  The non-linear evolution of the energy density of the radiation is shown to
induce 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.
