11 Conclusion

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:

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

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 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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