6.5 Full numerical solutions

UpdateJump To The Next Update Information In order to study scalar perturbations fully, we need to numerically solve the coupled bulk and brane equations for the master variable Ω. A thorough analysis was done in [69Jump To The Next Citation Point]. For this purpose, it is convenient to use the static coordinate where the bulk equation is simple and consider a moving FRW brane;
(5) 2 ℓ2- μ ν 2 ds = z2 [ημνdx dx + dz ]. (329 )
The bulk master variable satisfies the following wave equation (see Equation (278View Equation))
2 2 ( ) 0 = − ∂-Ω-+ ∂-Ω-+ 3∂-Ω + -1-− k2 Ω. (330 ) ∂τ2 ∂z2 z ∂z z2
From the junction condition, Ω satisfies a boundary condition on the brane
[ ( ) 3 ] ∂nΩ + 1- 1 + ρ- Ω + 6ρa--Δ = 0. (331 ) ℓ λ λk2 b
where ∂n is the derivative orthogonal to the brane
( ) 1- -∂- √ ------2-2-∂- ∂n = a − H ℓ∂τ + 1 + H ℓ ∂z , (332 )
and Δ is the density perturbation in the comoving gauge. The subscript b implies that the quantities are evaluated on the brane. On a brane, Δ satisfies a wave equation
[ ] d2Δ 2 dΔ 2 2 3ρa2 3ρ2a2 k4(1 + w )Ωb ---2 + (1 + 3cs − 6w )Ha ----+ csk + ---2-A + --2-2-B Δ = -------3-----, dη dη λℓ λ ℓ 3ℓa A = 6c2s − 1 − 8w + 3w2, B = 3c2s − 9w − 4,


where we consider a perfect fluid on a brane with an equation state w and cs is a sound speed for perturbations. The above ordinal differential equation, the bulk wave equation (330View Equation) and the boundary condition (331View Equation) comprise a closed set of equations for Δ and Ωc.

On a brane, we take the longitudinal gauge

ds2 = − (1 + 2ψ)dt2 + (1 + 2ℛ )δ dxidxj. (334 ) b ij
Using the expressions for metric perturbations in terms of the master variable Ω (see Equation (277View Equation)) and the junction condition Equation (331View Equation), ψ and ℛ are written in terms of Δ and Ω as
2 ( 2 2 2) ℛ = 3a-ρ(ρ-+-λ-)Δ + 3H--a--+-k-- Ωb − -H---dΩb-, (335a ) k2ℓ2λ2 6ℓa3 2ℓa2 d η 3ρa2(3w ρ + 4ρ + λ) [(3w + 4)ρ2 (5 + 3w )ρ k2 ] ψ = − --------22--2------Δ − ----3---2-- + ----3-----+ ---3- Ωb k ℓλ 2 ℓa λ 2ℓ aλ 3ℓa 3H--d-Ωb --1--d2Ωb- + 2ℓa2 dη − 2 ℓa3 dη2 . (335b )

Other quantities of interest are the curvature perturbation on uniform density slices,

[ ] HaV Δ 1 3 ρa2(wλ − λ − ρ ) Δ Ha dΔ k2 ζ = ℛ − -----+ ---------= --− -------2-2-2------ ------+ --2-----------+ ----3Ωb, (336 ) k 3(1 + w) 3 k ℓ λ 1 + w k (1 + w ) dη 6 ℓa
where the velocity perturbation V is also written by Δ and Ωb. There are two independent numerical codes that can be used to solve for Δ and Ωb. The first is the pseudo-spectral (PS) method used in [200Jump To The Next Citation Point] and the second is the characteristic integration (CI) algorithm developed in [70Jump To The Next Citation Point].

