Observations of the present universe establish that, on sufficiently
large scales, galaxies and clusters of galaxies are distributed
homogeneously and they are expanding isotropically. On the assumption
that this is true for the large scale universe throughout its evolution
(at least back to redshifts ~ 10^{10}, when the universe was a
few hundred milliseconds old), the relation between space-time points
may be described uniquely by the Robertson - Walker Metric

(1) |

where *r* is a *comoving* radial coordinate and
and are
*comoving* spherical coordinates related by

(2) |

A useful alternative to the comoving radial coordinate
*r* is ,
defined by

(3) |

The 3-space curvature is described by
, the curvature constant.
For closed (bounded), or "spherical" universes,
> 0; for
open (unbounded), or "hyperbolic" models,
< 0; when
= 0, the universe is
spatially flat or "Euclidean". It is the
"scale factor", *a* = *a*(*t*), which describes how
physical distances
between comoving locations change with time. As the universe expands,
*a* increases while, for comoving observers, *r*,
, and
remain fixed. The growth of the separation between comoving observers
is solely due to the growth of *a*. Note that **neither**
*a* **nor**
is observable since a
rescaling of
can always
be compensated by a rescaling of *a*.

Photons and other massless particles travel on geodesics: *ds* = 0;
for them (see eq. 1)
*d* = ±
*cdt* / *a*(*t*). To
illustrate the significance of this result consider a photon travelling
from emission at time *t*_{e} to observation at a later
time *t*_{o}. In the course of its journey through the
universe the photon traverses a comoving radial distance
, where

(4) |

Some special choices of *t*_{e} or *t*_{o} are
of particular interest. For
*t*_{e}
0,
_{H}(*t*_{o})
is the comoving radial distance to the "Particle Horizon" at time
*t*_{0}.
It is the comoving distance a photon could have travelled (in the absence
of scattering or absorption) from the beginning of the expansion of the
universe until the time *t*_{o}. The "Event Horizon",
_{E}(*t*_{e}), corresponds to the limit
*t*_{o}
(provided
that _{E} is
finite!). It is the comoving radial distance
a photon will travel for the entire future evolution of the universe,
after it is emitted at time *t*_{e}.