### 7.2 Full numerical solutions

Update
Full numerical solutions for the tensor perturbations have been obtained by the two methods – the
pseudo-spectral (PS) method used in [200, 199] and the characteristic integration (CI) algorithm
developed in [70]. It was shown that both methods give identical results and the behaviour of
gravitational waves on a brane is quite insensitive to the initial conditions in the bulk as for the
scalar perturbations [378]. Here we summarize the results obtained in [199]. It is convenient
to use the static bulk metric and consider a moving brane. The simplest initial condition is
= const when the mode is outside the hubble horizon on the brane. If the brane is
static, this would give a zero-mode solution where is an initial time.
However, due to the motion of the brane, which causes the expansion of the brane universe, KK
modes are excited and this solution is modified. Figure 14 demonstrate this effect. Once the
perturbation enters the horizon, non-trivial waves are excited in the bulk and the amplitude of the
tensor perturbation is damped. Figure 15 shows the behaviour of gravitational waves for two
different wave numbers. Here at the time when the mode re-enters the horizon. As for
scalar perturbations, we can define the effective 4D solutions by ignoring the bulk as a reference
only takes into account the effect of the high-energy modification of the Friedmann equation.
Figure 15 shows that the full solution has an additional suppression of the amplitude compared with .
This suppression is caused by the excitations of KK modes at the horizon crossing as is seen in Figure 14.
The suppression is stronger for modes that enter the horizon earlier and it becomes negligible at
low energies .
The ratio evaluated at the low-energy regime long after the horizon re-entry time
monotonically decreases with the frequency and the suppression of amplitude becomes significant
above the critical frequency given by

This corresponds to a frequency of the mode that enters the horizon when (cf. [202]). The ratio
obtained from numerical solutions is fitted as
with and .
There are two important effects on the spectrum in the high-energy regime. Let us first consider the
non-standard cosmological expansion due to the -term. The spectrum of the stochastic gravitational
waves is modified to

where is an equation of state. This is because gravitational waves re-enter the horizon when the
-term dominates at high frequencies . In the high-energy radiation dominated phase, the
spectrum of the stochastic gravitational waves is modified to
The other effect is the KK-mode excitations. Taking account of the KK-mode excitations, the spectrum is
calculated as
where we used the fact . Combining it with the result (384), the spectrum becomes nearly
flat above the critical frequency:
which is shown in filled squares in Figure 16. In this figure, the spectrum calculated from the reference
gravitational waves is also shown in filled circles. Note that the normalization factor of the spectrum
is determined as from the CMB constraint. The short-dashed line and the solid line
represent asymptotic behaviors in the high-frequency region. The spectrum taking account of the two
high-energy effects seems almost indistinguishable from the standard four-dimensional prediction shown in
long-dashed line in the figure. In other words, while the effect due to the non-standard cosmological
expansion enhances the spectrum, the KK-mode effect reduces the GW amplitude, which results in the
same spectrum as the one predicted in the four-dimensional theory. Note that the amplitude taking account
of the two effects near is slightly suppressed, which agrees with the results in the previous
study for using the Gaussian-normal coordinates [201] discussed in the previous
subsection.
This cancelation of two high energy effects is valid only for . For other equations of state, the
final spectrum at high frequencies are different from 4D predictions. For example for ,
for while the 4D theory predicts .