Editing Hubble's law (section)

=== Redshift velocity and recessional velocity ===
Redshift can be measured by determining the wavelength of a known transition, such as hydrogen α-lines for distant quasars, and finding the fractional shift compared to a stationary reference. Thus, redshift is a quantity unambiguously acquired from observation. Care is required, however, in translating these to recessional velocities: for small redshift values, a linear relation of redshift to recessional velocity applies, but more generally the redshift-distance law is nonlinear, meaning the co-relation must be derived specifically for each given model and epoch.<ref name="Harrison">{{cite journal |last=Harrison |first=E. |date=1992 |title=The redshift-distance and velocity-distance laws |journal=[[The Astrophysical Journal]] |volume=403 |pages=28–31 | bibcode=1993ApJ...403...28H |doi=10.1086/172179|doi-access=free }}</ref>

==== Redshift velocity ====
The redshift {{mvar|z}} is often described as a ''redshift velocity'', which is the recessional velocity that would produce the same redshift {{em|if}} it were caused by a linear [[Doppler effect]] (which, however, is not the case, as the velocities involved are too large to use a non-relativistic formula for Doppler shift). This redshift velocity can easily exceed the speed of light.<ref name="Madsen">{{cite book |last=Madsen |first=M. S. |date=1995 |title=The Dynamic Cosmos |url=https://books.google.com/books?id=_2GeJxVvyFMC&pg=PA35 |page=35 |publisher=[[CRC Press]] |isbn=978-0-412-62300-4 }}</ref> In other words, to determine the redshift velocity {{math|''v''<sub>rs</sub>}}, the relation:

<math display="block"> v_\text{rs} \equiv cz,</math>

is used.<ref name="Dekel">{{cite book |last1=Dekel |first1=A. |last2=Ostriker |first2=J. P. |date=1999 |title=Formation of Structure in the Universe |url=https://books.google.com/books?id=yAroX6tx-l0C&pg=PA164 |page=164 |publisher=[[Cambridge University Press]] |isbn=978-0-521-58632-0 }}</ref><ref name="Padmanabhan">{{cite book |last=Padmanabhan |first=T. |date=1993 |title=Structure formation in the universe | url=https://books.google.com/books?id=AJlOVBRZJtIC&pg=PA58 |page=58 |publisher=[[Cambridge University Press]] |isbn=978-0-521-42486-8 }}</ref> That is, there is {{em|no fundamental difference}} between redshift velocity and redshift: they are rigidly proportional, and not related by any theoretical reasoning. The motivation behind the "redshift velocity" terminology is that the redshift velocity agrees with the velocity from a low-velocity simplification of the so-called [[Relativistic Doppler effect|Fizeau–Doppler formula]]<ref name="Sartori">{{cite book |last=Sartori |first=L. |date=1996 |title=Understanding Relativity |page=163, Appendix 5B |publisher=[[University of California Press]] |isbn=978-0-520-20029-6 }}</ref>

<math display="block">z = \frac{\lambda_\text{o}}{\lambda_\text{e}}-1 = \sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}-1 \approx \frac{v}{c}.</math>

Here, {{math|''λ''<sub>o</sub>}}, {{math|''λ''<sub>e</sub>}} are the observed and emitted wavelengths respectively. The "redshift velocity" {{math|''v''<sub>rs</sub>}} is not so simply related to real velocity at larger velocities, however, and this terminology leads to confusion if interpreted as a real velocity. Next, the connection between redshift or redshift velocity and recessional velocity is discussed.<ref name="L_Sartori">{{cite book|last=Sartori |first=L. |date=1996 |title=Understanding Relativity |pages=304–305 |publisher=[[University of California Press]] |isbn=978-0-520-20029-6 }}</ref>

==== 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]].