Past exam of the astrophysics course of the University of Oxford / 2018 / 3 / c / Solution

Keep both component masses constant during a circular, gravitational-radiation-driven inspiral. Since $J=\mu \sqrt{GMA}$, differentiation gives $\dot{J}/J=\dot{A}/(2A)$. Combining this with the supplied angular-momentum loss yieldsThe expression $A/\dot{A}$ printed in the source is negative during inspiral. A positive characteristic shrinking timescale isThe actual remaining coalescence time follows by integrating $A^{3}dA=-Kdt$:This constant factor does not change the requested separation scaling.

For an old, approximately coeval population, let $dN/d{A}_0=C$ over the relevant separation interval, and write the positive delay as $t_m=B{A}_0^4$. Conservation of event counts under this change of variable givesThereforeThe scaling applies between the minimum and maximum delays of the assumed separation distribution. It is a rate per delay interval, not the number per logarithmic delay interval.