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6.3 Density perturbations on large scales

In the covariant approach, we define matter density and expansion (velocity) perturbation scalars, as in 4D general relativity,
2 Δ = a-⃗∇2 ρ, Z = a2⃗∇2 Θ. (288 ) ρ
Then we can define dimensionless KK perturbation scalars [218Jump To The Next Citation Point],
2 a--⃗ 2 a-⃗ 2 ℰ 1-⃗2 ℰ U = ρ ∇ ρℰ, Q = ρ ∇ q , Π = ρ∇ π , (289 )
where the scalar potentials ℰ q and ℰ π are defined by Equations (251View Equation, 252View Equation). The KK energy density (dark radiation) produces a scalar fluctuation U which is present even if ρℰ = 0 in the background, and which leads to a non-adiabatic (or isocurvature) mode, even when the matter perturbations are assumed adiabatic [122Jump To The Next Citation Point]. We define the total effective dimensionless entropy Stot via
p S = a2 ⃗∇2p − c2 a2⃗∇2ρ , (290 ) tot tot tot tot tot
where c2tot = ˙ptot∕ρ˙tot is given in Equation (103View Equation). Then
9[c2 − 1 + (2 + w + c2) ρ∕λ] [4 ρ ] Stot = -----------------s---3----3--------s-------------------- ---ℰ-Δ − (1 + w )U . (291 ) [3(1 + w)(1 + ρ∕λ ) + 4ρ ℰ∕ρ][3w + 3(1 + 2w)ρ∕2 λ + ρℰ∕ρ] 3 ρ

If ρℰ = 0 in the background, then U is an isocurvature mode: Stot ∝ (1 + w)U. This isocurvature mode is suppressed during slow-roll inflation, when 1 + w ≈ 0.

If ρℰ ⁄= 0 in the background, then the weighted difference between U and Δ determines the isocurvature mode: S ∝ (4ρ ∕3 ρ)Δ − (1 + w )U tot ℰ. At very high energies, ρ ≫ λ, the entropy is suppressed by the factor λ∕ρ.

The density perturbation equations on the brane are derived by taking the spatial gradients of Equations (253View Equation), (255View Equation), and (257View Equation), and using Equations (254View Equation) and (256View Equation). This leads to [122Jump To The Next Citation Point]

˙Δ = 3wH Δ − (1 + w )Z, (292 ) ( 2 ) [ ( 2 ) ] ˙ --cs-- ⃗ 2 2 1- 2 ρ- --4cs- ρℰ- Z = − 2HZ − 1 + w ∇ Δ − κ ρU − 2 κ ρ 1 + (4 + 3w )λ − 1 + w ρ Δ, (293 ) ( 2 ) ( ) ( ) U˙= (3w − 1)HU + -4cs-- ρℰ- H Δ − 4-ρℰ Z − a⃗∇2Q, (294 ) 1 + w ρ 3ρ 1 2 1 [ ( 4c2 ) ρ ρ ] Q˙= (3w − 1)HQ − --U − --a⃗∇2 Π + --- ----s- -ℰ− 3(1 + w)-- Δ. (295 ) 3a 3 3μ 1 + w ρ λ
The KK anisotropic stress term Π occurs only via its Laplacian, ⃗∇2 Π. If we can neglect this term on large scales, then the system of density perturbation equations closes on super-Hubble scales [218]. An equivalent statement applies to the large-scale curvature perturbations [191Jump To The Next Citation Point]. KK effects then introduce two new isocurvature modes on large scales (associated with U and Q), and they modify the evolution of the adiabatic modes as well [122Jump To The Next Citation Point, 201Jump To The Next Citation Point].

Thus on large scales the system of brane equations is closed, and we can determine the density perturbations without solving for the bulk metric perturbations.

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Figure 8: The evolution of the covariant variable Φ, defined in Equation (298View Equation) (and not to be confused with the Bardeen potential), along a fundamental world-line. This is a mode that is well beyond the Hubble horizon at N = 0, about 50 e-folds before inflation ends, and remains super-Hubble through the radiation era. A smooth transition from inflation to radiation is modelled by w = 13 [(2 − 32ε)tanh (N − 50) − (1 − 32ε)], where ε is a small positive parameter (chosen as ε = 0.1 in the plot). Labels on the curves indicate the value of ρ0∕ λ, so that the general relativistic solution is the dashed curve (ρ ∕λ = 0 0). (Figure taken from [122].)

