Simple brane-world models provide a rich phenomenology for exploring some of the ideas that are emerging from M theory. The higher-dimensional 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.

This review has attempted to show some of the key features of brane-world gravity from the perspective of astrophysics and cosmology, emphasizing a geometric approach to dynamics and perturbations. It has focused mainly on 1-brane RS-type brane-worlds, but also considered the DGP brane-world models. The RS-type models have some attractive features:

- They provide a simple 5D phenomenological realization of the Horava–Witten 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 self-gravity 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.

The review has highlighted both the successes and some remaining open problems of the RS models and their generalizations. The open problems stem from a common basic difficulty, i.e., understanding and solving for the gravitational interaction between the bulk and the brane (which is nonlocal from the brane viewpoint). The key open problems of relevance to astrophysics and cosmology are

- 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 large-scale structure, and to impose observational constraints from high-precision data.

The RS-type models are the simplest brane-worlds with curved extra dimension that allow for a meaningful approach to astrophysics and cosmology. One also needs to consider generalizations that attempt to make these models more realistic, or that explore other aspects of higher-dimensional gravity which are not probed by these simple models. Two important types of generalization are the following:

- 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].
For two-brane 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 Brans–Dicke type [155].) In particular, such models will allow some fundamental problems to be addressed:

- The hierarchy problem of particle physics.
- An extra-dimensional mechanism for initiating inflation (or the hot radiation era with super-Hubble 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 extra-dimensional explanation for the dark energy (and possibly also dark matter) puzzles: Could dark energy or late-time 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 Einstein–Hilbert action, including the
following:
- Higher-order curvature invariants, which arise in the AdS/CFT correspondence as
next-to-leading order corrections in the CFT. The Gauss–Bonnet combination in particular has
unique properties in 5D, giving field equations which are second-order in the bulk
metric (and linear in the second derivatives), and being ghost-free. The action is
where is the Gauss–Bonnet coupling constant, related to the string
scale. The cosmological dynamics of these brane-worlds 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
Gauss–Bonnet term. In this sense, the Gauss–Bonnet correction removes an unstable and
singular solution.
In the early universe, the Gauss–Bonnet corrections to the Friedmann equation have the dominant form

at the highest energies. If the Gauss–Bonnet term is a small correction to the Einstein–Hilbert term, as may be expected if it is the first of a series of higher-order corrections, then there will be a regime of RS-dominance as the energy drops, when . Finally at energies well below the brane tension, the general relativity behaviour is recovered. - 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 brane-world is the DGP model [131, 95, 344, 391], which we
investigated in this review as an alternative to the RS-type models. Another viewpoint
is to see the induced-gravity term in the action as a correction to the RS action:
where is a positive coupling constant.
The cosmological models have been analyzed in [117, 238, 126, 230, 118, 372, 392, 370, 397, 7, 309, 297, 337, 186, 240]. (Brane-world black holes with induced gravity are investigated in [241].) Unlike RS-type models, DGP models lead to 5D behaviour on large scales rather than small scales. Then on an FRW brane, the late-universe 5D behaviour of gravity can naturally produce a late-time acceleration, even without dark energy, although the self-accelerating 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.

- Higher-order curvature invariants, which arise in the AdS/CFT correspondence as
next-to-leading order corrections in the CFT. The Gauss–Bonnet combination in particular has
unique properties in 5D, giving field equations which are second-order in the bulk
metric (and linear in the second derivatives), and being ghost-free. The action is
where is the Gauss–Bonnet coupling constant, related to the string
scale. The cosmological dynamics of these brane-worlds 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
Gauss–Bonnet term. In this sense, the Gauss–Bonnet correction removes an unstable and
singular solution.

The RS and DGP models are 5-dimensional phenomenological models, and so a key issue is how to realize such models in 10-dimensional string theory. Some progress has been made. 6-dimensional cascading brane-worlds are extensions of the DGP model. 10-dimensional 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 Dirac–Born–Infeld action and this gives the possibility of generating a large non-Gaussianity 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 4-dimensional approach to deal with extra dimensions. For example, the stabilization mechanism, which is necessary to fix moduli fields in string theory, exploits non-perturbative effects and they are often added in the 4-dimensional effective theory. Then it is not clear whether the resultant 4-dimensional effective theory is consistent with the 10-dimensional 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 10-dimensional supergravity [31, 30, 28, 29]. This will provide us with a very interesting bridge between phenomenological brane-world models, where dynamics of higher-dimensional gravity is studied in detail, and string theory approaches, where 4D effective theory is intensively used. It is crucial to identify the higher-dimensional signature of the models in order to test a fundamental theory like string theory.

In summary, brane-world gravity opens up exciting prospects for subjecting M theory ideas to the increasingly stringent tests provided by high-precision astronomical observations. At the same time, brane-world models provide a rich arena for probing the geometry and dynamics of the gravitational field and its interaction with matter.

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