Shan--Chen interacting vacuum cosmology
  In this paper, we introduce a novel class of interacting vacuum models, based
on recasting the equation of state originally developed in the context of
lattice kinetic theory by Shan \& Chen as the coupling between the vacuum and
cold dark matter (CDM). This coupling allows the vacuum to evolve and is
nonlinear around a characteristic energy scale $\rho_*$, changing into a linear
coupling with a typical power law evolution at scales much lower and much
higher than $\rho_*$. Focusing on the simplest sub-class of models where the
interaction consists only of an energy exchange and the CDM remains geodesic,
we first illustrate the various possible models that can arise from the
Shan--Chen coupling, with several different behaviours at both early and late
times depending on the values of the model parameters selected. We then place
the first observational constraints on this Shan--Chen interacting vacuum
scenario, performing an MCMC analysis to find those values of the model and
cosmological parameters which are favoured by observational data. We focus on
models where the nonlinearity of the coupling is relevant at late times,
choosing for the reference energy scale $\rho_*$, the critical energy density
in $\Lambda$CDM. We show that the observational data we use are compatible with
a wide range of models which result in different cosmologies. However, we also
show that $\Lambda$CDM is preferred over all of the Shan--Chen interacting
vacuum models that we study, and comment on the inability of these models to
relax the $H_0$ and $\sigma_8$ tensions.
