Dynamical Casimir effect for gravitons in bouncing braneworlds

Marcus Ruser and Ruth Durrer

Phys. Rev. D 76, 104014 – Published 9 November 2007

Abstract

We consider a two-brane system in five-dimensional anti–de Sitter space-time. We study particle creation due to the motion of the physical brane which first approaches the second static brane (contraction) and then recedes from it (expansion). The spectrum and the energy density of the generated gravitons are calculated. We show that the massless gravitons have a blue spectrum and that their energy density satisfies the nucleosynthesis bound with very mild constraints on the parameters. We also show that the Kaluza-Klein modes cannot provide the dark matter in an anti–de Sitter braneworld. However, for natural choices of parameters, backreaction from the Kaluza-Klein gravitons may well become important. The main findings of this work have been published in the form of a Letter [R. Durrer and M. Ruser, Phys. Rev. Lett. 99, 071601 (2007)].

27 More

Received 6 April 2007

DOI:https://doi.org/10.1103/PhysRevD.76.104014

Authors & Affiliations

Marcus Ruser^* and Ruth Durrer^†

Département de Physique Théorique, Université de Genève, 24 quai Ernest Ansermet, 1211 Genève 4, Switzerland

^*marcus.ruser@physics.unige.ch
^†ruth.durrer@physics.unige.ch

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 76, Iss. 10 — 15 November 2007

