5 Brane-World Cosmology: Dynamics

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 AdS5 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 [342, 375, 193, 386, 390, 290, 347, 180].

The generalization of AdS5 that preserves 4-isotropy and solves the vacuum 5D Einstein equation (22View Equation) is Schwarzschild–AdS5, 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–AdS5, with the Z2-symmetric FRW brane at the boundary. (Note that FRW branes can also be embedded in non-vacuum generalizations, e.g., in Reissner–Nordström–AdS5 and Vaidya–AdS5.)

In natural static coordinates, the bulk metric is

2 ( 2 ) (5)ds2 = − F (R )dT2 +-dR---+ R2 ---dr--- + r2dΩ2 , (184 ) F (R ) 1 − Kr2 R2 m F (R ) = K + ---− ---, (185 ) ℓ2 R2
where K = 0,±1 is the FRW curvature index, and m is the mass parameter of the black hole at R = 0 (recall that the 5D gravitational potential has R− 2 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 R = a(T ), where a is the FRW scale factor, and the junction conditions determine the velocity via the Friedmann equation for a [336, 45]. Thus one can interpret the expansion of the universe as motion of the brane through the static bulk. In the special case m = 0 and da∕dT = 0, 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 [38Jump To The Next Citation Point]:

[ 2 ] (5)ds2 = − N 2(t,y)dt2 + A2 (t,y) --dr---- + r2dΩ2 + dy2. (186 ) 1 − Kr2
Here a(t) = A (t,0) is the scale factor on the FRW brane at y = 0, and t may be chosen as proper time on the brane, so that N (t,0 ) = 1. In the case where there is no bulk black hole (m = 0), the metric functions are
˙A(t,y)- N = ˙a(t) , (187 ) [ ( ) { } ( ) ] A = a(t) cosh y- − 1 + ρ(t) sinh |y| . (188 ) ℓ λ ℓ
Again, the junction conditions determine the Friedmann equation. The extrinsic curvature at the brane is
( ) μ N ′ A′ A ′ A ′ K ν = diag N-, A-,A--,A-- . (189 ) brane
Then, by Equation (68View Equation),
N ′| κ2 ---|| = -5(2ρ + 3p − λ ), (190 ) N |brane 6 A-′| κ25- A |brane = − 6 (ρ + λ). (191 )
The field equations yield the first integral [38Jump To The Next Citation Point]
′2 A2- ˙2 Λ5- 4 (AA ) − N 2A + 6 A + m = 0, (192 )
where m is constant. Evaluating this at the brane, using Equation (191View Equation), gives the modified Friedmann equation (194View Equation).

The dark radiation carries the imprint on the brane of the bulk gravitational field. Thus we expect that ℰμν for the Friedmann brane contains bulk metric terms evaluated at the brane. In Gaussian normal coordinates (using the field equations to simplify),

A ′′| Λ ( 1 ) ℰ00 = 3---|| + --5, ℰ ij = − --ℰ00 δij. (193 ) A brane 2 3

Either form of the cosmological metric, Equation (184View Equation) or (186View Equation), 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 [89, 211, 65]. 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 (140View Equation) to find the modified Friedmann equation:

2 ( ) H2 = κ-ρ 1 + -ρ- + m--+ 1-Λ − K--, (194 ) 3 2λ a4 3 a2
on using Equation (124View Equation), where
κ2 4 m = 3-ρℰ0a0. (195 )
The covariant Raychauhuri equation (129View Equation) yields
2 ( ) H˙ = − κ--(ρ + p ) 1 + ρ- − 2m--+ K-, (196 ) 2 λ a4 a2
which also follows from differentiating Equation (194View Equation) and using the energy conservation equation.

When the bulk black hole mass vanishes, the bulk geometry reduces to AdS5, and ρℰ = 0. In order to avoid a naked singularity, we assume that the black hole mass is non-negative, so that ρℰ0 ≥ 0. (By Equation (185View Equation), it is possible to avoid a naked singularity with negative m when K = − 1, provided |m | ≤ ℓ2∕4.) This additional effective relativistic degree of freedom is constrained by nucleosynthesis and CMB observations to be no more than ∼ 5% of the radiation energy density [271Jump To The Next Citation Point, 24Jump To The Next Citation Point, 208, 47]:

| ρℰ-|| ρrad| ≲ 0.05. (197 ) nuc
The other modification to the Hubble rate is via the high-energy correction ρ∕λ. 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
4 4 λ > (1 MeV ) ⇒ M5 > 10 GeV. (198 )
This is much weaker than the limit imposed by table-top experiments, Equation (42View Equation). Since ρ2∕λ decays as −8 a during the radiation era, it will rapidly become negligible after the end of the high-energy regime, ρ = λ.

If ρℰ = 0 and K = 0 = Λ, then the exact solution of the Friedmann equations for w = p ∕ρ = const. is [38]

1∕3(w+1) Mp −1 − 9 a = const.× [t(t + tλ)] , tλ = √------------ ≲ (1 + w ) 10 s, (199 ) 3π λ(1 + w )
where w > − 1. If ρℰ ⁄= 0 (but K = 0 = Λ), then the solution for the radiation era (w = 13) is [24Jump To The Next Citation Point]
√ -- a = const. × [t(t + tλ)]1∕4, tλ = -√-----3Mp------. (200 ) 4 π λ(1 + ρℰ∕ρ)
For t ≫ tλ we recover from Equations (199View Equation) and (200View Equation) the standard behaviour, 2∕3(w+1) a ∝ t, whereas for t ≪ tλ, we have the very different behaviour of the high-energy regime,
ρ ≫ λ ⇒ a ∝ t1∕3(w+1). (201 )

When w = − 1 we have ρ = ρ0 from the conservation equation. If K = 0 = Λ, we recover the de Sitter solution for ρℰ = 0 and an asymptotically de Sitter solution for ρℰ > 0:

------------- ∘ ρ0 ( ρ0 ) a = a0exp [H0 (t − t0)], H0 = κ --- 1 + --- , for ρℰ = 0, (202 ) ∘ ---- 3 2λ 2 -m- a = H sinh [2H0 (t − t0)] for ρℰ > 0. (203 ) 0
A qualitative analysis of the Friedmann equations is given in [68, 67].

 5.1 Brane-world inflation
 5.2 Brane-world instanton
 5.3 Models with non-empty bulk

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