How many GW events do we need to measure the Hubble parameter with a precision 1-5%? How do we calculate that number?
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2 Answers
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Consider the following figure from Ezquiaga and Zumalacárregui (https://arxiv.org/abs/1807.09241).
These authors project that with 15 multi-messenger binary neutron star detections (like GW170817), the uncertainty on the Hubble constant from standard sirens will be roughly comparable to HST+GAIA2 in 2018. (about 5%). That forecast is based on the following paper by Nissanke et al: https://arxiv.org/abs/1307.2638. There is a careful discussion in that paper about the assumed properties of BNS signals that could be found as multi-messenger detections; it is important to understand that these kinds of assumptions about the population of BNS signals that can be detected are built into the estimate shown in the figure.
Exactly when 15 multi-messenger BNS events is very hard to predict; it will depend on the operating schedules and sensitivities of Advanced LIGO and Virgo, it will depend on the unknown distribution of BNS systems in the Universe, and it will depend on what specific events Nature provides from those distributions during the time LIGO and Virgo are observing. According to Table 5 of https://dcc.ligo.org/LIGO-P1200087/public, between 0 and 62 BNS events would be excepted in the next observing run (O4) which is due to start in 2023; only some fraction of those would be multi-messenger events. Suffice to say, it is theoretically possible, but would require quite a bit of luck, to reach 15 multi-messenger BNS detections during O4. Beyond that it becomes increasingly hard to make predictions.
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Measure of $H_0$ is relatively straight forward of a gravitational wave event has a well defined electromagnetic counterpart allowing measurement of the redshift. @Andrew's answer deals with the ``bright siren'' scenario.
If no counterpart is found for a gravitational wave event, it is still possible to try to infer the value of $H_0$. There various approaches to this, but the simplest (an easiest to explain) is the so called statistical dark siren approach. In this approach, the localization of the gravitational wave event is cross-correlated with redshift data from galaxies catalogues, to assign a redshift. If done correctly (see 2212.08694 for an in depth discussion), this can provide an unbiased measurement of $H_0$.
Of course, per event this method is a lot less potent then the bright siren approach. However, this is in part compensated by the fact that we expect a lot more gravitational wave events without EM counterpart then with. The statistical error of this method depends not only on the number of events, but also on how well those event can be localized, and how complete/accurate the used galaxy catalogues are. One can try to provide projections for those, as done in this paper. (Under optimistic assumptions) they estimate an uncertainty that scales as $40\%/\sqrt{N}$ where $N$ is the number of GW170817-like events. To get an uncertainty of 5%, we would need $8^2 = 64$ events. To reach 1% would need $40^2 = 1600$ events.
The general expectation is that measurement of $H_0$ using bright sirens will out pace the dark siren methods. Whether that ends up being the case depends on how lucky we got to find a counterpart with the first neutron star merger we found.
