Non-Gaussian likelihood of weak lensing power spectra

The power spectrum of weak lensing fluctuations has a non-Gaussian distribution due to its quadratic nature. On small scales the Central Limit Theorem acts to Gaussianize this distribution but non-Gaussianity in the signal due to gravitational collapse is increasing and the functional form of the likelihood is unclear. Analyses have traditionally assumed a Gaussian likelihood with non-linearity incorporated into the covariance matrix; here we provide the theory underpinning this assumption. We calculate, for the first time, the leading-order correction to the distribution of angular power spectra from non-Gaussianity in the underlying signal and study the transition to Gaussianity. Our expressions are valid for an arbitrary number of correlated maps and correct the Wishart distribution in the presence of weak (but otherwise arbitrary) non-Gaussianity in the signal. Surprisingly, the resulting distribution is not equivalent to an Edgeworth expansion. The leading-order effect is to broaden the covariance matrix by the usual trispectrum term, with residual skewness sourced by the trispectrum and the square of the bispectrum. Using lognormal lensing maps we demonstrate that our likelihood is uniquely able to model both large and mildly non-linear scales. We provide easy-to-compute statistics to quantify the size of the non-Gaussian corrections. We show that the full non-Gaussian likelihood can be accurately modelled as a Gaussian on small, non-linear scales. On large angular scales non-linearity in the lensing signal imparts a negligible correction to the likelihood, which takes the Wishart form in the full-sky case. Our formalism is equally applicable to any kind of projected field.