On the assumption of Gaussianity for cosmological two-point statistics and parameter dependent covariance matrices

Abstract

In this note we revisit the Fisher information content of cosmological power spectra or two-point functions of Gaussian fields in order to comment on the assumption of Gaussian estimators and the use of parameter dependent covariance matrices for parameter inference in the context of precision cosmology. We discuss that despite the fact that the assumption of a Gaussian likelihood is motivated by the central limit theorem, it leads if used consistently to a Fisher information content that violates the Cramer-Rao bound, due to the presence of independent but artificial information from the parameter dependent covariance matrix. At any fixed multipole, this artificial term is shown to become dominant in the limit of a large number of correlated fields. While the distribution of the estimators does indeed tend to a Gaussian with a large number of modes, it is shown, however, that its Fisher information content does not, in the sense that their covariance matrix never carries independent information content, precisely because of the non-Gaussian shape of the distribution. We discuss in this light the use of parameter dependent covariance matrices with Gaussian likelihoods for parameter inference from two-point statistics. As a rule of thumb, Gaussian likelihoods should always be used with a covariance matrix fixed in parameter space, since only this guarantees that a conservative information content is assigned to the observables, as well as preventing the apparition of biases.