Physical Foundations of Cosmology

44 Propagation of light and horizons

h

χ

final singularity

π

particle horizon

event horizon

initial singularity

χ = const

h = const

π

Fig. 2.1.

χ=0isgivenby

χp(η)=η−ηi=η. (2.24)

In the radiation-dominated universe, the particle horizon spans the whole space
whenη→π,that is, just as the universe recollapses. At this last moment of time,
all points in space become visible. The light that reaches an observer from the
most remote point,χ=π,reveals information about the state of the universe at the
beginning of expansion. In the dust-dominated universe, the whole universe also
becomes entirely visible atη=π.However, here this corresponds to the moment
of maximum expansion. There remains enough time for light to make a second trip
across the whole space before the universe recollapses.
The event horizon is given by

χe(η)=ηmax−η. (2.25)

In the radiation-dominated universe, there exists an event horizon for anyηsince
ηmax=π. In contrast, for the dust-dominated universe, whereηmax= 2 π, the event
horizon exists only during the contraction phase whenη>π. All events that occur
atη<π, no matter how far away, can be seen before the universe recollapses.
In summary, as shown in Figures 2.1 and 2.2, particle and event horizons ex-
ist at eachηin the closed radiation-dominated universe. In the matter-dominated