BraneWorld Gravity
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Abstract
The observable universe could be a 1+3surface (the “brane”) embedded in a 1+3+ddimensional 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+10dimensional M theory encompasses the known 1+9dimensional 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 higherdimensional way. This introduces significant changes to gravitational dynamics and perturbations, with interesting and potentially testable implications for highenergy astrophysics, black holes, and cosmology. Braneworld 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 braneworld models for cosmology and astrophysics, mainly focusing on warped 5dimensional braneworlds based on the RandallSundrum models. We also cover the simplest braneworld models in which 4dimensional gravity on the brane is modified at low energies — the 5dimensional DvaliGabadadzePorrati models. Then we discuss codimension two branes in 6dimensional models.
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At high enough energies, Einstein’s theory of general relativity breaks down, and will be superceded by a quantum gravity theory.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 gravitational 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 braneworlds (mainly the RS 1brane model) and their generalizations, 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 stringtheory 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 examples in cosmology for achieving latetime acceleration of the universe without dark energy. Finally, we give brief overviews of 6D models, in which the brane has codimension two, introducing very different features to the 5D case with codimension one branes.
1.1 Heuristics of higherdimensional gravity
One of the fundamental aspects of string theory is the need for extra spatial dimensions^{1}. This revives the original higherdimensional ideas of Kaluza and Klein in the 1920s, but in a new context of quantum gravity. An important consequence of extra dimensions is that the
1.2 Braneworlds and M theory
String theory thus incorporates the possibility that the fundamental scale is much less than the Planck scale felt in 4 dimensions. There are five distinct 1+9dimensional superstring theories, all giving quantum theories of gravity. Discoveries in the mid1990s of duality transformations that relate these superstring theories and the 1+10dimensional supergravity theory, led to the conjecture that all of these theories arise as different limits of a single theory, which has come to be known as M theory. The 11th dimension in M theory is related to the string coupling strength; the size of this dimension grows as the coupling becomes strong. At low energies, M theory can be approximated by 1+10dimensional supergravity.
In the HoravaWitten solution [203], gauge fields of the standard model are confined on two 1+9branes located at the end points of an S^{1}/Z_{2} orbifold, i.e., a circle folded on itself across a diameter. The 6 extra dimensions on the branes are compactified on a very small scale close to the fundamental scale, and their effect on the dynamics is felt through “moduli” fields, i.e., 5D scalar fields. A 5D realization of the HoravaWitten theory and the corresponding braneworld cosmology is given in [300, 301, 302].
In the ADD models, more than one extra dimension is required for agreement with experiments, and there is “democracy” amongst the equivalent extra dimensions, which are typically flat. By contrast, the RS models have a “preferred” extra dimension, with other extra dimensions treated as ignorable (i.e., stabilized except at energies near the fundamental scale). Furthermore, this extra dimension is curved or “warped” rather than flat: The bulk is a portion of antide Sitter (AdS_{5}) spacetime. As in the HoravaWitten solutions, the RS branes are Z_{2}symmetric (mirror symmetry), and have a tension, which serves to counter the influence of the negative bulk cosmological constant on the brane. This also means that the selfgravity of the branes is incorporated in the RS models. The novel feature of the RS models compared to previous higherdimensional models is that the observable 3 dimensions are protected from the large extra dimension (at low energies) by curvature rather than straightforward compactification.
The RS braneworlds and their generalizations (to include matter on the brane, scalar fields in the bulk, etc.) provide phenomenological models that reflect at least some of the features of M theory, and that bring exciting new geometric and particle physics ideas into play. The RS models also provide a framework for exploring holographic ideas that have emerged in M theory. Roughly speaking, holography suggests that higherdimensional gravitational dynamics may be determined from knowledge of the fields on a lowerdimensional boundary. The AdS/CFT correspondence is an example, in which the classical dynamics of the higherdimensional gravitational field are equivalent to the quantum dynamics of a conformal field theory (CFT) on the boundary. The RS model with its AdS_{5} metric satisfies this correspondence to lowest perturbative order [129] (see also [342, 375, 193, 386, 390, 290, 347, 180] for the AdS/CFT correspondence in a cosmological context).
Before turning to a more detailed analysis of RS braneworlds, We discuss the notion of KaluzaKlein (KK) modes of the graviton.
1.3 Heuristics of KK modes

a 4D spin2 graviton h_{ ij } (2 polarizations),

a 4D spin1 gravivector (graviphoton) ∑_{ i } (2 polarizations), and

a 4D spin0 graviscalar β.
In the general case of d extra dimensions, the number of degrees of freedom in the graviton follows from the irreducible tensor representations of the isometry group as \({1 \over 2}(d + 1)(d + 4)\).
2 RandallSundrum BraneWorlds
We will concentrate mainly on RS 1brane from now on, referring to RS 2brane occasionally. The RS 1brane models are in some sense the most simple and geometrically appealing form of a braneworld model, while at the same time providing a framework for the AdS/CFT correspondence [129, 342, 375, 193, 386, 390, 290, 347, 180]. The RS 2brane introduce the added complication of radion stabilization, as well as possible complications arising from negative tension. However, they remain important and will occasionally be discussed.
2.1 KK modes in RS 1brane
2.2 RS model in string theory
3 Covariant Approach to BraneWorld Geometry and Dynamics
The RS models and the subsequent generalization from a Minkowski brane to a FriedmannRobertsonWalker (FRW) brane [38, 260, 216, 226, 183, 330, 209, 145, 154] were derived as solutions in particular coordinates of the 5D Einstein equations, together with the junction conditions at the Z_{2}symmetric brane^{2}. A broader perspective, with useful insights into the interplay between 4D and 5D effects, can be obtained via the covariant ShiromizuMaedaSasaki approach [388], in which the brane and bulk metrics remain general. The basic idea is to use the GaussCodazzi equations to project the 5D curvature along the brane. (The general formalism for relating the geometries of a spacetime and of hypersurfaces within that spacetime is given in [423].)
3.1 Field equations on the brane
 \({{\mathcal S}_{\mu \nu}} \sim {({T_{\mu \nu}})^2}\) is the highenergy correction term, which is negligible for ρ ≪ λ, but dominant for ρ ≫ λ:$${{\vert {\kappa ^2}{{\mathcal S}_{\mu \nu}}/\lambda \vert} \over {\vert {\kappa ^2}{T_{\mu \nu}}\vert}}\sim{{\vert {T_{\mu \nu}}\vert} \over \lambda}\sim{\rho \over \lambda}.$$(80)

\({{\mathcal E}_{\mu \nu}}\) is the projection of the bulk Weyl tensor on the brane, and encodes corrections from 5D graviton effects (the KK modes in the linearized case). From the braneobserver viewpoint, the energymomentum corrections in \({{\mathcal S}_{\mu \nu}}\) are local, whereas the KK corrections in \({{\mathcal E}_{\mu \nu}}\) are nonlocal, since they incorporate 5D gravity wave modes. These nonlocal corrections cannot be determined purely from data on the brane. In the perturbative analysis of RS 1brane which leads to the corrections in the gravitational potential, Equation (41), the KK modes that generate this correction are responsible for a nonzero \({{\mathcal E}_{\mu \nu}}\); this term is what carries the modification to the weakfield field equations. The 9 independent components in the tracefree \({{\mathcal E}_{\mu \nu}}\) are reduced to 5 degrees of freedom by Equation (79); these arise from the 5 polarizations of the 5D graviton. Note that the covariant formalism applies also to the twobrane case. In that case, the gravitational influence of the second brane is felt via its contribution to \({{\mathcal E}_{\mu \nu}}\).
3.2 5dimensional equations and the initialvalue problem
3.3 The brane viewpoint: A 1 + 3covariant analysis
Following [303], a systematic analysis can be developed from the viewpoint of a branebound observer. (See also [228].) The effects of bulk gravity are conveyed, from a brane observer viewpoint, via the local \({{\mathcal S}_{\mu \nu}}\) and nonlocal \({{\mathcal E}_{\mu \nu}}\) corrections to Einstein’s equations. (In the more general case, bulk effects on the brane are also carried by \({{\mathcal F}_{\mu \nu}}\), which describes any 5D fields.) The \({{\mathcal E}_{\mu \nu}}\) term cannot in general be determined from data on the brane, and the 5D equations above (or their equivalent) need to be solved in order to find \({{\mathcal E}_{\mu \nu}}\).

The KK (or Weyl) anisotropic stress \(\pi _{\mu \nu}^{\mathcal E}\) incorporates the scalar or spin0 (“Coulomb”), the vector (transverse) or spin1 (gravimagnetic), and the tensor (transverse traceless) or spin2 (gravitational wave) 4D modes of the spin2 5D graviton.

