Thermodynamics of an universe with Decaying Cold Dark Matter

In this work we focus on the thermodynamics consistency of a new set of solutions emerging from a cosmology in which dark matter is able to decay into relativistic particles within the dark sector. It is important to stress that the lifetime of dark matter is larger than the age of the universe in order to be consistent with observations. Given that the corresponding decay rate is small, this one can be used as a perturbative parameter and it is possible to construct analytic solutions from a perturbative analysis for the densities of the species and the scale factor. The decay of dark matter is an irreversible process since it occurs out of chemical equilibrium and therefore the entropy per comoving volume increases considerably, as a consequence the temperature does not scale as $a^{-1}$ in contrast to an adiabatic expansion. We take into account two scenarios: a) The case in which both species making up the fluid end up in thermal equilibrium and therefore their temperature is the same. b) A second instance in which the species do not reach thermal equilibrium and therefore they have different temperatures. We verify that the second law of thermodynamics is satisfied in any case.