is the comoving distance, related to the luminosity distance $D_L(z)$ by $D_c(z)=D_L(z)/(1+z)$: see [@hogg] for our notation and conventions.

The merger rate ${\cal R} (z_m)$ is a convolution of the star formation rate and the number density of binaries per unit star forming mass $M_f$ per unit time delay between formation and merger $\tau$: \[Rzm\] [R]{} (z\_m) &=& \_0\^[t\_m]{} \_0\^[t\_[det]{}]{} (z\_f) (t\_f; m\_1, m\_2, )\
&& (t\_m-t\_f-) ddt\_f, where ${\rm SFR} = \frac{dM_f}{dV_c dt_f} (z_f)$ is the star formation rate per unit comoving volume per unit time $t_f$ at formation redshift $z_f$.

The distribution of binaries in mass and time delay space, $\frac{dN} {dM_f dm_1
  dm_2 d\tau} (t_f; m_1, m_2, \tau)$, is obtained with the [StarTrack]{} population synthesis code, taking into account the metallicity distribution at the formation redshift as described in Section \[binevol\]. Since [StarTrack]{} simulations produce a set of merging binaries with specific component masses and time delays sampling the desired distribution, the integrals above are easily computed via Monte Carlo over the simulated systems. For computational efficiency the outer integral over the time of formation in Eq. (\[Rzm\]) is binned over $\Delta t_f = 100$ Myr segments, while the integral over the merger redshift $z_m$ in Eq. 