Inadequacy of internal covariance estimation for super-sample covariance
  We give an analytical interpretation of how subsample-based internal
covariance estimators lead to biased estimates of the covariance, due to
underestimating the super-sample covariance (SSC). This includes the jackknife
and bootstrap methods as estimators for the full survey area, and subsampling
as an estimator of the covariance of subsamples. The limitations of the
jackknife covariance have been previously presented in the literature because
it is effectively a rescaling of the covariance of the subsample area. However
we point out that subsampling is also biased, but for a different reason: the
subsamples are not independent, and the corresponding lack of power results in
SSC underprediction. We develop the formalism in the case of cluster counts
that allows the bias of each covariance estimator to be exactly predicted. We
find significant effects for a small-scale area or when a low number of
subsamples is used, with auto-redshift biases ranging from 0.4% to 15% for
subsampling and from 5% to 75% for jackknife covariance estimates. The
cross-redshift covariance is even more affected; biases range from 8% to 25%
for subsampling and from 50% to 90% for jackknife. Owing to the redshift
evolution of the probe, the covariances cannot be debiased by a simple
rescaling factor, and an exact debiasing has the same requirements as the full
SSC prediction. These results thus disfavour the use of internal covariance
estimators on data itself or a single simulation, leaving analytical prediction
and simulations suites as possible SSC predictors.
