Gravitational waves from bubble dynamics: Beyond the Envelope

TL;DR

This work analyzes gravitational waves from bubble dynamics during a cosmic first-order phase transition beyond the envelope approximation. It employs a thin-wall model with a damping function to capture post-collision wall propagation and derives analytic expressions for the GW spectrum from the energy-momentum tensor two-point correlator, yielding Δ^(s) and Δ^(d). The main findings show that, in the long-lasting wall limit, the low-frequency spectrum grows as Ω_GW ∝ f for small f and saturates at late times, with a notable enhancement relative to the envelope-based results and a transition of the infrared slope from f^3 to f. These insights apply to both scalar-field bubble collisions and fluid sound waves, offering guidance for predicting signals in future GW detectors and highlighting the need to address nonlinear plasma dynamics in the high terminal velocity regime.

Abstract

We study gravitational-wave production from bubble dynamics (bubble collisions and sound waves) during a cosmic first-order phase transition with an analytic approach. We first propose modeling the system with the thin-wall approximation but without the envelope approximation often adopted in the literature, in order to take bubble propagation after collisions into account. The bubble walls in our setup are considered as modeling the scalar field configuration and/or the bulk motion of the fluid. We next write down analytic expressions for the gravitational-wave spectrum, and evaluate them with numerical methods. It is found that, in the long-lasting limit of the collided bubble walls, the spectrum grows from $\propto f^3$ to $\propto f^1$ in low frequencies, showing a significant enhancement compared to the one with the envelope approximation. It is also found that the spectrum saturates in the same limit, indicating a decrease in the correlation of the energy-momentum tensor at late times. We also discuss the implications of our results to gravitational-wave production both from bubble collisions (scalar dynamics) and sound waves (fluid dynamics).

Figures (39)

• Figure 1: Rough sketch of the bubble dynamics with the envelope approximation. The bubble walls, denoted by the black lines, accumulate energy and momentum as they expand and then lose them instantly when they collide with each other. Compare this figure with Fig. \ref{['fig:BeyondEnv']}. This figure is the same as in Ref. Jinno:2016vai.
• Figure 2: Rough sketch of the bubble dynamics without the envelope approximation. The collided bubble walls, denoted by the gray lines, gradually lose their energy and momentum densities after collisions.
• Figure 3: The total spectrum $\Delta$ as a function of the duration time $\tau$ for $v = 1$. The blue, red, yellow and green lines correspond to $k = (1, 0.6, 0.4, 0.2) \times 10^{-n}$ ($n \in \mathbb{Z}$), respectively.
• Figure 4: The total spectrum $\Delta$ as a function of wavenumber $k$ for $v = 1$. Each data point corresponds to $\tau = (0.01, 0.03, 0.1, 0.3, 1, 3, 10) \times (1/k)$, while the black line shows the spectrum with the envelope approximation obtained in Ref. Jinno:2016vai.
• Figure 5: The total spectrum $\Delta$ as a function of wavenumber $k$ for $v = 1$. Each colored line corresponds to $\tau = 1, 3, 10, 30, 100$ from bottom to top, while the black-dashed line corresponds to the data points for $\tau = 10 \times (1/k)$. The black-solid line is the same as Fig. \ref{['fig:kDeltaT_v=1']}.