Paradoxes of the Expansion 155far from what we see today: matter is separated into galaxies of mass 10^12 푀⊙.The
size of present superclusters is so large that their mass must have been assembled
from vast regions of the Universe which were outside the particle horizon at푡=2ms.
But then they must have been formed quite recently, in contradiction to the age of the
quasars and galaxies they contain. This paradox is the horizon problem.
The lesson of Equations (7.4)–(7.9) is that we can get rid of the horizon problem
by choosing physical conditions where the net pressure is negative, either by having
a large curvature term or a dominating cosmological term or some large scalar field
which acts as an effective cosmological term. We turn to the latter case in Section 7.2.
GUT Phase Transition. Even more serious problems emerge as we approach very
early times. At GUT time, the temperature of the cosmic background radiation was
푇GUT≃ 1. 2 × 1028 K, or a factor
푇GUT
푇 0≃ 4. 4 × 1027
greater than today. This is the factor by which the linear scale푎(푡)has increased since
the time푡GUT. If we take the present Universe to be of size 2000ℎ−^1 Mpc= 6 × 1025 m,
its linear size was only 2cm at GUT time.
Note, however, that linear size and horizon are two different things. The horizon
size depends on the time perspective back to some earlier time. Thus the particle
horizon today has increased since푡GUTby almost the square of the linear scale factor,
or by
푡 0
푡GUT=
(
푔∗(푇GUT)
푔∗(푇 0 )
) 1 ∕ 2 (
푇GUT
푇LSS
) 2 (
푇LSS
푇 0
) 3 ∕ 2
≃ 2. 6 × 1054. (7.13)
At GUT time the particle horizon was only 2× 10 −^29 m. It follows that to arrive at the
present homogeneous Universe, the homogeneity at GUT time must have extended
out to a distance 5× 1026 times greater than the distance of causal contact! Why did
the GUT phase transition happen simultaneously in a vast number of causally discon-
nected regions? Concerning even earlier times, one may ask the same question about
the Big Bang. Obviously, this paradox poses a serious problem to the standard Big
Bang model.
In all regions where the GUT phase transition was completed, several important
parameters—such as the coupling constants, the charge of the electron, and the
masses of the vector bosons and Higgs bosons—obtained values which would charac-
terize the present Universe. Recall that the coupling constants are functions of energy,
and the same is true for particle masses. One may wonder why they obtained the same
value in all causally disconnected regions.
The Higgs field had to take the same value everywhere, because this is uniquely
dictated by what is its ground state. But one might expect that there would be domains
where the phase transition was not completed, so that certain remnant symmetries
froze in. The Higgs field could then settle to different values, causing some parameter
values to be different. The physics in these domains would then be different, and so