What is the super-sample covariance? A fresh perspective for second-order shear statistics

Cosmological analyses of second-order weak lensing statistics require precise and accurate covariance estimates. These covariances are impacted by two sometimes neglected terms: a negative contribution to the Gaussian covariance due to a finite survey area, and the super-sample covariance (SSC), which for the power spectrum contains the impact of Fourier modes larger than the survey window. We show here that these two effects are connected and can be seen as correction terms to the 'large-field-approximation', the asymptotic case of an infinitely large survey area. We describe the two terms collectively as "finite-field terms". We derive the covariance of second-order shear statistics from first principles. For this, we use an estimator in real space without relying on an estimator for the power spectrum. The resulting covariance does not scale inversely with the survey area, as might naively be assumed. This scaling is only correct under the large-field approximation when the contribution of the finite-field terms tends to zero. Furthermore, all parts of the covariance, not only the SSC, depend on the power spectrum and trispectrum at all modes, including those larger than the survey. We also show that it is generally impossible to transform an estimate of the power spectrum covariance into the covariance of a real-space statistic. Such a transformation is only possible in the asymptotic case of the large-field approximation. Additionally, we find that the total covariance of a real-space statistic can be calculated using correlation function estimates on spatial scales smaller than the survey window. Consequently, estimating covariances of real-space statistics, in principle, does not require information on spatial scales larger than the survey area. We demonstrate that this covariance estimation method is equivalent to the standard sample covariance method.