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. 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 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 discusses the geometry, dynamics and perturbations of simple braneworld models for cosmology and astrophysics, mainly focusing on warped 5dimensional braneworlds based on the RandallSundrum 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At high enough energies, Einstein’s theory of general relativity breaks down, and will be superceded by a quantum gravity theory.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 recent reviews see, e.g., [153, 263, 283]) and quantum geometry (loop quantum gravity; for recent reviews see, e.g., [272, 306]). 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, and mainly the 5dimensional warped braneworlds.
1.1 Heuristics of higherdimensional gravity
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 [143], 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 [215, 216, 217].
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 RS2 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 RS2 model with its AdS_{5} metric satisfies this correspondence to lowest perturbative order [87] (see also [254, 282, 136, 289, 293, 210, 259, 125] for the AdS/CFT correspondence in a cosmological context).
In this review, I focus on RS braneworlds (mainly RS 1brane) and their generalizations, with the emphasis on geometry and gravitational dynamics (see [219, 228, 190, 316, 260, 185, 267, 79, 36, 189, 186] for previous reviews with a broadly similar approach). Other recent reviews focus on stringtheory aspects, e.g., [101, 231, 68, 264], or on particle physics aspects, e.g., [262, 273, 182, 112, 51]. Before turning to a more detailed analysis of RS braneworlds, I 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) Σ_{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 ½ (d + 1)(d + 4).
2 RandallSundrum BraneWorlds
 RS 2brane: There are two branes in this model [266], at y = 0 and y = L, with Z_{2}symmetry identificationsThe branes have equal and opposite tensions ±λ, where$$y \leftrightarrow  y,\quad \quad y + L \leftrightarrow L  y.$$(24)The positivetension brane has fundamental scale M_{5} and is “hidden”. Standard model fields are confined on the negative tension (or “visible”) brane. Because of the exponential warping factor, the effective scale on the visible brane at y = L is M_{p}, where$$\lambda = {{3M_{\rm{p}}^2} \over {4\pi {\ell ^2}}}.$$(25)So the RS 2brane model gives a new approach to the hierarchy problem. Because of the finite separation between the branes, the KK spectrum is discrete. Furthermore, at low energies gravity on the branes becomes BransDickelike, with the sign of the BransDicke parameter equal to the sign of the brane tension [105]. In order to recover 4D general relativity at low energies, a mechanism is required to stabilize the interbrane distance, which corresponds to a scalar field degree of freedom known as the radion [120, 305, 248, 202].$$M_{\rm{p}}^2 = M_5^3\ell \;\left[ {1  {e^{ 2L/\ell}}} \right].$$(26)
 RS 1brane: In this model [265], there is only one, positive tension, brane. It may be thought of as arising from sending the negative tension brane off to infinity, L → ∞. Then the energy scales are related viaThe infinite extra dimension makes a finite contribution to the 5D volume because of the warp factor:$$M_5^3 = {{M_{\rm{p}}^2} \over \ell}.$$(27)Thus the effective size of the extra dimension probed by the 5D graviton is ℓ.$$\int {{d^5}} X\sqrt {{ ^{(5)}}g} = 2\int {{d^4}} x\int\nolimits_0^\infty d y{e^{ 4y/\ell}} = {\ell \over 2}\int {{d^4}} x.$$(28)
3 Covariant Approach to BraneWorld Geometry and Dynamics
The RS models and the subsequent generalization from a Minkowski brane to a FriedmannRobertsonWalker (FRW) brane [27, 181, 155, 162, 128, 243, 149, 99, 104] were derived as solutions in particular coordinates of the 5D Einstein equations, together with the junction conditions at the Z_{2}symmetric brane. A broader perspective, with useful insights into the interplay between 4D and 5D effects, can be obtained via the covariant ShiromizuMaedaSasaki approach [291], 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 [315].)
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:$${{\vert {\kappa ^2}{\mathcal{S}_{\mu \nu}}/\lambda \vert} \over {\vert {\kappa ^2}{T_{\mu \nu}}\vert}}\sim {{\vert {T_{\mu \nu}}\vert} \over \lambda}\sim {\rho \over \lambda}.$$(74)

ɛ_{μν} 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 ɛ_{μν} 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 ɛ_{μν}; this term is what carries the modification to the weakfield field equations. The 9 independent components in the tracefree ɛ_{μν} are reduced to 5 degrees of freedom by Equation (73); 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 ɛ_{μν}.
