We make the first detailed MCMC likelihood study of cosmological constraints that are expected from some of the largest, ongoing and proposed, cluster surveys in different wave-bands and compare the estimates to the prevalent Fisher matrix forecasts. Mock catalogs of cluster counts
expected from the surveys — eROSITA, WFXT, RCS2, DES and Planck, along with a mock dataset
of follow-up mass calibrations are analyzed for this purpose.
A fair agreement between MCMC and Fisher results is found
only in the case of minimal models. However, for many cases, the marginalized constraints obtained from
Fisher and MCMC methods can differ by factors of 30-100%.
The discrepancy can be alarmingly large for a time dependent dark energy equation of state, w(a);
the Fisher methods are seen to under-estimate the constraints by as much as a factor of 4-5.
Typically, Fisher estimates become more and more inappropriate as we move away from ΛCDM, to a constant-w dark energy to varying-w dark energy cosmologies.
Fisher analysis, also, predicts incorrect parameter degeneracies. There are noticeable offsets in the likelihood contours obtained from
Fisher methods that is caused due to an asymmetry in the posterior likelihood distribution as seen through a MCMC
analysis. From the point of mass-calibration uncertainties, a
high value of unknown scatter about the mean mass-observable relation, and its redshift dependence, is seen to have large
degeneracies with the cosmological parameters σ8 and w(a) and can degrade the cosmological constraints considerably.
We find that the addition of mass-calibrated cluster datasets can improve dark energy and σ8 constraints by factors of 2-3 from what can be obtained from CMB+SNe+BAO only .
Finally, we show that a joint analysis of datasets of two (or more) different cluster surveys would significantly tighten cosmological constraints from using clusters only. Since, details of future cluster surveys are still being planned, we emphasize that optimal survey design must be done using MCMC analysis rather than Fisher forecasting.