Initial Kaluza-Klein fluctuations and inflationary gravitational waves in braneworld cosmology

We study the spectrum of gravitational waves generated from inflation in the Randall-Sundrum braneworld. Since the inflationary gravitational waves are of quantum-mechanical origin, the initial configuration of perturbations in the bulk includes Kaluza-Klein quantum fluctuations as well as fluctuations in the zero mode. We show, however, that the initial fluctuations in Kaluza-Klein modes have no significant effect on the late time spectrum, irrespective of the energy scale of inflation and the equation of state parameter in the post-inflationary stage. This is done numerically, using the Wronskian formulation.

Figure 1

Contribution of initial Kaluza-Klein fluctuations. The equation of state parameter is $w=1/3$ and the inflationary energy scale is given by $\ell H_i=100$ (lower line). $r_{\mathrm{KK}}$ is plotted also for $\ell H_i=10$ (upper line). (The previous estimate in Ref. 17 should be compared with the lower line.)

Figure 2

Spectrum of gravitational waves from inflation with $\ell H_i=100$, followed by a FRW phase with $w=0$. The total power spectrum is shown by circles, while crosses represent the contribution only from the initial zero mode. Dashed line indicates the four-dimensional result.

Figure 3

Same as Fig. 2, but $w=2/3$.

Figure 4

Same as Fig. 2, but $w=1$.

Figure 5

Fraction $r_{\mathrm{KK}}$ for different inflationary energy scales, showing (from top to bottom) $\ell H_i=10$, 50, and 100. The equation of state parameter is given by $w=0$.

Figure 6

Same as Fig. 5, but $w=1$.