7.2 Full numerical solutions

UpdateJump To The Next Update Information Full numerical solutions for the tensor perturbations have been obtained by the two methods – the pseudo-spectral (PS) method used in [200, 199Jump To The Next Citation Point] 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 [199Jump To The Next Citation Point]. It is convenient to use the static bulk metric and consider a moving brane. The simplest initial condition is f (z, τ) = const when the mode is outside the hubble horizon on the brane. If the brane is static, this would give a zero-mode solution f = cos(k(τ − τ )) 0 where τ 0 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 14View Image 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 15View Image shows the behaviour of gravitational waves for two different wave numbers. Here πœ–∗ = ρ βˆ•λ 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
2 ¨href + 3H Λ™href +--k--href = 0. (381 ) a (t)2
href only takes into account the effect of the high-energy modification of the Friedmann equation. Figure 15View Image shows that the full solution has an additional suppression of the amplitude compared with href. This suppression is caused by the excitations of KK modes at the horizon crossing as is seen in Figure 14View Image. The suppression is stronger for modes that enter the horizon earlier πœ–∗ > 1 and it becomes negligible at low energies πœ– < 1 ∗.
View Image

Figure 14: The evolution of gravitational waves. We set the comoving wave number to √ -- k = 3 βˆ•β„“ (πœ– = 1.0 ∗). The right panel depicts the projection of the three-dimensional waves of the left panel. Figure taken from [199Jump To The Next Citation Point].

The ratio |h5D βˆ•href| evaluated at the low-energy regime long after the horizon re-entry time monotonically decreases with the frequency and the suppression of amplitude h5D becomes significant above the critical frequency fcrit given by

fcrit = -1-acrit aeq (382 ) 2πβ„“ aeq a0 ( )− 1βˆ•2 ( )1βˆ•2( ) −1βˆ•4 = 5.6 × 10− 5 Hz ---β„“---- ------H0------- 1-+-zeq . (383 ) 0.1 mm 72 km βˆ•s ⋅ Mpc 3200
This corresponds to a frequency of the mode that enters the horizon when −1 H ∗ = β„“ (cf. [202]). The ratio |h5Dβˆ•href| obtained from numerical solutions is fitted as
|| || ( ) −β |h5D| = α -f-- (384 ) |href| fcrit
with α = 0.76 ± 0.01 and β = 0.67 ± 0.01.
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Figure 15: Squared amplitude of gravitational waves on the brane in the low-energy (left) and the high-energy (right) regimes. In both panels, solid lines represent the numerical solutions. The dashed lines are the amplitudes of reference gravitational waves href obtained from Equation (381View Equation). Figure taken from [199Jump To The Next Citation Point].

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 ρ2-term. The spectrum of the stochastic gravitational waves is modified to

{ 63ww−+21 Ωref = f 6w+2 forf < fcrit, (385 ) f 3w+2 forf > fcrit,
where w is an equation of state. This is because gravitational waves re-enter the horizon when the ρ2-term dominates at high frequencies f > fcrit. In the high-energy radiation dominated phase, the spectrum of the stochastic gravitational waves is modified to
4βˆ•3 Ωref ∝ f (fcrit < f). (386 )
The other effect is the KK-mode excitations. Taking account of the KK-mode excitations, the spectrum is calculated as
||h5D ||2 ΩGW = ||----|| Ωref, (387 ) href
where we used the fact ΩGW ∝ h2f 2. Combining it with the result (384View Equation), the spectrum becomes nearly flat above the critical frequency:
Ω ∝ f 0, (388 ) GW
which is shown in filled squares in Figure 16View Image. In this figure, the spectrum calculated from the reference gravitational waves Ωref is also shown in filled circles. Note that the normalization factor of the spectrum is determined as Ω = 10−14 GW 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 f ≈ fcrit is slightly suppressed, which agrees with the results in the previous study for πœ–∗ ≤ 0.3 using the Gaussian-normal coordinates [201Jump To The Next Citation Point] discussed in the previous subsection.
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Figure 16: The energy spectrum of the stochastic background of gravitational wave around the critical frequency fc in radiation dominated epoch. The filled circles represent the spectrum caused by the non-standard cosmological expansion of the universe. Taking account of the KK-mode excitations, the spectrum becomes the one plotted by filled squares. In the asymptotic region depicted in the solid line, the frequency dependence becomes almost the same as the one predicted in the four-dimensional theory (long-dashed line). Figure taken from [199].
This cancelation of two high energy effects is valid only for w = 1βˆ•3. For other equations of state, the final spectrum at high frequencies are different from 4D predictions. For example for w = 1, ΩGW ∝ f2βˆ•5 for f > fc while the 4D theory predicts ΩGW ∝ f1.


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