Constraining galaxy cluster velocity field with the tSZ-kSZ-kSZ bispectrum

The Sunyaev-Zel'dovich (SZ) effects are produced by the interaction of cosmic microwave background (CMB) photons with the ionized and diffuse gas of electrons inside galaxy clusters integrated along the line of sight. The two main effects are the thermal SZ (tSZ) produced by thermal pressure inside galaxy clusters and the kinetic SZ (kSZ) produce by peculiar motion of galaxy clusters compared to CMB rest-frame. The kSZ effect is particularly challenging to measure as it follows the same spectral behavior as the CMB, and consequently can not be separated from the CMB using spectral considerations. In this paper, we explore the feasibility of detecting the kSZ through the computation of the tSZ-CMB-CMB cross-correlation bispectrum for current and future CMB experiments. We conclude that next generation of CMB experiments will offer the possibility to detect at high S/N the tSZ-kSZ-kSZ bispectrum. This measurement will constraints the intra-cluster dynamics and the velocity field of galaxy cluster that is extremely sensitive to the growth rate of structures and thus to dark energy properties. Additionally, we also demonstrate that the tSZ-kSZ-kSZ bispectrum can be used to break the degeneracies between the mass-observable relation and the cosmological parameters to set tight constraints, up to 4\%, on the $Y-M$ relation calibration.