Extrinsic curvature effects in brane-world scenarios

TL;DR

This paper derives a modified Klein-Gordon equation for fields on curved space-times embedded in higher dimensions, revealing how extrinsic curvature influences field dynamics and cosmological models.

Contribution

It introduces a novel derivation of field equations incorporating extrinsic curvature effects, extending standard models without relying on ad hoc assumptions.

Findings

Derived an induced potential depending on intrinsic and extrinsic curvature.

Computed the potential for Schwarzschild and Robertson-Walker embeddings.

Proposed an extended inflation model with a power-law scaling solution.

Abstract

We consider models of bosons on curved 3+1 dimensional space-time embedded in a higher dimensional flat ambient space. We propose to derive (rather than postulate) equations of motions by assuming that a standard Klein-Gordon field on ambient space is restricted to space-time by a strong confining potential. This leads to a modified Klein-Gordon equation on space-time which includes, in addition to the standard terms, a term with a so-called induced potential which depends on intrinsic- and extrinsic curvature of the embedded space-time but not on the details of the confining potential. We compute this induced potential for natural, simple embeddings of Schwarzschild- and Robertson-Walker space-times. We also discuss possible observable implications of our results and, in particular, propose and study an extension of a standard model of cosmological inflation taking into account extrinsic curvature effects. We show that the modified model allows for a solution where the scaling function vanishes like a power law with exponent 0.6830.. at some initial time.