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

where is the FRW curvature index, and is the mass parameter of the black hole at (recall that the 5D gravitational potential has behaviour). The bulk black hole gives rise to dark radiation on the brane via its Coulomb effect. The FRW brane moves radially along the 5th dimension, with , where is the FRW scale factor, and the junction conditions determine the velocity via the Friedmann equation for [32, 249]. Thus one can interpret the expansion of the universe as motion of the brane through the static bulk. In the special case and , the brane is fixed and has Minkowski geometry, i.e., the original RS 1-brane brane-world is recovered in different coordinates.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 [27]:

Here is the scale factor on the FRW brane at , and may be chosen as proper time on the brane, so that . In the case where there is no bulk black hole (), the metric functions are Again, the junction conditions determine the Friedmann equation. The extrinsic curvature at the brane is Then, by Equation (62), The field equations yield the first integral [27] where is constant. Evaluating this at the brane, using Equation (185), gives the modified Friedmann equation (188).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:

on using Equation (118), where The covariant Raychauhuri equation (123) yields which also follows from differentiating Equation (188) and using the energy conservation 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]:

The other modification to the Hubble rate is via the high-energy correction . In order to recover the observational successes of general relativity, the high-energy regime where significant deviations occur must take place before nucleosynthesis, i.e., cosmological observations impose the lower limit This is much weaker than the limit imposed by table-top experiments, Equation (42). Since decays as during the radiation era, it will rapidly become negligible after the end of the high-energy regime, .If and , then the exact solution of the Friedmann equations for is [27]

where . If (but ), then the solution for the radiation era () is [19] For we recover from Equations (193) and (194) the standard behaviour, , whereas for , we have the very different behaviour of the high-energy regime,When we have from the conservation equation. If , we recover the de Sitter solution for and an asymptotically de Sitter solution for :

A qualitative analysis of the Friedmann equations is given in [47, 48].

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