The KK momentum density \(q_\mu ^{\mathcal E}\) incorporates spin0 and spin1 modes, and defines a velocity \(\upsilon _\mu ^{\mathcal E}\) of the Weyl fluid relative to u^{ μ } via \(q_\mu ^{\mathcal E} = \rho {\mathcal E}\upsilon _\mu ^{\mathcal E}\).

The KK energy density \({\rho _{\mathcal E}}\), often called the “dark radiation”, incorporates the spin0 mode.
3.4 Conservation equations
The absence of bulk source terms in the conservation equations is a consequence of having Λ_{5} as the only 5D source in the bulk. For example, if there is a bulk scalar field, then there is energymomentum exchange between the brane and bulk (in addition to the gravitational interaction) [311, 21, 322, 143, 274, 144, 48].

Inhomogeneous and anisotropic effects from the 4D matterradiation distribution on the brane are a source for the 5D Weyl tensor, which nonlocally “backreacts” on the brane via its projection \({{\mathcal E}_{\mu \nu}}\).

There are evolution equations for the dark radiative (nonlocal, Weyl) energy (\({\rho _{\mathcal E}}\)) and momentum \(q_\mu ^{\mathcal E}\) densities (carrying scalar and vector modes from bulk gravitons), but there is no evolution equation for the dark radiative anisotropic stress (\(\pi _{\mu \nu}^{\mathcal E}\)) (carrying tensor, as well as scalar and vector, modes), which arises in both evolution equations.
In particular cases, the Weyl anisotropic stress \(\pi _{\mu \nu}^{\mathcal E}\) may drop out of the nonlocal conservation equations, i.e., when we can neglect \({\sigma ^{\mu \nu}}\pi _{\mu \nu}^{\mathcal E},\;{\vec \nabla ^\nu}\pi _{\mu \nu}^{\mathcal E}\) and \({A^\nu}\pi _{\mu \nu}^{\mathcal E}\) This is the case when we consider linearized perturbations about an FRW background (which remove the first and last of these terms) and further when we can neglect gradient terms on large scales (which removes the second term). This case is discussed in Section 6. But in general, and especially in astrophysical contexts, the \(\pi _{\mu \nu}^{\mathcal E}\) terms cannot be neglected. Even when we can neglect these terms, \(\pi _{\mu \nu}^{\mathcal E}\) arises in the field equations on the brane.
All of the matter source terms on the right of these two equations, except for the first term on the right of Equation (118), are imperfect fluid terms, and most of these terms are quadratic in the imperfect quantities q_{ μ } and π_{ μν }. For a single perfect fluid or scalar field, only the \({\vec \nabla _\mu}\rho\) term on the right of Equation (118) survives, but in realistic cosmological and astrophysical models, further terms will survive. For example, terms linear in π_{ μν } will carry the photon quadrupole in cosmology or the shear viscous stress in stellar models. If there are two fluids (even if both fluids are perfect), then there will be a relative velocity υ_{ μ } generating a momentum densityq_{ μ } = ρυ_{ μ }, which will serve to source nonlocal effects.
In general, the 4 independent equations in Equations (117) and (118) constrain 4 of the 9 independent components of \({{\mathcal E}_{\mu \nu}}\) on the brane. What is missing is an evolution equation for \(\pi _{\mu \nu}^{\mathcal E}\), which has up to 5 independent components. These 5 degrees of freedom correspond to the 5 polarizations of the 5D graviton. Thus in general, the projection of the 5dimensional field equations onto the brane does not lead to a closed system, as expected, since there are bulk degrees of freedom whose impact on the brane cannot be predicted by brane observers. The KK anisotropic stress \(\pi _{\mu \nu}^{\mathcal E}\) encodes the nonlocality.
In special cases the missing equation does not matter. For example, if \(\pi _{\mu \nu}^{\mathcal E} = 0\) by symmetry, as in the case of an FRW brane, then the evolution of \({{\mathcal E}_{\mu \nu}}\) is determined by Equations (117) and (118). If the brane is stationary (with Killing vector parallel to u^{ μ }), then evolution equations are not needed for \({{\mathcal E}_{\mu \nu}}\), although in general \(\pi _{\mu \nu}^{\mathcal E}\) will still be undetermined. However, small perturbations of these special cases will immediately restore the problem of missing information.

if \({{\mathcal E}_{\mu \nu}} = 0\) and the brane energymomentum tensor has perfect fluid form, then the density ρ must be homogeneous, \({\vec \nabla _\mu}\rho = 0\);

