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

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.