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Recessional velocity

Suppose R (t) is called the scale factor of the universe, and increases as the universe expands in a manner that depends upon the cosmological model selected. Its meaning is that all measured proper distances D (t) between co-moving points increase proportionally to R. (The co-moving points are not moving relative to each other except as a result of the expansion of space.) In other words:

D (t) D (t 0) = R (t) R (t 0), {\displaystyle {\frac {D(t)}{D(t_{0})}}={\frac {R(t)}{R(t_{0})}},}

where t is some reference time. If light is emitted from a galaxy at time t and received by us at t, it is redshifted due to the expansion of space, and this redshift z is simply:

z = R (t 0) R (t e) − 1. {\displaystyle z={\frac {R(t_{0})}{R(t_{\text{e}})}}-1.}

Suppose a galaxy is at distance D, and this distance changes with time at a rate d D. We call this rate of recession the "recession velocity" v:

v r = d t D = d t R R D. {\displaystyle v_{\text{r}}=d_{t}D={\frac {d_{t}R}{R}}D.}

We now define the Hubble constant as

H ≡ d t R R, {\displaystyle H\equiv {\frac {d_{t}R}{R}},}

and discover the Hubble law:

v r = H D. {\displaystyle v_{\text{r}}=HD.}

From this perspective, Hubble's law is a fundamental relation between (i) the recessional velocity contributed by the expansion of space 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 z approximately by making a Taylor series expansion:

z = R (t 0) R (t e) − 1 ≈ R (t 0) R (t 0) (1 + (t e − t 0) H (t 0)) − 1 ≈ (t 0 − t e) H (t 0), {\displaystyle 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}),}

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:

z ≈ (t 0 − t e) H (t 0) ≈ D c H (t 0), {\displaystyle z\approx (t_{0}-t_{\text{e}})H(t_{0})\approx {\frac {D}{c}}H(t_{0}),}

or

c z ≈ D H (t 0) = v r. {\displaystyle cz\approx DH(t_{0})=v_{r}.}

According to this approach, the relation cz = v is an approximation valid at low redshifts, to be replaced by a relation at large redshifts that is model-dependent. See velocity-redshift figure.