Quantum-mechanical generation of gravitational waves in a braneworld

We study the quantum-mechanical generation of gravitational waves during inflation on a brane embedded in a five-dimensional anti de Sitter bulk. To make the problem well posed, we consider the setup in which both initial and final phases are given by a de Sitter brane with different values of the Hubble expansion rate. Assuming that the quantum state is in a de Sitter invariant vacuum in the initial de Sitter phase, we numerically evaluate the amplitude of quantum fluctuations of the growing solution of the zero mode in the final de Sitter phase. We find that the vacuum fluctuations of the initial Kaluza-Klein gravitons as well as of the zero mode gravitons contribute to the final amplitude of the zero mode on small scales, and the power spectrum is quite well approximated by what we call the rescaled spectrum, which is obtained by rescaling the standard four-dimensional calculation following a simple mapping rule. Our results confirm the speculation raised in Ref. [T. Kobayashi, H. Kudoh, and T. Tanaka, Phys. Rev. D 68, 044025 (2003).] before.

Figure 1

Numerical (backward) evolution scheme.

Figure 2

The Hubble parameter and the slow-roll parameter $ϵ$ in our model. This is a plot for the model of Fig. 9.

Figure 3

Top panel: power spectra of gravitational waves normalized by $(2C^2(\ell H_i)/M_{\mathrm{Pl}}^2)(H_i/2\pi {)}^2$. Red diamonds (upper ones) indicate the result including the contributions from the initial Kaluza-Klein modes, while green diamonds (lower ones) represent the contribution from the initial zero mode. Although the rescaled four-dimensional spectrum is shown by blue diamonds, they are almost hidden by the red ones since the result of the five-dimensional calculation is well approximated by the rescaled four-dimensional spectrum. Bottom panel: difference between the five-dimensional and rescaled four-dimensional spectra.

Figure 4

Same as Fig. 3, but the parameters are different.

Figure 5

Same as Fig. 3, but the parameters are different.

Figure 6

Same as Fig. 3, but the parameters are different.

Figure 7

Same as Fig. 3, but the parameters are different.

Figure 8

Same as Fig. 3, but the parameters are different.

Figure 9

Same as Fig. 3, but the parameters are different.

Figure 10

Same as Fig. 3, but the parameters are different.

Figure 11

Wronskian $|({\varphi }_{\nu }·\Phi {)}_i{|}^2$ as a function of $\nu$. The parameters are given by those of Fig. 9 and $k/a({\eta }_0)\overline{H}=10$.