Solving the main cosmological puzzles with a generalized time varying vacuum energy

We study the dynamics of the FLRW flat cosmological models in which the vacuum energy density varies with time, $\mathrm{\Lambda }(t)$. In particular, we investigate the dynamical properties of a generalized vacuum model and we find that under certain conditions the vacuum term in the radiation era varies as $\mathrm{\Lambda }(z)\propto (1+z{)}^4$, while in the matter era we have $\mathrm{\Lambda }(z)\propto (1+z{)}^3$ up to $z\simeq 3$ and $\mathrm{\Lambda }(z)=\mathrm{\Lambda }$ for $z\le 3$. The confirmation of such a behavior would be of paramount importance because it could provide a solution to the cosmic coincidence problem as well as to the fine tuning problem, without changing the well known (from the concordance $\mathrm{\Lambda }$ cosmology) Hubble expansion.