Exploring the cosmological degeneracy between decaying dark matter model and viscous $\Lambda$CDM

In the context of a homogeneous and isotropic universe, we consider the degeneracy condition at the background level between two scenarios in which processes out of equilibrium are possible; this consideration allows us to deviate from the perfect fluid description, and in this case bulk viscosity represents a viable candidate to describe entirely such effects. The cosmological model describing an unstable dark matter sector is mapped into a slight modification of the $\Lambda$CDM model characterized by a viscous dark matter sector; under this consideration our description does not depend on a specific formulation of viscous effects and these can be fully reconstructed and characterized by the parameter that determines the decay ratio of dark matter. However, in this new scenario the cosmic expansion is influenced by the viscous pressure and the dark energy sector given by the cosmological constant is translated into a dynamical one. As a consequence of our formulation, the test against observations of the model indicates consistency with quintessence dark energy, but the crossing of the phantom divide can be accessible.