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 variableOther quantities of interest are the curvature perturbation on uniform density slices,
where the velocity perturbation 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 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
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
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:
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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