Gravitational Wave Production by Collisions: More Bubbles

TL;DR

The paper revisits gravitational-wave production from bubble collisions during a first-order phase transition using the envelope approximation to model many-bubble dynamics with thin-wall energy transfer. It derives the spectrum from the stress-energy tensor and expresses it in terms of a frequency-dependent function Δ(ω/β, v_b), incorporating the bubble nucleation rate and wall velocity. Numerical simulations show that the GW spectrum scales as $Ω_{GW*} ∝ ω^3$ at low frequencies and $∝ ω^{-1}$ at high frequencies, a slower fall-off than earlier two-bubble results, with peak features that depend modestly on the wall velocity. The findings have direct implications for the detectability of electroweak-scale phase-transition signals with space-based detectors like LISA and BBO, and help reconcile analytic approaches with multi-bubble dynamics.

Abstract

We reexamine the production of gravitational waves by bubble collisions during a first-order phase transition. The spectrum of the gravitational radiation is determined by numerical simulations using the "envelope approximation". We find that the spectrum rises as f^3.0 for small frequencies and decreases as f^-1.0 for high frequencies. Thus, the fall-off at high frequencies is significantly slower than previously stated in the literature. This result has direct impact on detection prospects for gravity waves originating from a strong first-order electroweak phase transition at space-based interferometers, such as LISA or BBO. In addition, we observe a slight dependence of the peak frequency on the bubble wall velocity.

Figures (4)

• Figure 1: The left panel shows the fraction of gravitational radiation for a small simulation, $L_U=3 v_b / \beta$, 7 bubbles. For small wall velocities, significant boundary effects are visible. The right simulation is relatively large, $L_U=7 v_b / \beta$, 109 bubbles. Both simulations have been integrated over $N_k=32$ directions. Velocities decrease from top to bottom as $v_b= \{1, \, 0.1, \, 0.01\}$.
• Figure 2: The left panel shows the spectrum of gravitational radiation for a simulation with $L_U=7 v_b / \beta$, integrated over $N_k=32$ directions. The error bars result from averaging over eight different scenarios. Velocities decrease from top to bottom as $v_b= \{1, \, 0.1, \, 0.01\}$. The right panel shows the parameters $\tilde{\Delta}$ and $\tilde{f}_*/\beta$ as functions of $v_b$.
• Figure 3: Several spectra of gravitational radiation according to the old and new formulas. The parameters are taken from ref. KonHub and given in table \ref{['tab_spectrum']} with $\alpha$ decreasing from top to bottom. In the shaded region, the sensitivity of LISA and BBO is expected to drop considerably.
• Figure 4: The figure depicts the case of a bubble that intersects with four neighboring bubbles. The data structure we use contains the information which bubbles constitute the boundaries of the uncollided regions in each segment. This improves the performance greatly especially at the end of the phase transition when virtually all bubbles intersect with each other but only a few bubbles are relevant in each segment.