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 bigbang radiation era, as in the “ekpyrotic” or cyclic scenario [154, 163, 193, 231, 251, 301, 307], or in colliding bubble scenarios [29, 107, 108]. (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).
Highenergy braneworld modifications to the dynamics of inflation on the brane have been investigated [23, 24, 25, 63, 74, 135, 156, 184, 204, 221, 222, 233, 234, 240, 270, 302]. Essentially, the highenergy corrections provide increased Hubble damping, since implies that is larger for a given energy than in 4D general relativity. This makes slowroll inflation possible even for potentials that would be too steep in standard cosmology [70, 145, 206, 222, 226, 258, 277].
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 highenergy limit , we have . When the only matter in the universe is a selfinteracting scalar field, the condition for inflation becomes which reduces to when .In the slowroll approximation, we get
The braneworld 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 braneworld effects will reinforce slowroll at the same potential energy. We can see this by defining slowroll parameters that reduce to the standard parameters in the lowenergy limit: Selfconsistency of the slowroll approximation then requires . At low energies, , the slowroll 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 slowroll parameters. In particular, this means that steep potentials which do not give inflation in general relativity, can inflate the braneworld at high energy and then naturally stop inflating when drops below . These models can be constrained because they typically end inflation in a kineticdominated regime and thus generate a blue spectrum of gravitational waves, which can disturb nucleosynthesis [70, 206, 226, 258, 277]. They also allow for the novel possibility that the inflaton could act as dark matter or quintessence at low energies [4, 43, 70, 206, 208, 226, 239, 258, 277, 285].The number of efolds during inflation, , is, in the slowroll approximation,
Braneworld 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 efolds for a smaller initial inflaton value . For , Equation (206) 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 braneworld 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 [102, 122, 191, 218]. For 1brane RStype models, there is no scalar zeromode of the bulk graviton, and in the extreme slowroll (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 slowroll to neglect the KK effects carried by when computing the density perturbations.

To quantify the amplitude of scalar (density) perturbations we evaluate the usual gaugeinvariant 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 slowroll 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 reenter the Hubble scale during the matter dominated era which is given by . Using the slowroll 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 scaledependence of the perturbations is described by the spectral tilt
where the slowroll parameters are given in Equations (203) and (204). Because these slowroll 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 braneworld case is closer to scaleinvariance 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 highenergy 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 superPlanckian 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 slowroll approximation (using Equations (201) and (202)) , and this yields
In order to estimate the value of when scales corresponding to largeangle 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 highenergy approximation used above, which provides an excellent fit at low brane tension relative to .It must be emphasized that in comparing the highenergy braneworld case to the standard 4D case, we implicitly require the same potential energy. However, precisely because of the highenergy effects, largescale 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 highenergy braneworld 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:
The massive KK modes of tensor perturbations remain in the vacuum state during slowroll inflation [121, 192]. The evolution of the superHubble zero mode is the same as in general relativity, so that highenergy braneworld effects in the early universe serve only to rescale the amplitude. However, when the zero mode reenters the Hubble horizon, massive KK modes can be excited.
Braneworld effects on largescale isocurvature perturbations in 2field inflation have also been considered [12]. Braneworld (p)reheating after inflation is discussed in [5, 67, 309, 310, 321].
http://www.livingreviews.org/lrr20047 
© Max Planck Society
Problems/comments to 