Euclid preparation. Non-Gaussianity of 2-pt statistics likelihood: Parameter inference with a non-Gaussian likelihood in Fourier and configuration space

Abstract

In this work we account for this skewness in parameter inference by modelling the likelihood through an Edgeworth expansion which involves the complete skewness tensor, composed of 1-point, 2-point, and 3-point correlators. To simplify the calculations of this expansion we perform a change of basis which reduces the precision matrix to the identity. In this basis, the off-diagonal elements of the skewness tensor are consistent with zero, while the amplitude of its diagonal match the level expected for a Gaussian underlying field. We perform parameter inference with this likelihood model and find that including only the diagonal part of the skewness is sufficient, while incorporating the full skewness tensor injects noise without improving accuracy. Despite the estimated excess skewness in the original basis, the cosmological constraints remain effectively unchanged when adopting a Gaussian likelihood or considering the more complete Edgeworth expansion, with variations in the figure of merit of cosmological parameters between the two cases below $5\%$. This result remains unchanged against variations of the survey volume and geometry, scale-cut, and 2-point statistic (power spectrum or correlation function). Using $10\, 000$ cloned \Euclid large mocks based on realistic galaxy catalogues with characteristics close to future \Euclid data, we find no detectable excess skewness on intermediate scales, due to the level of shot noise expected for the \Euclid spectroscopic sample. We conclude that the Gaussian likelihood assumption is robust for \Euclid 2-point statistics analyses in both Fourier and configuration space.

Figures (15)

• Figure 1: Validation of the forward modelling of the window function on the power spectrum for all geometries. Top: Mean of the power spectra estimated from the $10\, 000$COVMOS realisations for each geometry (squares with error bars) and the model convolved with the respective window function (lines). The vertical dashed lines represent the effective fundamental mode for each geometry. Bottom: Relative difference between the mean COVMOS power spectra and the convolved models for $z=0$ (solid lines) and $z=0.5$ (dashed lines). The error bars represent the error on the mean over the $10\, 000$ realisations (we only show it for $z=0$ to improve readability). The green area represents $10$% of the error on a single realisation for S200 (we only show it for this case).
• Figure 2: Validation of the forward modelling of the window function on the 2PCF for all geometries. Top: Mean over the 2PCFs estimated from the $10\, 000$COVMOS realisations for each geometry (squares with error bars) and the model (line), in absolute value. The error bars represent the error on the mean over the $10\, 000$ realisations. Bottom: Residual in number of $\sigma$ considering $10$% of the error on one realisation for each geometry.
• Figure 3: Comparison of the estimated skewness and the expectation in the case of a Gaussian field. Top: Skewness of the power spectrum distribution estimated on the $10\, 000$COVMOS realisations for each of the four geometries at $z=0$. The dashed lines show the skewness predicted in the case of a Gaussian field. Bottom: Difference between the estimated skewness and the prediction for a Gaussian field.
• Figure 4: Skewness of the 2PCF distribution estimated from the $10\, 000$COVMOS realisations for each of the four geometries, at $z=0$.
• Figure 5: Matrix of the reduced $2$-point correlators at order $3$ of the power spectrum (left panel) and 2PCF (right panel), estimated with Eq. \ref{['estimk']}, for the S200 geometry. In each panel the upper left elements are showing the correlators in the standard basis ($S_3^{ijj}=\langle x_ix_j^2\rangle_\mathrm{c}\,\sigma_i^{-1}\,\sigma_j^{-2}$) while the lower right shows the corresponding matrix when estimated after turning to the basis in which the covariance matrix is the identity ($S_3^{ijj}=\langle y_iy_j^2\rangle_\mathrm{c}\,\sigma_i^{-1}\,\sigma_j^{-2}$).