Figure 10View Image shows the output of the PS and CI codes for a typical simulation of a mode with ρ∕λ = 50 at the horizon re-enter. As expected we have excellent agreement between the two codes, despite the fact that they use different initial conditions. Note that for all simulations, we recover that Δ and ζ are phase-locked plane waves,

kη Δ (η) ∝ cos√---, Δ (η) ≈ 4ζ(η), (337 ) 3
at sufficiently late times kη ≫ 1, which is actually the same behaviour as seen in GR. Figure 11View Image illustrates how the ordinary superhorizon behaviour of perturbations in GR is recovered for modes entering the Hubble horizon in the low energy era. We see how Δ, ψ and ℛ smoothly interpolate between the non-standard high-energy behaviour to the usual expectations in GR. Also shown in this plot is the behaviour of the KK anisotropic stress, which steadily decays throughout the simulation. These results confirm that at low energies, we recover GR solutions smoothly.
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Figure 10: Comparison between typical results of the PS and CI codes for various brane quantities (left); and the typical behaviour of the bulk master variable (right) as calculated by the CI method. Very good agreement between the two different numerical schemes is seen in the left panel, despite the fact that they use different initial conditions. Also note that on subhorizon scales, Δ and ζ undergo simple harmonic oscillations, which is consistent with the behaviour in GR. The bulk profile demonstrates our choice of initial conditions: We see that the bulk master variable Ω is essentially zero during the early stages of the simulation, and only becomes “large” when the mode crosses the horizon. Figure taken from [69Jump To The Next Citation Point].
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Figure 11: The simulated behaviour of a mode on superhorizon scales. On the left we show how the Δ gauge invariant switches from the high-energy behaviour predicted to the familiar GR result as the universe expands through the critical epoch. We also show how the KK anisotropic stress 2 κ4δπℰ steadily decays throughout the simulation, which is typical of all the cases we have investigated. On the right, we show the metric perturbations ψ (= Ψ ) and ℛ (= Φ) as well as the curvature perturbation ζ. Again, note how the GR result Φ ≈ − Ψ ≈ − 2ζ∕3 is recovered at low energy. Figure taken from [69Jump To The Next Citation Point].

At high energies ρ > λ, there are two separate effects to consider: First, there is the modification of the universe’s expansion at high energies and the 𝒪 (ρ∕λ) corrections to the perturbative equations of motion. Second, there is the effect of the bulk degrees of freedom encapsulated by the bulk master variable Ω (or, equivalently, the KK fluid ℰ μν). To separate out the two effects, it is useful to introduce the 4-dimensional effective theory where all 𝒪(ρ∕ λ) corrections to GR are retained, but the bulk effects are removed by artificially setting Ω = 0. In the case of radiation domination, we obtain equations for the effective theory density contrast Δref and curvature perturbation ζref from Equations (336View Equation) and (333) with Ωb = 0:

2 ( 2 2 2 2) d-Δref k-- 4-ρa- 18ρ-a-- 0 = dη2 + 3 − λℓ2 − λ2ℓ2 Δref, (338a ) ( 2 2 2 ) ζref = 1-+ -3ρa---+ -9ρ-a--- Δref + 3Ha-dΔref. (338b ) 4 2λk2ℓ2 4λ2k2 ℓ2 4k dη

These give a closed set of ODEs on the brane that describe all of the 𝒪 (ρ∕λ ) corrections to GR.

Since in any given model we expect the primordial value of the curvature perturbation to be fixed by inflation, it makes physical sense to normalize the waveforms from each theory such that ζ5D ≈ ζref ≈ ζGR ≈ 1 for a ≪ a ∗. We can define a set of “enhancement factors”, which are functions of k that describe the relative amplitudes of Δ after horizon crossing in the various theories. Let the final amplitudes of the density perturbation with wavenumber k be 𝒞5D(k), 𝒞ref(k) and 𝒞GR(k) for the 5-dimensional, effective and GR theories, respectively, given that the normalization ζ5D ≈ ζref ≈ ζGR ≈ 1 holds. Then, we define enhancement factors as

