= Solution
A <gravitational wave> is a propagating tidal perturbation of spacetime. In the weak-field limit its transverse polarizations alternately stretch and compress the separation of freely falling test bodies. A laser <interferometer> such as <LIGO> measures the differential optical phase of orthogonal arms. The mirrors approximate free test <masses> over the detection band; arm cavities enhance the phase response, while suspension, seismic isolation and optical control reduce instrumental disturbances. A strain $h$ produces relative arm-length changes of order $hL$. Coincidence and waveform consistency between detectors distinguish candidate astrophysical signals from local noise.
Potential sources include black-hole and neutron-star binary inspirals and mergers, asymmetric <core-collapse supernovae>, nonaxisymmetric spinning <neutron stars>, and a stochastic background from many unresolved sources or early-Universe processes. Their detectability depends on frequency, strain, duration and detector noise; a source producing gravitational radiation need not be observable in a terrestrial interferometer's band.
For slowly varying approximately isotropic <stellar wind> loss, decreasing the binding <mass> widens the orbit. The prescribed <Jeans-mode mass loss> equation gives
$$
\frac{da}{a}=-\frac{dM}{M},\qquad \boxed{aM=\mathrm{constant}.}
$$
Hot massive stars have metal-line-driven winds. Higher <metallicity> generally strengthens the radiative coupling and <mass> loss, increasing orbital expansion and leaving less massive <black holes>. The supplied initial-contact formula also assigns larger zero-age radii and initial minimum separations at higher <metallicity>.
The initial total <mass> is $2M^0$, while the final total <mass> is $2M_{\rm BH}$. Neglecting supernova <mass> loss as requested, their ratio has no additional factor of two. The <wind-limited black-hole binary merger window> therefore has lower boundary
$$
\boxed{a_{\min}^{V}=a_{\min}^0\frac{M^0}{M_{\rm BH}}
=2R_\odot\left(\frac Z{Z_\odot}\right)^{0.4}\frac{(M^0/M_\odot)^2}{M_{\rm BH}/M_\odot}.}
$$
A nonempty allowed initial-separation interval requires $a_{\min}^{V}<a_{\max}^{\rm merger}$. Substituting the supplied final <masses> gives the following separation bounds, all in units of $R_\odot$:
$$
\begin{array}{c|rr|rr}
Z/Z_\odot&a_{\min}^{V}(30)&a_{\max}(30)&a_{\min}^{V}(100)&a_{\max}(100)\\\hline
1/4&39.76&30.62&191.45&57.33\\
1/10&25.59&32.37&113.74&64.35\\
1/20&18.73&33.23&75.43&71.13
\end{array}
$$
The arguments 30 and 100 indicate the initial <mass> of each star in solar units.
\Image[/past-exam-of-the-astrophysics-course-of-the-university-of-oxford/2016-metallicity-binary-separations.png]
{title=Wind-expanded minimum separations compared with merger upper bounds}
{description=Original calculation at the six supplied initial-mass and metallicity combinations. Joining segments are visual guides between tabulated points, not a predicted interpolation of final black-hole mass.}
Under these particular approximate formulae, only the $30M_\odot$ initial systems at $Z_\odot/10$ and $Z_\odot/20$ have overlapping bounds. The $100M_\odot$, $Z_\odot/20$ system is a near miss, not formally allowed: $75.43>71.13$. Given approximate evolutionary scalings, that small mismatch should not be treated as a precise astrophysical exclusion.
Lower <metallicity> favors merging remnants here through weaker winds, smaller initial stellar radii and larger final black-hole <masses>. More massive binaries can also produce stronger detectable signals over a suitable frequency range, although the detection volume and band limitations must be considered. The observed rate further depends on formation efficiency, <metallicity> history, delay times and binary survival. Common-envelope evolution, stellar rotation, supernova <mass> loss and natal kicks could alter this simplified channel; the wind-only inequalities do not establish that such mergers are impossible through other channels.
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