Extend the pairwise surrogate to time-dependent nucleation histories

Extend the pairwise gravitational-wave surrogate to bubbles nucleated at different times and therefore having different ages and sizes at collision, including geometric weights based on each bubble’s age, radiation integrals defined in the common transition frame, direction-dependent transformations before angular averaging, and validation against unequal-time lattice simulations.

Background

The presented surrogate assumes identical bubbles nucleated simultaneously. Under that assumption, the isolated two-bubble scalar dynamics admit the symmetry reduction used to construct the response library, and the geometric screening weights can be computed from equal-age spherical bubble walls.

A realistic transition can involve a time-dependent nucleation history, producing bubbles with different ages, sizes, and collision frames. The authors explicitly identify extending the reconstruction to this case, together with comparison to unequal-time lattice simulations, as work left unresolved.

References

An important future direction is to include time-dependent nucleation histories, allowing bubbles of different ages and sizes to collide. For ideal vacuum bubbles with spacelike-separated nucleation events, a boost along the pair axis makes nucleation simultaneous and preserves the $(1+1)$-dimensional scalar evolution . The reconstruction would then require geometric weights based on each bubble's age and radiation integrals with the cutoff defined in the common transition frame. In addition, the direction-dependent radiation must be transformed to that frame before angular averaging. We leave this extension and its comparison with unequal-time lattice simulations to future work.

— A Pairwise Surrogate for Gravitational-Wave Spectra from Highly Relativistic Vacuum Bubble Collisions  (2610.12389 - Buschmann et al., 8 Oct 2026) in Section 8, Discussion and conclusions, final paragraph