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 ~ 1010, when the universe was a few hundred milliseconds old), the relation between space-time points may be described uniquely by the Robertson - Walker Metric
where r is a comoving radial coordinate and and are comoving spherical coordinates related by
A useful alternative to the comoving radial coordinate r is , defined by
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 te to observation at a later time to. In the course of its journey through the universe the photon traverses a comoving radial distance , where
Some special choices of te or to are of particular interest. For te 0, H(to) is the comoving radial distance to the "Particle Horizon" at time t0. 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 to. The "Event Horizon", E(te), corresponds to the limit to (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 te.