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 (259), (261), and (263), and using Equations (260) and (262). This leads to [177]

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 [303]. An equivalent statement applies to the large-scale curvature perturbations [271]. 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 [177, 282].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 (140) 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 [282] 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 (308). 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 [187]

The dark radiation serves to reduce the final value of , leaving an imprint on , unlike the case, Equation (308). 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.http://www.livingreviews.org/lrr-2010-5 |
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