Density perturbations on large scales

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,

Then we can define dimensionless KK perturbation scalars [218

where the scalar potentials $ℰ q$ and $ℰ π$ are defined by Equations (251

, 252

). 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 [122

]. We define the total effective dimensionless entropy $Stot$ via

where $c2tot = ˙ptot∕ρ˙tot$ is given in Equation (103

). Then

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 (253 ), (255 ), and (257 ), and using Equations (254 ) and (256 ). This leads to [122 ]

˙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 [191

]. 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 [122

, 201

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.

Figure 8:

The evolution of the covariant variable $Φ$ , defined in Equation (298

) (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 (134 ) leads to a scalar covariant curvature perturbation variable,

It follows that $C$ is locally conserved (along $uμ$ flow lines):

We can further simplify the system of equations via the variable

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 [201

] 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

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,

At very high energies, for $w ≥ − 13$ , we get

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 (302

). For slow-roll inflation, when $1 + w ∼ ε$ , with $0 < ε ≪ 1$ and $−1 ′ H |˙ε| = |ε | ≪ 1$ , we get

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 8

. The early (growing) and late time (constant) attractor solutions are seen explicitly in the plots.

Figure 9:

The evolution of $Φ$ in the radiation era, with dark radiation present in the background. (Figure taken from [131

].)

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 (302

). In the very high energy limit,

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 9