# Is there a limit to the distance of detectable gravitational wave events?

This is somewhat a follow up question to this one.

This article indicates that the binaries detected have a distance between 320 and 2840 Mpc.

With the current technology, is there a limit to the distance of coalescing compact binaries that can be detected through gravitational waves? Or could we potentially see these mergers up to the edge of the cosmos (if there were any)?

Several factors influence whether a source of gravitational waves at a certain distance is observable by a certain instrument. One way to compute the limit to the distance is explained in Abadie et al 2010 and is as follows:

• Distance. The amplitude of gravitational waves decreases roughly with the inverse of the luminosity distance $$\propto D^{-1}$$. So sources that are further away will be more difficult to observe and at some point they will be less/not visible.

• Power of the source. The stronger the source the more easy it's signal can be detected. For binary systems with larger total mass $$M$$ and larger reduced mass $$\mu$$ you will observe higher amplitude waves. The amplitude of the signal $$\vert \tilde{h}(f) \vert$$ can be expressed as (the expression is from Abadie et al 2010 see Finn and Thorne 2010 for a derivation)

$$\vert \tilde{h}(f)\vert = \frac{2c}{D} \left(\frac{5 G \mu}{96 c^3} \right)^{1/2} \left( \frac{GM}{\pi^2c^3} \right)^{1/3} f^{-7/6}$$

• Sensitivity. The detector can be more or less sensitive. The more sensitive the detector the lower the luminosity or distance of the objects that it can observe.

The sensitivity can be expressed by the noise power density $$S_n(f)$$ (as function of frequency $$f$$) which is specific for the instrument (and you will be able to see graphs of this in many publications). A signal can be observed if it is stronger than the noise. Abadie et al 2010 use as limit a (conservative) signal-to-noise of $$\rho$$ =8, which means that the signal must be 8 or more times stronger than the background noise in order to be detected.

This signal-to-noise ratio is determined by an integral of the ratio of the frequency-domain waveform amplitude $$\vert \tilde{h}(f) \vert$$ and the noise power density $$S_n(f)$$.

$$\rho = \sqrt{4 \int_0^{f_{ISCO}} \frac{\vert \tilde{h}(f) \vert^2}{S_n(f)} \text{d}f }$$ where $$f_{ISCO}$$ is the frequency of the innermost stable circular orbit of the binary system

In that article (Abadie et al 2010), the limit of the distance for the detection of wave events from binary blackholes with mass $$10 M_{\odot}$$ was estimated at 2187 Mpc which is pretty close to the distance of 2840 Mpc estimated for GW170729 (which is heavier).

Note that the limits for binary neutron stars are more often reported and are easier to find. For instance in Moore et al 2015 you can read in more detail about the increase of the limit for LIGO from 80 to 100 Mpc in the recent years. The first image shows plots of $$S_n(f)$$ as function of $$f$$ and of $$D$$ as function of time (during the experiments improvements have been made and the distance was changing).

• Rate of occurances. When certain events have a higher probability to occur then it may also be more likely to observe them at a far away certain distance. The computation for these rates include the aspects of distance. Also as stated by Abadie et al

The real detection range of the network is a function of the data quality and the detection pipeline, and can only be obtained empirically.

In Abbott et al 2016 a computation is performed to determine the probability to observe a particular event at a certain distance. The distance, for $$40-40 M_{\odot}$$, ranges up to a roughly $$z=0.6$$ (or using $$d \approx z c / H_0 \approx 0.6 \times 3 \times 10^5 / 74.2 \approx 2.5 Gpc$$), which is plotted in the last figure of that reference.

Conclusion: the observation of GW170729 at about 3Gpc is about the limit of the current instruments

### References

As an addendum to Sexti Empirici answer note that LIGO and Virgo have not yet reached their full "design" sensitivity. Each run their sensitivity is further improved. The current observation run (O3) is already quite a bit more sensitive than the preceeding (O2) run. Some public alerts for candidate observations claim distances up to 6.5 Gpc.

The next generation of GW instruments like Einstein Telescope in Europe, Cosmic Explorer in the US and LISA in space. Will make a major step in sensitivity, and will be able to observe essential all black hole mergers in their frequency range (i.e. mergers of stellar mass BHs for ground based detectors, and supermassive BHs for LISA), going back to the formation of the first stars.

Such massive steps are possible because the sensitivity to GWs drops off inversely proportional to the distance, rather than the distance squared for EM observations.