Cosmology with Radion and Bulk Scalar Field in Two Branes Model

TL;DR

This paper studies the cosmological evolution of scalar fields and the radion in a five-dimensional two-brane model, deriving effective four-dimensional theories and exploring conditions for stabilization and inflation.

Contribution

It derives the low-energy effective four-dimensional equations from a five-dimensional two-brane model with a bulk scalar field, including stabilization effects and inflationary scenarios.

Findings

Effective four-dimensional bi-scalar tensor theory is obtained.

Radion stabilization leads to Brans-Dicke theory with specific parameter.

Acceptable late-time cosmology requires small dilaton coupling and brane separation.

Abstract

We investigate cosmological evolutions of the bulk scalar field $\phi(t)$ and the radion $d(t)$ in five-dimensional dilatonic two branes model. The bulk potential for the scalar field is taken as the exponential function $V_{bulk} \propto \exp(-2 \sqrt{2} b\phi)$, where $b$ is the parameter of the theory. This model includes Randall-Sundrum model (with $b=0$) and five-dimensional Ho\v{r}ava-Witten theory (with $b=1$). We consider matter on both branes and arbitrary potentials on the branes and in the bulk. These matter and potentials induce the cosmological expansion of the brane as well as the time evolution of the bulk scalar field and the radion. Starting with full five-dimensional equations, we derive four-dimensional effective equations which govern the low-energy dynamics of brane worlds. A correspondent five-dimensional geometry is also obtained. The effective four-dimensional theory on a positive tension brane is described by bi-scalar tensor theory. If the radion is stabilized, the effective theory becomes Brans-Dicke (BD) theory with BD parameter $1/2 b^2$. On the other hand, if the scalar field is stabilized, the effective theory becomes scalar-tensor theory with BD parameter $\frac{3}{2(3b^2+1)}\frac{\phi(t)}{1-\phi(t)}$ where $\phi$ is the BD field defined by radion $d(t)$. If we do not introduce the stabilization mechanism for these moduli fields, the acceptable late time cosmology can be realized only if the dilaton coupling $b$ is small ($b^2 < 1.6 \times 10^{-4}$) and the negative tension brane is sufficiently away from the positive tension brane. We also construct several models for inflationary brane worlds driven by potentials on the brane and in the bulk.