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 dS4 in AdS5 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:
It is more useful to reformulate Equation (363) as a Schrödinger-type equation,4 brane [225, 396].)
The non-zero value of the Hubble parameter implies the existence of a mass gap ,4 brane(s) with bulk scalar field, a universal lower bound on the mass gap of the KK tower is   .
This 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 . 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[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 :  confirms the effect of massive mode generation and consequent damping of the zero-mode.
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