Establish the domain of validity of the inverse-power-law time-delay distribution

Establish the domain of validity of the commonly assumed inverse-power-law time-delay distribution for isolated binary black-hole formation, including how its validity depends on metallicity, stellar winds, mass transfer, and natal kicks.

Background

For isolated binaries undergoing mass transfer, the time-delay distribution is often modeled as p(t_d) proportional to t_d^{-1}, based on an assumed inverse distribution of binary semi-major axes at formation. The paper notes that this assumption may not apply universally and that several uncertain physical ingredients can alter the resulting delay-time distribution.

The unresolved issue concerns determining when the inverse-power-law approximation is physically justified and quantifying its dependence on metallicity, stellar winds, mass-transfer physics, and natal-kick prescriptions. This uncertainty directly affects predictions for merger rates and gravitational-wave clustering.