Hubble Units:

The Hubble constant H_0 is the relative rate of expansion of the universe (or the rate of expansion of the universe per unit length). It is a fundamental parameter of modern cosmology. The Hubble units are characteristic quantities for the observable universe dervied the Hubble constant. Characteristic quantities are quantities that characterize a system. A characteristic quantity may be just a vague scale size (except for its precise definition) or it can have a precise meaning (beyond what its definition suggests). The Hubble units are of the former class, except that for the critical density and except for certain Friedmann-equation (FE) models where a precise meaning many occur.

The Hubble units (except that for the critical density) are of less interest since circa 1998 when a standard model of cosmology (SMC, Λ-CDM model) appeared for which characterizing parameters with precise meanings exist. Even if the Λ-CDM model needs revision or replacement (which is may due to the Hubble tension), those characterizing parameters are likely to endure.

What is the Hubble tension? Beginning circa and continuing to the present (circa 2021), there has been an disagreement in the determination of the Hubble constant between its indirect measurement (H_0 ≅ 68 (km/s)/Mpc) and its direct measurement (H_0 ≅ 74 (km/s)/Mpc) (see Wikipedia: Hubble's law: Measured values of the Hubble constant). Circa 2021 the disagreement is ∼ 4 standard deviations (i.e., 4) σ). The disagreement is called the Hubble tension. Resolving the Hubble tension may as aforesaid require revising or replacing the Λ-CDM model. For more details, see hubble_tension.html and Wikipedia: Hubble's law: Measured values of the Hubble constant.

Given the Hubble tension, it seems best in calculating Hubble units to adopt a fiducial value for Hubble constant that CANNOT be wrong by more than a few percent and is a nice round number. So we adopt H_0_fiducial=70 (km/s)/Mpc. We then define the reduced Hubble constant h_70 = H_0/[70 (km/s)/Mpc] and write the Hubble units with explicit h_70 factors.

1. Hubble constant H_0 = 70*h_70 (km/s)/Mpc: relative rate of expansion of the universe (or the rate of expansion of the universe per unit length).
2. Hubble time = 1/H_0 = (4.4081*10**17 s)/h_70 = (13.968 Gyr)/h_70: For cosmological models with Big Bang singularity, the Hubble time should be of order the age of the observable universe. In the Λ-CDM model, current best value for age of the observable universe = 13.797(23) Gyr (Planck 2018). It is actually just a coincidence that Hubble time and the current value are so close.
3. Hubble length = L_H = c/H_0 = 4.2827 Gpc/h_70 = 13.968 Gly/h_70: For cosmological models with Big Bang singularity, the Hubble length should be of order the proper radius of the observable universe. In the Λ-CDM model, current best value for proper radius of the observable universe = 14.25 Gpc = 46.48 Gly which, in fact, is 3.327 times the Hubble length. So there no close agreement between Hubble length and proper radius of the observable universe.
4. critical density ρ_c = 3H_0**2/(8πG) = (9.20387*10**(-27))*h_70**2 kg/m**3 = (1.35983*10**11)*h_70**2 M_☉/Mpc**3: In the Friedmann-equation (FE) models (which include the Λ-CDM model), the ratio of average density ρ of all the mass-energy of the universe (i.e., baryonic matter, electromagnetic radiation (EMR) (most importantly the cosmic background radiation), cosmic neutrino background, dark matter) to the critical density determines the spatial geometry. This ratio is the density parameter Ω = ρ/ρ_c:
1. Ω < 1: hyperbolic space or negative curvature space.
2. Ω = 1: Euclidean space or flat space.
3. Ω > 1: hyperspherical space or positive curvature space.
If the Friedmann-equation (FE) models are taken to apply to the whole universe (which formally they do) rather than to a part (in which the observable universe is embedded), then negative curvature space and flat space are imply an infinite universe or open universe and the positive curvature space implies a finite universe or closed universe, but with NO boundary: the 2-dimensional analogue is the surface of a sphere. Currently, the best observed Ω = 0.000(5) (Planck 2015) which is means the observable universe has flat space to within observational error, but the possibility of slight negative or positive curvature is open.