grids can be computed once and for all, given a waveform model and a detector’s power spectral density (PSD). Evaluating such a grid typically involves $100\times
100=10^4$ SNR evaluations, and it is much faster than the (impractical) evaluation of millions of SNR integrals such as Eq. (\[SNR\]), one for each representative binary produced by [ StarTrack]{}. The conversion between detectability at optimal location and optimal orientation and detectability at generic orientations involves a simple geometrical factor $p_{\rm det}$, as discussed below.

Rate calculation {#sec:fullrates}
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The detection rate is \[Rdet\] R\_[det]{} = \_0\^ (z\_m) p\_[det]{} dz\_m dm\_1 dm\_2, where ${\cal R} (z_m) \equiv \frac{dN}{dm_1 dm_2 dV_c dt_m}$ is the binary merger rate per unit component mass per unit comoving volume $V_c$ per unit time $t_m$ as measured in the source frame at merger redshift $z_m$, the term $\frac{dt_m}{dt_{\rm det}} = \frac{1}{1+z_m}$ accounts for the difference in clock rates at the merger and at the detector, and $p_{\rm det}=p_{\rm det} (z_m; m_1, m_2)$ is the probability (over isotropic sky locations and orientations) that a source with given masses at a given redshift will be detectable. The quantity =, with $E(z)=\sqrt{\Omega_{\rm M}(1+z)^3+\Omega_{\rm k}(1+z)^2+\Omega_{\Lambda}}$, is the comoving volume per unit redshift, and D\_c(z)=\_0\^z 