Caption: A schematic diagram illustrating expansion of the universe is the homogeneous and isotropic.
Features:
At some point beyond the observable universe, most cosmologists suspect that homogeneity and isotropy must fail somehow.
Bound systems do NOT expand: you, me, planets, stars, galaxies, and probably most galaxy groups and clusters don't expand.
Most galaxy superclusters may be unbound and so may expand.
They scale with cosmic time by the cosmic scale factor a(t).
Thus,
r_p = a(t)*r_c ,where r_c is (cosmological) comoving distance which is a time-independent coordinate for space.
By convention, a_0 = a(t=present) = 1, and so r_p(t=present) = r_c.
v_p = dr_p/dt = (da/dt) * r_c v_p = [(da/dt)/a] * r_p v_p = H * r_p ,where v_p is a recession velocity, H is the Hubble parameter, and v_p = H * r_p is the general Hubble's law.
Note recession velocity is NOT an ordinary velocity. It is the rate of growth of space.
The recession velocity does exceed vacuum light speed for cosmological redshift ≥ ∼ 1.
The present Hubble's law is
v = H_0*r ,where we drop the subscripts on v and r since we know what we mean and want a clean simple looking formula.
The Hubble constant has the physical dimension of inverse time.
The conventional unit for the Hubble constant is (km/s)/Mpc since recession velocities almost always expressed in kilometers per second (km/s) and proper distances megaparsecs (Mpc).
h_70 = H_0/[ 70 (km/s)/Mpc ] is the Hubble constant written in terms of the natural unit for Hubble constant, 70 (km/s)/Mpc.
The natural unit is just because we know nowadays that H_0 is 70 (km/s)/Mpc to within ∼ 10 % (see Wikipedia: Hubble's law: Determining the Hubble constant).
The Hubble time is a characteristic time of Friedmann-Lemaitre models that approximates the age of the universe to order of magnitude for plausible versions of Friedmann-Lemaitre models.
The Λ-CDM model gives an age of the observable universe = 13.797(23) Gyr (Planck 2018) (see also Wikipedia: Λ-CDM model parameters) which coincidently is very close to the Hubble time.
The Hubble length is a characteristi length of Friedmann-Lemaitre models that approximates the observable universe radius to order of magnitude for plausible versions of Friedmann-Lemaitre models.
The Λ-CDM model gives a observable universe radius of 46.6 Glyr = 14.3 Gpc (see Wikipedia: Observable universe) which is ∼ 3 times larger than the Hubble length.
lec031 file: expansion_same.html.