MODERN COSMOLOGY

Observations and horizons 141

hence via the Hawking–Penrose singularity theorems, leads to the prediction

of the existence of a spacetime singularity in our past [68].

3.8.2 Particle horizons and visual horizons

For ordinary equations of state, because causal influences can travel at most at
the speed of light, there is both aparticle horizon[110, 124], limiting causal
communication since the origin of the universe and avisual horizon[50], limiting
visual communication since the decoupling of matter and radiation. The former
depends on the equation of state of matter at early times, and can be changed
drastically by an early period of inflation; however the latter depends only on
the equation of state since decoupling, and is unaffected by whether inflation
took place or not. From (3.75), at an arbitrary time of observationt 0 ,the radial
comoving coordinate values corresponding to the particle and event horizons,
respectively, of an observer at the origin of coordinates are:

uph(t 0 )=

∫t 0

0

dt

S(t)

, uvh(t 0 )=

∫t 0

td

dt

S(t)

, (3.96)

where we have assumed the initial singularity occurred att=0 and decoupling
att=td. We cannot have had causal contact with objects lying at a coordinate
valuergreater thanuph(t 0 ), and cannot have received any type of electromagnetic
radiation from objects lying at a coordinate valuergreater thanuvh(t 0 ).
It is fundamental to note, then, that no object can leave either of these
horizons once it has entered it: once two objects are in causal or visual
contact, that contact cannot be broken, regardless of whether inflation or an
accelerated expansion takes place or not. This follows immediately from (3.96):
t 1 >t 0 ⇒uph(t 1 )>uph(t 0 )(the integrand betweent 0 andt 1 is positive, so
duph(t)/dt= 1 /S(t)>0.) Furthermore the physical scales associated with these
horizons cannot decrease while the universe is expanding. These are

Dph(t)=S(t)uph(t), Dvh(t)=S(t)uvh(t)

respectively, at timet; hence for example d(Dph(t))/dt= 1 +H(t)Dph(t)>0.
Much of the literature on inflation is misleading in this regard.

3.8.3 Small universes

The one case where visual horizons do not occur is when the universe has compact
spatial sections whose physical size is less than the Hubble radius; consider, for
example, the case of ak=0 model universe of toroidal topology, with a length
scale of identification of, say, 300 Mpc. In that case we can see right round
the universe, with many images of each galaxy, and indeed many images of our
own galaxy [48]. There are some philosophical advantages in such models [32],
but they may or may not correspond to physical reality. If this is indeed the