Gravitational Waves from Warped Spacetime

TL;DR

The paper analyzes a first-order cosmological phase transition between the AdS-Schwarschild phase and the RS1 geometry, stabilized by the Goldberger-Wise radion, and shows it can produce a strong relic gravitational-wave background detectable by LISA for TeV-scale transitions. The authors extend prior work to negative $\epsilon$ and nonzero $\delta T_1$, emphasize thick-wall nucleation (with possible $O(4)$ tunneling) and compute the nucleation conditions in terms of $S_3/T$ and $S_4$, identifying viable regions up to $N \approx 12$ under perturbativity. They compute the GW spectrum in terms of $\alpha$ and $\beta/H_*$, finding that large $\alpha$ (even $>1$) and moderate $\beta/H_*$ yield strong LISA signals, with peak frequencies scaling with the IR scale $\mu_{\rm TeV}$. Perturbativity and back-reaction constraints restrict the parameter space, but a substantial region—especially with $\epsilon<0$—remains viable, making a detectable RS1-associated GW signature a robust prospect for upcoming space-based detectors.

Abstract

We argue that the RSI model can provide a strong signature in gravitational waves. This signal is a relic stochastic background generated during the cosmological phase transition from an AdS-Schwarschild phase to the RS1 geometry that should occur at a temperature in the TeV range. We estimate the amplitude of the signal in terms of the parameters of the potential stabilizing the radion and show that over much of the parameter region in which the phase transition completes, a signal should be detectable at the planned space interferometer, LISA.

Figures (7)

• Figure 1: Lines delimiting the region in $(\epsilon, v_1)$ parameter space where the phase transition completes (we have to be above the dashed line for nucleation to take place). The upper line comes from using the approximate eq (\ref{['S4actionapprox']}) while the lower line results from solving eq. (\ref{['mu4']}).
• Figure 2: Comparison of the thin and thick wall approximations (dotted lines) with the exact solutions obtained by solving for the bounce numerically (solid lines).
• Figure 3: Top: comparison between thick wall, thin wall and exact solutions at fixed $\epsilon$ and $\delta T_1$; bottom left: exact results for different values of $N$. bottom right: Typical evolution of the radion potential with temperature. The height of the barrier falls off as $T$ goes down. For $T$ below $T_c/2$, it is a very good approximation to use the zero temperature potential to compute the bounce.
• Figure 4: $\alpha$ is the ratio of the latent heat to the radiation energy density in the CFT phase at the time of nucleation, given by Eq. (\ref{['alphaformula']}). It increases as the ratio of the nucleation temperature $T_n$ to the critical temperature $T_c$ decreases. Second plot is $\beta/H$ from Eq. (\ref{['betaformula']}) where $\beta^{-1}$ can be understood as the duration of the phase transition. The amplitude of the gravitational wave signal increases with $\alpha$ and decreases as $(\beta/H)^{-2}$.
• Figure 5: $\alpha$ and $\beta/H$ determine the spectrum of gravitational waves. They are calculated for some benchmark point: $\epsilon=-0.25$, $N=12$, $\delta T_1=-0.5 v_1^2$. The smallest values of $v_1/N$ correspond to a large amount of supercooling i.e. a small value of the ratio $T_n/T_c$. This ratio varies between 0.23 for $v_1/N=0.7$ to 0.87 for $v_1/N=1.5$. Assuming that $\mu_{\hbox{\tiny TeV}}=5$ TeV, the corresponding nucleation temperatures are in the range 490 GeV -- 2700 GeV. We also show the peak frequency $f_{\hbox{\tiny peak}}$ of the gravitational wave signal from turbulence and $\Omega_{\hbox{\tiny peak}} h^2$. The peak frequency depends on $\mu_{\hbox{\tiny TeV}}$, while $\alpha$, $\beta/H$ and $\Omega_{\hbox{\tiny peak}} h^2$ do not. As shown in Fig. \ref{['negativespectrum']}, this can lead to a spectacular signal at LISA and/or BBO.