Gravitational waves from bubble collisions: analytic derivation

TL;DR

The paper develops an analytic framework to predict gravitational-wave spectra from bubble collisions during cosmological first-order phase transitions, leveraging thin-wall and envelope approximations in flat spacetime to express the spectrum through the two-point energy-momentum tensor correlator. By computing the TT-projected correlator and decomposing it into single- and double-bubble contributions, the authors derive semi-analytic expressions for the spectrum and show that single-bubble contributions dominate; the high-frequency fall-off scales as $f^{-1}$. They provide numerical estimates and fitting formulas, including a peak with $\Delta_{\rm peak}=0.043$ at $f_{\rm peak}/\beta \approx 0.20$, and extend the analysis to finite wall velocity, clarifying how the spectrum depends on $v$ and approaching known limits. The work offers a practical, analytic pathway to predict cosmological GW signals from strong phase transitions and suggests straightforward extensions beyond the envelope approximation and for expanding backgrounds.

Abstract

We consider gravitational wave production by bubble collisions during a cosmological first-order phase transition. In the literature, such spectra have been estimated by simulating the bubble dynamics, under so-called thin-wall and envelope approximations in a flat background metric. However, we show that, within these assumptions, the gravitational wave spectrum can be estimated in an analytic way. Our estimation is based on the observation that the two-point correlator of the energy-momentum tensor $\langle T(x)T(y)\rangle$ can be expressed analytically under these assumptions. Though the final expressions for the spectrum contain a few integrations that cannot be calculated explicitly, we can easily estimate it numerically. As a result, it is found that the most of the contributions to the spectrum come from single-bubble contribution to the correlator, and in addition the fall-off of the spectrum at high frequencies is found to be proportional to $f^{-1}$. We also provide fitting formulae for the spectrum.

Figures (11)

• Figure 1: Rough sketch of how the phase transition looks with the envelope approximation. Collided walls are neglected as a source of GWs, and all spacial points are passed by bubble walls only once.
• Figure 2: Schematic picture of single- and double-bubble contributions to the correlator $\langle T(x)T(y) \rangle$. The red line (the one w/o dashed lines) corresponds to the wall of one single bubble, while the blue line (the one w/ dashed lines) corresponds to intersecting two bubble walls. The dashed lines are neglected in the envelope approximation. Note that, though this figure shows $t_x = t_y$ case for simplicity, contributions from $t_x \neq t_y$ exist in the calculation of $\langle T(x) T(y) \rangle$.
• Figure 3: Notations for quantities on the past light cones of $x$ and $y$.
• Figure 4: How the light cones in Fig. \ref{['fig:LightCone']} look like in $2 + 1$ dimensions. The yellow circles represent the nucleation points for single-bubble (on the red central arrows) and double-bubble (on the blue separate arrows) contributions. The red line along the intersection of the two light cones shows $\delta V_{xy}$ in Fig. \ref{['fig:LightCone']}.
• Figure 5: Notations for quantities on the constant-time hypersurface $\Sigma_t$. The red diamond-shaped region denotes the one where a bubble nucleate in single-bubble spectrum (see Sec. \ref{['subsec:single']}), while the outer blue regions between the dotted lines denote the ones where bubbles nucleate in double-bubble spectrum (see Sec. \ref{['subsec:double']}).