Nonlinear resonance of Kolmogorov–Arnold–Moser tori in bouncing universes

Abstract

The dynamics of closed Friedmann–Robertson–Walker (FRW) universes with a massiveinflaton field is examined where Friedmann equations are corrected by the introduction of apotential term that implements non-singular bounces in the early evolution of theuniverse. This potential term arises from quantum gravity/high-energy corrections tocosmological scenarios near the singularity and is semiclassical in nature, being effectiveonly when the scale factor is very small. For certain windows in the parameterspace labelled by the scalar field mass and the conserved Hamiltonian, nonlinearresonance phenomena take place. Nonlinear resonance may induce the destruction ofKolmogorov–Arnold–Moser (KAM) tori that trap the inflaton, leading to a rapid growth ofthe scale factor and the scalar field, with disruption of metastable states and consequentescape of the universe into inflation. We make a numerical/analytical approach tothe nonlinear resonance phenomena, characterizing a particular resonance by itscharacteristic periodic orbits and by the structure of the associated diffusion pattern.The diffusion occurs when the orbit escapes through a Cantorus in the borderof primary KAM islands that encloses the characteristic periodic orbits of theresonance. The windows of parametric resonance, characterized by an integern ≥ 2 (associated with the ratio of the frequencies in the scale factor/scalar field degrees offreedom) are the ones that strongly favour inflation in the system. We discuss how genericthis behaviour is for inflationary models, and its possible consequences for structureformation.