Editing Hubble's law (section)

==== Recessional velocity ====

Suppose {{math|''R''(''t'')}} is called the ''scale factor'' of the universe, and increases as the universe expands in a manner that depends upon the [[Physical cosmology|cosmological model]] selected. Its meaning is that all measured proper distances {{math|''D''(''t'')}} between co-moving points increase proportionally to {{mvar|R}}. (The co-moving points are not moving relative to their local environments.) In other words:

<math display="block">\frac {D(t)}{D(t_0)} = \frac{R(t)}{R(t_0)},</math>

where {{math|''t''<sub>0</sub>}} is some reference time.<ref>Matts Roos, ''Introduction to Cosmology''</ref> If light is emitted from a galaxy at time {{math|''t''<sub>e</sub>}} and received by us at {{math|''t''<sub>0</sub>}}, it is redshifted due to the expansion of the universe, and this redshift {{mvar|z}} is simply:

<math display="block">z = \frac {R(t_0)}{R(t_\text{e})} - 1. </math>

Suppose a galaxy is at distance {{mvar|D}}, and this distance changes with time at a rate {{mvar|d<sub>t</sub>D}}. We call this rate of recession the "recession velocity" {{math|''v''<sub>r</sub>}}:

<math display="block">v_\text{r} = d_tD = \frac {d_tR}{R} D. </math>

We now define the Hubble constant as

<math display="block">H \equiv \frac{d_tR}{R}, </math>

and discover the Hubble law:

<math display="block"> v_\text{r} = H D. </math>

From this perspective, Hubble's law is a fundamental relation between (i) the recessional velocity associated with the expansion of the universe and (ii) the distance to an object; the connection between redshift and distance is a crutch used to connect Hubble's law with observations. This law can be related to redshift {{mvar|z}} approximately by making a [[Taylor series]] expansion:

<math display="block"> z = \frac {R(t_0)}{R(t_e)} - 1 \approx \frac {R(t_0)} {R(t_0)\left(1+(t_e-t_0)H(t_0)\right)}-1 \approx (t_0-t_e)H(t_0), </math>

If the distance is not too large, all other complications of the model become small corrections, and the time interval is simply the distance divided by the speed of light:

<math display="block"> z \approx (t_0-t_\text{e})H(t_0) \approx \frac {D}{c} H(t_0), </math>

or

<math display="block"> cz \approx D H(t_0) = v_r. </math>

According to this approach, the relation {{math|1=''cz'' = ''v''<sub>r</sub>}} is an approximation valid at low redshifts, to be replaced by a relation at large redshifts that is model-dependent. See [[#redshift|velocity-redshift figure]].