A 1+4-dimensional spacetime with spatial 4-isotropy (4D spherical/plane/hyperbolic symmetry) has a natural foliation into the symmetry group orbits, which are 1+3-dimensional surfaces with 3-isotropy and 3-homogeneity, i.e., FRW surfaces. In particular, the bulk of the RS brane-world, which admits a foliation into Minkowski surfaces, also admits an FRW foliation since it is 4-isotropic. Indeed this feature of 1-brane RS-type cosmological brane-worlds underlies the importance of the AdS/CFT correspondence in brane-world cosmology [125, 136, 210, 254, 259, 282, 289, 293].
The generalization of that preserves 4-isotropy and solves the vacuum 5D Einstein equation (22) is Schwarzschild–, and this bulk therefore admits an FRW foliation. It follows that an FRW brane-world, the cosmological generalization of the RS brane-world, is a part of Schwarzschild–, with the -symmetric FRW brane at the boundary. (Note that FRW branes can also be embedded in non-vacuum generalizations, e.g., in Reissner–Nordström– and Vaidya–.)
In natural static coordinates, the bulk metric is
The velocity of the brane is coordinate-dependent, and can be set to zero. We can use Gaussian normal coordinates, in which the brane is fixed but the bulk metric is not manifestly static :
The dark radiation carries the imprint on the brane of the bulk gravitational field. Thus we expect that for the Friedmann brane contains bulk metric terms evaluated at the brane. In Gaussian normal coordinates (using the field equations to simplify),
Either form of the cosmological metric, Equation (178) or (180), may be used to show that 5D gravitational wave signals can take “short-cuts” through the bulk in travelling between points A and B on the brane [45, 59, 151]. 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.
Instead of using the junction conditions, we can use the covariant 3D Gauss–Codazzi equation (134) to find the modified Friedmann equation:
When the bulk black hole mass vanishes, the bulk geometry reduces to , and . In order to avoid a naked singularity, we assume that the black hole mass is non-negative, so that . (By Equation (179), it is possible to avoid a naked singularity with negative when , provided .) This additional effective relativistic degree of freedom is constrained by nucleosynthesis and CMB observations to be no more than of the radiation energy density [19, 34, 148, 191]:
If and , then the exact solution of the Friedmann equations for is 
When we have from the conservation equation. If , we recover the de Sitter solution for and an asymptotically de Sitter solution for :
© Max Planck Society