The wave equation (355) cannot be solved analytically except if the background metric functions are
separable, and this only happens for maximally symmetric branes, i.e., branes with constant Hubble rate
. This includes the RS case already treated in Section 2. The cosmologically relevant case is
the de Sitter brane, . We can calculate the spectrum of gravitational waves generated during brane
inflation [272, 176, 148, 232], if we approximate slow-roll inflation by a succession of de Sitter
phases. The metric for a de Sitter brane dS_{4} in AdS_{5} is given by Equations (186, 187, 188) with

The linearized wave equation (355) is separable. As before, we separate the amplitude as where is the 4D mass, and this leads to:

The general solutions for are where is a linear combination of Bessel functions, is a linear combination of associated Legendre functions, andIt is more useful to reformulate Equation (363) as a Schrödinger-type equation,

using the conformal coordinate and defining . The potential is given by (see Figure 13) where the last term incorporates the boundary condition at the brane. The “volcano” shape of the potential shows how the 5D graviton is localized at the brane at low energies. (Note that localization fails for an dSThe non-zero value of the Hubble parameter implies the existence of a mass gap [154],

between the zero mode and the continuum of massive KK modes. This result has been generalized: For dSThis enhanced zero mode produced by brane inflation remains frozen outside the Hubble radius, as in general relativity, but when it re-enters the Hubble radius during radiation or matter domination, it will no longer be separated from the massive modes, since will not be constant. Instead, massive modes will be excited during re-entry. In other words, energy will be lost from the zero mode as 5D gravitons are emitted into the bulk, i.e., as massive modes are produced on the brane. A phenomenological model of the damping of the zero mode due to 5D graviton emission is given in [281]. Self-consistent low-energy approximations to compute this effect are developed in [201, 136]. Update

At zero order, the low-energy approximation is based on the following [321, 323, 27]. In the radiation era, at low energy, the background metric functions obey

To lowest order, the wave equation therefore separates, and the mode functions can be found analytically [321, 323, 27]. The massive modes in the bulk, , are the same as for a Minkowski brane. On large scales, or at late times, the mode functions on the brane are given in conformal time by where marks the start of the low-energy regime (), and denotes a linear combination of Bessel functions. The massive modes decay on super-Hubble scales, unlike the zero-mode. Expanding the wave equation in , one arrives at the first order, where mode-mixing arises. The massive modes on sub-Hubble scales are sourced by the initial zero mode that is re-entering the Hubble radius [136]: where is a transfer matrix coefficient. The numerical integration of the equations [201] confirms the effect of massive mode generation and consequent damping of the zero-mode.http://www.livingreviews.org/lrr-2010-5 |
This work is licensed under a Creative Commons License. Problems/comments to |