  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.

General Relativity and Quantum Cosmology;High Energy Physics - Theory