The effect on cosmological parameter estimation of a parameter dependent
  covariance matrix
  Cosmological large-scale structure analyses based on two-point correlation
functions often assume a Gaussian likelihood function with a fixed covariance
matrix. We study the impact on cosmological parameter estimation of ignoring
the parameter dependence of this covariance matrix, focusing on the particular
case of joint weak-lensing and galaxy clustering analyses. Using a Fisher
matrix formalism (calibrated against exact likelihood evaluation in particular
simple cases), we quantify the effect of using a parameter dependent covariance
matrix on both the bias and variance of the parameters. We confirm that the
approximation of a parameter-independent covariance matrix is exceptionally
good in all realistic scenarios. The information content in the covariance
matrix (in comparison with the two point functions themselves) does not change
with the fractional sky coverage. Therefore the increase in information due to
the parameter dependent covariance matrix becomes negligible as the number of
modes increases. Even for surveys covering less than $1\%$ of the sky, this
effect only causes a bias of up to ${\cal O}(10\%)$ of the statistical
uncertainties, with a misestimation of the parameter uncertainties at the same
level or lower. The effect will only be smaller with future large-area surveys.
Thus for most analyses the effect of a parameter-dependent covariance matrix
can be ignored both in terms of the accuracy and precision of the recovered
cosmological constraints.
