Quantum corrections to cosmic perturbations for a bouncing background

TL;DR

This paper calculates quantum corrections to cosmological perturbations in a bouncing universe model within Loop Quantum Cosmology, revealing a small scale-dependent spectral modification and a damping mechanism post-bounce.

Contribution

It introduces a second-order quantum correction framework for scalar perturbations in LQC with a bouncing background, including quantum correlations and effective equations.

Findings

Quantum corrections produce a scale-dependent enhancement of the power spectrum.

A damping mechanism suppresses scalar perturbations after the bounce.

Corrections are suppressed by the sixth power of the Planck length, consistent with observations.

Abstract

We compute second-order quantum corrections, as quantum dispersions and correlations, to a cosmological model coupling a single scalar perturbation mode to a bouncing background within Loop Quantum Cosmology (LQC). Using an effective quantization approach in which quantum moments extend the classical phase space as new dynamical degrees of freedom, and incorporating the cosmic bounce through holonomy corrections in the $\mu_0$ scheme, we derive a coupled set of effective equations of motion for the expectation values and second-order quantum moments of both the gravitational and scalar sectors evolving with respect to a clock scalar field. Within the test-field approximation and for a vanishing scalar potential, the quantum moment equations reduce to a third-order ordinary differential equation for the mean squared deviation $G^{vv}$ of the Mukhanov-Sasaki variable in a de Sitter background with LQC bounce. Treating the effect of bounce as a perturbation of the solution, we construct the corresponding correction to the dimensionless curvature power spectrum. The leading correction is suppressed by the sixth power of the Planck length, producing a scale-dependent enhancement $\delta P_{R} \propto (k \ell_{\rm Pl})^6$ that modifies the spectral index by $\delta n_s \sim 6(k \ell_{\rm Pl})^6 \ll 1$ for all cosmologically observable modes, in full consistency with current observational constraints. Numerical evolution of the full coupled system reveals a conditional ultraviolet regularization of the bounce-induced spectrum: the gravitational quantum moments generate a damping mechanism that suppresses the scalar perturbation amplitude after the bounce. Including cross-sector quantum correlations amplifies perturbation modes and introduces numerical instabilities at high wavenumbers, signaling the limits of the second-order truncation in the ultraviolet.