Brane World Sum Rules

TL;DR

This paper derives consistency conditions, called sum rules, for brane world models with periodic internal spaces, providing a tool to verify model validity and analyzing specific models like Randall-Sundrum and Goldberger-Wise.

Contribution

It introduces sum rules for brane world scenarios, offering a new consistency check and analyzing their implications for various models including supersymmetric cases.

Findings

Randall-Sundrum model satisfies the sum rules.

Smooth brane generalizations are not compatible with the constraints.

Supersymmetric models automatically satisfy the sum rules.

Abstract

A set of consistency conditions is derived from Einstein equations for brane world scenarios with a spatially periodic internal space. In particular, the sum of the total tension of the flat branes and the non-negative integral of the gradient energy of the bulk scalars must vanish. This constraint allows us to make a simple consistency check of several models. We show that the two-brane Randall-Sundrum model satisfies this constraint, but it does not allow a generalization with smooth branes (domain walls), independently of the issue of supersymmetry. The Goldberger-Wise model of brane stabilization has to include the backreaction on the metric and the fine tuning of the cosmological constant to satisfy the constraints. We check that this is achieved in the DeWolfe-Freedman-Gubser-Karch scenario. Our constraints are automatically satisfied in supersymmetric brane world models.