  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 CMB surveys, e.g., ACT, 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 et. al (2003) 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, Omega_Lambda could be measured to within 0.1 and 0.2 respectively at
one sigma, but only upper limits on the equation of state parameter w 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.
