Global portraits of nonminimal inflation

Abstract

We reconsider the dynamical systems approach to analyze inflationary universe in the Jordan frame models of scalar field nonminimally coupled to curvature. The adopted set of variables allows us to clearly distinguish between different asymptotic states in the phase space, including the kinetic and inflationary regimes. Inflation is realized as a heteroclinic trajectory originating either at infinity from a nonhyperbolic asymptotic de Sitter point or from a regular saddle de Sitter point. We also present a comprehensive picture of possible initial conditions leading to sufficient inflationary expansion and show their extent on the phase diagrams. In addition we comment on the slow roll conditions applicable in the Jordan frame and show how they approximate the leading inflationary “attractor solution”. As particular examples we portrait quadratic and quartic potential models and note that increasing the nonminimal coupling diminishes the range of good initial conditions in the quadratic case, but enlarges is in the quartic case.

Global portraits of nonminimal teleparallel inflation

“…To realize inflation by a heteroclinic orbit starting from either a regular or asymptotic fixed point in the phase space [66][67][68][69][70][71], that point of origin must be a saddle with the eigenvalue corresponding to the outgoing direction having positive real part. This would correspond to a local or asymptotic maximum 1 Another approach with different definitions can be found in Refs.…”

“…To ensure sufficient measure of inflationary expansion and generate nearly scale invariant spectrum of perturbations, the system must evolve in a nearly de Sitter like regime, which typically implies a slowly varying regime for the scalar field. The corresponding slow roll conditions can be read off from the Friedmann equation ( 5) and the scalar field equation ( 13) in the scalar-curvature case [66],…”

“…In these variables, the scalar-curvature system ( 5)-( 7) was studied in detail in Refs. [66,74], and we will not repeat the details here. In the scalar-torsion case the system ( 8)-( 10) can be represented as…”

“…The aim of the present paper is to use the methods of dynamical systems [8] to study the cosmological evolution of scalar field models nonminimally coupled to torsion in spatially flat FLRW backgrounds following the approach of Ref. [66], and compare the results with models nonminimally coupled to curvature. The adopted variables (φ, φ H ) have the benefit of clearly distinguishing all asymptotic states, and we confirm the insight about the central role of heteroclinic orbits in the phase space for the realization of inflation [66][67][68][69][70][71] in the torsion case as well.…”

“…[66], and compare the results with models nonminimally coupled to curvature. The adopted variables (φ, φ H ) have the benefit of clearly distinguishing all asymptotic states, and we confirm the insight about the central role of heteroclinic orbits in the phase space for the realization of inflation [66][67][68][69][70][71] in the torsion case as well. Taking positive quadratic nonminimal couplings with quadratic and quartic potentials as examples, we plot the local and global phase portraits, indicate the leading slow roll trajectory of inflation, and mark the path of the last 50 e-folds of accelerated expansion as well as the range of initial conditions leading to it.…”