Paradoxes of the Expansion 153Inserting this dependence into the integral on the right-hand side of Equation (7.4)
we find
휎ph∝
∫푎 0푎mind푎
푎^2√
푎−^2
=
∫
푎 0푎mind푎
푎, (7.9)
an integral which does not converge at the limit푎min=0. Thus the particle horizon is
not finite in this case. But it is still true that an observer living at a time푡 1 <푡 0 would
see a particle horizon that is smaller by ln푎 0 −ln푎 1.
Horizon Problem. A consequence of the finite age푡 0 of the Universe is that the par-
ticle horizon today is finite and larger than at any earlier time푡 1. Also, the spatial
width of the past light cone has grown in proportion to the longer time perspective.
Thus the spatial extent of the Universe is larger than that our past light cone encloses
today; with time we will become causally connected with new regions as they move
in across our horizon. This renders the question of the full size of the whole Uni-
verse meaningless—the only meaningful size being the diameter of its horizon at a
given time.
In Chapter 6 we argued that thermal equilibrium could be established throughout
the Universe during the radiation era because photons could traverse the whole Uni-
verse and interactions could take place in a time much shorter than a Hubble time.
However, there is a snag to this argument: the conditions at any space-time point can
only be influenced by events within its past light cone, and the size of the past light
cone at the time of last scattering (푡LSS) would appear to be far too small to allow the
currently observable Universe to come into thermal equilibrium.
Since the time of last scattering, the particle horizon has grown with the expansion
in proportion to the^23 -power of time (actually this power law has been valid since the
beginning of matter domination at푡eq,but푡LSSand푡eqare nearly simultaneous). The
net effect is that the particle horizon we see today covers regions which were causally
disconnected at earlier times.
At the time of last scattering, the Universe was about 1090 times smaller than it is
now (푧LSS≈1065), and the time perspective back to the Big Bang was only the fraction
푡LSS∕푡 0 ≃ 2. 3 × 10 −^5 of our perspective. The last scattering surface is now at comov-
ing radius 14 Gpc, but at the epoch of recombination it was at a 1+푧=1089 timess
smaller radial distance. If we assume that the Universe was radiation dominated for
all the time prior to푡LSS, then, from Equation (5.40),푅(푡)∝
√
푡. The particle horizon
at the LSS,휎ph, is obtained by substituting푎(푡)∝(푡LSS∕푡)−^1 ∕^2 into Equation (2.38) and
integrating from zero time to푡LSS:
휎ph∝
∫푡LSS0d푡(
푡LSS
푡
) 1 ∕ 2
= 2 푡LSS. (7.10)
It is not very critical what we call ‘zero’ time: the lower limit of the integrand has
essentially no effect even if it is chosen as late as 10−^4 푡LSS.
The event horizon at the time of last scattering,휎eh, represents the extent of the
Universe we can observe today as light from the LSS (cf. Figures 2.1 and 7.1), since we
can observe no light from before the LSS. On the other hand, the particle horizon휎ph