Conclusion

9 Conclusion

Simple brane-world models of RS type 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 on 1-brane RS-type brane-worlds which 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 the 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 [38 , 91 , 142 , 177 , 268 , 290 , 298 , 299 , 300 , 320 ]) and manageable numerical codes (building on [38 , 91 , 142 , 177 , 268 ]) 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 [13, 16, 33, 35, 97, 98, 100, 103, 138, 139, 140, 141, 158, 165, 180, 194, 195, 225, 229, 236, 238, 276, 304, 318]). 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, 38, 42, 69, 124, 152, 174, 214, 261, 268]. 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 [202, 248, 305]. (By contrast, in the absence of a bulk scalar, low energy gravity is of Brans–Dicke type [106].) 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 [21 , 29 , 69 , 90 , 107 , 108 , 154 , 157 , 163 , 193 , 214 , 231 , 251 , 301 , 307 ]).
- 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 $α$ is the Gauss–Bonnet coupling constant related to the string scale. The cosmological dynamics of these brane-worlds is investigated in [14 , 26 , 55 , 56 , 80 , 83 , 84 , 112 , 126 , 209 , 211 , 224 , 232 , 253 , 255 , 256 ]. In [15 ] it is shown that the black string solution of the form of Equation (138 ) 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 $H2 ∝ ρ2$ . 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 [66 , 89 , 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 zero-mode of the bulk graviton), and there is no brane tension. Another viewpoint is to see the induced-gravity term in the action as a correction to the RS action:
  
  where $β$ is a positive coupling constant. Unlike the other brane-worlds discussed, these models lead to 5D behaviour on large scales rather than small scales. The cosmological models have been analyzed in [3 , 76 , 77 , 85 , 130 , 164 , 171 , 213 , 223 , 250 , 278 , 279 , 294 , 297 ]. (Brane-world black holes with induced gravity are investigated in [173 ].)
  The late-universe 5D behaviour of gravity can naturally produce a late-time acceleration, even without dark energy, although the fine-tuning problem is not evaded.
  
  The effect of the induced-gravity correction at early times is to restore the standard behaviour $H2 ∝ ρ$ to lowest order at the highest energies. As the energy drops, but is still above the brane tension, there may be an RS regime, $2 2 H ∝ ρ$ . In the late universe at low energies, instead of recovering general relativity, there may be strong deviations from general relativity, and late-time 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., Gauss–Bonnet and induced gravity, suppress the Randall–Sundrum type high-energy modifications to the Friedmann equation when the energy reaches a critical level. (Cosmologies with both induced-gravity and Gauss–Bonnet corrections to the RS action are considered in [172 ].)

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.