the converse does not hold, i.e., homogeneous density does not in general imply vanishing \({{\mathcal E}_{\mu \nu}}\).
If \({{\mathcal E}_{\mu \nu}} = 0\), then the field equations on the brane form a closed system. Thus for perfect fluid branes with homogeneous density and \({{\mathcal E}_{\mu \nu}} = 0\), the brane field equations form a consistent closed system. However, this is unstable to perturbations, and there is also no guarantee that the resulting brane metric can be embedded in a regular bulk.
3.5 Propagation and constraint equations on the brane
 Generalized Raychaudhuri equation (expansion propagation):$$\dot \Theta + {1 \over 3}{\Theta ^2} + {\sigma _{\mu \nu}}{\sigma ^{\mu \nu}}  2{\omega _\mu}{\omega ^\mu}  {\vec \nabla ^\mu}{A_\mu}  {A_\mu}{A^\mu} + {{{\kappa ^2}} \over 2}(\rho + 3p)  \Lambda =  {{{\kappa ^2}} \over 2}(2\rho + 3p){\rho \over \lambda}  {\kappa ^2}{\rho _{\mathcal E}}.$$(129)
 Vorticity propagation:$${\dot \omega _{\langle \mu \rangle}} + {2 \over 3}\Theta {\omega _\mu} + {1 \over 2}{\rm{curl}}\,{A_\mu}  {\sigma _{\mu \nu}}{\omega ^\nu} = 0.$$(130)
 Shear propagation:$${\dot \sigma _{\langle \mu \nu \rangle}} + {2 \over 3}\Theta {\sigma _{\mu \nu}} + {E_{\mu \nu}}  {\vec \nabla _{\langle \mu}}{A_{\nu \rangle}} + {\sigma _{\alpha \langle \mu}}{\sigma _{\nu \rangle}}^\alpha + {\omega _{\langle \mu}}{\omega _{\nu \rangle}}  {A_{\langle \mu}}{A_{\nu \rangle}} = {{{\kappa ^2}} \over 2}\pi _{\mu \nu}^{\mathcal E}.$$(131)
 Gravitoelectric propagation (MaxwellWeyl Edot equation):$$\begin{array}{*{20}c} {{{\dot E}_{\langle \mu \nu \rangle}} + \Theta {E_{\mu \nu}}  {\rm{curl}}\,{H_{\mu \nu}} + {{{\kappa ^2}} \over 2}(\rho + p){\sigma _{\mu \nu}}  2{A^\alpha}{\varepsilon _{\alpha \beta (\mu}}{H_{\nu)}}^\beta  3{\sigma _{\alpha \langle \mu}}{E_{\nu \rangle}}^\alpha + {\omega ^\alpha}{\varepsilon _{\alpha \beta (\mu}}{E_{\nu)}}^\beta =} \\ { {{{\kappa ^2}} \over 2}(\rho + p){\rho \over \lambda}{\sigma _{\mu \nu}}\quad \quad \quad \quad \quad \quad \quad \,\,\,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ { {{{\kappa ^2}} \over 6}\left[ {4\rho {\mathcal E}{\sigma _{\mu \nu}} + 3\dot \pi _{\langle \mu \nu \rangle}^{\mathcal E} + \Theta \pi _{\mu \nu}^{\mathcal E} + 3{{\vec \nabla}_{\langle \mu}}q_{\nu \rangle}^{\mathcal E} + 6{A_{\langle \mu}}q_{\nu \rangle}^{\mathcal E} + 3{\sigma ^\alpha}_{\langle \mu}\pi _{\nu \rangle \alpha}^{\mathcal E} + 3{\omega ^\alpha}{\varepsilon _{\alpha \beta (\mu}}\pi _{\nu)}^{\mathcal E}{}^\beta} \right].\quad} \\ \end{array}$$(132)
 Gravitomagnetic propagation (MaxwellWeyl Hdot equation):$$\begin{array}{*{20}c} {{{\dot H}_{\langle \mu \nu \rangle}} + \Theta {H_{\mu \nu}} + {\rm{curl}}\,{E_{\mu \nu}}  3{\sigma _{\alpha \langle \mu}}{H_{\nu \rangle}}^\alpha + {\omega ^\alpha}{\varepsilon _{\alpha \beta (\mu}}{H_{\nu)}}^\beta + 2{A^\alpha}{\varepsilon _{\alpha \beta (\mu}}{E_{\nu)}}^\beta =} \\ {\qquad \qquad {{{\kappa ^2}} \over 2}\left[ {{\rm{curl}}\,\pi _{\mu \nu}^{\mathcal E}  3{\omega _{\langle \mu}}q_{\nu \rangle}^{\mathcal E} + {\sigma _{\alpha (\mu}}{\varepsilon _{\nu)}}^{\alpha \beta}q_\beta ^{\mathcal E}} \right].\quad \quad \quad \quad} \\ \end{array}$$(133)
 Vorticity constraint:$${\vec \nabla ^\mu}{\omega _\mu}  {A^\mu}{\omega _\mu} = 0.$$(134)
 Shear constraint:$${\vec \nabla ^\nu}{\sigma _{\mu \nu}}  {\rm{curl}}\,{\omega _\mu}  {2 \over 3}{\vec \nabla _\mu}\Theta + 2{\varepsilon _{\mu \nu \alpha}}{\omega ^\nu}{A^\alpha} =  {\kappa ^2}q_\mu ^{\mathcal E}.$$(135)
 Gravitomagnetic constraint:$${\rm{curl}}\,{\sigma _{\mu \nu}} + {\vec \nabla _{\langle \mu}}{\omega _{\nu \rangle}}  {H_{\mu \nu}} + 2{A_{\langle \mu}}{\omega _{\nu \rangle}} = 0.$$(136)
 Gravitoelectric divergence (MaxwellWeyl divE equation):$$\begin{array}{*{20}c} {{{\vec \nabla}^\nu}{E_{\mu \nu}}  {{{\kappa ^2}} \over 3}{{\vec \nabla}_\mu}\rho  {\varepsilon _{\mu \nu \alpha}}{\sigma ^\nu}_\beta {H^{\alpha \beta}} + 3{H_{\mu \nu}}{\omega ^\nu} = \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \,\,} \\ {\,{{{\kappa ^2}} \over 3}{\rho \over \lambda}{{\vec \nabla}_\mu}\rho + {{{\kappa ^2}} \over 6}\left({2{{\vec \nabla}_\mu}{\rho _{\mathcal E}}  2\Theta q_\mu ^{\mathcal E}  3{{\vec \nabla}^\nu}\pi _{\mu \nu}^{\mathcal E} + 3{\sigma _\mu}^\nu q_\nu ^{\mathcal E}  9{\varepsilon _\mu}^{\nu \alpha}{\omega _\nu}q_\alpha ^{\mathcal E}} \right).} \\ \end{array}$$(137)
 Gravitomagnetic divergence (MaxwellWeyl divH equation):$$\begin{array}{*{20}c} {{{\vec \nabla}^\nu}{H_{\mu \nu}}  {\kappa ^2}(\rho + p){\omega _\mu} + {\varepsilon _{\mu \nu \alpha}}{\sigma ^\nu}_\beta {E^{\alpha \beta}}  3{E_{\mu \nu}}{\omega ^\nu} = \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {{\kappa ^2}(\rho + p){\rho \over \lambda}{\omega _\mu} + {{{\kappa ^2}} \over 6}\left({8{\rho _{\mathcal E}}{\omega _\mu}  3{\rm{curl}}\,q_\mu ^{\mathcal E}  3{\varepsilon _\mu}^{\nu \alpha}{\sigma _\nu}^\beta \pi _{\alpha \beta}^{\mathcal E}  3\pi _{\mu \nu}^{\mathcal E}{\omega ^\nu}} \right).} \\ \end{array}$$(138)
 GaussCodazzi equations on the brane (with ω_{ μ } = 0):$$R_{\langle \mu \nu \rangle}^ \bot + {\dot \sigma _{\langle \mu \nu \rangle}} + \Theta {\sigma _{\mu \nu}}  {\vec \nabla _{\langle \mu}}{A_{\nu \rangle}}  {A_{\langle \mu}}{A_{\nu \rangle}} = {\kappa ^2}\pi _{\mu \nu}^{\mathcal E},$$(139)where \(R_{\mu \nu}^ \bot\) is the Ricci tensor for 3surfaces orthogonal to u^{ μ } on the brane, and \({R^ \bot} = {h^{\mu \nu}}R_{\mu \nu}^ \bot\).$${R^ \bot} + {2 \over 3}{\Theta ^2}  {\sigma _{\mu \nu}}{\sigma ^{\mu \nu}}  2{\kappa ^2}\rho  2\Lambda = {\kappa ^2}{{{\rho ^2}} \over \lambda} + 2{\kappa ^2}{\rho _{\mathcal E}},$$(140)
The standard 4D general relativity results are regained when λ^{−1} → 0 and \({{\mathcal E}_{\mu \nu}} = 0\), which sets all right hand sides to zero in Equations (129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140). Together with Equations (119, 120, 121, 122), these equations govern the dynamics of the matter and gravitational fields on the brane, incorporating both the local, highenergy (quadratic energymomentum) and nonlocal, KK (projected 5D Weyl) effects from the bulk. Highenergy terms are proportional to ρ/λ, and are significant only when ρ > λ. The KK terms contain \({\rho _{\mathcal E}},\, q_\mu ^{\mathcal E}\), and \(\pi _{\mu \nu}^{\mathcal E}\), with the latter two quantities introducing imperfect fluid effects, even when the matter has perfect fluid form.
Bulk effects give rise to important new driving and source terms in the propagation and constraint equations. The vorticity propagation and constraint, and the gravitomagnetic constraint have no direct bulk effects, but all other equations do. Highenergy and KK energy density terms are driving terms in the propagation of the expansion Θ. The spatial gradients of these terms provide sources for the gravitoelectric field E_{ μν }. The KK anisotropic stress is a driving term in the propagation of shear σ_{ μν } and the gravitoelectric/gravitomagnetic fields, E and H_{ μν } respectively, and the KK momentum density is a source for shear and the gravitomagnetic field. The 4D MaxwellWeyl equations show in detail the contribution to the 4D gravitoelectromagnetic field on the brane, i.e., (E_{ μν }, E_{ μν }), from the 5D Weyl field in the bulk.
The system of propagation and constraint equations, i.e., Equations (119, 120, 121, 122) and (129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140), is exact and nonlinear, applicable to both cosmological and astrophysical modelling, including stronggravity effects. In general the system of equations is not closed: There is no evolution equation for the KK anisotropic stress \(\pi _{\mu \nu}^{\mathcal E}\).
4 Gravitational Collapse and Black Holes on the Brane
The physics of braneworld compact objects and gravitational collapse is complicated by a number of factors, especially the confinement of matter to the brane, while the gravitational field can access the extra dimension, and the nonlocal (from the brane viewpoint) gravitational interaction between the brane and the bulk. Extradimensional effects mean that the 4D matching conditions on the brane, i.e., continuity of the induced metric and extrinsic curvature across the 2surface boundary, are much more complicated to implement [160, 119, 422, 159]. Highenergy corrections increase the effective density and pressure of stellar and collapsing matter. In particular this means that the effective pressure does not in general vanish at the boundary 2surface, changing the nature of the 4D matching conditions on the brane. The nonlocal KK effects further complicate the matching problem on the brane, since they in general contribute to the effective radial pressure at the boundary 2surface. Gravitational collapse inevitably produces energies high enough, i.e., ρ ≫ λ, to make these corrections significant.
We expect that extradimensional effects will be negligible outside the highenergy, shortrange regime. The corrections to the weakfield potential, Equation (41), are at the second postNewtonian (2PN) level [164, 210]. However, modifications to Hawking radiation may bring significant corrections even for solarsized black holes, as discussed below.
4.1 The black string
Thus the “obvious” approach to finding a brane black hole fails. An alternative approach is to seek solutions of the brane field equations with nonzero \({{\mathcal E}_{\mu \nu}}\) [105]. Brane solutions of static black hole exteriors with 5D corrections to the Schwarzschild metric have been found [105, 160, 119, 422, 224, 73, 223], but the bulk metric for these solutions has not been found. Numerical integration into the bulk, starting from static black hole solutions on the brane, is plagued with difficulties [389, 78].
4.2 Taylor expansion into the bulk
4.3 The “tidal charge” black hole
The tidalcharge black hole metric does not satisfy the farfield r^{−3} correction to the gravitational potential, as in Equation (41), and therefore cannot describe the endstate of collapse. However, Equation (159) shows the correct 5D behaviour of the potential (∝ r^{−2}) at short distances, so that the tidalcharge metric could be a good approximation in the strongfield regime for small black holes.
4.4 Realistic black holes

Numerical simulations of highly relativistic static stars on the brane [427] indicate that general relativity remains a good approximation.

