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I am trying to get the Hubble constant from gravitational wave events. On the GWOSC website, for all events, redshift and Luminosity distances are mentioned. Now, by this information, I can get my Hubble constant by the equation $cz=H_0 D_l$. By the above logic, I applied it to some events, but I am getting $H_0$ to be around $47$ km/s Mpc, which is very far away from around its actual predicted value. For example taking the GW190413_134308 event, on GWOSC, the redshift for this event is given as $0.71$ and the luminosity distance is given as $4450$. By the above equation, I am getting the $H_0$ value to be $47.86$ km/s Mpc.

GWOSC website: https://www.gw-openscience.org/eventapi/html/GWTC-2/GW190413_134308/v1/

I know I am doing something horribly wrong, can anyone please point this out to to me?

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You can't estimate Hubble's constant using the method you propose, for two reasons.

(1) Redshift and luminosity distance are only linearly related by the Hubble parameter at small redshifts.

(2) You need an independent estimate of the redshift (e.g. from a spectrum of a host galaxy or something - see here for example or Abbott et al. 2017) to get the Hubble parameter from individual gravitational wave sources. That is because the inspiralling gravitational wave signature looks the same whatever the redshift of the source but does depend on the chirp mass, binary orientation and luminosity distance. For the vast majority of gravitational wave events an optical counterpart isn't available and so the redshift is actually estimated from the luminosity distance!

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  • $\begingroup$ This answer could be improved by explaining why redshift cannot be determined directly from a gravitational wave event. $\endgroup$
    – mmeent
    Nov 18 '21 at 9:51
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The comoving radial distance goes into Hubble's law,

A redshift of 0.71 and a Luminosity distance of 4450 Mpc corresponds to a comoving distance of about 2590 Mpc (according to Ned Smiths cosmology calculator, and assuming a flat cosmology). Which should give you something closer to H₀=0.7

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