Reformulating Scalar-Tensor Field Theories as Scalar-Scalar Field Theories Using Lorentzian Cofinsler Spaces

TL;DR

This paper introduces Lorentzian Cofinsler geometry, a new geometric framework using scalar fields, to reformulate scalar-tensor theories and derive self-inflating universe solutions with potential cosmological applications.

Contribution

It develops Lorentzian Cofinsler geometry from Finsler geometry and applies it to scalar-tensor theories, producing second-order Lagrangians and novel self-inflating universe solutions.

Findings

Lorentzian Cofinsler geometry constructed from scalar fields.

Reformulation of Horndeski Lagrangians to second order.

Existence of self-inflating universe solutions.

Abstract

In this paper I shall show how notions of Finsler geometry can be used to construct a new type of geometry using a scalar field, f, on the cotangent bundle of a differentiable manifold, M. This new geometry will be called Lorentzian Cofinsler geometry. This geometry will enable me to use the second vertical derivatives of f, along with the differential of the scalar field, phi on M, to construct a Lorentzian metric tensor on M, that depends upon phi. f will be chosen so that the resultant metric on M has the form of a FLRW metric, with the t equal constant slices being flat. When the Horndeski Lagrangians are evaluated for this choice of geometry the quartic and quintic Lagrangians are of third order, but reduce to non-degenerate second-order Lagrangians plus a divergence. Upon varying phi in these "scalarized" Horndeski Lagrangians, equations will be obtained which admit self-inflating universe solutions, provided that the coefficient functions appearing in the Horndeski Lagrangians are chosen suitably. This approach is also used to study solutions of the most general conformally invariant scalar-tensor field theory which is flat space compatible (i.e., such that the Lagrangians of the field theory are well-defined when either the space is flat or the scalar field is constant). There too the coefficient functions can be chosen to give self-inflating universes. Arguments will be presented to show that it is possible to construct model universes that begin explosively, and then settle down to a period of much quieter acceleration which either continues forever, or stops and collapses to the models original, pre-expansion, state.