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As far as I can tell, groups that measure the Hubble Constant usually measure galaxy recession speeds $v$ and galaxy distances $d$. Then they plot one versus the other and the best-fit slope is the Hubble Constant (or its inverse).

Except what they plot isn't really $v$ vs $d$ (or $d$ vs $v$), it's $cz$ versus $d$ (or $d$ versus $cz$), where $c$ is the speed of light and $z$ the galaxy redshift.

But $v=cz$ only at small $z$ values. So a straight line in $cz$ vs $d$ is expected only at small $z$ values.

At large $z$ value, $v(z)$ is a function of the "curvature of the universe", which isn't exactly known.

Does this mean that when experimental cosmologists try to measure the Hubble constant using a $v$ vs $d$ plot method, they must restrict themselves to low $z$ values?

Does this also mean that when high $z$ values are used to determine the Hubble constant, the results are biased due to the non- linearity of $v(z)$? And below what values of $z$ is it safe to assume that this bias is insignificant?

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Yes. The Hubble constant refers specifically to the current value of the Hubble parameter, which is not a constant on cosmological time scales. As such, its measurement is indeed restricted to low redshifts.

Other measurements are possible, such as measurements on the CMB. Such measurements place restrictions on cosmological parameters generally. But they are not direct measurements of Hubble's constant and they leave open the possibility of different models possible under the assumptions of general relativity, including the possibility of void models in which the Hubble constant is specifically a local parameter. See The effects of turbulence generated in Big Bang nucleosynthesis and references therein

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Do all measurements of the Hubble Constant have to be made at low redshift values?

No. Take for example the cosmic microwave background (CMB) with a redshift of roughly $z \approx 1100$. The Planck satellite has measured the anisotropies from the CMB across the full sky. Assuming a flat $\Lambda$CDM cosmology you can estimate the cosmological parameters from the Planck data with Monte Carlo simulations. For the results checkout table 1 in Planck 2018 results. VI. Cosmological parameters. The Hubble parameter can be derived from the sampling parameters and was determined to be $H_0 = (67.36 \pm 0.54)\,\mathrm{km/s/Mpc}$.

At large $z$ value, $v(z)$ is a function of the "curvature of the universe", which isn't exactly known.

This isn't really the issue. You could assume a flat (zero curvature) cosmology and determine the Hubble parameter under that assumption, or you could keep the curvature density parameter $\Omega_{K,0}$ as a free parameter that you sample over jointly with the Hubble parameter. To get $H_0$ you can then marginalise over $\Omega_{K,0}$ which will increase the uncertainty on $H_0$.

Does this mean that when experimental cosmologists try to measure the Hubble constant using a $v$ vs $d$ plot method, they must restrict themselves to low $z$ values?

"must restrict", no, but depending on what you call "low $z$" you might be naturally restricted by the maximum age of the objects. For that kind of method you typically require some type of astrophysical object, such as supernovae (SN1a), cepheids, stars from the tip of the red giant branch (TRGB) etc. Even if your observations were not limited by the observation depth (since you won't see the objects if they are too faint), all these astrophysical objects obviously are limited in their redshift range, since non of these even existed prior to redshift $z \sim 10$.

Yet another option to determine the Hubble parameter $H_0$ is via baryon acoustic oscillations (BAO), which just like the CMB reaches back to a redshift $z \approx 1100$.

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  • $\begingroup$ -1 without any comment to specify what's wrong? Wow, thanks... $\endgroup$
    – Zaus
    Aug 8, 2020 at 19:06

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