3.2 5dimensional equations and the initialvalue problem
3.3 The brane viewpoint: A 1 + 3covariant analysis
Following [218], a systematic analysis can be developed from the viewpoint of a branebound observer. The effects of bulk gravity are conveyed, from a brane observer viewpoint, via the local \(({{\mathcal S}_{\mu \nu}})\) and nonlocal (ɛ_{μν}) 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 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 ɛ_{μν}.

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 ρ_{ε}, 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 Λ5 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) [225, 16, 236, 97, 194, 98, 35].

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 ɛ_{μν}.

There are evolution equations for the dark radiative (nonlocal, Weyl) energy (ρ_{ε}) 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 (112), 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 (112) 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 density q_{μ} = ρυ_{μ}, which will serve to source nonlocal effects.
In general, the 4 independent equations in Equations (111) and (112) constrain 4 of the 9 independent components of ɛ_{μν} 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 ɛ_{μν} is determined by Equations (111) and (112). If the brane is stationary (with Killing vector parallel to u^{μ}), then evolution equations are not needed for ɛ_{μν}, 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 ɛ_{μν} = 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 ɛ_{μν}.
If ɛ_{μν} = 0, then the field equations on the brane form a closed system. Thus for perfect fluid branes with homogeneous density and ɛ_{μν} = 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):$$\dot \Theta + {1 \over 3}{\Theta ^2} + {\sigma _{\mu \nu}}{\sigma ^{\mu \nu}}  2{\omega _\mu}{\omega ^\mu}  {\vec \nabla ^\mu}{A_\mu} + {A_\mu}{A^\mu} + {{{\kappa ^2}} \over 2}(\rho + 3p)  \Lambda =  {{{\kappa ^2}} \over 2}(2\rho + 3p){\rho \over \lambda}  {\kappa ^2}{\rho _\mathcal{E}}.$$(123)
 Vorticity propagation:$${\dot \omega _{\langle \mu \rangle}} + {2 \over 3}\Theta {\omega _\mu} + {1 \over 2}{\rm{curl}}\;{A_\mu}  {\sigma _{\mu \nu}}{\omega ^\nu} = 0.$$(124)
 Shear propagation:$${\dot \sigma _{\langle \mu \nu \rangle}} + {2 \over 3}\Theta {\sigma _{\mu \nu}} + {E_{\mu \nu}}  {\vec \nabla _{\langle \mu}}{A_{\nu \rangle}} + {\sigma _{\alpha \langle \mu}}{\sigma _{\nu \rangle}}^\alpha + {\omega _{\langle \mu}}{\omega _{\nu \rangle}}  {A_{\langle \mu}}{A_{\nu \rangle}} = {{{\kappa ^2}} \over 2}\pi _{\mu \nu}^\mathcal{E}.$$(125)
 Gravitoelectric propagation (MaxwellWeyl Edot equation):$$\begin{array}{*{20}c} {{{\dot E}_{\langle \mu \nu \rangle}} + \Theta {E_{\mu \nu}}  {\rm{curl}}\;{H_{\mu \nu}} + {{{\kappa ^2}} \over 2}(\rho + p){\sigma _{\mu \nu}}  2{A^\alpha}{\varepsilon _{\alpha \beta (\mu}}{H_{\nu)}}^\beta  3{\sigma _{\alpha \langle \mu}}{E_{\nu \rangle}}^\alpha + {\omega ^\alpha}{\varepsilon _{\alpha \beta (\mu}}{E_{\nu)}}^\beta =} \\ {\quad  {{{\kappa ^2}} \over 2}(\rho + p){\rho \over \lambda}{\sigma _{\mu \nu}}} \\ {\quad  {{{\kappa ^2}} \over 6}\left[ {4{\rho _\mathcal{E}}{\sigma _{\mu \nu}} + 3\dot \pi _{\langle \mu \nu \rangle}^\mathcal{E} + \Theta \pi _{\mu \nu}^\mathcal{E} + 3{{\vec \nabla}_{\langle \mu}}q_{\nu \rangle}^\mathcal{E} + 6{A_{\langle \mu}}q_{\nu \rangle}^\mathcal{E} + 3{\sigma ^\alpha}_{\langle \mu}\pi _{\nu \rangle \alpha}^\mathcal{E} + 3{\omega ^\alpha}{\varepsilon _{\alpha \beta (\mu}}\pi {{_{\nu)}^\mathcal{E}}^\beta}} \right].} \\ \end{array}$$(126)
