

5 Brane-World Cosmology:
Dynamics
A
-dimensional spacetime with spatial 4-isotropy (4D
spherical/plane/hyperbolic symmetry) has a natural foliation into
the symmetry group orbits, which are
-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 [254, 282, 136, 289, 293, 210, 259, 125].
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
[249, 32]. 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 [59, 151, 45]. 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 [191
, 19
, 148, 34]:
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 [48, 47].

