

5.1 Brane-world
inflation
In 1-brane RS-type brane-worlds, where the bulk has only a vacuum
energy, inflation on the brane must be driven by a 4D scalar field
trapped on the brane. In more general brane-worlds, where the bulk
contains a 5D scalar field, it is possible that the 5D field
induces inflation on the brane via its effective
projection [165
, 138
, 100
, 276
, 141
, 140
, 304
, 318
, 180
, 195
, 139
, 33
, 238
, 158
, 229
, 103
, 12
].
More exotic possibilities arise from the
interaction between two branes, including possible collision, which
is mediated by a 5D scalar field and which can induce either
inflation [90
, 157
] or a hot big-bang
radiation era, as in the “ekpyrotic” or cyclic scenario [163
, 154
, 251
, 301
, 193
, 230
, 307
], or in colliding
bubble scenarios [29
, 106
, 107
]. (See
also [21
, 69
, 214
] for colliding
branes in an M theory approach.) Here we discuss the simplest
case of a 4D scalar field
with potential
(see [207] for a review).
High-energy brane-world modifications to the
dynamics of inflation on the brane have been
investigated [222
, 156, 63, 302, 234, 233, 74, 204, 23, 24, 25, 240, 135, 184, 270, 221]. Essentially, the
high-energy corrections provide increased Hubble damping, since
implies that
is larger for a given energy
than in 4D general relativity. This makes slow-roll inflation
possible even for potentials that would be too steep in standard
cosmology [222
, 70
, 226
, 277
, 258
, 206
, 145
].
The field satisfies the Klein-Gordon equation
In 4D general relativity, the condition for inflation,
, is
, i.e.,
, where
and
. The modified Friedmann equation leads
to a stronger condition for inflation: Using Equation (188), with
, and Equation (198), we find that
where the square brackets enclose the brane correction to the
general relativity result. As
, the 4D
result
is recovered, but for
,
must be more negative for inflation. In
the very high-energy limit
, we have
. When the only matter in the universe
is a self-interacting scalar field, the condition for inflation
becomes
which reduces to
when
.
In the slow-roll approximation, we get
The brane-world correction term
in
Equation (201) serves to enhance the
Hubble rate for a given potential energy, relative to general
relativity. Thus there is enhanced Hubble ‘friction’ in
Equation (202), and brane-world
effects will reinforce slow-roll at the same potential energy. We
can see this by defining slow-roll parameters that reduce to the
standard parameters in the low-energy limit:
Self-consistency of the slow-roll approximation then requires
. At low energies,
, the slow-roll
parameters reduce to the standard form. However at high energies,
, the extra contribution to the Hubble expansion
helps damp the rolling of the scalar field, and the new factors in
square brackets become
:
where
are the standard general relativity
slow-roll parameters. In particular, this means that steep
potentials which do not give inflation in general relativity, can
inflate the brane-world at high energy and then naturally stop
inflating when
drops below
. These models can be
constrained because they typically end inflation in a
kinetic-dominated regime and thus generate a blue spectrum of
gravitational waves, which can disturb nucleosynthesis [70
, 226
, 277
, 258
, 206
]. They also allow
for the novel possibility that the inflaton could act as dark
matter or quintessence at low energies [70
, 226
, 277
, 258
, 206
, 4, 239, 208, 43, 285].
The number of e-folds during inflation,
, is, in the slow-roll
approximation,
Brane-world effects at high energies increase the Hubble rate by a
factor
, yielding more inflation between any
two values of
for a given potential. Thus we can
obtain a given number of e-folds for a smaller initial inflaton
value
. For
, Equation (206) becomes
The key test of any modified gravity theory
during inflation will be the spectrum of perturbations produced due
to quantum fluctuations of the fields about their homogeneous
background values. We will discuss brane-world cosmological
perturbations in the next Section 6. In general, perturbations on the
brane are coupled to bulk metric perturbations, and the problem is
very complicated. However, on large scales on the brane, the
density perturbations decouple from the bulk metric
perturbations [218
, 191
, 122
, 102
]. For 1-brane
RS-type models, there is no scalar zero-mode of the bulk graviton,
and in the extreme slow-roll (de Sitter) limit, the massive scalar
modes are heavy and stay in their vacuum state during
inflation [102
]. Thus it seems a
reasonable approximation in slow-roll to neglect the KK effects
carried by
when computing the density
perturbations.
To quantify the amplitude of scalar (density)
perturbations we evaluate the usual gauge-invariant quantity
which reduces to the curvature perturbation
on uniform density hypersurfaces (
). This is conserved on large scales for purely
adiabatic perturbations as a consequence of energy conservation
(independently of the field equations) [317
]. The curvature
perturbation on uniform density hypersurfaces is given in terms of
the scalar field fluctuations on spatially flat hypersurfaces
by
The field fluctuations at Hubble crossing (
) in the slow-roll limit are given by
, a result for a massless field in de
Sitter space that is also independent of the gravity
theory [317]. For a single scalar
field the perturbations are adiabatic and hence the curvature
perturbation
can be related to the density
perturbations when modes re-enter the Hubble scale during the
matter dominated era which is given by
. Using the slow-roll equations and Equation (209), this gives
Thus the amplitude of scalar perturbations is increased relative to the standard result at
a fixed value of
for a given potential.