 Gravitomagnetic propagation (MaxwellWeyl Hdot equation):$$\begin{array}{*{20}c} {{{\dot H}_{\langle \mu \nu \rangle}} + \Theta {H_{\mu \nu}} + {\rm{curl}}\;E  3{\sigma _{\alpha \langle \mu}}{H_{\nu \rangle}}^\alpha + {\omega ^\alpha}{\varepsilon _{\alpha \beta (\mu}}{H_{\nu)}}^\beta + 2{A^\alpha}{\varepsilon _{\alpha \beta (\mu}}{E_{\nu)}}^\beta =} \\ {\quad \quad \quad {{{\kappa ^2}} \over 2}\left[ {{\rm{curl}}\;\pi _{\mu \nu}^\mathcal{E}  3{\omega _{\langle \mu}}q_{\nu \rangle}^\mathcal{E} + {\sigma _{\alpha (\mu}}{\varepsilon _{\nu)}}^{^{\alpha \beta}}q_\beta ^\mathcal{E}} \right].} \\ \end{array}$$(127)
 Vorticity constraint:$${\vec \nabla ^\mu}{\omega _\mu}  {A^\mu}{\omega _\mu} = 0.$$(128)
 Shear constraint:$${\vec \nabla ^\nu}{\sigma _{\mu \nu}}  {\rm{curl}}\;{\omega _\mu}  {2 \over 3}{\vec \nabla _\mu}\Theta + 2{\varepsilon _{\mu \nu \alpha}}{\omega ^\nu}{A^\alpha} =  {\kappa ^2}q_\mu ^\mathcal{E}.$$(129)
 Gravitomagnetic constraint:$${\rm{curl}}\;{\sigma _{\mu \nu}} + {\vec \nabla _{\langle \mu}}{\omega _{\nu \rangle}}  {H_{\mu \nu}} + 2{A_{\langle \mu}}{\omega _{\nu \rangle}} = 0.$$(130)
 Gravitoelectric divergence (MaxwellWeyl divE equation):$$\begin{array}{*{20}c} {{{\vec \nabla}^\nu}{E_{\mu \nu}}  {{{\kappa ^2}} \over 3}{{\vec \nabla}_\mu}\rho  {\varepsilon _{\mu \nu \alpha}}{\sigma ^\nu}_\beta {H^{\alpha \beta}} + 3{H_{\mu \nu}}{\omega ^\nu} =} \\ {\quad \quad \quad {{{\kappa ^2}} \over 3}{\rho \over \lambda}{{\vec \nabla}_\mu}\rho + {{{\kappa ^2}} \over 6}\left({2{{\vec \nabla}_\mu}{\rho _\mathcal{E}}  2\Theta q_\mu ^\mathcal{E}  3{{\vec \nabla}^\nu}\pi _{\mu \nu}^\mathcal{E} + 3{\sigma _\mu}^\nu q_\nu ^\mathcal{E}  9{\varepsilon _\mu}^{\nu \alpha}{\omega _\nu}q_\alpha ^\mathcal{E}} \right).} \\ \end{array}$$(131)
 Gravitomagnetic divergence (MaxwellWeyl divH equation):$$\begin{array}{*{20}c} {{{\vec \nabla}^\nu}{H_{\mu \nu}}  {\kappa ^2}(\rho + p){\omega _\mu} + {\varepsilon _{\mu \nu \alpha}}{\sigma ^\nu}_\beta {E^{\alpha \beta}}  3{E_{\mu \nu}}{\omega ^\nu} =} \\ {\quad \quad \quad {\kappa ^2}(\rho + p){\rho \over \lambda}{\omega _\mu} + {{{\kappa ^2}} \over 6}(8{\rho _\mathcal{E}}{\omega _\mu}  3\;{\rm{curl}}\;q_\mu ^\mathcal{E}  3{\varepsilon _\mu}^{\nu \alpha}{\sigma _\nu}^\beta \pi _{\alpha \beta}^\mathcal{E}  3\pi _{\mu \nu}^\mathcal{E}{\omega ^\nu}).} \\ \end{array}$$(132)
 GaussCodazzi equations on the brane (with ω_{μ} = 0):$$R_{\langle \mu \nu \rangle}^ \bot + {\dot \sigma _{\langle \mu \nu \rangle}} + \Theta {\sigma _{\mu \nu}}  {\vec \nabla _{\langle \mu}}{A_{\nu \rangle}}  {A_{\langle \mu}}{A_{\nu \rangle}} = {\kappa ^2}\pi _{\mu \nu}^\mathcal{E},$$(133)where \(R_{\mu \nu}^ \bot\) is the Ricci tensor for 3surfaces orthogonal to u^{μ} on the brane, and \({R^ \bot} = {h^{\mu \nu}}R_{\mu \nu}^ \bot\).$${R^ \bot} + {2 \over 3}{\Theta ^2}  {\sigma _{\mu \nu}}{\sigma ^{\mu \nu}}  2{\kappa ^2}\rho  2\Lambda = {\kappa ^2}{{{\rho ^2}} \over \lambda} + 2{\kappa ^2}{\rho _\mathcal{E}},$$(134)
