A light ray emitted by an event 
reaches an observer at the origin at time t0 , given by
with 
. The second equality follows from substituting 
and using the Friedmann equation (3.60). If the t (or a) integral converges, as 
, to a value 
then there may exist particles (fundamental observers) for which 
and from which no communication can have reached the origin by time 
. Such particles (or values of r) are said to lie beyond the particle horizon of the origin at time . From the form of the last integral in Eq. (1) it is clear that convergence requires 
. Thus particle horizons exist in a universe which is matter or radiation dominated at the earliest times, but do not exist in a universe which was initially dominated by vacuum energy density. As increases, 
becomes bigger, all particle horizons expand, and signals can be received from more and more distant particles.
If the t integral in Eq. (3.93) converges as 
or as t0 approaches the recollapse time for a universe with a finite lifetime), there may exist events which the observer at the origin will never see, and which therefore can never influence him/her by any physical means. Such events are said to lie beyond the event horizon of this observer. Event horizons exist in closed models and in models that are vacuum dominated at late times, but do not exist in flat or open universes with zero cosmological constant. In the latter case, therefore, any event will eventually be able to influence every fundamental observer in the Universe.
The existence of particle horizons in the Big Bang model has important implications, because it means that many parts of the presently observable Universe may not have been in causal contact at early times. This gives rise to certain difficulties.
