The tensor perturbations are given by Equation (267), i.e., (for a flat background brane),
The transverse traceless satisfies Equation (287), which implies, on splitting into Fourier modes with amplitude , By the transverse traceless part of Equation (273), the boundary condition is where is the tensor part of the anisotropic stress of matterradiation on the brane.The wave equation (334) 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 [102, 121, 167, 192], if we approximate slowroll inflation by a succession of de Sitter phases. The metric for a de Sitter brane in is given by Equations (180, 181, 182) with
where .

The linearized wave equation (334) 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 (342) as a Schrödingertype equation,
using the conformal coordinate and defining . The potential is given by (see Figure 10) 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 brane [161, 296].)The nonzero value of the Hubble parameter implies the existence of a mass gap [105],
between the zero mode and the continuum of massive KK modes. This result has been generalized: For brane(s) with bulk scalar field, a universal lower bound on the mass gap of the KK tower is [102] The massive modes decay during inflation, according to Equation (343), leaving only the zero mode, which is effectively a 4D gravitational wave. The zero mode, satisfying the boundary condition is given by where the normalization condition implies that the function is given by [192] At low energies () we recover the general relativity amplitude: . At high energies, the amplitude is considerably enhanced: The factor determines the modification of the gravitational wave amplitude relative to the standard 4D result: The modifying factor can also be interpreted as a change in the effective Planck mass [102].This enhanced zero mode produced by brane inflation remains frozen outside the Hubble radius, as in general relativity, but when it reenters 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 reentry. 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 [200]. Selfconsistent lowenergy approximations to compute this effect are developed in [91, 142].

At zero order, the lowenergy approximation is based on the following [22, 235, 237]. 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 [22, 235, 237]. 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 lowenergy regime (), and denotes a linear combination of Bessel functions. The massive modes decay on superHubble scales, unlike the zeromode. Expanding the wave equation in , one arrives at the first order, where modemixing arises. The massive modes on subHubble scales are sourced by the initial zero mode that is reentering the Hubble radius [91]: where is a transfer matrix coefficient. The numerical integration of the equations [142] confirms the effect of massive mode generation and consequent damping of the zeromode, as shown in Figure 11.
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