Probing the cosmic microwave background temperature using the Sunyaev-Zeldovich effect

We discuss the possibility to constrain the relation between redshift and temperature of the cosmic microwave background (CMB) using multifrequency Sunyaev-Zeldovich (SZ) observations. We have simulated a catalog of clusters of galaxies detected through their SZ signature assuming the sensitivities that will be achieved by the {\it Planck} satellite at 100, 143 and 353 GHz, taking into account the instrumental noise and the contamination from the Cosmic Infrared Background and from unresolved radiosources. We have parametrized the cosmological temperature-redshift law as $T\propto (1+z)^{(1-a)}$. Using two sets of SZ flux density ratios (100/143 GHz, which is most sensitive to the parametrization of the $T-z$ law, and 143/353 GHz, which is most sensitive to the peculiar velocities of the clusters) we show that it is possible to recover the $T-z$ law assuming that the temperatures and redshifts of the clusters are known. From a simulated catalog of $\sim 1200$ clusters, the parameter $a$ can be recovered to an accuracy of 10$^{-2}$. Sensitive SZ observations thus appear as a potentially useful tool to test the standard law. Most cosmological models predict a linear variation of the CMB temperature with redshift. The discovery of an alternative law would have profound implications on the cosmological model, implying creation of energy in a manner that would still maintain the black-body shape of the CMB spectrum at redshift zero.