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-Zel ’ dovich 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 _ { C _ { S } ^ { 2 } } { { } _ { < } \atop { } ^ { \sim } } ( 300 { km / s } ) ^ { 2 } .
This can be done around z > 0.3 with clusters of flux above 200 mJy which will be detected by 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 ALMA array or the NSF South Pole telescope and other surveys .
By measuring the cosmic sound speed and the bulk flow in , e.g. , 4 spheres of \sim 100 h ^ { -1 } Mpc at z = 0.3 , we could have a direct measurement of the matter density 0.21 < \Omega _ { m } < 0.47 at 95 % confidence level .