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Two branes in an ${AdS}_{5}$ space-time, with $y$ denoting the fifth dimension and $L$ the AdS-curvature scale. The physical brane is on the left at time-dependent position $y_{b} (t)$ . While it is approaching the static brane its scale factor is decreasing and when it moves away from the static brane it is expanding [cf. Eq. (2.3)]. The value of the scale factor of the brane metric as function of the extra dimension $y$ is also indicated.Reuse & Permissions
Figure 2
Evolution of $Ψ_{1}^{2} (t, y) = ϕ_{1}^{2} (t, y) / y^{3}$ corresponding to the probability to find the first KK graviton at time $t$ at the position $y$ in the AdS bulk. The static brane is at $y_{s} = 10 L$ and the maximal brane velocity is given by $v_{b} = 0.1$ .Reuse & Permissions
Figure 3
Evolution of $Ψ_{1}^{2} (t, y)$ as in Fig. 2 but zoomed into the bulk region close to the moving brane.Reuse & Permissions
Figure 4
Evolution of $Ψ_{1}^{2} (t, y)$ for $y_{s} = L$ and $v_{b} = 0.1$ .Reuse & Permissions
Figure 5
Localization of four-dimensional gravity on a moving brane: Evolution of $Ψ_{0}^{2} (t, y)$ for $y_{s} = L = 1$ and $v_{b} = 0.1$ which should be compared with $Ψ_{1}^{2} (t, y)$ shown in Fig. 4.Reuse & Permissions
Figure 6
Evolution of the graviton number $N_{α, k, •} (t)$ for the zero mode and the first ten KK modes for three-momentum $k = 0.01$ and $v_{b} = 0.1$ , $y_{s} = 10$ .Reuse & Permissions
Figure 7
$N_{n, k, •} (t)$ for the zero mode and the first ten KK modes for the parameters of Fig. 6, but without coupling of the zero mode to the KK modes, i.e. $M_{i 0} \equiv 0$ .Reuse & Permissions
Figure 8
Time evolution of the number of created zero-mode gravitons $N_{0, k, •} (t)$ and of the zero-mode power spectrum (4.8): (a) for the entire integration time; (b) for $t > 0$ only. Parameters are $k = 0.01$ , $y_{s} = 10$ , and $v_{b} = 0.1$ . Initial and final time of integration are given by $N_{in} = 10$ and $t_{out} = 4000$ , respectively. The power spectrum is shown with and without the term $O_{0, k, •}^{N}$ , i.e. before and after averaging, respectively, and compared with the analytical results.Reuse & Permissions
Figure 9
Numerical results for the time evolution of the number of created zero-mode gravitons $N_{0, k, •} (t)$ after the bounce $t > 0$ for different three-momenta $k$ . The maximal brane velocity at the bounce is $v_{b} = 0.1$ and the second brane is positioned at $y_{s} = 10$ . In the final particle spectrum the numerical values are compared with the analytical prediction Eq. (6.17). Initial and final time of integration are given by $N_{in} = 5$ and $t_{out} = 20000$ , respectively.Reuse & Permissions
Figure 10
Evolution of the zero-mode power spectrum after the bounce $t > 0$ corresponding to the values and parameters of Fig. 9. The numerical results are compared to the analytical predictions Eqs. (6.20, 6.22).Reuse & Permissions
Figure 11
Dependence of the zero-mode particle number on interbrane distance and initial integration time for momentum $k = 0.01$ , maximal brane velocity $v_{b} = 0.1$ in comparison with the analytical expression Eq. (6.16). (a) Evolution of the instantaneous particle number $N_{0, k, •} (t)$ with initial integration time given by $N_{in} = 5$ for $y_{s} = 0.35$ , 0.5, and 1. (b) Final zero-mode graviton spectrum $N_{0, k, •} (t_{out} = 2000)$ for various values of $y_{s}$ and $N_{in}$ . (c) Close-up view of (b) for large $y_{s}$ .Reuse & Permissions
Figure 12
Final-state KK-graviton spectra for $k = 0.001$ , $y_{s} = 100$ , different maximal brane velocities $v_{b}$ and $N_{in} = 1$ , $t_{out} = 400$ . The numerical results are compared with the analytical prediction Eq. (6.34) (dashed line).Reuse & Permissions
Figure 13
Final-state KK-graviton spectra for $k = 0.01$ , $y_{s} = 100$ , different $v_{b}$ and $N_{in} = 1$ , $t_{out} = 400$ . The numerical results are compared with the analytical prediction Eq. (6.34) (dashed line). For $v_{b} = 0.3$ , 0.5 the spectra obtained without KK intermode and self-couplings ( $M_{i j} \equiv 0 \forall i, j$ ) are shown as well.Reuse & Permissions
Figure 14
Final-state KK-graviton spectra for $k = 0.01$ , $y_{s} = 10$ , different maximal brane velocities $v_{b}$ and $N_{in} = 2$ , $t_{out} = 400$ . The numerical results are compared with the analytical prediction Eq. (6.34) (dashed line).Reuse & Permissions
Figure 15
Final-state KK-graviton spectra for $k = 0.1$ , $y_{s} = 10$ , different maximal brane velocities $v_{b}$ and $N_{in} = 2$ , $t_{out} = 400$ . The numerical results are compared with the analytical prediction Eq. (6.34) (dashed line).Reuse & Permissions
Figure 16
Upper panel: Final-state KK-particle spectra for $k = 0.01$ , $v_{b} = 0.1$ , and different $y_{s} = 3$ , 10, 30, and 100. The analytical prediction Eq. (6.34) is shown as well (dashed line). Lower panel: Energy $ω_{n, k}^{out} N_{n, k, •}^{out}$ of the produced final-state gravitons binned in mass intervals $Δ m = 1$ for $y_{s} = 10$ , 30, 100.Reuse & Permissions
Figure 17
Upper panel: Final-state KK-particle spectra for $k = 0.1$ , $v_{b} = 0.1$ , and different $y_{s} = 3$ , 10, 30, and 100. The analytical prediction Eq. (6.34) is shown as well (dashed line). Lower panel: Energy $ω_{n, k}^{out} N_{n, k, •}^{out}$ of the produced final-state gravitons binned in mass intervals $Δ m = 1$ for $y_{s} = 10$ , 30, 100.Reuse & Permissions
Figure 18
KK-particle spectra for three-momentum $k = 0.1$ , maximum brane velocity $v_{b} = 0.1$ and $y_{s} = 3$ and 10 with different couplings taken into account. The dashed lines indicates again the analytical expression Eq. (6.34).Reuse & Permissions
