A halo approach to the evaluation of the cross-correlation between the SZ sky and galaxy survey

Using a purely analytic approach to gaseous and dark matter halos, we study the cross-correlation between the Sunyaev-Zel'dovich (SZ) sky and galaxy survey under flat sky approximation, in an attempt to acquire the redshift information of the SZ map. The problem can be greatly simplified when it is noticed that the signals of the SZ-galaxy correlation arise only from hot gas and galaxies inside the same massive halos (i.e. clusters), and field galaxies make almost no contribution to the cross-correlation. Under the assumption that both the hot gas and galaxies trace the common gravitational potential of dark halos, we calculate the expected cross SZ-galaxy power spectra for the WMAP/Planck SZ maps and the SDSS galaxy sample at small scales $100<l<1000$. It turns out, however, that it is not presently feasible to measure such small angular cross power spectra because of the high noise levels at $l>400$ with the WMAP/Planck experiments. Future SZ observations with better angular resolutions and sufficiently wide sky coverages will be needed if this technique is applied for the statistical measurement of redshift distribution of the SZ sources.