

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
and
are defined by Equations (251, 252). The KK energy
density (dark radiation) produces a scalar fluctuation
which is present even if
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
via
where
is given in
Equation (103). Then
If
in the background, then
is an isocurvature mode:
.
This isocurvature mode is suppressed during slow-roll inflation,
when
.
If
in the background, then the
weighted difference between
and
determines the isocurvature mode:
. At very high energies,
, the entropy is suppressed by the factor
.
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
]
The KK anisotropic stress term
occurs only via its
Laplacian,
. 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
and
), 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.
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
is locally conserved (along
flow lines):
We can further simplify the system of equations via the variable
This should not be confused with the Bardeen metric perturbation
variable
, although it is the covariant analogue
of
in the general relativity limit. In the brane-world,
high-energy and KK effects mean that
is a complicated
generalization of this expression [201
] involving
, but the simple
above is still useful to
simplify the system of equations. Using these new variables, we
find the closed system for large-scale perturbations:
If there is no dark radiation in the background,
, then
and the above system reduces to a single equation for
. At low energies, and for constant
, the non-decaying attractor is the general relativity
solution,
At very high energies, for
, we get
where
, so that the isocurvature mode has an
influence on
. Initially,
is suppressed by the
factor
, but then it grows, eventually reaching
the attractor value in Equation (302). For slow-roll
inflation, when
, with
and
,
we get
where
, so that
has a growing-mode
in the early universe. This is different from general relativity,
where
is constant during slow-roll inflation. Thus more
amplification of
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.
The presence of dark radiation in the background
introduces new features. In the radiation era (
), the non-decaying low-energy attractor
becomes [131]
The dark radiation serves to reduce the final value of
, leaving an imprint on
, unlike the
case, Equation (302). In the very high
energy limit,
Thus
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.

