I am looking this paper by LIGO, and it claims at the very beginning,

At nearby distances $(d \lesssim 50 \text{ Mpc})$ [the Hubble constant] is well approximated by the expression $$v_H = H_0d, \tag{1}$$ where $v_H$ is the local “Hubble flow” velocity of a source, and $d$ is the distance to the source.

Are they implying that Eq. (1) is not valid at larger distances, and if so, why is this true? My cosmology book (Ryden) defines $H_0$ as $\dot{a}/a$, where $a$ is the scale factor, so it seems like Eq. (1) above must be true at all distances by that definition.

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    $\begingroup$ Ryden is wrong. $H_0 = \dot{a}_0/a_0$. $\endgroup$ – Rob Jeffries May 8 at 7:03

You need to think carefully about what is meant by $v$ and $d$ in your equation.

Whilst it is true that Hubble's law holds true at all distances, those distances need to be proper distances and the velocity measured needs to be the rate of change of proper distance.

A proper distance is one that establishes the separation between two objects at the same cosmic epoch. In practice we are always looking back in time and measuring the distance and velocity of an object in the past.

There are various ways of measuring and expressing distances in cosmology. The gravitational wave "standard siren" technique described in the paper referred to actually gives a "luminosity distance" to the source, which in an expanding universe will be larger than the proper distance by a factor of $\sim (1+z)$.

For the paper in question, they argue that given the other uncertainties (the GW standard-siren distance is uncertain by $\sim 10$%) that this difference is unimportant for $d<50$ Mpc because $z < 0.012$. At larger distances and higher redshifts then a plot of $v$ vs luminosity distance would be a curve, rather than a straight line, that depended on the cosmological parameters $\Omega_m$, $\Lambda$ etc.


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