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A certain number of gravitational wave events have been detected. Is it possible to know how far away the mergers that caused those gravitational wave events are?

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Yes, it is possible to calculate (within an error range) the distance of observed gravitational wave events. It is known that a variety of parameters will affect how the amplitude and frequency of the observed gravitational waves will change over time as recorded in the "chirp" event from the interferometers: the parameters include distance of the event, the mass of each of the colliding objects, the angular momentum of each of the colliding objects, the orientation of the objects' angular momentum vectors with respect to each other and with their orbital plane. With general relativity, you can build a model that calculates the expected "chirp" given a value for all these parameters; when a chirp is observed, it is possible to determine the combination of these parameters that result in a chirp that best matches the observation.

The effect of a larger distance parameter is to decrease the amplitude of the expected waves from colliding objects of a given mass, as well as to "slow down" the entire event due to cosmological red shift.

From GWTC-1: A Gravitational-Wave Transient Catalog of Compact Binary Mergers Observed by LIGO and Virgo during the First and Second Observing Runs

Gravitational waves from compact binaries carry information about the properties of the source such as the masses and spins. These can be extracted via Bayesian inference by using theoretical models of the GW signal that describe the inspiral, merger, and ringdown of the final object for BBH [23–30] and the inspiral (and merger) for BNS [31–33]. Such models are built by combining post-Newtonian calculations [34–38], the effective-one-body formalism [39–44], and numerical relativity [45–50].

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    $\begingroup$ Decreasing amplitude -- is this analogous to sounds getting softer due to distance, or is it some other process? $\endgroup$ – Barmar Jan 15 at 18:13
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    $\begingroup$ @Barmar yes, the normal thing — the further away you are, the more area the wavefront has to spread out over. $\endgroup$ – hobbs Jan 16 at 5:52
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    $\begingroup$ Thought so, but just making sure there's not some additional GR effect involved. $\endgroup$ – Barmar Jan 16 at 5:53
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Yes, it's possible, but less straightforward than for "normal" objects.

If the optical counterpart of the GW signal is located, as in the case of GW170817, the distance can be inferred by standard methods of observing the redshift of its host galaxy.

If not, the luminosity distance $d_L$ can still be inferred because the amplitude of the GW signal scales inversely with the $d_L$. This can then be converted to a redshift, assuming some cosmology. This was done for the first ever GW detection GW150914 (Abbott et al. 2016).

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    $\begingroup$ So there are GW events we don't know the distance of? $\endgroup$ – usernumber Jan 14 at 16:23
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    $\begingroup$ @usernumber We always have some clue of the luminosity distance (as far as I know), but they typically have huge error bars, of the order of 50% or so. I think the problem is that the distance correlates with the orbital plane, but system typically precesses during the inspiral, so it's difficult to get an unambiguous value. $\endgroup$ – pela Jan 14 at 19:52
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    $\begingroup$ @Pela I believe that multiple detectors can significantly clear up the orientation ambiguity. $\endgroup$ – Rob Jeffries Jan 17 at 8:12
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    $\begingroup$ @RobJeffries Ah yes, I hadn't thought of that. It makes sense, although it doesn't really seem like the uncertainties went down when VIRGO joined. $\endgroup$ – pela Jan 17 at 13:44
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    $\begingroup$ If anything determining the distance of a GW event is a lot more straightforward then for EM events. $\endgroup$ – mmeent Jan 18 at 0:05
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To answer the question in your title (by following the links in the other answers):

GW170817 (two neutron stars): 40 Mpc

GW150914 (two black holes): 410 (+160 or -180) Mpc

antlersoft's link (GWTC-1: A Gravitational-Wave Transient Catalog of Compact Binary Mergers Observed by LIGO and Virgo during the First and Second Observing Runs): distances range from 320 (+120 -110) Mpc to 2840 (+1400 -1360) Mpc for binary black hole mergers.

One Mpc (megaparsec) is about 3.26 million light years.

