Abstract

We construct global phase portraits of inflationary dynamics in teleparallel gravity models with a scalar field nonminimally coupled to torsion scalar. The adopted set of variables can clearly distinguish between different asymptotic states as fixed points, including the kinetic and inflationary regimes. The key role in the description of inflation is played by the heteroclinic orbits that run from the asymptotic saddle points to the late time attractor point and are approximated by nonminimal slow roll conditions. To seek the asymptotic fixed points, we outline a heuristic method in terms of the “effective potential” and “effective mass”, which can be applied for any nonminimally coupled theories. As particular examples, we study positive quadratic nonminimal couplings with quadratic and quartic potentials and note how the portraits differ qualitatively from the known scalar-curvature counterparts. For quadratic models, inflation can only occur at small nonminimal coupling to torsion, as for larger coupling, the asymptotic de Sitter saddle point disappears from the physical phase space. Teleparallel models with quartic potentials are not viable for inflation at all, since for small nonminimal coupling, the asymptotic saddle point exhibits weaker than exponential expansion, and for larger coupling, it also disappears.

Highlights

  • While early introductions of a nonminimal coupling between the scalar field and curvature were motivated by, e.g., Mach’s principle [1] or conformal invariance [2], the nonminimal coupling appears naturally as a result of quantum corrections to the scalar field on curved spacetime [3] as well as in the effective actions of higher dimensional constructions [4,5]

  • We construct global phase portraits of inflationary dynamics in teleparallel gravity models with a scalar field nonminimally coupled to torsion scalar

  • Inflation can only occur at small nonminimal coupling to torsion, as for larger coupling, the asymptotic de Sitter saddle point disappears from the physical phase space

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Summary

Introduction

While early introductions of a nonminimal coupling between the scalar field and curvature were motivated by, e.g., Mach’s principle [1] or conformal invariance [2], the nonminimal coupling appears naturally as a result of quantum corrections to the scalar field on curved spacetime [3] as well as in the effective actions of higher dimensional constructions [4,5]. The aim of the present paper is to use the methods of dynamical systems [9] to study the cosmological evolution of scalar field models nonminimally coupled to torsion in spatially flat FLRW backgrounds following the approach of Reference [68] and compare the results with models nonminimally coupled to curvature. In the torsion coupling case, the “effective potential” was introduced in Refence [52] to explain the existence and stability of de Sitter fixed points, and it was noted how these points correspond to the “balanced solutions” of Reference [48]. The point could be nonhyperbolic with the real part of the corresponding eigenvalue zero, but in such a case, the higher corrections must still make the direction repulsive, one needs to resort to center manifold theory to carry out a precise analysis

Slow Roll
Quadratic Potential Let us take as the first example
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