A potentially pure test of cosmic geometry: galaxy clusters and the real space Alcock-Paczyński test

We investigate the possibility of probing dark energy by measuring the isotropy of the galaxy cluster autocorrelation function (an Alcock-Paczynski test). The correlation function is distorted in redshift space because of the cluster peculiar velocities, but if these are known and can be subtracted, the correlation function measurement becomes in principle a pure test of cosmic geometry. Galaxy cluster peculiar velocities can be measured using the kinetic Sunyaev-Zel'dovich (kSZ) effect. Upcoming cosmic microwave background (CMB) surveys, for example, Atacama Cosmology Telescope (ACT), South Pole Telescope (SPT), Planck, are expected to do this with varying levels of accuracy, dependent on systematic errors due to cluster temperature measurements, radio point sources, and the primary CMB anisotropy. We use the Hubble volume N-body simulation and the hydrodynamic simulation results of Nagai, Kravtsov & Kosowsky to simulate various kSZ surveys. We find by model fitting that a measurement of the correlation function distortion can be used to recover the cosmological parameters that have been used to generate the simulation. However, the low space density of galaxy clusters requires larger surveys than are taking place at present to place tight constraints on cosmology. For example, with the SPT and ACT surveys, Ω Λ could be measured to within 0.1 and 0.2, respectively, at la, but only upper limits on the equation of state parameter ω will be possible. Nevertheless, with accurate measurements of the kSZ effect, this test can eventually be used to probe the dark energy equation of state and its evolution with redshift, with different systematic errors than other methods.