Inflationary quantum spectrum of the quasi-isotropic Universe

We investigate the quantum dynamics of the quasi-isotropic inflationary solution. This is achieved by deriving the Lagrangian and Hamiltonian for both the FLRW background and the inhomogeneous correction, via an expansion of the Einstein–Hilbert action up to second order in the perturbation amplitudes. Then we implement a semiclassical WKB scenario for which the inhomogeneous component of the Universe is treated as a “small” quantum subsystem, evolving on the classical isotropic background. Starting from the Wheeler–DeWitt equation, we recover a Schrödinger dynamics for the perturbations, in which the time dependence of the wave function emerges thanks to the classicality of the background, and we solve it for an inflationary phase. The main result of this paper is to show that, while the scalar component of the power spectrum has the standard scale invariant profile, the tensor one results to be not constrained by the inflationary expansion, apart from an overall normalization factor which guarantees a small tensor-to-scalar ratio. This means that the spatial distribution of the quasi-isotropic correction to the metric remains preserved, with the consequence that some information about the pre-inflationary Universe survives to the de Sitter expansion.