Physical Foundations of Cosmology

2.3 Conformal diagrams 49

In aflat de Sitter universe, the scale factor grows asa(t ̄)=H−^1 exp(H ̄t),
where the physical time ̄tis related to the conformal time ̄ηvia

exp(H ̄t)=− 1 /η. ̄ (2.34)

Hence, in conformal coordinates the metric becomes

ds^2 =

1

H^2 η ̄^2

(

dη ̄^2 −dχ ̄^2 −χ ̄^2 d 

2

)

, (2.35)

where 0>η> ̄ −∞and+∞>χ> ̄ 0. Unlike the case of a closed de Sitter uni-
verse, here ̄η,χ ̄have infinite ranges and to draw the conformal diagram, we must
first transform to coordinates which range over finite intervals. Fortunately, there
is a natural choice for such coordinates: we simply use theη,χcoordinates of the
closed de Sitter universe. The relation between ̄η,χ ̄andη,χcoordinates immedi-
ately follows from (1.99) if we expresstandt ̄in terms ofηand ̄ηrespectively. The
result is

η ̄=

sinη

cosη+cosχ

, χ ̄=

sinχ

cosη+cosχ

. (2.36)

Using these relations, one can draw the hypersurfaces of constant ̄ηand ̄χ(the
coordinates in (2.35)) in theη–χplane, as shown in Figure 2.5. We find that when
η, ̄ χ ̄coordinates run over their semi-infinite ranges, they cover only half of de Sitter

η

χ

π

i^0

i−

−π

I−

χ ̄= const

η ̄= const

Fig. 2.5.