5.1 Problem of initial conditions 227Initially the size of this domain was smaller by the ratio of the corresponding scale
factors,ai/a 0. Assuming that inhomogeneity cannot be dissolved by expansion, we
may safely conclude that the size of the homogeneous, isotropic region from which
our universe originated att=tiwas larger than
li∼ct 0
ai
a 0. (5.1)
It is natural to compare this scale to the size of a causal regionlc∼cti:
li
lc∼
t 0
tiai
a 0. (5.2)
To obtain a rough estimate of this ratio we note that if the primordial radiation
dominates atti∼tPl, then its temperature isTPl∼ 1032 K. Hence
(ai/a 0 )∼(T 0 /TPl)∼ 10 −^32and we obtain
li
lc∼
1017
10 −^43
10 −^32 ∼ 1028. (5.3)
Thus, at the initial Planckian time, the size of our universe exceeded the causality
scale by 28 orders of magnitude. This means that in 10^84 causally disconnected
regions the energy density was smoothly distributed with a fractional variation not
exceedingδε/ε∼ 10 −^4. Because no signals can propagate faster than light, no
causal physical processes can be responsible for such an unnaturally fine-tuned
matter distribution.
Assuming that the scale factor grows as some power of time, we can use an
estimatea/t∼a ̇and rewrite (5.2) as
li
lc∼
a ̇i
a ̇ 0. (5.4)
Thus, the size of our universe was initially larger than that of a causal patch by
the ratio of the corresponding expansion rates. Assuming that gravity was always
attractive and hence was decelerating the expansion, we conclude from (5.4) that
the homogeneity scale was always larger than the scale of causality. Therefore, the
homogeneity problem is also sometimes called thehorizonproblem.
Initial velocities (flatness) problemLet us suppose for a minute that someone has
managed to distribute matter in the required way. The next question concerns initial
velocities. Only after they are specified is the Cauchy problem completely posed
and can the equations of motion be used to predict the future of the universe
unambiguously. The initial velocities must obey the Hubble law because otherwise
the initial homogeneity is very quickly spoiled. That this has to occur in so many