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, , is , i.e., , where and . The modified Friedmann equation leads to a stronger condition for inflation: Using Equation (194), with , and Equation (204), 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 (207) serves to enhance the Hubble rate for a given potential energy, relative to general relativity. Thus there is enhanced Hubble ‘friction’ in Equation (208), 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 [99, 312, 369, 346, 286]. They also allow for the novel possibility that the inflaton could act as dark matter or quintessence at low energies [99, 312, 369, 346, 286, 8, 327, 288, 56, 380].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 (212) becomesThe 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 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) [425]. The curvature perturbation on uniform density hypersurfaces is given in terms of the scalar field fluctuations on spatially flat hypersurfaces by If one makes the assumption that backreaction due to metric perturbations in the bulk can be neglected, 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 [425]. 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 (215), this gives Thus the amplitude of scalar perturbations is increased relative to the standard result at a fixed value of for a given potential. UpdateA 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) and (210). 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 (212) 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 (206) becomes . In the slow-roll approximation (using Equations (207) and (208)) , 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 (218) and . The second term on the right of Equation (218) dominates, and we obtain Imposing the COBE normalization on the curvature perturbations given by Equation (216) requires Substituting in the value of given by Equation (220) 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 [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:

- 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 [272]:
Equation (224) 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 [205]:
This consistency relation persists when Z
_{2}symmetry is dropped [206] (and in a two-brane model with stabilized radion [172]). It holds only to lowest order in slow-roll, as in general relativity, but the reason for this [381] and the nature of the corrections [64] are not settled.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.

- 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 [52]. 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 [17]. Brane-world (p)reheating after inflation is discussed in [414, 429, 9, 415, 96].

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