Brane-World Cosmology: Dynamics

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 [125 , 136 , 210 , 254 , 259 , 282 , 289 , 293 ].

The generalization of $AdS5$ that preserves 4-isotropy and solves the vacuum 5D Einstein equation (22 ) 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 , (178 ) F (R ) 1 − Kr2 R2 m F (R ) = K + ---− ---, (179 ) 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 $a$ [32 , 249 ]. 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 [27 ]:

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) , (181 ) [ ( ) { } ( ) ] A = a(t) cosh y- − 1 + ρ(t) sinh |y| . (182 ) ℓ λ ℓ

Again, the junction conditions determine the Friedmann equation. The extrinsic curvature at the brane is

Then, by Equation (62

N ′| κ2 ---|| = -5(2ρ + 3p − λ ), (184 ) N |brane 6 A-′| κ25- A |brane = − 6 (ρ + λ). (185 )

The field equations yield the first integral [27

]

where $m$ 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 $ℰμν$ 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 $AdS5$ , and $ρ ℰ = 0$ . In order to avoid a naked singularity, we assume that the black hole mass is non-negative, so that $ρℰ0 ≥ 0$ . (By Equation (179 ), 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 [19 , 34 , 148 , 191 ]:

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

This is much weaker than the limit imposed by table-top experiments, Equation (42

). 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 [27 ]

where $w > − 1$ . If $ρℰ ⁄= 0$ (but $K = 0 = Λ$ ), then the solution for the radiation era ( $w = 13$ ) is [19

]

For $t ≫ tλ$ we recover from Equations (193

) and (194

) the standard behaviour, $2∕3(w+1) a ∝ t$ , whereas for $t ≪ tλ$ , we have the very different behaviour of the high-energy regime,

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, (196 ) ∘ ---- 3 2λ 2 -m- a = H sinh [2H0 (t − t0)] for ρℰ > 0. (197 ) 0

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

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