Perturbações escalares pré-inflacionárias em universos fechados na cosmologia quântica de laços

Abstract

The perturbations in the inflaton scalar field within the primordial universe are analyzed for the Starobinsky model, considering a closed universe with a background described by loop quantum cosmology (LQC), including an inflationary regime consistent with constraints imposed by cosmic microwave background (CMB) observations. The background dynamics, governed by an effective Hamiltonian in LQC, resolve the initial Big Bang singularity by inducing a bounce. Numerical analyses reveal that different choices of initial conditions and curvature values affect the duration of the inflationary regime, leading to different classes of solutions, referred to as type I, II, and III. Type III solutions are of greatest interest, as they represent a background where spatial curvature and quantum gravity effects play a significant role, whereas type I solutions characterize flat approximations of the background, and type II solutions depict a classical bounce. The initial conditions are set based on the inflationary cosmological parameters, inferred from PLANCK mission observations (2018). The evolution of linear perturbations is described within the ADM formalism by determining the expansion of the Hamiltonian up to quadratic order in the perturbations in the spatially spherical gauge, ensuring well-behaved evolution coefficients at the bounce. The initial state is chosen as a fourth-order adiabatic vacuum, fixed at a moment prior to the bounce. Perturbations are then numerically evolved through the inflationary period for different modes k, which are subsequently related to the comoving curvature perturbation to obtain the power spectrum, with different values of curvature ΩK considered. Quantum gravity and spatial curvature effects influence the largest scales of the spectra, introducing corrections such as oscillations and power amplification. The results are compared against standard inflationary predictions in a flat universe. For curvature parameter values |ΩK| > 10^-8, the average of the spectral oscillations deviates from the inflationary prediction, becoming incompatible with observations. The curvature values that yield spectrum corrections consistent with observations are constrained to |ΩK| < 10^-8.