3.2.3 The current matter density of the universe ₀ and its normalized value ₀

The matter density in space is of particular interest in relativistic universes, because it is closely related to the scale factor.

With the basic family of models introduced above (general relativity, Friedmann-Robertson-Walker metric) and the assumption = 0, and a presently matter dominated universe (p₀ = 0), ₀ relates in simple ways scale factor, deceleration parameter and Riemannian space curvature.

The value of k is either +1, 0, -1 according to whether

It was from this formula, with k = 0, using the then accepted Hubble constant of 500 km sec^-1 Mpc^-1 that Einstein and de Sitter derived the value

₀ = 4 x 10^-28 g cm^-3

in their famous paper of 1932.

Hubble (1934) determined the density of luminous matter in space from his counts of 44,000 nebulae with

log Mass = log b + 0.4 (5.7 - M)

with b = mass/luminosity ratio
M = absolute magnitude.

log b is estimated from nearby systems to be of order unity, M is estimated as -13.8 to -14.5. The result is given as

log ₀ = -29.8 or -29.9

with an uncertainty probably less than 0.5 (units of ₀ are g cm^-3):

``The discussion, of course, ignores the existence of internebular matter, the density of which, even in an optimal form, might be several thousand times greater without introducing appreciable absorption. Since absorption depends upon the state of material (the density for large meteorites, for instance, might surpass that of the galactic system without introducing appreciable obscuration), upper limits can be assigned to the density of internebular space only from dynamical considerations.''

An earlier value (Hubble 1926) is

₀ = 1.5 x 10^-31 g cm^-3.

The matter density ₀ is frequently replaced by the dimensionless quantity ₀ which measures ₀ in terms of the Hubble constant, according to the above formula:

For = 0, ₀ = 1, corresponding to the Euclidean universe (e.g. Borner 1988).

₀ can be determined directly from dynamical evidence, including the `cosmic virial theorem', and from the abundance of elements formed in the early universe.