On Conformal Transformation with Multiple Scalar Fields and Geometric Property of Field Space with Einstein-like Solutions

TL;DR

This paper explores conditions under which theories with multiple non-minimally coupled scalar fields can be transformed into a quasi-canonical form with conformally flat field space, providing solutions useful for model building.

Contribution

It derives conditions and solutions for transforming multi-scalar theories into a quasi-canonical form with conformally flat field space, including specific models like the Starobinsky model.

Findings

Solutions to nonlinear PDEs for scalar field transformations

Conditions for conformal flatness in multi-scalar theories

Existence of conformal flatness in modified gravity models

Abstract

Multiple scalar fields appear in vast modern particle physics and gravity models. When they couple to gravity non-minimally, conformal transformation is utilized to bring the theory into Einstein frame. However, the kinetic terms of scalar fields are usually not canonical, which makes analytic treatment difficult. Here we investigate under what conditions the theories can be transformed to the quasi-canonical form, in which case the effective metric tensor in field space is conformally flat. We solve the relevant nonlinear partial differential equations for arbitrary number of scalar fields and present several solutions that may be useful for future phenomenological model building, including the $\sigma$-model with a particular non-minimal coupling. We also find conformal flatness can always be achieved in some modified gravity theories, for example, Starobinsky model.