The non-linear evolution of the energy density of the radiation induces spectral distortions of the cosmic microwave background both at recombination and during the reionization era. This distortion has the same spectral signature as the one produced by the re-scattering of photons by non-relativistic hot electrons, the thermal Sunyaev-Zeldovich effect, whose amplitude is quantified by a Compton y parameter. A diffuse y-sky is then expected to emerge from mode couplings in the non-linear evolution of the cosmological perturbations and to superimpose to the point source contributions of galaxy clusters. The equations describing the evolution of the y field and a hierarchy governing its angular multipoles are derived from the second order Boltzmann equation. These equations are then integrated numerically to obtain the first predicted power spectrum of the diffuse y-sky. It is found to be a remarkable tracer of the reionization history of the Universe.