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

Abstract

In this paper 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 coordinate. In this formalism, we achieve also a geometrical unification of inflation and the present accelerating 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. In this model, the 4D inflationary period is governed by a power law expansion and for certain values of some parameters of the model, we obtain an spectral index satisfying \(0.9607\le n_s\le 0.9691\) 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 and its contribution is necessary so that \(n_s\) and r can fit the observational data. We also show that in 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.