  If the accelerated expansion of the universe is due to a modification of
general relativity at late times, it is likely that the growth of structure on
large scales would also display deviations from the standard cosmology. We
investigate the statistics of the distribution of galaxy cluster-sized halos as
a probe of gravity. We analyze the output of several matched N-body simulations
with the same initial conditions and expansion histories but using both DGP and
f (R) gravity with various parameters. From each simulation we extract the
cluster mass function, power spectrum, and mean pairwise velocity at redshifts
1, 0.3, and 0. All three statistics display systematic differences between
gravity theories. The mean pairwise velocity provides an important consistency
test for any posited departure from general relativity suggested by
measurements of the power spectrum and cluster mass function. Upcoming
microwave background experiments, including Simons Observatory and CMB- S4,
will detect tens of thousands of galaxy clusters via the thermal
Sunyaev-Zeldovich effect, probe their masses with lensing of the microwave
background, and potentially measure velocities using the transverse lensing
effect or the kinematic Sunyaev-Zeldovich effect. These cluster measurements
promise to be a substantial probe of modified gravity.