The standard 4D general relativity results are regained when λ^{−1} → 0 and ɛ_{μν} = 0, which sets all right hand sides to zero in Equations (123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134). Together with Equations (113, 114, 115, 116), 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 ρε, \(q_\mu ^{\mathcal E}\), 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}, H_{μν}), from the 5D Weyl field in the bulk.
The system of propagation and constraint equations, i.e., Equations (113, 114, 115, 116) and (123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134), 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 [110, 78, 314, 109]. 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 [114, 150]. 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 ɛ_{μν} [73]. Brane solutions of static black hole exteriors with 5D corrections to the Schwarzschild metric have been found [73, 110, 78, 314, 160, 49, 159], 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 [292, 54].
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 (153) 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 [319] indicate that general relativity remains a good approximation.

Exact analysis of OppenheimerSnyder collapse on the brane shows that the exterior is nonstatic [109], and this is extended to general collapse by arguments based on a generalized AdS/CFT correspondence [303, 94].
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 ɛ_{μν} in order to be compatible with massive KK modes in the strongfield regime. In the endstate of collapse, we expect an ɛ_{μν} which goes to zero at large distances, recovering the Schwarzschild weakfield limit, but which grows at short range. Furthermore, 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 [109]. 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 Σ be continuous (there are no intrinsic stresses on Σ). 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 Σ.

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 [150, 61, 129, 227, 62, 287]. 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 [51, 116, 93]).
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 AdS_{5} 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 [254, 282, 136, 289, 293, 210, 259, 125].
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 (1 8) or (180), 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 [59, 151, 45]. 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 [165, 138, 100, 276, 141, 140, 304, 318, 180, 195, 139, 33, 238, 158, 229, 103, 12].
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 [90, 157] or a hot bigbang radiation era, as in the “ekpyrotic” or cyclic scenario [163, 154, 251, 301, 193, 230, 307], or in colliding bubble scenarios [29, 106, 107]. (See also [21, 69, 214] 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 [207] for a review).
Highenergy braneworld modifications to the dynamics of inflation on the brane have been investigated [222, 156, 63, 302, 234, 233, 74, 204, 23, 24, 25, 240, 135, 184, 270, 221]. 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 [222, 70, 226, 277, 258, 206, 145].
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 [218, 191, 122, 102]. 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 [102]. Thus it seems a reasonable approximation in slowroll to neglect the KK effects carried by ɛ_{μν} when computing the density perturbations.
