Cosmological Challenges in Theories with Extra Dimensions and Remarks on the Horizon Problem

TL;DR

This paper investigates cosmology on a 3-brane embedded in a five-dimensional, $Z_2$-symmetric spacetime inspired by Hořava–Witten theory. By solving the bulk Einstein equations with Israel boundary conditions, the authors show that the customary FRW relation $H^2 \propto \rho$ is not generically realized on the brane; the expansion is often governed by moduli and boundary data from the extra dimensions, with FRW only emerging for particular bulk stress-energy configurations. A broad and robust finding is that stabilizing the four-dimensional Planck mass (i.e., fixing the orbifold radius) typically requires extreme fine-tuning of the brane equation of state, unless the brane energy is dominated by a cosmological constant. The paper also discusses causality and horizon problem implications, showing that a time-varying Planck mass or alternative inflationary assumptions could modify the standard inflationary requirements and horizon-scale arguments in brane-world cosmologies.

Abstract

We consider the cosmology that results if our observable universe is a 3-brane in a higher dimensional universe. In particular, we focus on the case where our 3-brane is located at the $Z_2$ symmetry fixed plane of a $Z_2$ symmetric five-dimensional spacetime, as in the Hořava-Witten model compactified on a Calabi-Yau manifold. As our first result, we find that there can be substantial modifications to the standard Friedmann-Robertson-Walker (FRW) cosmology; as a consequence, a large class of such models is observationally inconsistent. In particular, any relationship between the Hubble constant and the energy density on our brane is possible, including (but not only) FRW. Generically, due to the existence of the bulk and the boundary conditions on the orbifold fixed plane, the relationship is not FRW, and hence cosmological constraints coming from big bang nucleosynthesis, structure formation, and the age of the universe difficult to satisfy. We do wish to point out, however, that some specific choices for the bulk stress-energy tensor components do reproduce normal FRW cosmology on our brane, and we have constructed an explicit example. As our second result, for a broad class of models, we find a somewhat surprising fact: the stabilization of the radius of the extra dimension and hence the four dimensional Planck mass requires unrealistic fine-tuning of the equation of state on our 3-brane. In the last third of the paper, we make remarks about causality and the horizon problem that apply to {\it any} theory in which the volume of the extra dimension determines the four-dimensional gravitational coupling. We point out that some of the assumptions that lead to the usual inflationary requirements are modified.