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Brane-World Gravity Living Rev. Relativity, 13, (2010), 5 LIVINGREVIEWS http://www.livingreviews.org/lrr-2010-5 (Update of lrr-2004-7) in relativity Brane-World Gravity Roy Maartens Institute of Cosmology & Gravitation University of Portsmouth Portsmouth PO1 3FX, U.K. email: [email protected] http://research.icg.port.ac.uk/members Kazuya Koyama Institute of Cosmology & Gravitation University of Portsmouth Portsmouth PO1 3FX, U.K. email: [email protected] http://research.icg.port.ac.uk/members Accepted on 6 September 2010 Published on 14 September 2010 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. 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 astro- physics, 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 analyzes the geometry, dynamics and perturbations of simple brane-world mod- els for cosmology and astrophysics, mainly focusing on warped 5-dimensional brane-worlds based on the Randall{Sundrum models. We also cover the simplest brane-world models in which 4-dimensional gravity on the brane is modified at low energies { the 5-dimensional Dvali{Gabadadze{Porrati models. Then we discuss co-dimension two branes in 6-dimensional models. This review is licensed under a Creative Commons Attribution-Non-Commercial-NoDerivs 3.0 Germany License. http://creativecommons.org/licenses/by-nc-nd/3.0/de/ Imprint / Terms of Use Living Reviews in Relativity is a peer reviewed open access journal published by the Max Planck Institute for Gravitational Physics, Am M¨uhlenberg 1, 14476 Potsdam, Germany. ISSN 1433-8351. This review is licensed under a Creative Commons Attribution-Non-Commercial-NoDerivs 3.0 Germany License: http://creativecommons.org/licenses/by-nc-nd/3.0/de/ Because a Living Reviews article can evolve over time, we recommend to cite the article as follows: Roy Maartens and Kazuya Koyama, \Brane-World Gravity", Living Rev. Relativity, 13, (2010), 5. [Online Article]: cited [<date>], http://www.livingreviews.org/lrr-2010-5 The date given as <date> then uniquely identifies the version of the article you are referring to. Article Revisions Living Reviews supports two ways of keeping its articles up-to-date: Fast-track revision A fast-track revision provides the author with the opportunity to add short notices of current research results, trends and developments, or important publications to the article. A fast-track revision is refereed by the responsible subject editor. If an article has undergone a fast-track revision, a summary of changes will be listed here. Major update A major update will include substantial changes and additions and is subject to full external refereeing. It is published with a new publication number. For detailed documentation of an article's evolution, please refer to the history document of the article's online version at http://www.livingreviews.org/lrr-2010-5. 14 September 2010: Inserted subsections on numerical solutions in Sections 6 and 7 and five figures. Added new Subsection 2.2 on the Randall{Sundrum model in string theory. Added new Section 9 on Dvali{Gabadadze{Porrati models and nine figures. Added new Section 10 on 6-dimensional models and one figure. Added about 100 new references. Page 7: Extended a paragraph moved here from the end of Section 1.2. Page 15: Added Subsection 2.2 \RS model in string theory". Page 43: Added two paragraphs discussing the neglection of backreaction due to metric per- turbations in the bulk. Page 60: Added Subsection 6.5 \Full numerical solutions". Page 69: Removed former Figure 11: Damping of brane-world gravity waves on horizon re- entry due to massive mode generation. Page 70: Added Subsection 7.2 \Full numerical solutions". Page 79: Added Section 9 \DGP Models". Page 90: Added Section 10 \6-Dimensional Models". Contents 1 Introduction 7 1.1 Heuristics of higher-dimensional gravity ........................ 7 1.2 Brane-worlds and M theory ............................... 8 1.3 Heuristics of KK modes ................................. 10 2 Randall{Sundrum Brane-Worlds 12 2.1 KK modes in RS 1-brane ................................ 13 2.2 RS model in string theory ................................ 15 3 Covariant Approach to Brane-World Geometry and Dynamics 17 3.1 Field equations on the brane .............................. 18 3.2 5-dimensional equations and the initial-value problem . 20 3.3 The brane viewpoint: A 1 + 3-covariant analysis .................... 21 3.4 Conservation equations .................................. 23 3.5 Propagation and constraint equations on the brane . 26 4 Gravitational Collapse and Black Holes on the Brane 30 4.1 The black string ..................................... 30 4.2 Taylor expansion into the bulk ............................. 31 4.3 The \tidal charge" black hole .............................. 32 4.4 Realistic black holes ................................... 33 4.5 Oppenheimer{Snyder collapse gives a non-static black hole . 33 4.6 AdS/CFT and black holes on 1-brane RS-type models . 35 5 Brane-World Cosmology: Dynamics 38 5.1 Brane-world inflation ................................... 40 5.2 Brane-world instanton .................................. 46 5.3 Models with non-empty bulk .............................. 47 6 Brane-World Cosmology: Perturbations 50 6.1 1 + 3-covariant perturbation equations on the brane . 51 6.2 Metric-based perturbations ............................... 52 6.3 Density perturbations on large scales .......................... 54 6.4 Curvature perturbations and the Sachs{Wolfe effect . 57 6.5 Full numerical solutions ................................. 60 6.6 Vector perturbations ................................... 65 6.7 Tensor perturbations ................................... 66 7 Gravitational Wave Perturbations in Brane-World Cosmology 67 7.1 Analytical approaches .................................. 67 7.2 Full numerical solutions ................................. 70 8 CMB Anisotropies in Brane-World Cosmology 73 8.1 The low-energy approximation ............................. 74 8.2 The simplest model .................................... 76 9 DGP Models: Modifying Gravity at Low Energies 79 9.1 `Self-accelerating' DGP ................................. 79 9.2 `Normal' DGP ...................................... 85 10 6-Dimensional Models 90 10.1 Supersymmetric Large Extra Dimensions (SLED) Model . 90 10.2 Cosmology in 6D brane-world models .......................... 92 10.3 Cascading brane-world model .............................. 92 11 Conclusion 94 12 Acknowledgments 97 References 98 Brane-World Gravity 7 1 Introduction 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 gravi- tational collapse and in the hot big bang will be removed by quantum gravity. But 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. Thus 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. Developing a quantum theory of gravity and a unified theory of all the forces and particles of nature are the two main goals of current work in fundamental physics. There is as yet no generally accepted (pre-)quantum gravity theory. Two of the main contenders are M theory (for reviews see, e.g., [214, 356, 377]) and quantum geometry (loop quantum gravity; for reviews see, e.g., [365, 409]). It is important to explore the astrophysical and cosmological predictions of both these approaches. This review considers only models that arise within the framework of M theory. In this review, we focus on RS brane-worlds (mainly the RS 1-brane model) and their gen- eralizations, with the emphasis on geometry and gravitational dynamics (see [304, 314, 269, 424, 348, 268, 360, 120, 49, 267, 270] for previous reviews with a broadly similar approach). Other reviews focus on string-theory aspects, e.g., [147, 316, 97, 357], or on particle physics aspects, e.g., [354, 366, 261, 151, 75]. We also discuss the 5D DGP models, which modify general relativity at low energies, unlike the RS models; these models have become important
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