Cosmic shear covariance: The log-normal approximation
  [Abridged] We seek approximations to the cosmic shear covariance that are as
easy to use as the common approximations based on normal statistics, but yield
more accurate covariance matrices and parameter errors. We derive expressions
for the cosmic shear covariance under the assumption that the underlying
convergence field follows log-normal statistics. We also derive a simplified
version of this log-normal approximation. We use numerical simulations of weak
lensing to study how well the normal, log-normal, and simplified log-normal
approximations as well as empirical corrections to the normal approximation
proposed in the literature reproduce shear covariances for cosmic shear
surveys. We find that the normal approximation substantially underestimates the
cosmic shear covariances and the inferred parameter confidence regions, in
particular for surveys with small fields of view and large galaxy densities,
but also for very wide surveys. In contrast, the log-normal approximation
yields more realistic covariances and confidence regions, but also requires
evaluating slightly more complicated expressions. However, the simplified
log-normal approximation, although as simple as the normal approximation,
yields confidence regions that are almost as accurate as those obtained from
the log-normal approximation. The empirical corrections to the normal
approximation do not yield more accurate covariances and confidence regions
than the (simplified) log-normal approximation. Moreover, they fail to produce
positive-semidefinite data covariance matrices in certain cases, rendering them
unusable for parameter estimation. The simplified log-normal approximation
should be used in favour of the normal approximation for parameter estimation
and parameter error forecasts. Any approximation to the cosmic shear covariance
should ensure a positive-semidefinite data covariance matrix.