Figure 19
Comparison of KK-particle spectra for $y_{s} = 3$ , $v_{b} = 0.1$ , and three-momentum $k = 0.01$ , 0.03, 0.1, and 1 demonstrating the independence of the spectrum on $k$ for large masses. $n_{\max } = 60$ KK modes have been taken into account in the simulations.Reuse & Permissions
Figure 20
KK-particle spectra for three-momentum $k = 0.1$ , maximum brane velocities $v_{b} = 0.1$ and $y_{s} = 3$ for $n_{\max } = 40$ obtained for different coupling combinations.Reuse & Permissions
Figure 21
KK-particle spectra for $y_{s} = 10$ , $v_{b} = 0.1$ , $n_{\max } = 100$ , and three-momentum $k = 0.01$ and 0.1 with different couplings taken into account. The thin dashed lines indicate Eq. (6.34) and the thick dashed line indicates Eq. (5.4).Reuse & Permissions
Figure 22
KK-particle spectra for three-momentum $k = 0.1$ , $y_{s} = 10$ , and maximum brane velocities $v_{b} = 0.03$ , 0.1, and 0.3 with $n_{\max } = 100$ . As in Fig. 21 different couplings have been taken into account, the thin dashed lines indicate Eq. (6.34), and the thick dashed line indicates Eq. (5.4).Reuse & Permissions
Figure 23
KK-particle spectra for three-momentum $k = 0.01$ , 0.03, 0.1, and 1 for $y_{s} = 3$ and maximum brane velocity $v_{b} = 0.1$ with different couplings taken into account where the notation is like in Fig. 22. From the crossing of the $M_{i i} = M_{i j} = 0$ and $M_{i i} = M_{i 0} = 0$ results we determine the $k$ dependence of $m_{c} (k, v_{b})$ . The thick dashed line indicates Eq. (5.4).Reuse & Permissions
Figure 24
KK-graviton spectra for three-momentum $k = 0.1$ , $y_{s} = 3$ and maximal brane velocities $v_{b} = 0.3$ , 0.2, 0.1, 0.08, 0.05, and 0.03 with different couplings taken into account where the notation is like in Fig. 22. From the crossing of the $M_{i i} = M_{i j} = 0$ and $M_{i i} = M_{i 0} = 0$ results we determine the $v_{b}$ dependence of $m_{c}$ .Reuse & Permissions
Figure 25
KK-particle spectra for $k = 0.1$ , $y_{s} = 3$ and maximal brane velocities $v_{b} = 0.01$ , 0.03, 0.1 up to KK masses $m_{n} ≃ 100$ compared with an $1 / m_{n}$ decline. The dashed lines indicate the approximate expression (5.6) which describes the asymptotic behavior of the final KK-particle spectra reasonably well, in particular, for $v_{b} < 0.1$ .Reuse & Permissions
Figure 26
Evolution of the zero-mode particle number $N_{0, k, •} (t)$ and final KK-graviton spectra $N_{n, k, •}^{out}$ for $y_{s} = 3$ , maximal brane velocity $v_{b} = 0.1$ and three-momenta $k = 10$ and 30. The dashed line in the upper plots indicates Eq. (6.17) (divided by two) demonstrating the value of the number of produced zero-mode gravitons without coupling to the KK modes.Reuse & Permissions
Figure 27
4D-graviton number $N_{0, k, •} (t)$ for $k = 3$ , 5, 10, 20, and 30 with $y_{s} = 3$ and maximal brane velocity $v_{b} = 0.1$ . The small plot shows the final graviton spectrum $N_{0, k, •}^{out}$ together with a fit to the inverse law $a / (k + b)$ (dashed line) and the analytical fitting formula Eq. (6.23) (solid line). For $k = 10$ and 30 the corresponding KK-graviton spectra are shown in Fig. 26.Reuse & Permissions
Figure 28
Zero-mode particle number $N_{0, k, •} (t)$ and corresponding final KK-particle spectra $N_{n, k, •}^{out}$ for $y_{s} = 3$ , $k = 5$ , 10, and different maximal brane velocities $v_{b}$ . $n_{\max } = 60$ guarantees numerically stable solutions for the zero mode.Reuse & Permissions
Figure 29
Zero-mode particle number $N_{0, k •} (t)$ and corresponding KK-graviton spectra for $k = 10$ , $v_{b} = 0.1$ , and 2nd brane positions $y_{s} = 3$ and 10.Reuse & Permissions
Figure 30
Final KK-particle spectra $N_{n, k, •}^{out}$ for $v_{b} = 0.1$ , $y_{s} = 3$ , and $k = 3$ , 5, 10, and 30 and different couplings. Circles correspond to the full coupling case, squares indicate the results if $M_{i j} = M_{i i} = 0$ , i.e. no KK-intermode couplings, and diamonds correspond to $M_{i 0} = 0$ , i.e. no coupling of KK modes to the zero mode.Reuse & Permissions
Figure 31
Final KK-particle spectra $N_{n, k, •}^{out}$ for $v_{b} = 0.1$ , $y_{s} = 3$ , and $k = 5$ , 10, 20, 30, and 40. The dashed lines indicate Eq. (6.35) for $k = 10$ , 20, 30, and 40. For $k \geq 20$ , the simple analytical expression (6.35) agrees quite well with the numerical results.Reuse & Permissions
Figure 32
KK-particle spectrum for $y_{s} = 3$ , $v_{b} = 0.1$ , and $k = 0.1$ for the bouncing as well as smooth motions with $t_{s} = 0.005$ , 0.015, and 0.05 to demonstrate the influence of the bounce. $n_{\max } = 60$ KK modes have been taken into account in the simulations and the result for the kink motion is shown as well.Reuse & Permissions
Figure 33
Comparison of the final KK-graviton spectrum $N_{n, k, •}^{out}$ with the expression $d_{n, k} (t_{out})$ describing to what accuracy the diagonal part of the Bogoliubov relation (d4) is satisfied. Left panel: $y_{s} = 3$ , $k = 0.1$ , $v_{b} = 0.03$ , and $n_{\max } = 100$ (cf. Fig. 25). Right panel: $y_{s} = 3$ , $k = 30$ , $v_{b} = 0.1$ , and $n_{\max } = 100$ (cf. Fig. 26).Reuse & Permissions
Figure 34
Spectra of massless scalar particles produced under the influence of the uniform motion $y (t) = 1 + v t$ for velocities $v = 0.01$ , 0.02, 0.05, and 0.1. The numerical results are compared to the expression $N_{n} = 0.035 v^{2} / n$ (dashed lines) which agrees with the analytical prediction $N_{n} \propto v^{2} / n$ .Reuse & Permissions

Physical Review D

covering particles, fields, gravitation, and cosmology