Physical Foundations of Cosmology

2.3 Conformal diagrams 47

η

χ

a= ∞

a= ∞

η = const

η = −π/ 2

amin = (^1) /H
π
particle horizon
χ = const
event horizon
V
−π
Fig. 2.3.
a square. In fact, it has the same shape as the diagram for a closed, radiation-
dominated universe, with the difference that there are no singularities atηi=−π
andηmax= 0 −see Figure 2.3. Moreover, in a de Sitter universe, the scale factor,
a(η)=− 1 /Hsinη,is infinite at the lower boundary of the diagram whereη→
−π, decreases asηchanges from−πto−π/ 2 ,reaches its minimum value 1/H,
and then grows to infinity again asη→0. The blowing up of the scale factor does
not signify a singularity. We have seen that all curvature invariants are constant in
de Sitter spacetime, and hence, the infinite growth of the scale factor is entirely a
coordinate effect.
As with the closed radiation-dominated universe, de Sitter spacetime has both a
particle horizon,
χp(η)=(η−ηi)=η+π, (2.29)
and an event horizon,
χe(η)=(ηmax−η)=−η, (2.30)
which exist at any timeη. In both the closed de Sitter and radiation-dominat-
ed universes, the physical size of the event horizonde(t)approaches the curvature
scaleH−^1 near the upper boundary of the conformal diagram. However, in de Sitter
spacetime,H,and consequently the size of the event horizon, remain constant; in a
radiation-dominated universe,Hincreases and the size of the event horizon shrinks.