The angular power spectrum of the thermal Sunyaev-Zeldovich (tSZ) effect is highly sensitive to cosmological parameters such as sigma_8 and Omega_m, but its use as a precision cosmological probe is hindered by the astrophysical uncertainties in modeling the gas pressure profile in galaxy groups and clusters. In this paper we assume that the relevant cosmological parameters are accurately known and explore the ability of current and future tSZ power spectrum measurements to constrain the intracluster gas pressure or the evolution of the gas mass fraction, f_gas. We use the CMB bandpower measurements from the South Pole Telescope and a Bayesian Markov Chain Monte Carlo (MCMC) method to quantify deviations from the standard, universal gas pressure model. We explore analytical model extensions that bring the predictions for the tSZ power into agreement with experimental data. We find that a steeper pressure profile in the cluster outskirts or an evolving f_gas have mild-to-severe conflicts with experimental data or simulations. Varying more than one parameter in the pressure model leads to strong degeneracies that cannot be broken with current observational constraints. We use simulated bandpowers from future tSZ survey experiments, in particular a possible 2000 deg^2 CCAT survey, to show that future observations can provide almost an order of magnitude better precision on the same model parameters. This will allow us to break the current parameter degeneracies and place simultaneous constraints on the gas pressure profile and its redshift evolution, for example.
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