Disconnected pseudo-$C_\ell$ covariances for projected large-scale
  structure data
  The disconnected part of the power spectrum covariance matrix (also known as
the "Gaussian" covariance) is the dominant contribution on large scales for
galaxy clustering and weak lensing datasets. The presence of a complicated sky
mask causes non-trivial correlations between different Fourier/harmonic modes,
which must be accurately characterized in order to obtain reliable cosmological
constraints. This is particularly relevant for galaxy survey data.
Unfortunately, an exact calculation of these correlations involves $O(\ell_{\rm
max}^6)$ operations that become computationally impractical very quickly. We
present an implementation of approximate methods to estimate the Gaussian
covariance matrix of power spectra involving spin-0 and spin-2 flat- and
curved-sky fields, expanding on existing algorithms. These methods achieve an
$O(\ell_{\rm max}^3)$ scaling, which makes the computation of the covariance
matrix as fast as the computation of the power spectrum itself. We quantify the
accuracy of these methods on large-scale structure and weak lensing data,
making use of a large number of Gaussian but otherwise realistic simulations.
We show that, using the approximate covariance matrix, we are able to recover
the true posterior distribution of cosmological parameters to high accuracy. We
also quantify the shortcomings of these methods, which become unreliable on the
very largest scales, as well as for covariance matrix elements involving cosmic
shear $B$ modes. The algorithms presented here are implemented in the public
code NaMaster (https://github.com/LSSTDESC/NaMaster).