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This is additional to the other answers. We now have three GW detectors (LIGO x2 + VIRGO). This allows the direction of the event to be deduced, by the relative timing of the arrival of the chirp, which is an effectively planar wave passing through the Earth at the speed on light. More accurately, deduce one of two possible directions: towards the event ot towards its celestial antipode (a fourth detector would eliminate this ambiguity).

I don't know the acccuracy to which this direction can be deduced. However, if one assumes that a black hole merger would not be taking place in intergalactic space, it might serve alongside the other information deduced from the chirp to identify the galaxy in which it took place, even if there were no visible light emission.


Example: GW170817 and the relation between direction and distance

There are two ways by which improved knowledge/estimation of the direction can improve estimates of distance. Both of these ways are demonstrated in the detection of GW170817, a signal from a binary neutron star merger.

  • 1) Follow up searches for sources emitting light. In the case of GW170817 the searches for a light signal helped to pinpoint the origin of the source (NGC 4994) more precisely. This allows to improve the estimates of distance by including estimates of distance based on light sources. (those searches for a light signal were helped by the estimates of the location based on the gravitational wave signals)
  • 2) Relation between source position and observed detector amplitude. The amplitude of the received signal is depending on several factors like, position of the source in the sky, power/energy of the source, and distance of the source. By the relationship between the amplitude of the received signal and the distance to the source an estimate of the source distance can be made, but the better the knowledge or estimates about the other factors involved (among which the position) the better the estimate of the distance will be.

    The amplitude of the waves will be larger when the source is closer, but also when the direction of the source is more perpendicular to the arms of the detector (and vice versa the amplitude will be smaller for further sources, but it also happens when the source is at an angle to the detector).

    This means that the amplitude of the signal is relating to (at least) two different unknown parameters. Being able to independently pinpoint one of those parameters (the location), will allow to better estimate the other parameter (the source distance).


Detailed article about pinpointing parameters: https://arxiv.org/abs/gr-qc/9402014

How using three detectors LIGO + VIRGO improved location for GW170817: https://www.ligo.caltech.edu/page/press-release-gw170817 (see the image for comparison with other sources that only used the two LIGO detectors and have an estimate of location in a ring shape)

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    $\begingroup$ Does this help to determine the distance at which the merger occurred? $\endgroup$ – usernumber Jan 16 at 10:51
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    $\begingroup$ It certainly does if there is only one galaxy at a plausible distance in the narrow cone in which the observed chirp might have emanated. Not if there are many. $\endgroup$ – nigel222 Jan 16 at 11:00
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    $\begingroup$ You're right that three detectors greatly reduces the positional accuracy, but we're still talking many degrees on the sky, covering thousands of galaxies. An optical counterpart is needed to really see the position. $\endgroup$ – pela Jan 16 at 13:39
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    $\begingroup$ @usernumber in a (not yet approved) edit I have added some information explaining how this (distance) helps to determine the distance. $\endgroup$ – Sextus Empiricus Jan 16 at 17:07
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    $\begingroup$ You haven't quite got it. Multiple detections can tell you the orientation of the binary by differing sensitivity to the two polarisation directions by detectors with differing orientations. $\endgroup$ – Rob Jeffries Jan 17 at 8:09
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The amplitude of a detected gravitational wave depends on a number of factors - the luminosity of the source (which in turn depends on the masses and orbital period of the merging binary system), the orientation of the binary system with respect to the line of sight (since gravitational waves are emitted highly anisotropically, the inclination of the binary system plays a crucial role), the direction of the GW source with respect to the detectors (since the maximum signal in the interferometer occurs when the source is "overhead" with respect to the plane of the interferometer) and finally, the reciprocal of the distance.

In practice all these things are fitted simultaneously based on the signals found in one or more detectors, but the principle of detection is the following:

Both the masses and period can be estimated simultaneously by following the time evolution of the GW signal. The signal instantaneously has a frequency twice that of the binary and the rate of change of frequency yields something called the "chirp mass", which is what the source luminosity depends on.

