The homogeneity and isotopy of the of the cosmic scale factor a in an accelerating universe

Caption: A schematic diagram illustrating expansion of the universe is the homogeneous and isotropic.

Features:

Homogeneous means the same everywhere and isotropic means the same in all directions.
Both terms apply to the observable universe.
At some point beyond the observable universe, most cosmologists suspect that homogeneity and isotropy must fail somehow.
Note the universal expansion applies to the space between bound systems.
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.
The universal expansion is a literal growth of space according to general relativity which is a key physical ingredient in standard cosmological models (i.e., the Friedmann-Lemaitre models which include the Λ-CDM model).
The growth of space is illustrated in the diagram.
All points that participate in the mean expansion of the universe are separated by (cosmological) proper distance r_p.
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.
The derivative (i.e., rate of change) with respect to cosmic time of the formula for r_p gives:
```
 
  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.
For the present instant in cosmic time, we have H(t=present)=H_0, where H_0 is Hubble constant.
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.
Hubble constant is often called the expansion rate of the observable universe, but actually it's the expansion rate per unit proper distance of the observable universe.
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).
The Hubble constant and derived quantities written in terms of Fiducial values are:
1. Hubble constant H_0 = [ 70 (km/s)/Mpc ] * h_70.
  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).
2. Hubble time t_H = ( 13.968 ... Gyr )/h_70.
  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.
3. Hubble length = L_H = c/H_0 = 4.2827 Gpc/h_70 = 13.968 Gly/h_70.
  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.

2004

lec031 file: expansion_same.html.