  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.
