The effective distance to which a source can be seen

Given a gravitational-wave detector with some known noise spectrum, it would clearly be useful to define an ``effective distance'' $D_{\rm eff}$ to which some given gravitational-wave source can be seen. To first order, any source located farther away from the detector than $D_{\rm eff}$ would be too weak to be detectable in the datastream. Sources closer than $D_{\rm eff}$ would be detectable.

This naive, heuristic picture of $D_{\rm eff}$ 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 $D_{\rm eff}$ 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, $dE_{\rm gw}/df$. 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 $\rho_0$ is then

$$D^{\rm FH}_{\rm eff}(\rho_0)^2 = {2(1+z)^2\over5\pi^2\rho_... ...^\infty df {1\over f^2 S_h(f)}{dE_{\rm gw}\over df}[(1+z)f]\;.$$ (6.32.153)

The distance $D^{\rm FH}_{\rm eff}$ so defined is actually a luminosity distance; assuming some set of cosmological parameters and using standard formulae, one can then easily convert $D^{\rm FH}_{\rm eff}$ to an effective redshift $z^{\rm FH}_{\rm eff}$, and thence compute the comoving volume $V_c(z^{\rm FH}_{\rm eff})$ that is contained to that distance. If one assumes that the event rate of sources locked into Hubble flow does not evolve with redshift, this allows one to simply convert from an event rate density $R$ [with units number/(Mpc$^3$ year)] to a detected event rate $N$ (with units number/year). (This assumption is clearly a rather bad one: the event rate will undoubtedly evolve with redshift. However, we don't currently know how it will so evolve. This simple, albeit stupid, assumption is a useful one for estimating event rates for gravitational-wave sources.)

Notice that the cosmological redshift $z$ 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 $M$ to $(1+z)M$.

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 $\rho_0$, they define an effective distance $D^{\rm FC}_{\rm eff}(\rho_0)$ as

$$N(\rho > \rho_0) = {4\pi\over3} D^{\rm FC}_{\rm eff}(\rho_0)^3 R\;.$$ (6.32.154)

In words, the detection rate of events with signal-to-noise ratio $\rho$ greater than the threshold $\rho_0$ is given the event rate density in space $R$ times the volume of a sphere of radius $D^{\rm FC}_{\rm eff}(\rho_0)$. Finn and Chernoff then calculate $D^{\rm FC}_{\rm eff}$ using Monte-Carlo integration; see [15] for details.

Thorne [16],[17] has shown that for a pair of $1.4\,M_\odot-1.4\,M_\odot$ neutron stars and for distances small enough that cosmological effects are negligible, the definitions given in Eqs. () and () are related by

$${D^{\rm FC}_{\rm eff}\over D^{\rm FH}_{\rm eff}} = 1.10\;.$$ (6.32.155)

For the purposes of GRASP, we will use an effective distance that is based on () because it is quick and simple to calculate, but correct using () in the hope that this will put us in reasonable agreement with the very careful calculations of Finn and Chernoff. (This factor of $1.10$ probably varies somewhat with total system mass and with cosmological effects.) The formula for $D_{\rm eff}(\rho_0)$ that we use is

$$D_{\rm eff}(\rho_0)^2 = (1.10)^2 \times{2(1+z)^2\over5\pi^2 ... ...\infty df {1\over f^2 S_h(f)}{dE_{\rm gw} \over df}[(1+z)f]\;.$$ (6.32.156)

The following routine calculates this effective distance, providing also the associated redshift and comoving volume.