On the Validity of Cosmological Fisher Matrix Forecasts
  We present a comparison of Fisher matrix forecasts for cosmological probes
with Monte Carlo Markov Chain (MCMC) posterior likelihood estimation methods.
We analyse the performance of future Dark Energy Task Force (DETF) stage-III
and stage- IV dark-energy surveys using supernovae, baryon acoustic
oscillations and weak lensing as probes. We concentrate in particular on the
dark-energy equation of state parameters w0 and wa. For purely geometrical
probes, and especially when marginalising over wa, we find considerable
disagreement between the two methods, since in this case the Fisher matrix can
not reproduce the highly non-elliptical shape of the likelihood function. More
quantitatively, the Fisher method underestimates the marginalized errors for
purely geometrical probes between 30%-70%. For cases including structure
formation such as weak lensing, we find that the posterior probability contours
from the Fisher matrix estimation are in good agreement with the MCMC contours
and the forecasted errors only changing on the 5% level. We then explore
non-linear transformations resulting in physically-motivated parameters and
investigate whether these parameterisations exhibit a Gaussian behaviour. We
conclude that for the purely geometrical probes and, more generally, in cases
where it is not known whether the likelihood is close to Gaussian, the Fisher
matrix is not the appropriate tool to produce reliable forecasts.
