= Hubble's law
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{title2=$\dot d=H(t)d$}
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= Hubble law
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<Hubble's law> states that the instantaneous rate of change of <proper distance in cosmology> between <comoving observers> is proportional to that distance:
$$
v_{\rm rec}(t):=\dot d(t)=H(t)d(t).
$$
This is exact for fixed <comoving distance> $\chi$ in an <FLRW metric>: differentiating $d=a\chi$ gives $\dot d=\dot a\chi=(\dot a/a)d$. The dot uses <cosmic time>, and both distances are compared on spatial slices at the same <cosmic time>. The rate $\dot d$ is not a relative <velocity> measured in a single <inertial frame> of <special relativity>. Thus the instantaneous relation is not an exact observed <redshift>-distance formula. The associated <Hubble flow> describes the ideal expansion; <peculiar velocities> describe departures from it.
For nearby sources, the observed <redshift> approximately satisfies
$$
cz\simeq H_0d
$$
when source and observer <peculiar velocities> are small compared with $H_0d$, additional <gravitational redshift> is negligible, and the changes in the <scale factor> and <Hubble parameter> during light travel are small. Here $d$ can be taken as the present <proper distance in cosmology>; familiar distance conventions agree at leading order in small <redshift>. Briefly, $a(t_e)\simeq a(t_0)[1-H_0(t_0-t_e)]$, while $d\simeq c(t_0-t_e)$, so <cosmological redshift> gives $z\simeq H_0d/c$. At greater <redshift>, use an expansion history and an explicit distance convention instead.
Stipulate the illustrative <Hubble constant> $H_0=70\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ and <speed of light> $c=299792.458\,\mathrm{km\,s^{-1}}$. A $\mathrm{Mpc}$ is a million <parsecs>. The following are computed examples, not a catalogue of observed <galaxies>:
* At $d=10\,\mathrm{Mpc}$, $H_0d=700\,\mathrm{km\,s^{-1}}$ and the approximate <cosmological redshift> is $z\simeq0.002335$.
* At $d=100\,\mathrm{Mpc}$, $H_0d=7000\,\mathrm{km\,s^{-1}}$ and the approximate <cosmological redshift> is $z\simeq0.02335$.
* At $d=200\,\mathrm{Mpc}$, $H_0d=14000\,\mathrm{km\,s^{-1}}$ and the approximate <cosmological redshift> is $z\simeq0.04670$.
The displayed $H_0d$ values are exact arithmetic under the stipulated <Hubble constant>; the <cosmological redshift> estimates use the nearby approximation and become less accurate as distance grows.
\Image[/hubble-law-and-redshift.png]
{title=Computed <Hubble's law> examples and a <wavelength> doubling from <cosmological redshift>}
{description=Left: the stipulated <Hubble constant> gives the instantaneous rates $700,7000,14000\,\mathrm{km\,s^{-1}}$; the annotated <cosmological redshifts> use $z\simeq H_0d/c$. Right: a schematic of successive <wave> crests shows a <wavelength> of $500\,\mathrm{nm}$ becoming $1000\,\mathrm{nm}$ when the <scale factor> doubles. The right panel is not a <velocity> conversion or a continuation of the nearby approximation.}
<Hubble's law> turns a nearby observed <redshift> into an approximate distance indicator and makes the uniformity of <cosmic expansion> tangible: twice the <proper distance in cosmology> means twice the instantaneous separation rate. Recentring the ideal <Hubble flow> on another <comoving observer> preserves the same proportionality, as illustrated under <scale factor>, so recession in every direction does not select a central observer.
In his 1929 PNAS paper, Edwin Hubble presented a relation between the distances and <radial velocities> of extragalactic objects. His fits gave coefficients of $465$ and $513\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ for 24 individual objects and nine groups respectively, and he used about $500\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ as a rounded representative value. Those historical fits used the paper's distance estimates; they are distinct from the stipulated modern-scale illustration above. https://www.astro.puc.cl/~rparra/tools/PAPERS/hubble_1929.pdf[Hubble (1929), original PNAS paper, pp. 170-171]{ref}.
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