### 6.5 Full numerical solutions

In order to study scalar perturbations fully, we need to numerically solve the coupled bulk and brane equations for the master variable . A thorough analysis was done in [69]. For this purpose, it is convenient to use the static coordinate where the bulk equation is simple and consider a moving FRW brane;
The bulk master variable satisfies the following wave equation (see Equation (278))
From the junction condition, satisfies a boundary condition on the brane
where is the derivative orthogonal to the brane
and is the density perturbation in the comoving gauge. The subscript b implies that the quantities are evaluated on the brane. On a brane, satisfies a wave equation
 (333a) (333b)

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

On a brane, we take the longitudinal gauge

Using the expressions for metric perturbations in terms of the master variable (see Equation (277)) and the junction condition Equation (331), and are written in terms of and as

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

where the velocity perturbation is also written by and . There are two independent numerical codes that can be used to solve for and . The first is the pseudo-spectral (PS) method used in [200] and the second is the characteristic integration (CI) algorithm developed in [70].

Figure 10 shows the output of the PS and CI codes for a typical simulation of a mode with 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 and are phase-locked plane waves,

at sufficiently late times , which is actually the same behaviour as seen in GR. Figure 11 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 , and 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.

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

These give a closed set of ODEs on the brane that describe all of the 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 for . We can define a set of “enhancement factors”, which are functions of that describe the relative amplitudes of after horizon crossing in the various theories. Let the final amplitudes of the density perturbation with wavenumber be , and for the 5-dimensional, effective and GR theories, respectively, given that the normalization holds. Then, we define enhancement factors as

It follows that represents the enhancement to the density perturbation, gives the magnification due to KK modes, while 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 . Since 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 , which implies that the amplitude magnification due to the corrections is always larger than that due to the KK modes. Interestingly, the -factors appear to approach asymptotically constant values for large :
where is the comoving wavenumber of the mode that enters the horizon when .

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 that will tell us how the initial spectrum of curvature perturbations maps onto the spectrum of density perturbations at some low energy epoch within the radiation era. It is customary to normalize transfer functions such that , which leads us to the following definition

Here, is the primordial value of the curvature perturbation and is the maximum amplitude of the density perturbation in the epoch of interest. As demonstrated in Figure 11, we know that we recover the GR result in the extreme small scale limit , which gives the transfer function the correct normalization. In the righthand panel of Figure 12, we show the transfer functions derived from GR, the effective theory and the 5-dimensional simulations. As expected, the for each formulation match one another on subcritical scales . However, on supercritical scales we have .

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

Now, the spectrum of density fluctuations at any point in the radiation era is given by
Using Equation (342), we see that the RS matter power spectrum (evaluated in the low-energy regime) is times bigger than the GR prediction on scales given by .

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]).