Qualitative dynamics and inflationary attractors in loop cosmology

Abstract

Qualitative dynamics of three different loop quantizations of spatially flat isotropic and homogeneous models is studied using an effective spacetime description of the underlying quantum geometry. These include the standard loop quantum cosmology (LQC), its recently revived modification (referred to as mLQC-I), and another related modification of LQC (mLQC-II) of which the dynamics is studied in detail for the first time. Various features of LQC, including quantum bounce and preinflationary dynamics, are found to be shared with the mLQC-I and mLQC-II models. We study universal properties of dynamics for chaotic inflation, fractional monodromy inflation, Starobinsky potential, nonminimal Higgs inflation, and an exponential potential. We find various critical points and study their stability, which reveal various qualitative similarities in the postbounce phase for all these models. The prebounce qualitative dynamics of LQC and mLQC-II turns out to be very similar but is strikingly different from that of mLQC-I. In the dynamical analysis, some of the fixed points turn out to be degenerate. For these fixed points, center manifold theory is used. For all these potentials, nonperturbative quantum gravitational effects always result in a nonsingular inflationary scenario with a phase of superinflation succeeded by the conventional inflation. We show the existence of inflationary attractors and obtain scaling solutions in the case of the exponential potential. Since all of the models agree with general relativity at late times, our results are also of use in classical theory where qualitative dynamics of some of the potentials has not been studied earlier.