• Variance: Linked to the power spectrum ($P$) and the trispectrum ($T$).
• Skewness (3rd Cumulant): Linked to the bispectrum ($B$), trispectrum ($T$), and pentaspectrum ($Q$).
• Kurtosis (4th Cumulant): Linked to even higher-order statistics.
• They defined the reduced skewness $S_3^{(\ell)}$ and kurtosis $S_4^{(\ell)}$ to compare against Gaussian predictions.

Simulations (COVMOS)

To test these analytical predictions, the authors used the COVMOS method (a fast, approximate mock catalog generator) rather than computationally expensive N-body simulations.

• Scale: Generated 100,000 realizations of dark matter density fields.
• Configurations:
  • Cosmologies: $\Lambda$CDM, Massive Neutrinos ($16\nu$), and Dynamical Dark Energy ($w9p3$).
  • Redshifts: $z = 0, 0.5, 1, 1.5, 2$.
  • Spaces: Both Real Space and Redshift Space (including Redshift Space Distortions, RSD).
  • Volumes: Periodic boxes ($1 \text{ Gpc}^3/h^3$) and masked "cone" geometries to simulate survey footprints.
• Validation: The COVMOS results were validated against a subset of 12,288 N-body simulations (DEUS-PUR), showing robust agreement up to $k \sim 0.8 \, h/\text{Mpc}$.

Estimators

Power spectra were estimated using NBodyKit, testing various settings:

• Mass Assignment Schemes (CIC vs. PCS).
• Aliasing correction (Interlacing).
• Grid resolution ($N_{mesh} = 128$ vs. $512$).
• Fourier mode binning ($\Delta k$).

3. Key Contributions & Results

A. Dominance of Higher-Order Statistics

The analysis of the skewness decomposition (Eq. 18) revealed:

• Bispectrum ($B$): The contribution of the bispectrum to the skewness is negligible (orders of magnitude smaller than other terms) because the shell-averaging process filters out most triangular configurations, leaving only nearly equilateral ones.
• Trispectrum ($T$): Dominates the non-Gaussian skewness at intermediate scales and high redshifts.
• Pentaspectrum ($Q$): Becomes the dominant driver of skewness at small scales (high $k$) and low redshifts ($z=0$).
• Conclusion: The excess skewness is primarily driven by the pentaspectrum contribution on non-linear scales, surpassing the trispectrum.

B. Impact of Survey Settings

• Redshift Space Distortions (RSD): RSD significantly suppresses the excess skewness of the monopole (by a factor of $\sim 10$) but transfers the non-Gaussian signal to the quadrupole and hexadecapole multipoles.
• Cosmological Model: Varying cosmological parameters (massive neutrinos, dynamical dark energy) has no significant impact on the skewness, provided the power spectrum amplitude remains similar.
• Aliasing & Mass Assignment: Numerical settings (interlacing, mass assignment schemes, grid resolution) have no detectable effect on the skewness, confirming the results are physical, not numerical artifacts.
• Fourier Binning ($\Delta k$): Increasing the bin width increases the excess skewness (up to $\sim 20\%$) on intermediate scales ($0.15 < k < 0.4 \, h/\text{Mpc}$) because it increases the number of modes and the shell-averaged higher-order terms.
• Shot Noise: Increasing shot noise (lowering number density) reduces the excess skewness. When shot noise dominates, the distribution approaches a Poissonian limit where skewness decreases. For Euclid's expected density, skewness is reduced by $\sim 50\%$ at $k \sim 0.4 \, h/\text{Mpc}$.

C. Survey Geometry and Integral Constraint

• Survey Mask: Applying a survey mask (cone geometry) increases the skewness significantly compared to a periodic box. This is due to the correlation of Fourier modes induced by the window function.
• Integral Constraint (IC): The IC (arising from estimating the mean density within the survey volume) introduces an additional excess skewness on small scales. This effect is more pronounced for smaller survey volumes.
• Euclid Context: For Euclid's first data release (DR1), the survey volume is large enough ($\sim 1.5 \text{ Gpc}^3/h^3$) that the IC contribution to skewness is subdominant.

D. Origin of Non-Gaussianity

The authors investigated the distribution of individual Fourier modes ($X_i = |\delta_k|^2$).

• Finding: Even in a non-Gaussian density field, the distribution of individual $X_i$ follows an exponential distribution (consistent with a Gaussian field).
• Implication: The non-Gaussianity of the power spectrum (the sum of modes) is not caused by the non-Gaussianity of individual modes, but rather by the correlations between modes within a $k$-shell (captured by the trispectrum and pentaspectrum).

4. Significance and Conclusion

• Validation of COVMOS: The study confirms that COVMOS is a reliable tool for studying not just the covariance, but the full non-Gaussian distribution of power spectra, including high-order moments.
• Likelihood Modeling: The paper demonstrates that the likelihood of the matter power spectrum is significantly non-Gaussian on non-linear scales, particularly at low redshift. The deviation is driven by the pentaspectrum.
• Implications for Euclid:
  • While the distribution is non-Gaussian, the companion paper (GB-2025) indicates that assuming a Gaussian likelihood does not introduce significant biases in cosmological parameter estimation for Euclid-like datasets.
  • However, survey design choices (binning, mask size, shot noise) can modulate the level of non-Gaussianity.
  • The integral constraint must be accounted for in small-volume analyses to avoid overestimating skewness.

In summary, this work provides a rigorous analytical and numerical characterization of the non-Gaussian features of the power spectrum, identifying the pentaspectrum as the key driver of skewness on small scales and clarifying the role of survey geometry and noise in shaping the likelihood function.