How to know the distance of the black hole merger?

Question

I have a question about the detection of the gravitational waves by LIGO. When there is no electromagnetic wave coming from the black holes, then how do they know in which distance it is?


Answer 1

This is not discussed very clearly in the 2015 LIGO paper in which they announced the first observation of gravitational waves. A better explanation is in this review article at p. 20.

There is a quantity $M$ called the chirp mass, which can be expressed in the following three ways:

$$M=\operatorname{function}(m_1,m_2),\qquad[1]$$

$$M=\operatorname{function}(hr,f),\ \text{and}\qquad[2]$$

$$M=\operatorname{function}(f,\dot{f}),\qquad[3]$$

where $h$=amplitude, $r$=distance, and $f$=frequency. Since $h$, $f$, and $\dot{f}$ can all be measured, it follows that one can determine $r$ and constrain the masses. These equations are based on general relativity's predictions of how the process should go. I think these closed-form equations are low-frequency approximations, so in reality they actually estimate the parameters by comparing with numerical simulations. There is also a relation of the form

$$Mm_1m_2=\operatorname{function}(t_\text{chirp},R),\qquad[4]$$

where $t_\text{chirp}$ is the length of the chirp in time an $R$ is the distance between the black holes. I think this is what allows one to constrain the individual masses rather than just $M$. But note that the individual masses are not very well determined in the 2015 event. The physics of equation [4] is simply that if one mass is very small and the other is very big, then radiation is weak, so the orbit persists for a long time.