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 [192]:$$A_{\rm{t}}^2 \approx \left({{{32V} \over {75M_{\rm{p}}^2}}} \right)\left[ {{{3{V^2}} \over {4{\lambda ^2}}}} \right],$$(217)Equation (218) 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,$${{A_{\rm{t}}^2} \over {A_{\rm{s}}^2}} \approx \left({{{M_{\rm{p}}^2} \over {16\pi}}{{{V^{\prime 2}}} \over {{V^2}}}} \right)\left[ {{{6\lambda} \over V}} \right].$$(218)but remarkably the same consistency relation as in general relativity holds [145]:$${n_{\rm{t}}} =  3\epsilon ,$$(219)This consistency relation persists when Z_{2} symmetry is dropped [146] (and in a twobrane model with stabilized radion [118]). It holds only to lowest order in slowroll, as in general relativity, but the reason for this [286] and the nature of the corrections [44] are not settled. The massive KK modes of tensor perturbations remain in the vacuum state during slowroll inflation [192, 121]. 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.$${n_{\rm{t}}} =  2{{A_{\rm{t}}^2} \over {A_{\rm{s}}^2}}.$$(220)

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 [39]. 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 [13]. Braneworld (p)reheating after inflation is discussed in [309, 321, 5, 310, 67].
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 [17]. This has the form of Equation (178), withwhere q is 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 (44) 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 (178). The effect of the black hole charge on the brane arises via the junction conditions and leads to the modified Friedmann equation [17],$$F(R) = K + {{{R^2}} \over {{\ell ^2}}}  {m \over {{R^2}}} + {{{q^2}} \over {{R^4}}},$$(224)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 [144] for problems with such bounces).$${H^2} = {{{\kappa ^2}} \over 3}\rho \left({1 + {\rho \over {2\lambda}}} \right) + {m \over {{a^4}}}  {{{q^2}} \over {{a^6}}} + {1 \over 3}\Lambda  {K \over {{a^2}}}.$$(225)
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 tensor, while the former arises from nonvacuum stresses in the bulk via the \({{\mathcal F}_{\mu \nu}}\) tensor in Equation (63). 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 v = T + ∫ dR/F in Equation (178), so that v = const. are null surfaces, and$$^{(5)}d{s^2} =  F(R,\;v)d{v^2} + 2dvdR + {R^2}\left({{{d{r^2}} \over {1  K{r^2}}} + {r^2}d{\Omega ^2}} \right),$$(226)This model has a moving FRW brane in a 4isotropic bulk (which is not static), with either a radiating bulk black hole (dm/dv < 0), or a radiating brane (dm/dv > 0) [53, 198, 197, 199]. The metric satisfies the 5D field equations (44) with a nullradiation energymomentum tensor,$$F(R,\;v) = K + {{{R^2}} \over {{\ell ^2}}}  {{m(v)} \over {{R^2}}}.$$(227)where ψ ∝ dm/dv. It follows that$$^{(5)}{T_{AB}} = \psi {k_A}{k_B},\quad \quad {k_A}{k^A} = 0,\quad \quad {k_A}{u^A} = 1,$$(228)In this case, the same effect, i.e., a varying mass parameter m, contributes to both ɛ_{μν} 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:$${\mathcal{F}_{\mu \nu}} = \kappa _5^{ 2}\psi {h_{\mu \nu}}.$$(229)By Equations (68) and (228), we arrive at the matter conservation equations,$${H^2} = {{{\kappa ^2}} \over 3}\rho \left({1 + {\rho \over {2\lambda}}} \right) + {{m(t)} \over {{a^4}}} + {1 \over 3}\Lambda  {K \over {{a^2}}}.$$(230)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$${\nabla ^\nu}{T_{\mu \nu}} =  2\psi {u_\mu}.$$(231)The evolution of m is governed by the 4D contracted Bianchi identity, using Equation (229):$$\dot \rho + 3H(\rho + p) =  2\psi .