Inflationary Quantum Spectrum of the Quasi-Isotropic Universe

TL;DR

This paper develops a quantum framework for the quasi-isotropic inflationary universe, deriving the power spectra of scalar and tensor perturbations, revealing a standard scale-invariant scalar spectrum and an unconstrained tensor spectrum.

Contribution

It introduces a gauge-invariant quantum treatment of inhomogeneous perturbations during inflation using a semiclassical WKB approach, deriving the power spectra in a quasi-isotropic setting.

Findings

Scalar power spectrum is scale invariant.

Tensor spectrum remains unconstrained by inflation.

The spatial distribution of metric corrections is preserved.

Abstract

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. We do this in a gauge invariant fashion adopting the Mukhanov-Sasaki variable for the scalar degree of freedom. Then we construct the quantum dynamics in terms of 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 Schr\"odinger 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 a de Sitter inflationary phase. The main result of this paper is that the resulting scalar component of the power spectrum has the standard scale invariant profile, while the tensor one results to be not constrained by the inflationary expansion (modulo an overall normalization factor), therefore preserving the spatial distribution of the quasi-isotropic correction to the metric.