Abstract

The observable universe could be a 1 + 3-surface (the “brane”) embedded in a 1 + 3 + d-dimensional spacetime (the “bulk”), with Standard Model particles and fields trapped on the brane while gravity is free to access the bulk. At least one of the d extra spatial dimensions could be very large relative to the Planck scale, which lowers the fundamental gravity scale, possibly even down to the electroweak (∼ TeV) level. This revolutionary picture arises in the framework of recent developments in M theory. The 1 + 10-dimensional M theory encompasses the known 1 + 9-dimensional superstring theories, and is widely considered to be a promising potential route to quantum gravity. General relativity cannot describe gravity at high enough energies and must be replaced by a quantum gravity theory, picking up significant corrections as the fundamental energy scale is approached. At low energies, gravity is localized at the brane and general relativity is recovered, but at high energies gravity “leaks” into the bulk, behaving in a truly higher-dimensional way. This introduces significant changes to gravitational dynamics and perturbations, with interesting and potentially testable implications for high-energy astrophysics, black holes, and cosmology. Brane-world models offer a phenomenological way to test some of the novel predictions and corrections to general relativity that are implied by M theory. This review discusses the geometry, dynamics and perturbations of simple brane-world models for cosmology and astrophysics, mainly focusing on warped 5-dimensional brane-worlds based on the Randall-Sundrum models.

Highlights

  • At high enough energies, Einstein’s theory of general relativity breaks down, and will be superceded by a quantum gravity theory

  • The classical singularities predicted by general relativity in gravitational collapse and in the hot big bang will be removed by quantum gravity

  • Simple brane-world models of RS type provide a rich phenomenology for exploring some of the ideas that are emerging from M theory

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Summary

Introduction

Einstein’s theory of general relativity breaks down, and will be superceded by a quantum gravity theory. Even below the fundamental energy scale that marks the transition to quantum gravity, significant corrections to general relativity will arise. These corrections could have a major impact on the behaviour of gravitational collapse, black holes, and the early universe, and they could leave a trace – a “smoking gun” – in various observations and experiments. It is important to estimate these corrections and develop tests for detecting them or ruling them out In this way, quantum gravity can begin to be subject to testing by astrophysical and cosmological observations. This review considers only models that arise within the framework of M theory, and mainly the 5-dimensional warped brane-worlds

Heuristics of higher-dimensional gravity
Brane-worlds and M theory
Heuristics of KK modes
Randall–Sundrum Brane-Worlds
Covariant Approach to Brane-World Geometry and Dynamics
Field equations on the brane
Conservation equations
Propagation and constraint equations on the brane
Gravitational Collapse and Black Holes on the Brane
The black string
Taylor expansion into the bulk
The “tidal charge” black hole
Realistic black holes
Oppenheimer–Snyder collapse gives a non-static black hole
Brane-World Cosmology
Brane-world inflation
Brane-world instanton
Models with non-empty bulk
Metric-based perturbations
Density perturbations on large scales
Curvature perturbations and the Sachs–Wolfe effect
Vector perturbations
Tensor perturbations
Gravitational Wave Perturbations in Brane-World Cosmology
CMB Anisotropies in Brane-World Cosmology
The low-energy approximation
The simplest model
Conclusion
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