Models with non-empty bulk

5.3 Models with non-empty bulk

The single-brane cosmological model can be generalized to include stresses other than $Λ 5$ in the bulk:

The simplest example arises from considering a charged bulk black hole, leading to the Reissner–Nordström $AdS 5$ bulk metric [17 ]. This has the form of Equation (178 ), with

where $q$ is the “electric” charge parameter of the bulk black hole. The metric is a solution of the 5D Einstein–Maxwell equations, so that $(5)TAB$ in Equation (44 ) is the energy-momentum tensor of a radial static 5D “electric” field. In order for the field lines to terminate on the boundary brane, the brane should carry a charge $− q$ . Since the RN $AdS 5$ metric is 4-isotropic, it is still possible to embed a FRW brane in it, which is moving in the coordinates of Equation (178 ). The effect of the black hole charge on the brane arises via the junction conditions and leads to the modified Friedmann equation [17 ],

The field lines that terminate on the brane imprint on the brane an effective negative energy density $− 3q2∕(κ2a6)$ , which redshifts like stiff matter ( $w = 1$ ). The negativity of this term introduces the possibility that at high energies it can bring the expansion rate to zero and cause a turn-around or bounce (but see [144 ] for problems with such bounces).
Apart from negativity, the key difference between this “dark stiff matter” and the dark radiation term $4 m ∕a$ is that the latter arises from the bulk Weyl curvature via the $ℰμν$ tensor, while the former arises from non-vacuum stresses in the bulk via the $ℱ μν$ tensor in Equation (63 ). The dark stiff matter does not arise from massive KK modes of the graviton.
Another example is provided by the Vaidya– $AdS5$ metric, which can be written after transforming to a new coordinate $∫ v = T + dR ∕F$ in Equation (178 ), so that $v = const.$ are null surfaces, and

This model has a moving FRW brane in a 4-isotropic bulk (which is not static), with either a radiating bulk black hole ( $dm ∕dv < 0$ ), or a radiating brane ( $dm ∕dv > 0$ ) [53 , 197 , 198 , 199 ]. The metric satisfies the 5D field equations (44 ) with a null-radiation energy-momentum tensor,

where $ψ ∝ dm ∕dv$ . It follows that

In this case, the same effect, i.e., a varying mass parameter $m$ , contributes to both $ℰμν$ and $ℱμν$ in the brane field equations. The modified Friedmann equation has the standard 1-brane RS-type form, but with a dark radiation term that no longer behaves strictly like radiation:

By Equations (68 ) and (228 ), we arrive at the matter conservation equations,

This shows how the brane loses ( $ψ > 0$ ) or gains ( $ψ < 0$ ) energy in exchange with the bulk black hole. For an FRW brane, this equation reduces to

The evolution of $m$ is governed by the 4D contracted Bianchi identity, using Equation (229 ):

For an FRW brane, this yields

where $ρ = 3m (t)∕(κ2a4) ℰ$ .
A more complicated bulk metric arises when there is a self-interacting scalar field $Φ$ in the bulk [16 , 35 , 97 , 98 , 194 , 225 , 236 ]. In the simplest case, when there is no coupling between the bulk field and brane matter, this gives

where $Φ (x,y)$ satisfies the 5D Klein–Gordon equation,

The junction conditions on the field imply that

Then Equations (68 ) and (235 ) show that matter conservation continues to hold on the brane in this simple case:

From Equation (235 ) one finds that

where

so that the modified Friedmann equation becomes

When there is coupling between brane matter and the bulk scalar field, then the Friedmann and conservation equations are more complicated [16 , 35 , 97 , 98 , 194 , 225 , 236 ].