Physical Foundations of Cosmology

2.3 Conformal diagrams 41

comoving coordinates

χ>χe(η)=

∫ηmax

η

dη=ηmax−η. (2.12)

Hence, the physical size of the event horizon at timetis

de(t)=a(t)

∫tmax

t

dt

a

, (2.13)

where “max” refers to the final moment of time. If the universe expands forever,
thentmaxis infinite. However, the value ofηmax, and hencede,can be either infinite
or finite depending on the rate of expansion. In flat and open decelerating universes,
tmaxandηmaxare both infinite,χeanddediverge, and so there is no event horizon.
However, if the universe undergoes accelerated expansion, then the integral in (2.13)
converges and the radius of the event horizon is finite, even if the universe is flat or
open. In this case,ηapproaches a finite limitηmax,astmax→∞.
An important example is a flat de Sitter universe, where

de(t)=exp(Ht)

∫∞

t

exp(−Ht)dt=H−^1 , (2.14)

that is, the size of the event horizon is equal to the curvature scale. Every event
that occurs at a given moment of time at a distance greater thanH−^1 will never be
seen by an observer and cannot influence his future because the intervening space
is expanding too rapidly. For this reason, the situation is sometimes characterized
as “superluminal expansion.”
In a closed decelerating universe, the time available for future observations is
finite since the universe ultimately collapses. Therefore, there is both an event
horizon and a particle horizon.

Problem 2.4Verify that, in a closed, radiation-dominated universe, the curvature
scaleH−^1 is roughly equal to theparticle horizonsize at the beginning of expansion
but roughly coincides with the radius of theevent horizonduring the final stages
of collapse.

2.3 Conformal diagrams

The homogeneous, isotropic universe is a particular case of a spherically symmetric
space. The most general form of metric respecting spherical symmetry is

ds^2 =gab(xc)dxadxb−R^2 (xc)d 

2 , (2.15)