How is the Hubble constant determined from gravitational waves?

Question

We know there is a discrepancy between measurements of the Hubble constant, $H_0$. On one side there is the method of the Planck mission, where they use the CMB and the $\Lambda$CDM model to determine the Hubble constant. On the other side, they use standard candles, like Cepheid variables (for example: Riess et al., 2019) and red giants (for example: Freedman et al., 2020). The very precise measurements don’t overlap and thus we need a ‘solution’.

One way to improve our knowledge about $H_0$ is by using gravitational waves to determine the Hubble constant (Feeney et al., Arxiv version). They compare the measured ‘strength’ of the gravitational wave with the ‘strength’ of the wave when it was created to determine the distance to the source object. But how do they calculate/determine the ‘strength’ of the wave at the source? I know they also use an EMS observation for calculating the redshift (and that redshift converted to radial velocity divided by the distance gives $H_0$), but from where do they calculate the source-‘strength’ of the gravitational wave? How can they otherwise determine how much energy the wave has lost? I can’t seem to find the answer.

Answer 1

If you measure the gravitational waveform from an inspiralling binary, you can at any point measure the amplitude, instantaneous frequency and the rate of change of frequency. The last two give you the "chirp mass", which is related to the product and sum of the binary component masses.

The amplitude of the gravitational wave then depends on the chirp mass and the distance to the source. Thus we can estimate the (luminosity) distance.

If we then know the redshift of the source, or at least the redshift of the galaxy it is in, then an independent estimate of the Hubble parameter is possible. Therefore this can only be done for gravitational wave sources with identified host galaxies (so far this can only be done for GW sources that show an electromagnetic counterpart - merging neutron stars).

It is a bit more complicated than this, because the amplitude also depends on the inclination of the binary orbit to the line of sight. Fortunately, this can be constrained by measuring the relative amplitudes of the two possible gravitational wave polarisations (+ and ×) and this can be done if you have two interferometers with different orientation of their arms (like LIGO and VIRGO for example), which also helps to locate the GW source in the sky to look for electromagnetic counterparts.