The scale-dependence of the perturbations is
described by the spectral tilt
where the slow-roll parameters are given in Equations (203) and (204). Because these
slow-roll parameters are both suppressed by an extra factor
at high energies, we see that the spectral index is
driven towards the Harrison-Zel’dovich spectrum,
, as
; however, as explained
below, this does not necessarily mean that the brane-world case is
closer to scale-invariance than the general relativity case.
As an example, consider the simplest chaotic
inflation model
. Equation (206) gives the integrated
expansion from
to
as
The new high-energy term on the right leads to more inflation for a
given initial inflaton value
.
The standard chaotic inflation scenario requires
an inflaton mass
to match the observed level
of anisotropies in the cosmic microwave background (see below).
This corresponds to an energy scale
when the
relevant scales left the Hubble scale during inflation, and also to
an inflaton field value of order
. Chaotic inflation
has been criticised for requiring super-Planckian field values,
since these can lead to nonlinear quantum corrections in the
potential.
If the brane tension
is much below
, corresponding to
,
then the terms quadratic in the energy density dominate the
modified Friedmann equation. In particular the condition for the
end of inflation given in Equation (200) becomes
. In the slow-roll approximation (using
Equations (201) and (202))
, and this yields
In order to estimate the value of
when scales
corresponding to large-angle anisotropies on the microwave
background sky left the Hubble scale during inflation, we take
in Equation (212) and
. The second term on the right of Equation (212) dominates, and we
obtain
Imposing the COBE normalization on the curvature perturbations
given by Equation (210) requires
Substituting in the value of
given by Equation (214) shows that in the
limit of strong brane corrections, observations require
Thus for
, chaotic inflation can
occur for field values below the 4D Planck scale,
, although still above the 5D scale
. The relation determined by COBE constraints for
arbitrary brane tension is shown in Figure 5, together with the
high-energy approximation used above, which provides an excellent
fit at low brane tension relative to
.
It must be emphasized that in comparing the
high-energy brane-world case to the standard 4D case, we implicitly
require the same potential energy. However, precisely because of
the high-energy effects, large-scale perturbations will be
generated at different values of
than in the standard case,
specifically at lower values of
, closer to the reheating
minimum. Thus there are two competing effects, and it turns out
that the shape of the potential determines which is the dominant
effect [203
]. For the quadratic
potential, the lower location on
dominates, and the spectral
tilt is slightly further from scale invariance than in the standard
case. The same holds for the quartic potential. Data from WMAP and
2dF can be used to constrain inflationary models via their
deviation from scale invariance, and the high-energy brane-world
versions of the quadratic and quartic potentials are thus under
more pressure from data than their standard counterparts [203
], as shown in
Figure 6.
Other perturbation modes have also been investigated:
- High-energy inflation on the brane also
generates a zero-mode (4D graviton mode) of tensor perturbations,
and stretches it to super-Hubble scales, as will be discussed
below. This zero-mode has the same qualitative features as in
general relativity, remaining frozen at constant amplitude while
beyond the Hubble horizon. Its amplitude is enhanced at high
energies, although the enhancement is much less than for scalar
perturbations [192
]:
Equation (218) means that
brane-world effects suppress the large-scale tensor contribution to
CMB anisotropies. The tensor spectral index at high energy has a
smaller magnitude than in general relativity,
but remarkably the same consistency relation as in general
relativity holds [145]:
This consistency relation persists when
symmetry is
dropped [146] (and in a two-brane
model with stabilized radion [118]). It holds only to
lowest order in slow-roll, as in general relativity, but the reason
for this [286] and the nature of
the corrections [44] are not settled.
The massive KK modes of tensor perturbations
remain in the vacuum state during slow-roll inflation [192
, 121
]. The evolution of
the super-Hubble zero mode is the same as in general relativity, so
that high-energy brane-world effects in the early universe serve
only to rescale the amplitude. However, when the zero mode
re-enters the Hubble horizon, massive KK modes can be excited.
- Vector perturbations in the bulk metric can
support vector metric perturbations on the brane, even in the
absence of matter perturbations (see the next Section 6). However, there is no
normalizable zero mode, and the massive KK modes stay in the vacuum
state during brane-world inflation [39]. Therefore, as in
general relativity, we can neglect vector perturbations in
inflationary cosmology.
Brane-world effects on large-scale isocurvature
perturbations in 2-field inflation have also been
considered [13]. Brane-world (p)reheating after
inflation is discussed in [309, 321, 5, 310, 67].

