[Paper Review] Brane cosmology, varying speed of light and inflation in models with one or more extra dimensions

D. A. Steer, Matthew Parry|ArXiv.org|Jan 16, 2002

Black Holes and Theoretical Physics3 references3 citations

TL;DR

This paper introduces a 'mirage cosmology' framework treating branes as test particles moving through static higher-dimensional bulk spacetimes to derive modified Friedmann equations on the brane. It shows that in 10D bulk spacetimes like Sch-AdS₅×S⁵ and rotating black holes, new dark fluid terms emerge, including bulk-dependent cosmological constants and angular momentum-driven terms, with a critical brane angular momentum ℓc determining stable orbits and effective speed of light c_eff that increases asymptotically to a constant as the universe expands.

ABSTRACT

We summarise the approach to brane cosmology known as ``mirage cosmology'' and use it to determine the Friedmann equation on a 3-brane embedded in different bulk spacetimes all with one or more extra dimensions. Usually, when there is more than one extra dimension the junction conditions, central to the usual brane world scenarios, are difficult to apply. This problem does not arise in mirage cosmology because the brane is treated as a ``test particle'' in the background spacetime. We discuss in detail the dynamics of a brane embedded in two specific 10D bulk spacetimes, namely Sch-AdS$_5 imes$S$_5$ and a rotating black hole, and from the dynamics--which are now rather more complicated since the brane can move in all the extra dimensions--determine the new ``dark fluid'' terms in the brane Friedmann equation. Some of these, such as the cosmological constant term, are seen to be bulk dependent. However, for both bulks we show that there exists a critical brane angular momentum, $\ell_c$, and discuss its significance. We then show explicitly how this mirage cosmology approach matches with the familiar junction condition approach when there is just one extra dimension. The issue of a varying speed of light in mirage cosmology is reviewed and we find a scenario in which $c_{\bf eff}$ always increases, tending asymptotically to a constant $c_0$ as the universe expands. Finally some comments are made regarding brane inflation and limitations of the mirage cosmology approach are also discussed.

Motivation & Objective

• To develop a framework for brane cosmology in higher-dimensional spacetimes where standard junction conditions fail due to multiple extra dimensions.
• To investigate how brane dynamics in static 10D bulk spacetimes—specifically Sch-AdS₅×S⁵ and rotating black holes—generate effective cosmological terms on the brane.
• To determine the role of brane angular momentum ℓ and identify a critical value ℓc that separates stable and unstable cosmological evolution.
• To explore the implications for a varying effective speed of light c_eff in this mirage cosmology framework.

Proposed method

• Treat the 3-brane as a test particle moving through a static 10D bulk spacetime, avoiding the need to solve complex junction conditions in higher codimension.
• Use the Dirac-Born-Infeld (DBI) action with a Wess-Zumino term to describe the brane dynamics in type IIB string theory.
• Derive the effective Friedmann equation on the brane by analyzing the geodesic motion of the brane in the bulk metric, extracting dark fluid terms from the effective potential.
• For the Sch-AdS₅×S⁵ bulk, solve the effective potential to identify critical angular momentum ℓc where the effective potential has a degenerate maximum.
• In the rotating black hole bulk, analyze the brane trajectory using rescaled radial coordinates and identify conditions for trapped vs. unbound orbits based on energy E and angular momentum ℓ.
• Compute the effective speed of light c_eff from the brane's effective potential and study its time evolution as a function of ℓ and E.

Experimental results

Research questions

• RQ1 How do dark fluid terms in the brane Friedmann equation depend on the bulk geometry when there is more than one extra dimension?
• RQ2 What is the significance of a critical brane angular momentum ℓc in determining the stability and evolution of cosmological solutions in mirage cosmology?
• RQ3 How does the effective speed of light c_eff evolve over time in mirage cosmology, and under what conditions does it increase monotonically to a constant?
• RQ4 To what extent are the cosmological terms in the Friedmann equation sensitive to the bulk's properties, such as cosmological constant differences or black hole rotation?
• RQ5 Can the mirage cosmology approach reproduce standard brane world results in the one-extra-dimensional limit, and how does it compare to junction condition methods?

Key findings

• In the Sch-AdS₅×S⁵ bulk, a critical angular momentum ℓc exists such that for ℓ < ℓc, the effective potential allows unbound, expanding brane trajectories with c_eff increasing asymptotically to 1.
• For ℓ = ℓc, the effective speed of light c_eff tends to a constant c₀ < 1, indicating a non-standard late-time cosmological evolution.
• In the rotating black hole bulk, the effective cosmological constant on the brane depends on energy E and does not vanish for q = ±1 unless E = 1, indicating bulk-dependent dark energy.
• The brane dynamics in the rotating black hole bulk exhibit three distinct classes of trajectories depending on ℓ, with E < 1 leading to trapped orbits and E > 1 allowing unbound motion.
• For small black hole rotation (a ≪ 1), dark fluid terms proportional to a⁻¹⁰, a⁻⁸, a⁻⁶, and a⁻⁴ emerge, with the a⁻¹⁰ and a⁻⁶ terms surviving even for ℓ = 0 due to black hole angular momentum.
• In both bulk spacetimes, c_eff increases monotonically with cosmic time for ℓ < ℓc, tending asymptotically to a finite constant, suggesting a dynamically evolving speed of light in this cosmological framework.