2.5. Characteristic scales and horizons

The big bang Universe has two characteristic scales

• The Hubble time (or length) H^-1.
• The curvature scale a| k|^-1/2.

The first of these gives the characteristic timescale of evolution of a(t), and the second gives the distance up to which space can be taken as having a flat (Euclidean) geometry. As written above they are both physical scales; to obtain the corresponding comoving scale one should divide by a(t). The ratio of these scales gives a measure of the total density; from the Friedmann equation we find

(15)

A crucial property of the big bang Universe is that it possesses horizons; even light can only have travelled a finite distance since the start of the Universe t_*, given by

(16)

For example, matter domination gives d_H(t) = 3t = 2H^-1. In a big bang Universe, d_H(t₀) is a good approximation to the distance to the surface of last scattering (the origin of the observed microwave background, at a time known as `decoupling'), since t₀ >> t_dec.