Non-Linear Matter Power Spectrum Covariance Matrix Errors and
  Cosmological Parameter Uncertainties
  The covariance matrix of the matter power spectrum is a key element of the
statistical analysis of galaxy clustering data. Independent realisations of
observational measurements can be used to sample the covariance, nevertheless
statistical sampling errors will propagate into the cosmological parameter
inference potentially limiting the capabilities of the upcoming generation of
galaxy surveys. The impact of these errors as function of the number of
independent realisations has been previously evaluated for Gaussian distributed
data. However, non-linearities in the late time clustering of matter cause
departures from Gaussian statistics. Here, we address the impact of
non-Gaussian errors on the sample covariance and precision matrix errors using
a large ensemble of numerical N-body simulations. In the range of modes where
finite volume effects are negligible ($0.1\lesssim k\,[h\,{\rm
Mpc^{-1}}]\lesssim 1.2$) we find deviations of the estimated variance of the
sample covariance with respect to Gaussian predictions above $\sim 10\%$ level.
These reduce to about $\sim 5\%$ in the case of the precision matrix. Finally,
we perform a Fisher analysis to estimate the effect of covariance errors on the
cosmological parameter constraints. In particular, assuming Euclid-like survey
characteristics we find that a number of independent realisation larger than
$\gtrsim 5000$ is necessary to reduce the contribution of sample covariance
errors to the cosmological parameter uncertainties at sub-percent level. We
also show that restricting the analysis to large scales $k\lesssim0.2\,h\,{\rm
Mpc^{-1}}$ results in a considerable loss in constraining power, while using
the linear covariance to include smaller scales leads to an underestimation of
the errors on the cosmological parameters.
