Measuring the Mach number of the universe via the Sunyaev-Zeldovich effect

We introduce a new statistic to measure more accurately the cosmic sound speed of clusters of galaxies at different redshifts. This statistic is evaluated by cross-correlating cosmic microwave background (CMB) fluctuations caused by the Sunyaev-Zeldovich effect from observed clusters of galaxies with their redshifts. When clusters are distributed in redshift bins of narrow width, one could measure the mean squared cluster peculiar velocity with an error sigma(cs)(2) less than or similar to (300 km s(-1))(2). This can be done around z > 0.3 with clusters of flux above 200 mJy, which will be detected Planck, coupled with high- resolution microwave images to eliminate the cosmological part of the CMB fluctuations. The latter can be achieved with observations by the planned Atacama Large Millimeter Array or the NSF South Pole telescope and other surveys. By measuring the cosmic sound speed and the bulk flow in, for example, four spheres of similar to100 h(-1) Mpc at z = 0.3, we could have a direct measurement of the matter density 0.21 < Omega(m) < 0.47 at the 95% confidence level.