gravitational waves - Redshift of merging black holes

How did they found that the gravitational waves where emitted at redshift $z=0.09$?

I understand the measurement of redshift for an electromagnetic wave where we have measured in a lab various transitions and therefore we can make a comparison with the wavelength we receive.

But how can they manage to get the redshift for the emitter of gravitational waves, since we have no reference?

Answer

As stated in the LIGO discovery paper (pdf), the event is placed at $410^{+160}_{-180}\ \mathrm{Mpc}$ luminosity distance, equivalent to a redshift of $z = 0.09^{+0.03}_{-0.04}$. This gives a clue as to how one measures the distance for this event.

If we know how intrinsically luminous an object (like a star, or a supernova) is, we can compare this to how bright it seems and recover a distance via the standard inverse-square law. The distance we get is by definition the luminosity distance. For this detection, the same principle applies, since the simulations predict the intrinsic strength of the signal.

Actually, we also can leverage frequency information. Again, we have numerical simulations that predict waveforms, and the waveform itself will be redshifted in the same way as any other signal propagating at the speed of light.

In practice, one takes the entire waveform and a bank of numerical simulations, and does a statistical analysis to see how well the signal matches models, and what self-consistent distance/redshift make it fit. This is detailed in Veitch et al. 2015 Phys. Rev. D 91 042003.

Note there is some degeneracy with inclinations. The detectors are not monopole antennae, but at least with two of them we can sort of localize the source on the sky to figure out what fraction of the power is actually absorbed. A more stubborn degeneracy lies in the orientation of the astrophysical system with respect to our line of sight. Since gravitational waves are (at least) quadrupolar in order, an edge-on system nearby will be hard to distinguish from a face-on system further away. This is at least part of the reason for the large uncertainties.