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 [134, 220
] or a hot big-bang
radiation era, as in the “ekpyrotic” or cyclic scenario [229
, 215
, 339
, 403
, 273
, 317
, 412
], or in colliding
bubble scenarios [40
, 156
, 157
]. (See also [26
, 98
, 299
] for colliding branes in an M theory approach.)
Here we discuss the simplest case of a 4D scalar field
with potential
(see [287] for a
review).
High-energy brane-world modifications to the dynamics of inflation on the brane have been
investigated [308, 216, 92, 405, 320, 319, 106, 285
, 34, 35, 36, 328, 192, 264, 363, 307]. 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 [308
, 99
, 312
, 369
, 346
, 286
, 205
].
The field satisfies the Klein–Gordon equation
In 4D general relativity, the condition for inflation,In the slow-roll approximation, we get
The brane-world correction term The number of e-folds during inflation, , is, in the slow-roll approximation,
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 [303, 271
, 177
, 148
]. 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 [148
]. 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 A crucial assumption is that backreaction due to metric perturbations in the bulk can be
neglected. In the extreme slow-roll limit this is necessarily correct because the coupling between
inflaton fluctuations and metric perturbations vanishes; however, this is not necessarily the
case when slow-roll corrections are included in the calculation. Previous work [250, 253, 255]
has shown that such bulk effects can be subtle and interesting (see also [109, 114] for other
approaches). In particular, subhorizon inflaton fluctuations on a brane excite an infinite ladder of
Kaluza–Klein modes of the bulk metric perturbations at first order in slow-roll parameters,
and a naive slow-roll expansion breaks down in the high-energy regime once one takes into
account the backreaction of the bulk metric perturbations, as confirmed by direct numerical
simulations [200]. However, an order-one correction to the behaviour of inflaton fluctuations on
subhorizon scales does not necessarily imply that the amplitude of the inflaton perturbations receives
corrections of order one on large scales; one must consistently quantise the coupled brane inflaton
fluctuations and bulk metric perturbations. This requires a detailed analysis of the coupled brane-bulk
system [70
, 252
].
It was shown that the coupling to bulk metric perturbations cannot be ignored in the equations of motion. Indeed, there are order-unity differences between the classical solutions without coupling and with slow-roll induced coupling. However, the change in the amplitude of quantum-generated perturbations is at next-to-leading order [252] because there is still no mixing at leading order between positive and negative frequencies when scales observable today crossed the horizon, so the Bogoliubov coefficients receive no corrections at leading order. The amplitude of perturbations generated is also subject to the usual slow-roll corrections on super-horizon scales. The next-order slow-roll corrections from bulk gravitational perturbations are calculated in [254] and they are the same order as the usual Stewart–Lyth correction [404]. These results also show that the ratio of tensor-to-scalar perturbation amplitudes are not influenced by brane-bulk interactions at leading order in slow-roll. It is remarkable that the predictions from inflation theories should be so robust that this result holds in spite of the leading-order change to the solutions of the classical equations of motion.
The scale-dependence of the perturbations is described by the spectral tilt
where the slow-roll parameters are given in Equations (209 As an example, consider the simplest chaotic inflation model . Equation (212
) gives the
integrated expansion from
to
as
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 (206
) becomes
. In the
slow-roll approximation (using Equations (207
) and (208
))
, and this yields
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 [284
]. 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 [284
], as shown in
Figure 6
.
Other perturbation modes have also been investigated:
The massive KK modes of tensor perturbations remain in the vacuum state during slow-roll
inflation [272, 176
]. 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.
Brane-world effects on large-scale isocurvature perturbations in 2-field inflation have also been considered [17]. Brane-world (p)reheating after inflation is discussed in [414, 429, 9, 415, 96].
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