Exact analysis of OppenheimerSnyder collapse on the brane shows that the exterior is nonstatic [159], and this is extended to general collapse by arguments based on a generalized AdS/CFT correspondence [406, 139].
The 4D Schwarzschild metric cannot describe the final state of collapse, since it cannot incorporate the 5D behaviour of the gravitational potential in the strongfield regime (the metric is incompatible with massive KK modes). A nonperturbative exterior solution should have nonzero \({{\mathcal E}_{\mu \nu}}\) in order to be compatible with massive KK modes in the strongfield regime. In the endstate of collapse, we expect an \({{\mathcal E}_{\mu \nu}}\) which goes to zero at large distances, recovering the Schwarzschild weakfield limit, but which grows at short range. Furthermore, \({{\mathcal E}_{\mu \nu}}\) may carry a Weyl “fossil record” of the collapse process.
4.5 OppenheimerSnyder collapse gives a nonstatic black hole
The simplest scenario in which to analyze gravitational collapse is the OppenheimerSnyder model, i.e., collapsing homogeneous and isotropic dust [159]. The collapse region on the brane has an FRW metric, while the exterior vacuum has an unknown metric. In 4D general relativity, the exterior is a Schwarzschild spacetime; the dynamics of collapse leaves no imprint on the exterior.
The standard 4D DarmoisIsrael matching conditions, which we assume hold on the brane, require that the metric and the extrinsic curvature of ∑ be continuous (there are no intrinsic stresses on ∑). The extrinsic curvature is continuous if the metric is continuous and if Ṙ is continuous. We therefore need to match the metrics and Ṙ across ∑.

5D bulk graviton stresses, which transmit effects nonlocally from the interior to the exterior, and

the nonvanishing of the effective pressure at the boundary, which means that dynamical information from the interior can be conveyed outside via the 4D matching conditions.
The result suggests that gravitational collapse on the brane may leave a signature in the exterior, dependent upon the dynamics of collapse, so that astrophysical black holes on the brane may in principle have KK “hair”. It is possible that the nonstatic exterior will be transient, and will tend to a static geometry at late times, close to Schwarzschild at large distances.
4.6 AdS/CFT and black holes on 1brane RStype models
OppenheimerSnyder collapse is very special; in particular, it is homogeneous. One could argue that the nonstatic exterior arises because of the special nature of this model. However, the underlying reasons for nonstatic behaviour are not special to this model; on the contrary, the role of highenergy corrections and KK stresses will if anything be enhanced in a general, inhomogeneous collapse. There is in fact independent heuristic support for this possibility, arising from the AdS/CFT correspondence.

Quantum backreaction due to Hawking radiation in the 4D picture is described as classical dynamics in the 5D picture.

The black hole evaporates as a classical process in the 5D picture, and there is thus no stationary black hole solution in RS 1brane.

Primordial black holes in 1brane RStype cosmology have been investigated in [210, 185, 184, 313, 91, 384]. Highenergy effects in the early universe (see the next Section 5) can significantly modify the evaporation and accretion processes, leading to a prolonged survival of these black holes. Such black holes evade the enhanced Hawking evaporation described above when they are formed, because they are much smaller than ℓ.

Black holes will also be produced in particle collisions at energies ≳ M_{5}, possibly well below the Planck scale. In ADD braneworlds, where \(M_{4+d} = {\mathcal O}(\rm{TeV})\) is not ruled out by current observations if d > 1, this raises the exciting prospect of observing black hole production signatures in the nextgeneration colliders and cosmic ray detectors (see [75, 169, 138]).
5 BraneWorld Cosmology: Dynamics
A 1+4dimensional spacetime with spatial 4isotropy (4D spherical/plane/hyperbolic symmetry) has a natural foliation into the symmetry group orbits, which are 1+3dimensional surfaces with 3isotropy and 3homogeneity, i.e., FRW surfaces. In particular, the AdS5 bulk of the RS braneworld, which admits a foliation into Minkowski surfaces, also admits an FRW foliation since it is 4isotropic. Indeed this feature of 1brane RStype cosmological braneworlds underlies the importance of the AdS/CFT correspondence in braneworld cosmology [342, 375, 193, 386, 390, 290, 347, 180].
The generalization of AdS_{5} that preserves 4isotropy and solves the vacuum 5D Einstein equation (22) is SchwarzschildAdS_{5}, and this bulk therefore admits an FRW foliation. It follows that an FRW braneworld, the cosmological generalization of the RS braneworld, is a part of SchwarzschildAdS_{5}, with the Z_{2}symmetric FRW brane at the boundary. (Note that FRW branes can also be embedded in nonvacuum generalizations, e.g., in ReissnerNordströmAdS_{5} and VaidyaAdS_{5}.)
Either form of the cosmological metric, Equation (184) or (186), may be used to show that 5D gravitational wave signals can take “shortcuts” through the bulk in travelling between points A and B on the brane [89, 211, 65]. The travel time for such a graviton signal is less than the time taken for a photon signal (which is stuck to the brane) from A to B.
5.1 Braneworld inflation
In 1brane RStype braneworlds, where the bulk has only a vacuum energy, inflation on the brane must be driven by a 4D scalar field trapped on the brane. In more general braneworlds, where the bulk contains a 5D scalar field, it is possible that the 5D field induces inflation on the brane via its effective projection [231, 195, 146, 368, 198, 197, 407, 426, 259, 275, 196, 46, 325, 221, 315, 149, 18].
More exotic possibilities arise from the interaction between two branes, including possible collision, which is mediated by a 5D scalar field and which can induce either inflation [134, 220] or a hot bigbang radiation era, as in the “ekpyrotic” or cyclic scenario [229, 215, 339, 403, 273, 317, 412], or in colliding bubble scenarios [40, 156, 157]. (See also [26, 98, 299] for colliding branes in an M theory approach.) Here we discuss the simplest case of a 4D scalar field ϕ with potential V (ϕ) (see [287] for a review).
Highenergy braneworld modifications to the dynamics of inflation on the brane have been investigated [308, 216, 92, 405, 320, 319, 106, 285, 34, 35, 36, 328, 192, 264, 363, 307]. Essentially, the highenergy corrections provide increased Hubble damping, since ρ ≫ λ implies that H is larger for a given energy than in 4D general relativity. This makes slowroll inflation possible even for potentials that would be too steep in standard cosmology [308, 99, 312, 369, 346, 286, 205].
The key test of any modified gravity theory during inflation will be the spectrum of perturbations produced due to quantum fluctuations of the fields about their homogeneous background values. We will discuss braneworld cosmological perturbations in the next Section 6. In general, perturbations on the brane are coupled to bulk metric perturbations, and the problem is very complicated. However, on large scales on the brane, the density perturbations decouple from the bulk metric perturbations [303, 271, 177, 148]. For 1brane RStype models, there is no scalar zeromode of the bulk graviton, and in the extreme slowroll (de Sitter) limit, the massive scalar modes are heavy and stay in their vacuum state during inflation [148]. Thus it seems a reasonable approximation in slowroll to neglect the KK effects carried by \({{\mathcal E}_{\mu \nu}}\) when computing the density perturbations.
A crucial assumption is that backreaction due to metric perturbations in the bulk can be neglected. In the extreme slowroll limit this is necessarily correct because the coupling between inflaton fluctuations and metric perturbations vanishes; however, this is not necessarily the case when slowroll corrections are included in the calculation. Previous work [250, 253, 255] has shown that such bulk effects can be subtle and interesting (see also [109, 114] for other approaches). In particular, subhorizon inflaton fluctuations on a brane excite an infinite ladder of KaluzaKlein modes of the bulk metric perturbations at first order in slowroll parameters, and a naive slowroll expansion breaks down in the highenergy regime once one takes into account the backreaction of the bulk metric perturbations, as confirmed by direct numerical simulations [200]. However, an orderone correction to the behaviour of inflaton fluctuations on subhorizon scales does not necessarily imply that the amplitude of the inflaton perturbations receives corrections of order one on large scales; one must consistently quantise the coupled brane inflaton fluctuations and bulk metric perturbations. This requires a detailed analysis of the coupled branebulk system [70, 252].
It was shown that the coupling to bulk metric perturbations cannot be ignored in the equations of motion. Indeed, there are orderunity differences between the classical solutions without coupling and with slowroll induced coupling. However, the change in the amplitude of quantumgenerated perturbations is at nexttoleading order [252] because there is still no mixing at leading order between positive and negative frequencies when scales observable today crossed the horizon, so the Bogoliubov coefficients receive no corrections at leading order. The amplitude of perturbations generated is also subject to the usual slowroll corrections on superhorizon scales. The nextorder slowroll corrections from bulk gravitational perturbations are calculated in [254] and they are the same order as the usual StewartLyth correction [404]. These results also show that the ratio of tensortoscalar perturbation amplitudes are not influenced by branebulk interactions at leading order in slowroll. It is remarkable that the predictions from inflation theories should be so robust that this result holds in spite of the leadingorder change to the solutions of the classical equations of motion.
The standard chaotic inflation scenario requires an inflaton mass m ∼ 10^{13} GeV to match the observed level of anisotropies in the cosmic microwave background (see below). This corresponds to an energy scale ∼ 10^{16} GeV when the relevant scales left the Hubble scale during inflation, and also to an inflaton field value of order 3M_{p}. Chaotic inflation has been criticised for requiring superPlanckian field values, since these can lead to nonlinear quantum corrections in the potential.
 Highenergy inflation on the brane also generates a zeromode (4D graviton mode) of tensor perturbations, and stretches it to superHubble scales, as will be discussed below. This zeromode has the same qualitative features as in general relativity, remaining frozen at constant amplitude while beyond the Hubble horizon. Its amplitude is enhanced at high energies, although the enhancement is much less than for scalar perturbations [272]:$$A_{\rm{t}}^2 \approx \left({{{32V} \over {75M_{\rm{p}}^2}}} \right)\;\left[ {{{3{V^2}} \over {4{\lambda ^2}}}} \right],$$(223)Equation (224) means that braneworld effects suppress the largescale tensor contribution to CMB anisotropies. The tensor spectral index at high energy has a smaller magnitude than in general relativity,$${{A_{\rm{t}}^2} \over {A_{\rm{s}}^2}} \approx \left({{{M_{\rm{p}}^2} \over {16\pi}}{{{{V}\prime 2}} \over {{V^2}}}} \right)\;\left[ {{{6\lambda} \over V}} \right].$$(224)but remarkably the same consistency relation as in general relativity holds [205]:$${n_{\rm{t}}} =  3\epsilon,$$(225)This consistency relation persists when Z_{2} symmetry is dropped [206] (and in a twobrane model with stabilized radion [172]). It holds only to lowest order in slowroll, as in general relativity, but the reason for this [381] and the nature of the corrections [64] are not settled. The massive KK modes of tensor perturbations remain in the vacuum state during slowroll inflation [272, 176]. The evolution of the superHubble zero mode is the same as in general relativity, so that highenergy braneworld effects in the early universe serve only to rescale the amplitude. However, when the zero mode reenters the Hubble horizon, massive KK modes can be excited.$${n_{\rm{t}}} =  2{{A_{\rm{t}}^2} \over {A_{\rm{s}}^2}}.$$(226)

