Kaluza-Klein braneworld cosmology with static internal dimensions

TL;DR

This paper extends the Shiromizu-Maeda-Sasaki equations to higher dimensions and explores how static internal dimensions influence the effective Friedmann equations and Newton constant in Kaluza-Klein braneworld cosmology.

Contribution

It generalizes the SMS equations to higher dimensions and derives the effective Friedmann equation considering static internal dimensions in a Kaluza-Klein braneworld.

Findings

Friedmann equation depends on the equation of state, unlike in 5D models.

Effective Newton constant varies logarithmically with scale factor during radiation era.

Matter on the brane stabilizes the Newton constant after radiation-matter equality.

Abstract

We investigate the Kaluza-Klein braneworld cosmology from the point of view of observers on the brane. We first generalize the Shiromizu-Maeda-Sasaki (SMS) equations to higher dimensions. As an application, we study a (4+n)-dimensional brane with n dimensions compactified on the brane, in a (5+n)-dimensional bulk. By assuming that the size of the internal space is static, that the bulk energy-momentum tensor can be neglected, we determine the effect of the bulk geometry on the Kaluza-Klein braneworld. Then we derive the effective Friedmann equation on the brane. It turns out that the Friedmann equation explicitly depends on the equation of state, in contrast to the braneworld in a 5-dimensional bulk spacetime. In particular, in a radiation-dominated era, the effective Newton constant depends on the scale factor logarithmically. If we include a pressureless matter on the brane, this dependence disappears after the radiation-matter equality. This may be interpreted as stabilization of the Newton constant by the matter on the brane. Our findings imply that the Kaluza-Klein braneworld cosmology is quite different from the conventional Kaluza-Klein cosmology even at low energy.