94 Cosmological Models
Although the world is not devoid of matter and the cosmological constant is small,
the de Sitter universe may still be of more than academic interest in situations when휌
changes much more slowly than the scale푎. The de Sitter metric then takes the form
d푠^2 =( 1 −푟^2 퐻^2 )d푡^2 −( 1 −푟^2 퐻^2 )−^1 d푟^2 −푟^2 (d휃^2 +sin^2 휃d휙^2 ). (5.62)Not that the coordinates here violate the cosmological principle. There is aninside
region in the de Sitter space at푟<퐻−^1 , for which the metric tensor component
푔 00 is positive and푔 11 is negative. This resembles the regionoutsidea black hole of
Schwarzschild radius푟c=퐻−^1 ,at푟>푟c,where푔 00 is positive and푔 11 is negative. Out-
side the radius푟=퐻−^1 in de Sitter space and inside the Schwarzschild black hole
these components of the metric tensor change sign.
At푟=퐻−^1 ,푔 00 ==^1 −푟^2 퐻^2 vanishes and푔 11 =−(^1 −푟^2 퐻^2 )−^1 is singular. In the sub-
space defined by 0⩽푟⩽퐻−^1 we can transform the singularity away by the substitu-
tion푢^2 =퐻−^1 −푟. The new radial coordinate,푢, is then in the range 0⩽푢⩽퐻−^1 and
the nonsingular metric becomes
d푠^2 =( 1 −푟^2 퐻^2 )d푡^2 − 4 퐻−^1 ( 1 +rH)−^1 d푢^2 −푟^2 (d휃^2 +sin^2 휃d휙^2 ). (5.63)From this example we see that some singularities may be just the consequence of a
badly chosen metric and not a genuine property of the theory.
The interpretation of this geometry is that the de Sitter metric describes an expand-
ing space-time surrounded by a black hole. Inside the region푟=퐻−^1 no signal can
be received from distances outside퐻−^1 because there the metric corresponds to the
inside of a black hole! In an anti-de Sitter universe the constant attraction ultimately
dominates, so that the expansion turns into contraction. Thus de Sitter universes are
open and anti-de Sitter universes are closed.
Let us study the particle horizon푟Hin a de Sitter universe. Recall that this is defined
as the location of the most distant visible object, and that the light from it started on
its journey towards us at time푡H. From Equation (2.48) the particle horizon is at
푟H(푡)=푅(푡)휒ph=푅(푡)∫푡 0푡Hd푡′
푅(푡′). (5.64)
Let us choose푡Has the origin of time,푡H=0. The distance푟H(푡)as a function of the
time of observation푡then becomes
푟H(푡)=퐻−^1 eHt( 1 −e−Ht). (5.65)The comoving distance to the particle horizon,휒ph, quickly approaches the constant
value퐻−^1. Thus for a comoving observer in this world the particle horizon would
always be located at퐻−^1. Points which were inside this horizon at some time will be
able to exchange signals, but events outside the horizon cannot influence anything
inside this world.
The situation in a Friedmann universe is quite different. There the time dependence
of푎is not an exponential but a power of푡, Equation (5.38), so that the comoving
distance휒phis an increasing function of the time of observation, not a constant. Thus
points which were once at space-like distances, prohibited to exchange signals with
each other, will be causally connected later, as one sees in Figure 2.1.