Vector perturbations in the bulk metric can support vector metric perturbations on the brane, even in the absence of matter perturbations (see the next Section 6). However, there is no normalizable zero mode, and the massive KK modes stay in the vacuum state during braneworld inflation [52]. Therefore, as in general relativity, we can neglect vector perturbations in inflationary cosmology.
Braneworld effects on largescale isocurvature perturbations in 2field inflation have also been considered [17]. Braneworld (p)reheating after inflation is discussed in [414, 429, 9, 415, 96].
5.2 Braneworld instanton
5.3 Models with nonempty bulk
 The simplest example arises from considering a charged bulk black hole, leading to the ReissnerNordström AdS_{5} bulk metric [22]. This has the form of Equation (184), withwhere q the “electric” charge parameter of the bulk black hole. The metric is a solution of the 5D EinsteinMaxwell equations, so that ^{(5)}T_{ ab } in Equation (50) is the energymomentum tensor of a radial static 5D “electric” field. In order for the field lines to terminate on the boundary brane, the brane should carry a charge −q. Since the RNAdS_{5} metric is 4isotropic, it is still possible to embed a FRW brane in it, which is moving in the coordinates of Equation (184). The effect of the black hole charge on the brane arises via the junction conditions and leads to the modified Friedmann equation [22],$$F(R) = K + {{{R^2}} \over {{\ell ^2}}}  {m \over {{R^2}}} + {{{q^2}} \over {{R^4}}},$$(230)The field lines that terminate on the brane imprint on the brane an effective negative energy density −3q^{2}/(κ^{2}a^{6}), which redshifts like stiff matter (w = 1). The negativity of this term introduces the possibility that at high energies it can bring the expansion rate to zero and cause a turnaround or bounce (but see [204] for problems with such bounces).$${H^2} = {{{\kappa ^2}} \over 3}\rho \left({1 + {\rho \over {2\lambda}}} \right) + {m \over {{a^4}}}  {{{q^2}} \over {{a^6}}} + {1 \over 3}\Lambda  {K \over {{a^2}}}.$$(231)
Apart from negativity, the key difference between this “dark stiff matter” and the dark radiation term m/a^{4} is that the latter arises from the bulk Weyl curvature via the \({{\mathcal E}_{\mu \nu}}\) tensor, while the former arises from nonvacuum stresses in the bulk via the \({{\mathcal F}_{\mu \nu}}\) tensor in Equation (69). The dark stiff matter does not arise from massive KK modes of the graviton.
 Another example is provided by the VaidyaAdS_{5} metric, which can be written after transforming to a new coordinate υ = T + ∫ dR/F in Equation (184), so that υ = const. are null surfaces, and$$^{(5)}d{s^2} =  F(R,v)d{v^2} + 2dv\,dR + {R^2}\left({{{d{r^2}} \over {1  K{r^2}}} + {r^2}d{\Omega ^2}} \right),$$(232)This model has a moving FRW brane in a 4isotropic bulk (which is not static), with either a radiating bulk black hole (dm/dυ < 0), or a radiating brane (dm/d υ > 0) [77, 278, 277, 280]. The metric satisfies the 5D field equations (50) with a nullradiation energymomentum tensor,$$F(R,v) = K + {{{R^2}} \over {{\ell ^2}}}  {{m(v)} \over {{R^2}}}.$$(233)where ψ ∝ dm/dυ. It follows that$$^{(5)}{T_{AB}} = \psi {k_A}{k_B},\qquad {k_A}{k^A} = 0,\qquad {k_A}{u^A} = 1,$$(234)In this case, the same effect, i.e., a varying mass parameter m, contributes to both \({{\mathcal E}_{\mu \nu}}\) and \({{\mathcal F}_{\mu \nu}}\) in the brane field equations. The modified Friedmann equation has the standard 1brane RStype form, but with a dark radiation term that no longer behaves strictly like radiation:$${{\mathcal F}_{\mu \nu}} = \kappa _5^{ 2}\psi {h_{\mu \nu}}.$$(235)By Equations (74) and (234), we arrive at the matter conservation equations,$${H^2} = {{{\kappa ^2}} \over 3}\rho \left({1 + {\rho \over {2\lambda}}} \right) + {{m(t)} \over {{a^4}}} + {1 \over 3}\Lambda  {K \over {{a^2}}}.$$(236)This shows how the brane loses (ψ > 0) or gains (ψ < 0) energy in exchange with the bulk black hole. For an FRW brane, this equation reduces to$${\nabla ^\nu}{T_{\mu \nu}} =  2\psi {u_\mu}.$$(237)The evolution of m is governed by the 4D contracted Bianchi identity, using Equation (235):$$\dot \rho + 3H(\rho + p) =  2\psi .$$(238)For an FRW brane, this yields$${\nabla ^\mu}{{\mathcal E}_{\mu \nu}} = {{6{\kappa ^2}} \over \lambda}{\nabla ^\mu}{{\mathcal S}_{\mu \nu}} + {2 \over 3}\left[ {\kappa _5^2\left({\dot \psi + \Theta \psi} \right)  3{\kappa ^2}\psi} \right]{u_\mu} + {2 \over 3}\kappa _5^2{\vec \nabla _\mu}\psi .$$(239)where \({\rho _{\mathcal E}} = 3m(t)/({\kappa ^2}{a^4})\).$${\dot \rho _{\mathcal E}} + 4H{\rho _{\mathcal E}} = 2\psi  {2 \over 3}{{\kappa _5^2} \over {{\kappa ^2}}}\left({\dot \psi + 3H\psi} \right),$$(240)
 A more complicated bulk metric arises when there is a selfinteracting scalar field Φ in the bulk [311, 21, 322, 143, 274, 144, 48]. In the simplest case, when there is no coupling between the bulk field and brane matter, this giveswhere Φ(x,y) satisfies the 5D KleinGordon equation,$$^{(5)}{T_{AB}} = {\Phi _{,A}}{\Phi _{,B}}  {\;^{(5)}}{g_{AB}}\left[ {V(\Phi) + {1 \over 2}{\;^{(5)}}{g^{CD}}{\Phi _{,C}}{\Phi _{,D}}} \right],$$(241)The junction conditions on the field imply that$$^{(5)}\square\Phi  V{\prime}(\Phi) = 0{.}$$(242)Then Equations (74) and (241) show that matter conservation continues to hold on the brane in this simple case:$${\partial _y}\Phi (x,0) = 0{.}$$(243)From Equation (241) one finds that$${\nabla ^\nu}{T_{\mu \nu}} = 0.$$(244)where$${{\mathcal F}_{\mu \nu}} = {1 \over {4\kappa _5^2}}\left[ {4{\phi _{,\mu}}{\phi _{,\nu}}  {g_{\mu \nu}}\left({3V(\phi) + {5 \over 2}{g^{\alpha \beta}}{\phi _{,\alpha}}{\phi _{,\beta}}} \right)} \right],$$(245)so that the modified Friedmann equation becomes$$\phi (x) = \Phi (x,0),$$(246)When there is coupling between brane matter and the bulk scalar field, then the Friedmann and conservation equations are more complicated [311, 21, 322, 143, 274, 144, 48].$${H^2} = {{{\kappa ^2}} \over 3}\rho \left({1 + {\rho \over {2\lambda}}} \right) + {m \over {{a^4}}} + {{\kappa _5^2} \over 6}\left[ {{1 \over 2}{{\dot \phi}^2} + V(\phi)} \right] + {1 \over 3}\Lambda  {K \over {{a^2}}}.$$(247)
6 BraneWorld Cosmology: Perturbations
The background dynamics of braneworld cosmology are simple because the FRW symmetries simplify the bulk and rule out nonlocal effects. But perturbations on the brane immediately release the nonlocal KK modes. Then the 5D bulk perturbation equations must be solved in order to solve for perturbations on the brane. These 5D equations are partial differential equations for the 3D Fourier modes, with both initial and boundary conditions needed.
The theory of gaugeinvariant perturbations in braneworld cosmology has been extensively investigated and developed [303, 271, 24, 308, 99, 312, 369, 346, 286, 177, 272, 176, 331, 333, 190, 235, 265, 416, 258, 332, 417, 231, 266, 191, 123, 158, 234, 50, 127, 340, 385, 368, 285, 85, 90, 116, 41, 54, 364, 53, 362, 282, 281] and is qualitatively well understood. The key task is integration of the coupled branebulk perturbation equations. Special cases have been solved, where these equations effectively decouple [271, 24, 282, 281], and approximation schemes have been developed [398, 428, 387, 399, 400, 245, 361, 51, 201, 136, 321, 323, 27] for the more general cases where the coupled system must be solved. Below we will also present the results of full numerical integration of the 5D perturbation equations in the RS case.
As in 4D general relativity, there are various different, but essentially equivalent, ways to formulate linear cosmological perturbation theory. First we describe the covariant branebased approach.
6.1 1 + 3covariant perturbation equations on the brane
These equations are the basis for a 1+3covariant analysis of cosmological perturbations from the brane observer’s viewpoint, following the approach developed in 4D general relativity (for a review, see [137]). The equations contain scalar, vector, and tensor modes, which can be separated out if desired. They are not a closed system of equations until \(\pi _{\mu \nu}^{\mathcal E}\) is determined by a 5D analysis of the bulk perturbations. An extension of the 1+3covariant perturbation formalism to 1+4 dimensions would require a decomposition of the 5D geometric quantities along a timelike extension u^{ A } into the bulk of the brane 4velocity field u^{ μ }, and this remains to be done. The 1+3covariant perturbation formalism is incomplete until such a 5D extension is performed. The metricbased approach does not have this drawback.
6.2 Metricbased perturbations
In the following, we will discuss various perturbation problems, using either a 1+3covariant or a metricbased approach.
6.3 Density perturbations on large scales
If \({\rho _{\mathcal E}} = 0\) in the background, then U is an isocurvature mode: S_{tot} ∝ (1+w)U. This isocurvature mode is suppressed during slowroll inflation, when 1 + w ≈ 0.
If \({\rho _{\mathcal E}} = 0\) in the background, then the weighted difference between U and Δ determines the isocurvature mode: \({S_{{\rm{tot}}}} \propto (4{\rho _{\mathcal E}}/3\rho)\Delta  (1 + w)U\). At very high energies, ρ ≫ λ, the entropy is suppressed by the factor λ/ρ.
Thus on large scales the system of brane equations is closed, and we can determine the density perturbations without solving for the bulk metric perturbations.
6.4 Curvature perturbations and the SachsWolfe effect

