Extending $\Lambda(t)-$CDM to the inflationary epoch using dynamical foliations and a pre-inflationary vacuum energy from 5D geometrical vacuum as a unifying mechanism.

In this letter assuming a 5D quantum pre-inflationary vacuum energy, we propose a manner to extend some $\Lambda(t)$-CDM models to the inflationary period by using dynamical foliations of the five-dimensional (5D) Ricci-flat space-time manifold, regarding a non-compact extra space-like coor\-dinate. In this formalism we achieve also a geometrical unification of inflation and the present acce\-le\-rating epoch. In this approach inflation is generated by a pre-inflationary quantum vacuum energy that maintains the 5D classical vacuum on cosmological scales. We obtain from geometrical conditions that we can model the presence of the pre-inflationary vacuum energy in 4D as a dynamical cosmological constant. The 4D inflationary period results to be governed by a power law expansion and for certain values of some parameters of the model we obtain an spectral index satisfying $0.9607\leq n_s\leq 0.9691$ and a scalar to tensor ratio $r=0.098$, values that fit well according to Planck 2018 results. The 4D inflationary potential is induced for the 5D geometry and the 4D pre-inflationary potential is determined by the model. We show thatin this theoretical framework the present acceleration in the expansion of the universe can be explained due to a remanent of this pre-inflationary vacuum energy scaled to the present epoch and that its description can be done with the same $\Lambda(t)$. In this period we obtain a deceleration parameter in agreement with Planck 2018 data under certain restrictions of the parameters of the model. From the geometrical point of view $\Lambda(t)$ is depending on the dynamical foliation of the 5D space-time manifold.