Given a gravitational-wave detector with some known noise spectrum, it would clearly be useful to define an ``effective distance'' to which some given gravitational-wave source can be seen. To first order, any source located farther away from the detector than would be too weak to be detectable in the datastream. Sources closer than would be detectable.
This naive, heuristic picture of doesn't make much sense in the real world because the source does not emit isotropically, and the detector does not detect isotropically: there are positions on the sky for which the detector can ``see'' farther, and the source radiates more strongly into some angles than others. A useful definition of must therefore average over angles in a meaningful, well-understood way.
One simple way to average over angles is to use Eq. (2.30) of Ref. [13]. In that reference, Flanagan and Hughes show
that the signal-to-noise ratio, rms angle averaged over all source
orientations and all positions on the sky, depends only on the
spectrum of emitted gravitational-wave energy,
.
Rearranging their formula (2.30) slightly, the effective distance to
which a source can be seen with some rms angle-averaged
signal-to-noise ratio is then
Notice that the cosmological redshift explicitly appears in Eq. (). These factors enter in such a way that the mass ``imprinted'' on the gravitational waveform (i.e., the mass that gravitational-wave detections measure at the earth) will be redshift from to .
Finn and Chernoff [15] define an effective
distance in a somewhat different and more careful manner.
Given a noise spectrum and given a threshold signal-to-noise ratio
, they define an effective distance
as
Thorne [16],[17] has shown
that for a pair of
neutron stars and
for distances small enough that cosmological effects are negligible,
the definitions given in Eqs. () and
() are related by
The following routine calculates this effective distance, providing also the associated redshift and comoving volume.