A contribution from the KK entropy perturbation \({S_{\mathcal E}}\) that is similar to an extra isocurvature contribution.

The KK anisotropic stress \(\delta {\pi _{\mathcal E}}\) also contributes to the CMB anisotropies. In the absence of anisotropic stresses, the curvature perturbation ζ_{tot} would be sufficient to determine the metric perturbation \({\mathcal R}\) and hence the largeangle CMB anisotropies via Equations (324, 325, 326). However, bulk gravitons generate anisotropic stresses which, although they do not affect the largescale curvature perturbation ζ_{tot}, can affect the relation between ζ_{tot}, \({\mathcal R}\), and ψ, and hence can affect the CMB anisotropies at large angles.
A selfconsistent approximation is developed in [245], using the lowenergy 2brane approximation [398, 428, 387, 399, 400] to find an effective 4D form for \({{\mathcal E}_{\mu \nu}}\) and hence for \(\delta {\pi _{\mathcal E}}\). This is discussed below. In a single brane model in the AdS bulk, full numerical simulations were done to find the behaviour of \(\delta {\pi _{\mathcal E}}\) [69], as will be discussed in the next subsection.
6.5 Full numerical solutions
The amplitude enhancement of perturbations is important on comoving scales ≲ 10 AU, which are far too small to be relevant to presentday/cosmic microwave background measurements of the matter power spectrum. However, it may have an important bearing on the formation of compact objects such as primordial black holes and boson stars at very high energies, i.e., the greater gravitational force of attraction in the early universe will create more of these objects than in GR (different aspects of primordial black holes in RS cosmology in the context of various effective theories have been considered in [185, 184, 91, 384, 383, 382]).
6.6 Vector perturbations
Inflation will redshift away the vorticity and the KK mode. Indeed, the massive KK vector modes are not excited during slowroll inflation [53, 362].
6.7 Tensor perturbations
7 Gravitational Wave Perturbations in BraneWorld Cosmology
7.1 Analytical approaches
This enhanced zero mode produced by brane inflation remains frozen outside the Hubble radius, as in general relativity, but when it reenters the Hubble radius during radiation or matter domination, it will no longer be separated from the massive modes, since H will not be constant. Instead, massive modes will be excited during reentry. In other words, energy will be lost from the zero mode as 5D gravitons are emitted into the bulk, i.e., as massive modes are produced on the brane. A phenomenological model of the damping of the zero mode due to 5D graviton emission is given in [281]. Selfconsistent lowenergy approximations to compute this effect are developed in [201, 136].
7.2 Full numerical solutions
This cancelation of two high energy effects is valid only for w = 1/3. For other equations of state, the final spectrum at high frequencies are different from 4D predictions. For example for w = 1, Ω_{GW} ∝f^{2/5} for f > f_{ c } while the 4D theory predicts Ω_{GW} ∝ f^{1}.
8 CMB Anisotropies in BraneWorld Cosmology
The perturbation equations in the previous Section 7 form the basis for an analysis of scalar and tensor CMB anisotropies in the braneworld. The full system of equations on the brane, including the Boltzmann equation for photons, has been given for scalar [282] and tensor [281] perturbations. But the systems are not closed, as discussed above, because of the presence of the KK anisotropic stress \(\pi _{\mu \nu}^{\mathcal E}\), which acts a source term.
The formalism and machinery are ready to compute the temperature and polarization anisotropies in braneworld cosmology, once a solution, or at least an approximation, is given for \(\pi _{\mu \nu}^{\mathcal E}\). The resulting power spectra will reveal the nature of the braneworld imprint on CMB anisotropies, and would in principle provide a means of constraining or possibly falsifying the braneworld models. Once this is achieved, the implications for the fundamental underlying theory, i.e., M theory, would need to be explored.
However, the first step required is the solution for \(\pi _{\mu \nu}^{\mathcal E}\) This solution will be of the form given in Equation (249). Once \({\mathcal G}\) and F_{ k } are determined or estimated, the numerical integration in Equation (249) can in principle be incorporated into a modified version of a CMB numerical code. The full solution in this form represents a formidable problem, and one is led to look for approximations.
8.1 The lowenergy approximation
8.2 The simplest model
9 DGP Models: Modifying Gravity at Low Energies
9.1 ‘Selfaccelerating’ DGP
Most braneworld models modify general relativity at high energies. The RandallSundrum models discussed up to now are a typical example. At low energies, Hℓ ≪ 1, the zeromode of the graviton dominates on the brane, and general relativity is recovered to a good approximation. At high energies, Hℓ ≫ 1, the massive modes of the graviton dominate over the zeromode, and gravity on the brane behaves increasingly 5dimensional. On the unperturbed FRW brane, the standard energyconservation equation holds, but the Friedmann equation is modified by an ultraviolet correction, (Gℓρ)^{2}. At high energies, gravity “leaks” off the brane and H^{2} ∝ ρ^{2}.
By contrast, the braneworld model of DvaliGabadadzePorrati [135] (DGP), which was generalized to cosmology by Deffayet [115], modifies general relativity at low energies. This model produces ‘selfacceleration’ of the latetime universe due to a weakening of gravity at low energies. Like the RS model, the DGP model is a 5D model with infinite extra dimensions.^{3}
Observations based on structure formation provide further evidence of the difference between DGP and LCDM, since the two models suppress the growth of density perturbations in different ways [296, 295]. The distancebased observations draw only upon the background 4D Friedman equation (418) in DGP models — and therefore there are quintessence models in general relativity that can produce precisely the same supernova distances as DGP. By contrast, structure formation observations require the 5D perturbations in DGP, and one cannot find equivalent quintessence models [251]. One can find 4D general relativity models, with dark energy that has anisotropic stress and variable sound speed, which can in principle mimic DGP [263]. However, these models are highly unphysical and can probably be discounted on grounds of theoretical consistency.