$$(232)For an FRW brane, this yields$${\nabla ^\mu}{\mathcal{E}_{\mu \nu}} = {{6{\kappa ^2}} \over \lambda}{\nabla ^\mu}{\mathcal{S}_{\mu \nu}} + {2 \over 3}\left[ {\kappa _5^2\left({\dot \psi + \Theta \psi} \right)  3{\kappa ^2}\psi} \right]{u_\mu} + {2 \over 3}\kappa _5^2{\vec \nabla _\mu}\psi .$$(233)where ρε = 3m(t)/(κ^{2}a^{4}).$${\dot \rho _\mathcal{E}} + 4H{\rho _\mathcal{E}} = 2\psi  {2 \over 3}{{\kappa _5^2} \over {{\kappa ^2}}}\left({\dot \psi + 3H\psi} \right),$$(234)
 A more complicated bulk metric arises when there is a selfinteracting scalar field Φ in the bulk [225, 16, 236, 97, 194, 98, 35]. 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,$$^{(5)}{T_{AB}} = {\Phi _{,A}}{\Phi _{,B}}{ ^{(5)}}{g_{AB}}\left[ {V(\Phi) + {1 \over 2}(5){g^{CD}}{\Phi _C}{\Phi _D}} \right],$$(235)The junction conditions on the field imply that$$^{(5)}\square\Phi  V^{\prime}(\Phi) = 0.$$(236)then Equations (68) and (235) show that matter conservation continues to hold on the brane in this simple case:$${\partial _y}\Phi (x,\;0) = 0.$$(237)From Equation (235) one finds that$${\nabla ^\nu}{T_{\mu \nu}} = 0.$$(238)where$${\mathcal{F}_{\mu \nu}} = {1 \over {4\kappa _5^2}}\left[ {4{\phi _{,\mu}}{\phi _{,\nu}}  {g_{\mu \nu}}\left({3V(\phi) + {5 \over 2}{g^{\alpha \beta}}{\phi _{,\alpha}}{\phi _{,\beta}}} \right)} \right],$$(239)so that the modified Friedmann equation becomes$$\phi (x) = \Phi (x,\;0),$$(240)When there is coupling between brane matter and the bulk scalar field, then the Friedmann and conservation equations are more complicated [225, 16, 236, 97, 194, 98, 35].$${H^2} = {{{\kappa ^2}} \over 3}\rho \left({1 + {\rho \over {2\lambda}}} \right) + {m \over {{a^4}}} + {{\kappa _5^2} \over 6}\left[ {{1 \over 2}{{\dot \phi}^2} + V(\phi)} \right] + {1 \over 3}\Lambda  {K \over {{a^2}}}.$$(241)
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.
As in 4D general relativity, there are various different, but essentially equivalent, ways to formulate linear cosmological perturbation theory. First I 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 [92]). 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, I will discuss various perturbation problems, using either a 1 + 3covariant or a metricbased approach.
6.3 Density perturbations on large scales
If ρε = 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 ρε ≠ 0 in the background, then the weighted difference between U and Δ determines the isocurvature mode: S_{tot} ∝ (4ρ_{ε}/3_{ρ})Δ − (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 Sε that is similar to an extra isocurvature contribution.

The KK anisotropic stress δπ_{ε} 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 (318, 319, 320). 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 [177], using the lowenergy 2brane approximation [298, 320, 290, 299, 300] to find an effective 4D form for ɛ_{μν} and hence for δπ_{ɛ}. This is discussed below.
6.5 Vector perturbations
Inflation will redshift away the vorticity and the KK mode. Indeed, the massive KK vector modes are not excited during slowroll inflation [40, 269].
6.6 Tensor perturbations
7 Gravitational Wave Perturbations in BraneWorld Cosmology
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 [200]. Selfconsistent lowenergy approximations to compute this effect are developed in [142, 91].
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 [201] and tensor [200] 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.
