

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 [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 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 [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 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 [305, 248, 202]. (By contrast, in the
absence of a bulk scalar, low energy gravity is of Brans-Dicke
type [105].) 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 [90, 157, 163, 154, 251, 301, 193, 230, 307, 21, 69, 214, 29, 106, 107]).
- 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 [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
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 [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 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
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 [76, 171, 85, 164, 77, 279, 294, 278, 297, 3, 223, 213, 250, 130]. (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
to lowest order at the highest energies. As the
energy drops, but is still above the brane tension, there may be an
RS regime,
. 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.