On small scales, below the Vainshtein radius (which for cosmological purposes is roughly the scale of clusters), the spin0 scalar degree of freedom becomes strongly coupled, so that the general relativistic limit is recovered [256].
 On scales relevant for structure formation, i.e., between cluster scales and the Hubble radius, the spin0 scalar degree of freedom produces a scalartensor behaviour. A quasistatic approximation (as in the Newtonian approximation in standard 4D cosmology) to the 5D perturbations shows that DGP gravity is like a BransDicke theory with parameter [251]$${\omega _{BD}} = {3 \over 2}(\beta  1),$$(425)At late times in an expanding universe, when Hr_{ c } ≳ 1, it follows that β < 1, so that ω_{ BD } < 0. (This is a signal of the ghost pathology in DGP, which is discussed below.)$$\beta = 1 + 2{H^2}{r_c}{\left({{H^2} + {K \over {{a^2}}}} \right)^{ 1/2}}\left[ {1 + {{\dot H} \over {3{H^2}}} + {{2K} \over {3{a^2}{H^2}}}} \right].$$(426)

Although the quasistatic approximation allows us to analytically solve the 5D wave equation for the bulk degree of freedom, this approximation breaks down near and beyond the Hubble radius. On superhorizon scales, 5D gravity effects are dominant, and we need to solve numerically the partial differential equation governing the 5D bulk variable [71].
It is evident from Figure 23 that the DGP model, which provides a best fit to the geometric data (see Figure 21), is in serious tension with the WMAP5 data on large scales. The problem arises because there is a large deviation of ϕ_{−} = (ϕ − ψ)/2 in the DGP model from the LCDM model. This deviation, i.e., a stronger decay of ϕ_{−}, leads to an overstrong ISW effect (which is determined by \({\dot \phi _ }\)), in tension with WMAP5 observations.
As a result of the combined observations of background expansion history and largeangle CMB anisotropies, the DGP model provides a worse fit to the data than LCDM at about the 5σ level [142]. Effectively, the DGP model is ruled out by observations in comparison with the LCDM model.
In addition to the severe problems posed by cosmological observations, a problem of theoretical consistency arises from the fact that the latetime asymptotic de Sitter solution in DGP cosmological models has a ghost. The ghost is signaled by the negative BransDicke parameter in the effective theory that approximates the DGP on cosmological subhorizon scales: The existence of the ghost is confirmed by detailed analysis of the 5D perturbations in the de Sitter limit [246, 175, 81, 246]. The DGP ghost is a ghost mode in the scalar sector of the gravitational field — which is more serious than the ghost in a phantom scalar field. It effectively rules out the DGP, since it is hard to see how an ultraviolet completion of the DGP can cure the infrared ghost problem. However, the DGP remains a valuable toy model for illustrating the kinds of behaviour that can occur from a modification to Einstein’s equations — and for developing cosmological tools to test modified gravity and Einstein’s theory itself.
9.2 ‘Normal’ DGP
Perturbations in the normal branch have the same structure as those in the selfaccelerating branch, with the same regimes — i.e., below the Vainshtein radius (recovering a GR limit), up to the Hubble radius (BransDicke behaviour), and beyond the Hubble radius (strongly 5D behaviour). The quasistatic approximation and the numerical integrations can be simply repeated with the replacement r_{ c } → − r_{ c } (and the addition of Λ to the background). In the subHubble regime, the effective BransDicke parameter is still given by Equations (425) and (426), but now we have ω_{ BD } > 0 — and this is consistent with the absence of a ghost. Furthermore, a positive BransDicke parameter signals an extra positive contribution to structure formation from the scalar degree of freedom, so that there is less suppression of structure formation than in LCDM — the reverse of what happens in the selfaccelerating DGP. This is confirmed by computations, as illustrated in Figure 25.
10 6Dimensional Models
For braneworld models in 6dimensional spacetime, the codimension of a brane is two and the behaviour of gravity is qualitatively very different from the codimension one braneworld models. Here we briefly discuss some important examples and features of 6dimensional models.
10.1 Supersymmetric Large Extra Dimensions (SLED) Model
An unambiguous way to investigate this problem is to study the dynamical solutions directly in the 6D spacetime. However, once we consider the case where the tension becomes time dependent, we encounter a difficulty to deal with the branes [419]. This is because for codimension 2 branes, we encounter a divergence of metric near the brane if we put matter other than tension on a brane. Hence, without specifying how we regularize the branes, we cannot address the question what will happen if we change the tension. Is the selftuning mechanism at work and does it lead to another static solution? Or do we get a dynamical solution driven by the runaway behaviour of the modulus field?
There was a negative conclusion on the selftuning in this supersymmetric model for a particular kind of regularization [419, 420]. However, the answer could depend on the regularization of branes and the jury remains out. It is important to study the timedependent dynamics in the 6D spacetime and the regularization of the branes in detail [410, 60, 62, 411, 61, 33, 110, 338, 350, 100].
10.2 Cosmology in 6D braneworld models
It is much harder to obtain cosmological solutions in 6D models than in 5dimensional models. In 5D braneworld models, cosmological solutions can be obtained by considering a moving brane in a static bulk spacetime. This is because the motion of the homogeneous and isotropic brane does not change the bulk spacetime thanks to Birkoff’s theorem. However, this is no longer the case in 6D spacetime. Thus we need to find time dependent bulk solutions that are coupled to a motion of a brane, which requires us to solve nonlinear partial differential equations numerically. Moreover, as is mentioned above, there appears a curvature singularity if one considers an infinitely thin brane and puts matter other than tension on the brane [419, 420]. Thus we need some regularization scheme to find cosmological solutions.
One of the popular ways to regularize a brane is to promote a brane that is a pointlike object in two extradimensions to a 5D ring [353]. The ring is a codimension one object and it is possible to consider a motion of this ring. However, a problem is that this ring brane is not homogeneous as one of dimensions on the brane is compact and this breaks Birkhoff’s theorem. It has been shown that cosmology obtained by the motion of a brane in a static bulk shows pathological behaviour [351, 326]. This indicates that we need to find fully timedependent bulk solutions.
At the moment, the only accessible way is to solve the 6D bulk spacetime using the gradient expansion methods assuming physical scales on a brane are much lower than mass scales in the bulk [150, 16, 233]. It is still an open question what are high energy effects in 6D models.
As in 5D models, there are many generalizations such as the inclusion of the induced gravity term on a brane and the GaussBonnet term in the bulk [43, 222, 101, 84, 349, 239, 83, 82].
10.3 Cascading braneworld model
This model addresses several fundamental issues in induced gravity models in 6D spacetime. Without the induced gravity term on the 4brane, the 6D graviton propagator diverges logarithmically near the position of the 3brane [162]. On the other hand, the graviton propagator on the 3brane in this model behaves like G (p) → log(p^{(5)}r_{ c }) in the M_{5} → 0 limit where p is 4D momentum. Thus the crossover scale ^{(5)}r_{ c } acts as a cutoff for the propagator and it remains finite even at the position of the 3brane.
Cosmological solutions in the cascading brane model are again notoriously difficult to find because it is necessary to find 6dimensional solutions that depend on time and two extracoordinates [1]. The simplest de Sitter solutions have been obtained [324]. Interestingly, there exists a selfaccelerating solution for the 3brane even when the solution for the 4brane is in the normal branch. It is still not clear whether this selfaccelerating solution is stable or not and it is crucial to check the stability of this new selfaccelerating solution.
A similar class of models includes intersecting branes [217, 102, 103]. In this model, we have two 4branes that intersect and a 3brane sits at the intersection. Again there are selfaccelerating de Sitter solutions and cosmology has been studied by considering a motion of one of the 4branes. A model without a 4brane has been studied by regularizing a 3brane by promoting it to a 5D ring brane [219, 218].
11 Conclusion
Simple braneworld models provide a rich phenomenology for exploring some of the ideas that are emerging from M theory. The higherdimensional degrees of freedom for the gravitational field, and the confinement of standard model fields to the visible brane, lead to a complex but fascinating interplay between gravity, particle physics, and geometry, that enlarges and enriches general relativity in the direction of a quantum gravity theory.

