Mapping the temperature of the intra-cluster medium with the tSZ effect

The hot electrons in the intra-cluster medium produce a spectral distorsion of the cosmic microwave background (CMB) black body emission, the thermal Sunyaev-Zel'dovich effect (tSZ). This characteristic spectral distorsion is now commonly used to detect and characterize the properties of galaxy clusters. The tSZ effect spectral distorsion does not depends on the redshift, and is only slightly affected by the galaxy cluster properties via the relativistic corrections, when the electrons reach relativistic velocities. The present work proposes a linear component separation approach to extract the tSZ effect Compton parameter and relativistic corrections for next-generation CMB experiments. We demonstrated that relativistic corrections, if neglected, would induce a significant bias on galaxy cluster Compton parameter, tSZ scaling relation slope, and tSZ angular power spectrum shape measurements. We showed that tSZ relativistic corrections mapping can be achieved at high signal-to-noise ratio with a low level of contamination up to $\ell=3000$ for next generation CMB experiments. At smaller angular scales the contamination produced by infra-red emission will be a significant source of bias. Such tSZ relativistic corrections mapping enables the study of galaxy cluster temperature profile via the tSZ effect only.