Cross-correlating the Thermal Sunyaev-Zel'dovich Effect and the
  Distribution of Galaxy Clusters
  We present the analytical formulas, derived based on the halo model, to
compute the cross-correlation between the thermal Sunyaev-Zel'dovich (SZ)
effect and the distribution of galaxy clusters. By binning the clusters
according to their redshifts and masses, this cross-correlation, the so-called
stacked SZ signal, reveals the average SZ profile around the clusters. The
stacked SZ signal is obtainable from a joint analysis of an
arcminute-resolution cosmic microwave background (CMB) experiment and an
overlapping optical survey, which allows for detection of the SZ signals for
clusters whose masses are below the individual cluster detection threshold. We
derive the error covariance matrix for measuring the stacked SZ signal, and
then forecast for its detection from ongoing and forthcoming combined
CMB-optical surveys. We find that, over a wide range of mass and redshift, the
stacked SZ signal can be detected with a significant signal to noise ratio
(total S/N \gsim 10), whose value peaks for the clusters with intermediate
masses and redshifts. Our calculation also shows that the stacking method
allows for probing the clusters' SZ profiles over a wide range of scales, even
out to projected radii as large as the virial radius, thereby providing a
promising way to study gas physics at the outskirts of galaxy clusters.