The inclination of the binary system is estimated from the polarisation of the GW signal. GWs come in two polarisations, but these are not emitted isotropically, so the ratio tells you the inclination. The polarisation of the received signal is found by having detectors with interferometer arms rotated at different angles with respect to each other. The two LIGO detectors are nearly aligned, so poor at determining polarisation and inclination. Thus distance estimates based only on LIGO only have big error bars. The addition of VIRGO had made an enormous improvement.

The direction on the sky is also important (though not as important as inclination, factor of $2$ vs factor of $2\sqrt{2}$ in detected amplitude). Direction can also be roughly determined with two detectors from time delays between signals, but even better with three detectors and can be pinpointed exactly if an optical counterpart can be found.

So with all these things done, the distance is finally found. In the best cases, it is found to about 10% (three detectors, detecting it and an optical counterpart), for two detectors and no counterpart, the precision is more like a factor of two, mainly due to the inability to constrain the polarisation of the signal and the inclination of the binary.

Details:

The relationship between chirp mass, frequency and the rate of change of frequency is approximately given by $$\frac{df}{dt} = \left(\frac{96}{5}\right)\left(\frac{G\mathcal{M}_c}{c^3}\right)^{5/3}\pi^{8/3} f^{11/3}\, ,$$ where $f$ is the frequency and $\mathcal{M}_c$ is the chirp mass. Thus by measuring the frequency and rate of change of frequency (the chirpiness of the chirp!) we estimate the chirp mass.

GWs come in two polarisations (labelled as plus and cross). The amplitude of the signal received by a GW detectors in each of the two polarisations is given by $$h_+= \frac{2c}{D} \left(\frac{G \mathcal{M}_c}{c^3}\right)^{5/3} \left(\frac{f}{2\pi}\right)^{2/3}\left(1 + \cos^2 i\right) \cos 2\phi(t),$$ $$h_\times = \frac{4c}{D} \left(\frac{G \mathcal{M}_c}{c^3}\right)^{5/3} \left(\frac{f}{2\pi}\right)^{2/3}(\cos i) \sin 2\phi(t),$$ where $D$ is the distance to the source, $\phi(t)$ is the phase of the binary orbit, and $i$ is the orbital inclination of the binary to the line of sight ($i=0^{\circ}$ means a face-on orbital plane and both polarisations have equal amplitude). If $i = 90^{\circ}$ (edge-on) then only $h_+$ polarisation waves are emitted towards the observer and the amplitude of these is reduced by at least a factor of 2 with respect to the face-on case, depending on the detector orientation. Only by measuring the ratio of the amplitudes of the two different polarisations can $i$ be estimated and the measured amplitude be directly converted into a distance.

The way this is done is to have separate interferometers whose arms are not in the same spatial orientation. These will have different sensitivities to the plus and cross polarisations. For example if the arms were rotated by 45 degrees with respect to each other then a face-on binary would produce the same signal in both detectors, but if the orbit is viewed edge-on then a detector with arms at 45 degrees to the line defined by the projected orbital plane would see nothing.

If this polarisation information is unavailable, then one has to just guess. The guess is that binaries tend to be more likely to be edge on than face-on and in fact the average value of $i$ is about 60 degrees if the binary orientation is random.

The orientation of the detectors with respect to the line of sight to the source is also required. Imagine the plus polarisation. If the source is directly "overhead", then this will produce an equal response in both interferometer arms. If you now place the source in the plane of the detector instead, then it will only produce a response in one of the two arms of the interferometer leading to a factor of two reduction in signal.

A reasonably accessible account of all this can be found in Holz, Hughes & Schutz (2018).

A more technical discussion as applied to GW170817 (a merging neutron sar binary, seen by 3 detectors) is given by Abbott et al. (2017), where the distance was constrained from the gravitational wave signals alone to be $43.8^{+2.9}_{-6.9}$ Mpc. This paper notably contains the sentences

The measurement of the GW polarization is crucial for inferring the binary inclination.

One of the main sources of uncertainty in our measurement of H0 is due to the degeneracy between distance and inclination in the GW measurements. A face-on or face-off binary far away has a similar gravitational-wave amplitude to an edge-on binary closer in.

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