8.1 The low-energy approximation

The basic idea of the low-energy approximation [398Jump To The Next Citation Point, 428Jump To The Next Citation Point, 387Jump To The Next Citation Point, 399Jump To The Next Citation Point, 400Jump To The Next Citation Point] is to use a gradient expansion to exploit the fact that, during most of the history of the universe, the curvature scale on the observable brane is much greater than the curvature scale of the bulk (ℓ < 1 mm):
−1∕2 (5) − 1∕2 −1 −1 L ∼ |R μναβ| ≫ ℓ ∼ | RABCD | ⇒ |∇ μ| ∼ L ≪ |∂y| ∼ ℓ . (397 )
These conditions are equivalent to the low energy regime, since ℓ2 ∝ λ−1 and |R μναβ| ∼ |T μν|:
ℓ2 ρ ---∼ --≪ 1. (398 ) L2 λ
Using Equation (397View Equation) to neglect appropriate gradient terms in an expansion in ℓ2∕L2, the low-energy equations can be solved. However, two boundary conditions are needed to determine all functions of integration. This is achieved by introducing a second brane, as in the RS 2-brane scenario. This brane is to be thought of either as a regulator brane, whose backreaction on the observable brane is neglected (which will only be true for a limited time), or as a shadow brane with physical fields, which have a gravitational effect on the observable brane.

The background is given by low-energy FRW branes with tensions ± λ, proper times t ±, scale factors a±, energy densities ρ± and pressures p±, and dark radiation densities ρℰ±. The physical distance between the branes is ℓ¯d(t), and

d ¯ d ¯ ¯( ) ¯ ----= ed---, a− = a+e −d, H− = ed H+ − d˙¯ , ρℰ− = e4dρℰ+. (399 ) dt− dt+
Then the background dynamics is given by
2 H2 = ± κ-(ρ± ± ρℰ±), (400 ) ± 3 ¨ ˙ ˙2 κ2 [ 2d¯ ] d¯+ 3H+ ¯d − ¯d = --- ρ+ − 3p+ + e (ρ− − 3p − ) . (401 ) 6
(see [39, 276] for the general background, including the high-energy regime). The dark energy obeys ρℰ+ = C∕a4+, where C is a constant. From now on, we drop the +-subscripts which refer to the physical, observed quantities.

The perturbed metric on the observable (positive tension) brane is described, in longitudinal gauge, by the metric perturbations ψ and ℛ, and the perturbed radion is ¯ d = d + N. The approximation for the KK (Weyl) energy-momentum tensor on the observable brane is

[ 2( ) ( )] ℰ μν = ---2--- − κ-- T μ+ e−2dTμ − ∇ μ∇ νd + δμ∇2d − ∇ μd∇ νd + 1-δμ(∇d )2 , (402 ) e2d − 1 2 ν − ν ν 2 ν
and the field equations on the observable brane can be written in scalar-tensor form as
κ2 κ2 (1 − χ )2 1 ( ) ω(χ) ( 1 ) G μν = ---Tμν + -----------Tμ− ν + -- ∇ μ∇ νχ − δμν∇2 χ + ----- ∇μ χ∇ νχ − --δμν(∇ χ)2 , (403 ) χ χ χ χ2 2
where
3 χ χ = 1 − e−2d, ω (χ ) = -------. (404 ) 2 1 − χ

The perturbation equations can then be derived as generalizations of the standard equations. For example, the 0 δG 0 equation is

¯ ( ) ¯ 2 ˙ 1k2- 1- 2--e2d-- −4¯d 2-2 --e2d--- H ψ − H ℛ − 3a2 ℛ = − 6 κ e2¯d − 1 δρ + e δρ− + 3κ e2d¯− 1 ρℰN 1 [( ) ( )2 1k2 ] − ---¯--- d˙¯− H N˙ + d˙¯− H N − ˙¯d2ψ + 2H d˙¯ψ − ˙¯dℛ˙− ----N . e2d − 1 3a2 (405 )
The trace part of the perturbed field equation shows that the radion perturbation determines the crucial quantity, δπ ℰ:
ℛ + ψ = − ---2---N = − κ2a2δ πℰ, (406 ) e2d¯− 1
where the last equality follows from Equation (325View Equation). The radion perturbation itself satisfies the wave equation
( ) ( ) 2 ¨ ˙¯ ˙ ˙ 2 ˙¯2 ˙¯ ¨¯ k-- N + 3H − 2d N − 2H + 4H + 2d − 6H d − 2d N + a2 N ˙¯˙ ˙¯˙ ( ¨¯ ˙¯ ˙¯2) − dψ + 3dℛ + − 2d − 6H d + 2d ψ κ2 [ ] = --- δρ − 3δp + e−2¯d(δρ− − 3δp − ) . (407 ) 6

A new set of variables φ ,E ± turns out be very useful [244Jump To The Next Citation Point, 245Jump To The Next Citation Point]:

a2 1 ℛ = − φ+ − -2H E˙+ -E, k2 3 ψ = − φ+ − a-( ¨E + 2H ˙E), k2 a2 ˙ N = φ − − φ+ − k2¯dE˙. (408 )
Equation (406View Equation) gives
( ˙ ) 2 2¯d 2( ) E¨ + 3H + --2d¯-- E˙− 1k--E = − -2e----k-- φ − e− 2¯dφ . (409 ) e2¯d − 1 3a2 e2d¯− 1 a2 + −
The variable E determines the metric shear in the bulk, whereas φ ± give the brane displacements in transverse traceless gauge. The latter variables have a simple relation to the curvature perturbations on large scales [244, 245Jump To The Next Citation Point] (restoring the +-subscripts):
( ) H2± ˙φ± ζtot± = − φ± + ˙--- ----+ φ± , (410 ) H± H ±
where ˙f± ≡ df±∕dt ±.
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