They provide a simple 5D phenomenological realization of the HoravaWitten supergravity solutions in the limit where the hidden brane is removed to infinity, and the moduli effects from the 6 further compact extra dimensions may be neglected.

They develop a new geometrical form of dimensional reduction based on a strongly curved (rather than flat) extra dimension.

They provide a realization to lowest order of the AdS/CFT correspondence.

They incorporate the selfgravity of the brane (via the brane tension).

They lead to cosmological models whose background dynamics are completely understood and reproduce general relativity results with suitable restrictions on parameters.

to find the simplest realistic solution (or approximation to it) for an astrophysical black hole on the brane, and settle the questions about its staticity, Hawking radiation, and horizon; and

to develop realistic approximation schemes (building on recent work [398, 428, 387, 399, 400, 245, 361, 51, 201, 136]) and manageable numerical codes (building on [245, 361, 51, 201, 136]) to solve for the cosmological perturbations on all scales, to compute the CMB anisotropies and largescale structure, and to impose observational constraints from highprecision data.
 The inclusion of dynamical interaction between the brane(s) and a bulk scalar field, so that the action is(see [311, 21, 322, 143, 274, 144, 48, 231, 195, 146, 368, 198, 197, 407, 426, 259, 275, 196, 46, 325, 221, 315, 149, 18]). The scalar field could represent a bulk dilaton of the gravitational sector, or a modulus field encoding the dynamical influence on the effective 5D theory of an extra dimension other than the large fifth dimension [26, 98, 299, 361, 51, 55, 242, 212, 352, 179].$$\begin{array}{*{20}c} {S = {1 \over {2\kappa _5^2}}\int {{d^5}} x\sqrt { {}^{(5)}g} \left[ {{}^{(5)}R  \kappa _5^2{\partial _A}\Phi {\partial ^A}\Phi  2{\Lambda _5}(\Phi)} \right]}\\ {+ \int\nolimits_{{\rm{brane}}({\rm{s}})} {d^4}x\sqrt { g} \left[ { \lambda (\Phi) + {K \over {\kappa _5^2}} + {L_{{\rm{matter}}}}} \right]\quad \quad}\\ \end{array}$$(451)For twobrane models, the brane separation introduces a new scalar degree of freedom, the radion. For general potentials of the scalar field which provide radion stabilization, 4D Einstein gravity is recovered at low energies on either brane [408, 335, 283]. (By contrast, in the absence of a bulk scalar, low energy gravity is of BransDicke type [155].) In particular, such models will allow some fundamental problems to be addressed:

The hierarchy problem of particle physics.

An extradimensional mechanism for initiating inflation (or the hot radiation era with superHubble correlations) via brane interaction (building on the initial work in [134, 220, 229, 215, 339, 403, 273, 317, 412, 26, 98, 299, 40, 156, 157]).

An extradimensional explanation for the dark energy (and possibly also dark matter) puzzles: Could dark energy or latetime acceleration of the universe be a result of gravitational effects on the visible brane of the shadow brane, mediated by the bulk scalar field?

 The addition of stringy and quantum corrections to the EinsteinHilbert action, including the following:
 Higherorder curvature invariants, which arise in the AdS/CFT correspondence as nexttoleading order corrections in the CFT. The GaussBonnet combination in particular has unique properties in 5D, giving field equations which are secondorder in the bulk metric (and linear in the second derivatives), and being ghostfree. The action iswhere α is the GaussBonnet coupling constant, related to the string scale. The cosmological dynamics of these braneworlds is investigated in [121, 343, 345, 341, 161, 80, 289, 37, 318, 291, 181, 125, 19, 124, 79, 310]. In [20] it is shown that the black string solution of the form of Equation (144) is ruled out by the GaussBonnet term. In this sense, the GaussBonnet correction removes an unstable and singular solution.$$\begin{array}{*{20}c} {S = {1 \over {2\kappa _5^2}}\int {d^5}x\sqrt { {}^{(5)}g} \left[ {{}^{(5)}R  2{\Lambda _5} + \alpha \left({{}^{(5)}{R^2}  4{}^{(5)}{R_{AB}}{}^{(5)}{R^{AB}} + {}^{(5)}{R_{ABCD}}{}^{(5)}{R^{ABCD}}} \right)} \right]} \\ {+ \int\nolimits_{{\rm{brane}}} {d^4}x\sqrt { g} \left[ { \lambda + {K \over {\kappa _5^2}} + {L_{{\rm{matter}}}}} \right],\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \;} \\ \end{array}$$(452)In the early universe, the GaussBonnet corrections to the Friedmann equation have the dominant format the highest energies. If the GaussBonnet term is a small correction to the EinsteinHilbert term, as may be expected if it is the first of a series of higherorder corrections, then there will be a regime of RSdominance as the energy drops, when H^{2} ∝ ρ^{2}. Finally at energies well below the brane tension, the general relativity behaviour is recovered.$${H^2} \propto {\rho ^{2/3}}$$(453)
 Quantum field theory corrections arising from the coupling between brane matter and bulk gravitons, leading to an induced 4D Ricci term in the brane action. The original induced gravity braneworld is the DGP model [131, 95, 344, 391], which we investigated in this review as an alternative to the RStype models. Another viewpoint is to see the inducedgravity term in the action as a correction to the RS action:where β is a positive coupling constant.$$S = {1 \over {2\kappa _5^2}}\int {{d^5}} x\sqrt { {}^{(5)}g} \left[ {{}^{(5)}R  2{\Lambda _5}} \right] + \int\nolimits_{{\rm{brane}}} {{d^4}} x\sqrt { g} \left[ {\beta R  \lambda + {K \over {\kappa _5^2}} + {L_{{\rm{matter}}}}} \right],$$(454)
The cosmological models have been analyzed in [117, 238, 126, 230, 118, 372, 392, 370, 397, 7, 309, 297, 337, 186, 240]. (Braneworld black holes with induced gravity are investigated in [241].) Unlike RStype models, DGP models lead to 5D behaviour on large scales rather than small scales. Then on an FRW brane, the lateuniverse 5D behaviour of gravity can naturally produce a latetime acceleration, even without dark energy, although the selfaccelerating models suffer from a ghost. Nevertheless, the DGP model is a critical example of modified gravity models in cosmology that act as alternatives to dark energy.

The RS and DGP models are 5dimensional phenomenological models, and so a key issue is how to realize such models in 10dimensional string theory. Some progress has been made. 6dimensional cascading braneworlds are extensions of the DGP model. 10dimensional type IIB supergravity solutions have been found with the warped geometry that generalizes the RS geometry. These models have also been important for building inflationary models in string theory, based on the motion of D3 branes in the warped throat [63, 133] (see the reviews [292, 32] and references therein). The action for D3 branes is described by the DiracBornInfeld action and this gives the possibility of generating a large nonGaussianity in the Cosmic Microwave Background temperature anisotropies, which can be tested in future experiments [395, 213] (see the reviews [86, 248]).
These models reply on the effective 4dimensional approach to deal with extra dimensions. For example, the stabilization mechanism, which is necessary to fix moduli fields in string theory, exploits nonperturbative effects and they are often added in the 4dimensional effective theory. Then it is not clear whether the resultant 4dimensional effective theory is consistent with the 10dimensional equations of motion [107, 108, 237, 249]. Recently there has been a new development and it has become possible to calculate all significant contributions to the D3 brane potential in the single coherent framework of 10dimensional supergravity [31, 30, 28, 29]. This will provide us with a very interesting bridge between phenomenological braneworld models, where dynamics of higherdimensional gravity is studied in detail, and string theory approaches, where 4D effective theory is intensively used. It is crucial to identify the higherdimensional signature of the models in order to test a fundamental theory like string theory.
In summary, braneworld gravity opens up exciting prospects for subjecting M theory ideas to the increasingly stringent tests provided by highprecision astronomical observations. At the same time, braneworld models provide a rich arena for probing the geometry and dynamics of the gravitational field and its interaction with matter.
Footnotes
Notes
Acknowledgement
We thank our many collaborators and friends for discussions and sharing of ideas. R.M. is supported by the UK’s Science & Technology Facilities Council (STFC). K.K. is supported by the European Research Council, Research Councils UK and STFC.
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