Study of a viscous $\Lambda$WDM model: Near equilibrium condition, entropy production, and cosmological constraints

Extensions to a $\Lambda$DM model have been explored to face current tensions that occur within its framework, which encompasses broadening the nature of the dark matter (DM) component to include warmness and a non-perfect fluid description. In this paper, we investigated the late-time cosmological evolution of an exact solution recently found in [N. Cruz, E. Gonz\'alez, and J. Jovel, Phys. Rev. D \textbf{105}, 024047 (2022)], which describe a viscous warm $\Lambda$DM model ($\Lambda$WDM) with a DM component that obeys a polytropic equation of state (EoS), which experience dissipative effects with a bulk viscosity proportional to its energy density, with proportionality constant $\xi_{0}$. This solution has the particularity of having a very similar behavior to the $\Lambda$CDM model for small values of $\xi_{0}$, evolving also to a de Sitter type expansion in the very far future. We explore firstly the thermodynamic consistences of this solution in the framework of the Eckart's theory of non-perfect fluids, focusing on the fulfillment of the two following conditions: (i) the near equilibrium condition and (ii) the positiveness of the entropy production. We explore the range of parameters of the model that allow to fulfilling these two conditions at the same time, finding that a viscous WDM component is compatible with both ones, being in this sense, a viable model from the thermodynamic point of view. Besides, we constraint the free parameters of the model with the observational data coming form supernovae Ia (SNe Ia) and the observational Hubble parameter data (OHD).(Abridged)