  We investigate whether a Gaussian likelihood is sufficient to obtain accurate
parameter constraints from a Euclid-like combined tomographic power spectrum
analysis of weak lensing, galaxy clustering and their cross-correlation.
Testing its performance on the full sky against the Wishart distribution, which
is the exact likelihood under the assumption of Gaussian fields, we find that
the Gaussian likelihood returns accurate parameter constraints. This accuracy
is robust to the choices made in the likelihood analysis, including the choice
of fiducial cosmology, the range of scales included, and the random noise
level. We extend our results to the cut sky by evaluating the additional
non-Gaussianity of the joint cut-sky likelihood in both its marginal
distributions and dependence structure. We find that the cut-sky likelihood is
more non-Gaussian than the full-sky likelihood, but at a level insufficient to
introduce significant inaccuracy into parameter constraints obtained using the
Gaussian likelihood. Our results should not be affected by the assumption of
Gaussian fields, as this approximation only becomes inaccurate on small scales,
which in turn corresponds to the limit in which any non-Gaussianity of the
likelihood becomes negligible. We nevertheless compare against N-body weak
lensing simulations and find no evidence of significant additional
non-Gaussianity in the likelihood. Our results indicate that a Gaussian
likelihood will be sufficient for robust parameter constraints with power
spectra from Stage IV weak lensing surveys.
