v_redshift = H₀ r_observable ,

where v_redshift = zc is redshift velocity, r_observable is a direct observable kind of distance (usually a luminosity distance), and H_0 is Hubble constant. The subscripts can be dropped if you know what you mean.

By convention redshift velocity is measured in kilometers per second (km/s) and distance in megaparsec (Mpc).

The Hubble constant which is the slope of the best-fit line on the plot has units of (km/s)/Mpc.

1. Observational errors in measuring z and the direct observable distance.

2. The objects (e.g., galaxies and quasars) in the actual observable universe have peculiar velocities (which are ordinary physical velocities) superimposed on the recession velocities (which are growth rates of space between us and points participating in the mean expansion of the universe).

3. Objects in the Local Group of galaxies do NOT participate in the expansion of the universe relative to us (who are in the Local Group) since the Local Group is a gravitationally bound system.

   You, me, the Milky Way, the Local Group, and all other gravitationally bound systems throughout the observable universe do NOT expand. Only the space between gravitationally bound systems expands.

But actually, the expanding universe theory predicts a general scaling up of the observable universe. So every observer seems to see himself/herself at the center of expansion at first glance.

All distances between points participating in the mean expansion of the universe scale up with cosmic time (with zero time at the Big Bang) by the cosmic scale factor a(t).

The cosmic scale factor for our current epoch (conventionally cosmic time t_0) is conventionally set to 1 and given symbol a_0. Thus a(t=t_0) = a_0 = 1.

Albert Einstein (1879--1955) failed to discover the expanding universe theory.

It was discovered independently by Alexander Alexandrovich Friedmann (1888--1925) and Georges Lemaitre (1894--1966) in the 1920s.

Let r be a proper distance (i.e., a distance that can be measured at one instant in time with a ruler) between two points participating in the mean expansion of the universe. Now

      r = r₀*a(t)

where r_0 is the comoving distance (a time independent distance equal to the proper distance at the present cosmic time t_0), and a(t) is the cosmic scale factor. So all r's scale up and down with cosmic time as determined by a(t).

Taking the derivative of r with respect to time t (a calculus operation) gives

      v = dr/dt = r₀*(da/dt) ,

where v = dr/dt is the rate of change of r and da/dt is the rate of change of a(t). Note v = dr/dt is the recession velocity. It's NOT an ordinary velocity: it's a rate of growth of space.

Now r_0 = v/(da/dt) and substituting this into the second to last equation and rearranging gives

      v = [(da/dt)/a(t)]*r ,

The quantity [(da/dt)/a(t)] depends on cosmic time, but NOT on position. We evaluate it for the present cosmic time and call it the Hubble constant

      H₀ = [(da/dt)/a(t)]₀ ,

where subscript 0 means evaluated at the present cosmic time.

Now we have the general Hubble's law

      v = H₀r .

But there is a big problem. The recession velocity v and the proper distance are NOT in general direct observables.

However, it can be shown that to 1st order in small z, we obtain the ordinary Hubble's law given above.

There are many contemporary determinations of the Hubble constant. The values are converging to H_0 = 70 (km/s)/Mpc to within a few percent.

Nowadays H_0 = 70 (km/s)/Mpc is taken as the fiducial value for many calculations. Formulae containing the Hubble constant can be written as 70 (km/s)/Mpc * h_70, where h_70 = H_0/[70 (km/s)/Mpc] is left as a variable.

The evidence when first presented did convince people that the expanding universe was probably true. Other evidence since that time has confirmed the expanding universe theory.