**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
actually gives a measure of ;
from the Friedmann equation we find

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

For example, matter domination gives
*d*_{H}(*t*) = 3*t* = 2*H*^{-1}.
In a big bang Universe,
*d*_{H}(*t*_{0}) 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 *t*_{0} >> *t*_{dec}.