However, the first step required is the solution for \(\pi _{\mu \nu}^{\mathcal E}\). This solution will be of the form given in Equation (243). Once \({\mathcal G}\) and F_{k} are determined or estimated, the numerical integration in Equation (243) 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 Conclusion
Simple braneworld models of RS type 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 [298, 320, 290, 299, 300, 177, 268, 37, 142, 91]) and manageable numerical codes (building on [177, 268, 37, 142, 91]) 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 [225, 16, 236, 97, 194, 98, 35, 165, 138, 100, 276, 141, 140, 304, 318, 180, 195, 139, 33, 238, 158, 229, 103, 12]). 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 [21, 69, 214, 268, 37, 42, 174, 152, 261, 124]. 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 [305, 248, 202]. (By contrast, in the absence of a bulk scalar, low energy gravity is of BransDicke type [105].) In particular, such models will allow some fundamental problems to be addressed:$$S = {1 \over {2\kappa _5^2}}\int {{d^5}} x\sqrt {{ ^{(5)}}g} \left[ {^{(5)}R  \kappa _5^2{\partial _A}\Phi {\partial ^A}\Phi  2{\Lambda _5}(\Phi)} \right] + \int\nolimits_{{\rm{brane}}({\rm{s}})} {{d^4}} x\sqrt { g} \left[ { \lambda (\Phi) + {K \over {\kappa _5^2}} + {L_{{\rm{matter}}}}} \right]$$(386)

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 [90, 157, 163, 154, 251, 301, 193, 230, 307, 21, 69, 214, 29, 106, 107]).

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 [80, 256, 255, 253, 111, 56, 209, 26, 232, 211, 126, 84, 14, 83, 55, 224]. In [15] it is shown that the black string solution of the form of Equation (138) is ruled out by the GaussBonnet term.$$\begin{array}{*{20}c} {S = {1 \over {2\kappa _5^2}}\int {{d^5}} x\sqrt {{ ^{(5)}}g} \left[ {^{(5)}R  2{\Lambda _5} + \alpha \left({^{(5)}{R^2}  4{\;^{(5)}}{R_{AB}}{\;^{(5)}}{R^{AB}} + {\;^{(5)}}{R_{ABCD}}{\;^{(5)}}{R_{ABCD}}} \right)} \right]} \\ {\quad \;\; + \int\nolimits_{{\rm{brane}}} {{d^4}} x\sqrt { g} \left[ { \lambda + {K \over {\kappa _5^2}} + {L_{{\rm{matter}}}}} \right],} \\ \end{array}$$(387)In this sense, the GaussBonnet correction removes an unstable and singular solution. 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.$${H^2} \propto {\rho ^{2/3}}$$(388)
 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 [89, 66, 257, 295] was put forward as an alternative to the RS mechanism: The bulk is flat Minkowski 5D spacetime (and as a consequence there is no normalizable zeromode of the bulk graviton), and there is no brane tension. 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. Unlike the other braneworlds discussed, these models lead to 5D behaviour on large scales rather than small scales. The cosmological models have been analyzed in [76, 171, 85, 164, 77, 279, 294, 278, 297, 3, 223, 213, 250, 130]. (Braneworld black holes with induced gravity are investigated in [173].)$$S = {1 \over {2\kappa _5^2}}\int {{d^5}} x\sqrt {{ ^{(5)}}g} \left[ {^{(5)}R  2{\Lambda _5}} \right] + \int\nolimits_{{\rm{brane}}} {{d^4}} x\sqrt { g} \left[ {\beta R  \lambda + {K \over {\kappa _5^2}} + {L_{{\rm{matter}}}}} \right],$$(389)
The lateuniverse 5D behaviour of gravity can naturally produce a latetime acceleration, even without dark energy, although the finetuning problem is not evaded.
The effect of the inducedgravity correction at early times is to restore the standard behaviour H^{2} ∝ ρ to lowest order at the highest energies. As the energy drops, but is still above the brane tension, there may be an RS regime, H^{2} ∝ ρ^{2}. In the late universe at low energies, instead of recovering general relativity, there may be strong deviations from general relativity, and latetime acceleration from 5D gravity effects (rather than negative pressure energy) is typical.
Thus we have a striking result that both forms of correction to the gravitational action, i.e., GaussBonnet and induced gravity, suppress the RandallSundrum type highenergy modifications to the Friedmann equation when the energy reaches a critical level. (Cosmologies with both inducedgravity and GaussBonnet corrections to the RS action are considered in [172].)

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.
Notes
Acknowledgments
I thank my many collaborators and friends for discussions and sharing of ideas. My work is supported by PPARC.
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