go to next pagego upgo to previous page

6.2 Metric-based perturbations

An alternative approach to brane-world cosmological perturbations is an extension of the 4D metric-based gauge-invariant theory [170241]. A review of this approach is given in [40Jump To The Next Citation Point269Jump To The Next Citation Point]. In an arbitrary gauge, and for a flat FRW background, the perturbed metric has the form
|_ | _| - 2N 2y A2(@iB - Si) | N a { } | d (5)gAB = A2(@jB - Sj) A2 2Rdij + 2@i@jC + 2@(iFj) + fij | A2(@ib - xi) , (267) |_ ------------------------------------------------------|---------------- _| N a A2(@ b - x ) | 2n j j
where the background metric functions A, N are given by Equations (181View Equation, 182View Equation). The scalars y, R, C,a, b,n represent scalar perturbations. The vectors S i, F i, and x i are transverse, so that they represent 3D vector perturbations, and the tensor fij is transverse traceless, representing 3D tensor perturbations.

In the Gaussian normal gauge, the brane coordinate-position remains fixed under perturbation,

(5)ds2 = [g(0)(x,y) + dg (x,y)]dxmdxn + dy2, (268) mn mn
where g(0m)n is the background metric, Equation (180View Equation). In this gauge, we have
a = b = n = xi = 0. (269)

In the 5D longitudinal gauge, one gets

-B + C = 0 = - b + C'. (270)
In this gauge, and for an AdS5 background, the metric perturbation quantities can all be expressed in terms of a “master variable” _O_ which obeys a wave equation [244246]. In the case of scalar perturbations, we have for example
( ) 1 '' 1 /\5 R = 6A- _O_ - N-2 ¨_O_ - -3-_O_ , (271)
with similar expressions for the other quantities. All of the metric perturbation quantities are determined once a solution is found for the wave equation
( 1 ). (/\ k2) N ( N )' ----- _O_ + -5-+ --- --- _O_ = ---_O_' . (272) N A3 6 A2 A3 A3

The junction conditions (62View Equation) relate the off-brane derivatives of metric perturbations to the matter perturbations:

[ ] 2 1( (0)) 1-(0) @y dgmn = - k5 dTmn + 3 c - T dgmn- 3gmn dT , (273)
where
0 dT 0 = - dr, (274) dT 0i = a2qi, (275) i i i dT j = dp dj + dp j. (276)
For scalar perturbations in the Gaussian normal gauge, this gives
2 @yy(x, 0) = k-5(2dr + 3dp), (277) 6 @yB(x, 0) = k25dp, (278) 2 @yC(x,0) = - k5-dp, (279) 2 k25- i @yR(x, 0) = - 6 dr- @i@ C(x,0), (280)
where dp is the scalar potential for the matter anisotropic stress,
1- k dpij = @i@jdp - 3 dij @k@ dp. (281)
The perturbed KK energy-momentum tensor on the brane is given by
0 2 dE 0 = k drE, (282) dE0i = - k2a2qEi , (283) k2 dEij = - --drE dij- dpEij. (284) 3
The evolution of the bulk metric perturbations is determined by the perturbed 5D field equations in the vacuum bulk,
d (5)GA = 0. (285) B
Then the matter perturbations on the brane enter via the perturbed junction conditions (273View Equation).

For example, for scalar perturbations in Gaussian normal gauge, we have

{ ( ) [ ( ) ]} (5) y ' A' N ' ' A2 ' A N ' d G i = @i - y + A--- N-- y - 2R - 2N-2- B + 5A-- N-- B . (286)
For tensor perturbations (in any gauge), the only nonzero components of the perturbed Einstein tensor are
{ 2 ( ) ( ' ') } d (5)Gij = - 1- - -1-f¨ij + f''ij- k-f ij + -1- N--- 3A- f ij + N--+ 3 A-- f 'ij . (287) 2 N 2 A2 N 2 N A N A

In the following, I will discuss various perturbation problems, using either a 1 + 3-covariant or a metric-based approach.



go to next pagego upgo to previous page