𝒬ref(k) = 𝒞ref(k)-, 𝒬 ℰ(k) = 𝒞5D(k), 𝒬5D(k) = 𝒞5D(k). (339 ) 𝒞GR(k ) 𝒞ref(k) 𝒞GR(k)
It follows that 𝒬 (k) ref represents the 𝒪(ρ ∕λ) enhancement to the density perturbation, 𝒬 (k) ℰ gives the magnification due to KK modes, while 𝒬5D (k) gives the total 5-dimensional amplification over the GR case. They all increase as the scale is decreased, and that they all approach unity for k → 0. Since 𝒬 = 1 implies no enhancement of the density perturbations over the standard result, this means we recover general relativity on large scales. For all wavenumbers we see 𝒬ref > 𝒬 ℰ > 1, which implies that the amplitude magnification due to the 𝒪 (ρ∕λ ) corrections is always larger than that due to the KK modes. Interestingly, the 𝒬-factors appear to approach asymptotically constant values for large k:
𝒬ref(k ) ≈ 3.0, 𝒬ℰ(k ) ≈ 2.4, 𝒬5D(k) ≈ 7.1, k ≫ kc, (340 )
where kc is the comoving wavenumber of the mode that enters the horizon when −1 H = ℓ.

In cosmological perturbation theory, transfer functions are very important quantities. They allow one to transform the primordial spectrum of some quantity set during inflation into the spectrum of another quantity at a later time. In this sense, they are essentially the Fourier transform of the retarded Green’s function for cosmological perturbations. There are many different transfer functions one can define, but for our case it is useful to consider a function T (k) that will tell us how the initial spectrum of curvature perturbations 𝒫iζnf maps onto the spectrum of density perturbations 𝒫 Δ at some low energy epoch within the radiation era. It is customary to normalize transfer functions such that T (k;η) → 1(k → 0 ), which leads us to the following definition

[ ]−2 T (k;η) = 9- ----k----- Δk-(η). (341 ) 4 H (η)a(η) ζiknf
Here, inf ζk is the primordial value of the curvature perturbation and Δk (η) is the maximum amplitude of the density perturbation in the epoch of interest. As demonstrated in Figure 11View Image, we know that we recover the GR result in the extreme small scale limit (k → 0), which gives the transfer function the correct normalization. In the righthand panel of Figure 12View Image, we show the transfer functions derived from GR, the effective theory and the 5-dimensional simulations. As expected, the T(k; η) for each formulation match one another on subcritical scales k < kc. However, on supercritical scales we have T5D > Tref > TGR.
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Figure 12: Density perturbation enhancement factors (left) and transfer functions (right) from simulations, effective theory, and general relativity. All of the 𝒬 factors monotonically increase with k∕kc, and we see that the Δ amplitude enhancement due to 𝒪 (ρ∕λ ) effects 𝒬ref is generally larger than the enhancement due to KK effects 𝒬 ℰ. For asymptotically small scales k ≫ k c, the enhancement seems to level off. The transfer functions in the right panel are evaluated at a given subcritical epoch in the radiation dominated era. The T functions show how, for a fixed primordial spectrum of curvature perturbations 𝒫inζf, the effective theory predicts excess power in the Δ spectrum 𝒫 Δ ∝ T 2𝒫iζnf on supercritical/subhorizon scales compared to the GR result. The excess small-scale power is even greater when KK modes are taken into account, as shown by T (k;η) 5D. Figure taken from [69].

Note that if we are interested in the transfer function at some arbitrary epoch in the low-energy radiation regime Ha ≫ kc, it is approximately given in terms of the enhancement factor as follows:

{ 1, k < 3Ha, T5D(k;η) ≈ 2 (342 ) (3Ha ∕k ) 𝒬5D(k), k > 3Ha,
Now, the spectrum of density fluctuations at any point in the radiation era is given by
( ) 16 2 k 4 inf 𝒫 Δ(k;η ) = --T (k;η) ---- 𝒫ζ (k). (343 ) 81 Ha
Using Equation (342View Equation), we see that the RS matter power spectrum (evaluated in the low-energy regime) is ∼ 50 times bigger than the GR prediction on scales given by k ∼ 103kc.

The amplitude enhancement of perturbations is important on comoving scales ≲ 10 AU, which are far too small to be relevant to present-day/cosmic microwave background measurements of the matter power spectrum. However, it may have an important bearing on the formation of compact objects such as primordial black holes and boson stars at very high energies, i.e., the greater gravitational force of attraction in the early universe will create more of these objects than in GR (different aspects of primordial black holes in RS cosmology in the context of various effective theories have been considered in [185, 184, 91, 384, 383, 382]).

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