We can simplify the system as follows. The 3-Ricci tensor defined in Equation (134View Equation) leads to a scalar covariant curvature perturbation variable,

( ρ ) C ≡ a4⃗∇2R ⊥ = − 4a2HZ + 2κ2a2ρ 1 + --- Δ + 2κ2a2ρU. (296 ) 2λ
It follows that C is locally conserved (along uμ flow lines):
C = C0, C˙0 = 0. (297 )
We can further simplify the system of equations via the variable
2 2 Φ = κ a ρΔ. (298 )
This should not be confused with the Bardeen metric perturbation variable ΦH, although it is the covariant analogue of ΦH in the general relativity limit. In the brane-world, high-energy and KK effects mean that ΦH is a complicated generalization of this expression [201Jump To The Next Citation Point] involving Π, but the simple Φ above is still useful to simplify the system of equations. Using these new variables, we find the closed system for large-scale perturbations:
[ κ2ρ ( ρ) ] [ a2κ4 ρ2] [ κ2ρ] Φ˙= − H 1 + (1 + w) ---2- 1 + -- Φ − (1 + w)------- U + (1 + w )---- C0, (299 ) [ 2H ] λ [ 2H ] [ 4H ] 2κ2ρℰ 2 ρℰ ρ 6c2sH2 ρℰ U˙= − H 1 − 3w + ----2- U − --2---- 1 + --− --------2-- Φ + --2---- C0. (300 ) 3H 3a H ρ λ (1 + w)κ ρ 3a H ρ
If there is no dark radiation in the background, ρ ℰ = 0, then
( ∫ ) U = U exp − (1 − 3w )dN , (301 ) 0
and the above system reduces to a single equation for Φ. At low energies, and for constant w, the non-decaying attractor is the general relativity solution,
3(1 + w) Φlow ≈ 2(5-+-3w-)C0. (302 )
At very high energies, for w ≥ − 13, we get
[ ] 3-λ-- --C0--- --2&tidle;U0-- Φhigh → 2 ρ (1 + w) 7 + 6w − 5 + 6w , (303 ) 0
where &tidle;U0 = κ2a20ρ0U0, so that the isocurvature mode has an influence on Φ. Initially, Φ is suppressed by the factor λ∕ρ 0, but then it grows, eventually reaching the attractor value in Equation (302View Equation). For slow-roll inflation, when 1 + w ∼ ε, with 0 < ε ≪ 1 and −1 ′ H |˙ε| = |ε | ≪ 1, we get
3 λ Φhigh ∼ -ε---C0e3εN, (304 ) 2 ρ0
where N = ln (a ∕a0), so that Φ has a growing-mode in the early universe. This is different from general relativity, where Φ is constant during slow-roll inflation. Thus more amplification of Φ can be achieved than in general relativity, as discussed above. This is illustrated for a toy model of inflation-to-radiation in Figure 8View Image. The early (growing) and late time (constant) attractor solutions are seen explicitly in the plots.
View Image

Figure 9: The evolution of Φ in the radiation era, with dark radiation present in the background. (Figure taken from [131Jump To The Next Citation Point].)

The presence of dark radiation in the background introduces new features. In the radiation era (w = 13), the non-decaying low-energy attractor becomes [131]

Φlow ≈ C0-(1 − α ), (305 ) 3 α = ρℰ-≲ 0.05. (306 ) ρ
The dark radiation serves to reduce the final value of Φ, leaving an imprint on Φ, unlike the ρ = 0 ℰ case, Equation (302View Equation). In the very high energy limit,
[ ] ( )2 [ &tidle; ] Φ → -λ- 2-C − 4-&tidle;U + 16α λ-- C0--− 4U0- . (307 ) high ρ0 9 0 7 0 ρ0 273 539
Thus Φ is initially suppressed, then begins to grow, as in the no-dark-radiation case, eventually reaching an attractor which is less than the no-dark-radiation attractor. This is confirmed by the numerical integration shown in Figure 9View Image.
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