Linearly Polarized Gravitational Waves from Bubble Collisions

TL;DR

This work investigates a slow first-order phase transition in the early Universe that completes via the nucleation and collision of exactly two vacuum bubbles, producing a linearly polarized stochastic gravitational-wave background. Using TT gauge analysis for a two-bubble collision, the authors show that $h_{\times}=0$ so only $h_{+}$ is generated, yielding a fully linear polarization detectable by future triangular detectors. They model the transition in a radiation-dominated era with a thermal tunneling rate $\Gamma(t)=C(t)e^{-A(t)}$, derive the condition for two-bubble completion, and compute the GW spectrum, highlighting a polarization-dominated bubble-wall collision component that can fall within the sensitivity bands of LISA and the Einstein Telescope. Polarization diagnostics via frame-invariant Stokes parameters provide a clear observational handle to distinguish this scenario, with implications for probing physics of the Universe’s earliest moments. The results motivate further numerical simulations and explicit model-building to realize and test slow two-bubble transitions observationally.

Abstract

Physics beyond the Standard Model may give rise to first-order phase transitions proceeding via the nucleation of vacuum bubbles, whose subsequent collisions generate gravitational waves (GWs). Their detection would open the possibility of investigating the universe in its first instants. If the transition is slow enough, such that it completes with the nucleation and collision of only two bubbles, the resulting GW signal is linearly polarized. We show that in this case triangular interferometers such as LISA and the Einstein Telescope could be able to not only measure the magnitude of the GW but also establish its linear polarization. This would give a strong hint about the origin of the signal.

Figures (4)

• Figure 1: Range of the inverse phase-transition duration, $\beta_H$, for which the expected number of bubbles at completion is two, shown as a function of the bubble wall velocity $v_w$ (purple region). The dashed line indicates $\beta_H = 1$; below this value, nucleation proceeds more slowly than the cosmic expansion, implying that the bubble wall velocity must satisfy $v_w/c > 0.74$.
• Figure 2: Ratio of the mean bubble size to the Hubble radius at the completion time as a function of the wall velocity $v_w$. The upper (lower) black line corresponds to the lower (upper) bound of $\beta_H$, for which the expected number of bubbles at completion is two. The shaded purple region indicates the allowed values of $R_\star H_\star$, with $\beta_H$ increasing from top to bottom within the permitted range.
• Figure 3: Total stochastic GW background (black) from a radiation-dominated phase transition and its individual contributions from bubble-wall collisions (blue), sound waves (orange), and hydrodynamic turbulence (green), for $\alpha = 0.5$, $T_\star = 3.6 \times 10^3~\text{GeV}$ (solid) and $\alpha = 0.8$, $T_\star = 5.0 \times 10^3~\text{GeV}$ (dashed). The projected sensitivity curves of the future detectors LISA and Einstein Telescope (ET) are shown as grey regions.
• Figure 4: Parameter space of $\beta_H$ and $N_\text{tot}^\star$ for which the expected number of bubbles per Hubble volume at completion satisfies $2 \leq N(t_\star) < 3$ in a supercooled phase transition. The green region corresponds to cases where the completion time is determined by $I(t_\star) = 4.6$, while the purple region denotes the regime where it is set by $(3H)^{-1}dI/dt_{|_{t=t_\star}}=1$. The dashed red line marks $R_\star H_\star=1.8$.