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