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, $Λ(t)$. 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 $Λ(z) \propto (1+z)^{4}$, while in the matter era we have $Λ(z)\propto (1+z)^{3}$ up to $z\simeq 3$ and $Λ(z)=Λ$ 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 $Λ$ cosmology) Hubble expansion.

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