On the insufficiency of arbitrarily precise covariance matrices: non-Gaussian weak lensing likelihoods

Abstract: We investigate whether a Gaussian likelihood, as routinely assumed in the analysis of cosmological data, is supported by simulated survey data. We define test statistics, based on a novel method that first destroys Gaussian correlations in a dataset, and then measures the non-Gaussian correlations that remain. This procedure flags pairs of datapoints which depend on each other in a non-Gaussian fashion, and thereby identifies where the assumption of a Gaussian likelihood breaks down. Using this diagnostic, we find that non-Gaussian correlations in the CFHTLenS cosmic shear correlation functions are significant. With a simple exclusion of the most contaminated datapoints, the posterior for $s_8$ is shifted without broadening, but we find no significant reduction in the tension with $s_8$ derived from Planck Cosmic Microwave Background data. However, we also show that the one-point distributions of the correlation statistics are noticeably skewed, such that sound weak lensing data sets are intrinsically likely to lead to a systematically low lensing amplitude being inferred. The detected non-Gaussianities get larger with increasing angular scale such that for future wide-angle surveys such as Euclid or LSST, with their very small statistical errors, the large-scale modes are expected to be increasingly affected. The shifts in posteriors may then not be negligible and we recommend that these diagnostic tests be run as part of future analyses.