On the validity of cosmological Fisher matrix forecasts

Abstract

We present a comparison of Fisher matrix forecastsfor cosmological probes with Monte Carlo Markov Chain (MCMC)posterior likelihood estimation methods. We analyse the performanceof future Dark Energy Task Force (DETF) stage-III and stage-IVdark-energysurveys using supernovae, baryon acoustic oscillationsand weak lensing as probes. We concentrate in particular on thedark-energy equation of state parameters w0 and wa.For purely geometrical probes, and especially when marginalising overwa, we find considerable disagreement between the two methods, since in this case theFisher matrix can not reproduce the highly non-elliptical shape ofthe likelihood function. More quantitatively, the Fisher methodunderestimates the marginalized errors for purely geometrical probesbetween 30%-70%. For cases including structure formation suchas weak lensing, we find that the posterior probability contoursfrom the Fisher matrix estimation are in good agreement with theMCMC contours and the forecasted errors only changing on the 5% level.We then explore non-linear transformationsresulting in physically-motivated parameters and investigate whetherthese parameterisations exhibit a Gaussian behaviour.We concludethat for thepurely geometrical probes and, more generally, in caseswhere it is not known whether the likelihood is close to Gaussian, the Fisher matrix is notthe appropriate tool to produce reliable forecasts.