Euclid: Covariance of weak lensing pseudo-$C_\ell$ estimates.
  Calculation, comparison to simulations, and dependence on survey geometry
  An accurate covariance matrix is essential for obtaining reliable
cosmological results when using a Gaussian likelihood. In this paper we study
the covariance of pseudo-$C_\ell$ estimates of tomographic cosmic shear power
spectra. Using two existing publicly available codes in combination, we
calculate the full covariance matrix, including mode-coupling contributions
arising from both partial sky coverage and non-linear structure growth. For
three different sky masks, we compare the theoretical covariance matrix to that
estimated from publicly available N-body weak lensing simulations, finding good
agreement. We find that as a more extreme sky cut is applied, a corresponding
increase in both Gaussian off-diagonal covariance and non-Gaussian super-sample
covariance is observed in both theory and simulations, in accordance with
expectations. Studying the different contributions to the covariance in detail,
we find that the Gaussian covariance dominates along the main diagonal and the
closest off-diagonals, but further away from the main diagonal the super-sample
covariance is dominant. Forming mock constraints in parameters describing
matter clustering and dark energy, we find that neglecting non-Gaussian
contributions to the covariance can lead to underestimating the true size of
confidence regions by up to 70 per cent. The dominant non-Gaussian covariance
component is the super-sample covariance, but neglecting the smaller connected
non-Gaussian covariance can still lead to the underestimation of uncertainties
by 10--20 per cent. A real cosmological analysis will require marginalisation
over many nuisance parameters, which will decrease the relative importance of
all cosmological contributions to the covariance, so these values should be
taken as upper limits on the importance of each component.
