Determining the Hubble Constant the Bernard Schutz way
In my short post about Monday’s announcement of the detection of a pair of coalescing neutron stars (GW170817), I mentioned that one of the results that caught my eye in particular was the paper about using such objects to determine the Hubble constant.
Here is the key result from that paper, i.e. the posterior distribution of the Hubble constant H0 given the data from GW170817:
You can also see latest determinations from other methods, which appear to be in (slight) tension; you can read more about this here. Clearly the new result from GW170817 yields a fairly broad range for H0 but, as I said in my earlier post, it’s very impressive to be straddling the target with the first salvo.
Anyway, I just thought I’d mention here that the method of measuring the Hubble constant using coalescing binary neutron stars was invented by none other than Bernard Schutz of Cardiff University, who works in the Data Innovation Institute (as I do). The idea was first published in September 1986 in a Letter to Nature. Here is the first paragraph:
I report here how gravitational wave observations can be used to determine the Hubble constant, H 0. The nearly monochromatic gravitational waves emitted by the decaying orbit of an ultra–compact, two–neutron–star binary system just before the stars coalesce are very likely to be detected by the kilometre–sized interferometric gravitational wave antennas now being designed1–4. The signal is easily identified and contains enough information to determine the absolute distance to the binary, independently of any assumptions about the masses of the stars. Ten events out to 100 Mpc may suffice to measure the Hubble constant to 3% accuracy.
In in the paper, Bernard points out that a binary coalescence — such as the merger of two neutron stars — is a self calibrating `standard candle’, which means that it is possible to infer directly the distance without using the cosmic distance ladder. The key insight is that the rate at which the binary’s frequency changes is directly related to the amplitude of the gravitational waves it produces, i.e. how `loud’ the GW signal is. Just as the observed brightness of a star depends on both its intrinsic luminosity and how far away it is, the strength of the gravitational waves received at LIGO depends on both the intrinsic loudness of the source and how far away it is. By observing the waves with detectors like LIGO and Virgo, we can determine both the intrinsic loudness of the gravitational waves as well as their loudness at the Earth. This allows us to directly determine distance to the source.
It may have taken 31 years to get a measurement, but hopefully it won’t be long before there are enough detections to provide greater precision – and hopefully accuracy! – than the current methods can manage!
Above all, congratulations to Bernard for inventing a method which has now been shown to work very well!
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In the Dark 
October 19, 2017 at 1:19 pm
Thank you, Peter!
October 19, 2017 at 5:07 pm
Well, among other things, they built and serviced the Hubble Space Telescope 5 times to help do the Distance Ladder, so it’s maybe a “well-earned” tie…
October 19, 2017 at 8:37 pm
Wow. As a self-calibrating standard candle, such mergers are a bit like a Cepheid variable, is that right? Is Bernard the new Henrietta Leavitt?
October 20, 2017 at 8:37 am
Doesn’t this observation also use the EM counterpart and not exclusively with GW information alone?
October 20, 2017 at 2:43 pm
To measure the Hubble constant you need a distance and a redshift. The GW observation gives you the distance. But does it also tell you the redshift? Or do they need to spot the optical counterpart for that?
Do gravitational waves get redshifted by cosmic expansion the same way as light does? Does this introduce some degeneracy with the self-calibration (using the frequency of the gravitational waves to determine the intrinsic loudness of the merger)?
November 13, 2017 at 7:59 pm
Hmmm my data and stats is a bit rusty here but shouldn’t they perform a logarithm on that graph to try and remove some of that right skewness?
In response to Alex, Yes GW’s also get redshifted as the medium through which they travel undergoes the same expansion that affects EM radiation (presumably some form of quantum field).
Good question regarding the impact of frequency redshift on the accuracy of the Amplitude calculations. I think we need to read Bernards Schultz paper to know for sure, but let me take a run at this.
Basically, we know that the Energy of a Wave is proportional to its Amplitude squared (From memory this is derived from looking at complicated wave equations of the kinetic and potential energy of points in a standing wave, but would appreciate a source from others).
E ~ A^2
We also know that Amplitude drops off with 1/r^2 (inverse square law) and therefore,
A drops with 1/r.
We measure the amplitude of the GW at earth, and we make an estimate for the intrinsic amplitudes using the Frequency of the binary orbit (Schultz). However, that frequency is also red-shifted as Alex mentions above.
Reading this paper: https://www.darkenergysurvey.org/wp-content/uploads/2017/10/Pub_Hubble.pdf
They say “Our measurement combines the distance to the source inferred purely from the gravitational-wave signal with the recession velocity inferred from measurements of the redshift using electromagnetic data. This approach does not require
any form of cosmic “distance ladder” (Freedman et al. 2001)”
I find this hard to justify as if they are using the red-shift data from EM sources this does not make the GW estimates of H0 independent from the EM measurements made to date.
November 13, 2017 at 8:23 pm
More reading has turned up a paper by Del Pozzo where he outlines how H0 can be estimated to approx ~5% without any optical counterparts at all (just by looking at multiple GW sources).
https://arxiv.org/abs/1108.1317
June 25, 2020 at 6:16 am
GRB170817A/GW170817 continues to require modified afterglow models, which affect priors (often without this being considered by authors). Lamb et al. 2020 https://arxiv.org/pdf/2005.12426.pdf finds that the rate of decline of luminosity of the AT2017gfo putative source is declining. Due to GW170817 systematic errors and various degeneracies, the calculation of H0 utilizing LIGO-Virgo is like stone soup; Lamb et al 2020 mentions degeneracy in all light curves only once, by citing Nakar and Piran 2020 https://arxiv.org/pdf/2005.01754.pdf, who found 16° angle of inclination only determinable model-free value. H0 cannot be calculated from GW170817 alone, and it certainly cannot be calculated from GRB170817A/AT2017gfo light curves.
https://fulguritics.blogspot.com/2018/06/gw170817-occurs-at-green-bar.html
https://fulguritics.blogspot.com/2018/10/why-is-information-on-ngc-4993.html
https://fulguritics.blogspot.com/2018/10/a-short-grb-analogue-with-multi.html
March 22, 2021 at 12:49 pm
[…] time ago I blogged about how one can use gravitational waves to estimate the Hubble Constant, H0. Well, about a month ago the LIGO people produced a user-friendly update on progress in that regard […]