On the Validity of Cosmological Fisher Matrix Forecasts

TL;DR

This paper compares Fisher matrix forecasts with MCMC methods for future dark energy surveys, revealing significant discrepancies for geometrical probes and better agreement for structure formation probes, highlighting limitations of Fisher forecasts.

Contribution

It provides a detailed analysis of the reliability of Fisher matrix forecasts versus MCMC methods in cosmology, especially for geometrical probes and non-Gaussian likelihoods.

Findings

Fisher matrix underestimates errors by 30-70% for geometrical probes.

Good agreement between Fisher and MCMC for structure formation probes.

Fisher matrix is unreliable for non-Gaussian likelihoods in geometrical probes.

Abstract

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.