Annu. Rev. Astron. Astrophys. 1991. 29:
325-362
Copyright © 1991 by Annual Reviews. All rights reserved |

2.2.2. THE FLATNESS PROBLEM
*kc*^{2} /
*a*^{2} in Eq. 2 in considering the
very early universe. The use of the density parameter in Eq. 2 leads
to the relation

A comparison with the present epoch gives, assuming *k* 0,

Since, as *t* -> 0, || ->
, the convergence of *(t)* onto the flat value
1 is very rapid as we approach *t* = 0. Expressed as a function of
temperature, this relation becomes

The present astronomical observations place within a range of 0.1 to
2. Thus | - 1| is of order
unity. At the GUTs epoch, however, with *a*
*t*^{1/2}, we have ^{2}
*t*^{-1}. Putting in numbers, we find that _{0}^{2} / ^{2}
10^{-48}. In other words, | *(t)* - 1| is extremely fine-tuned to value zero.

Stated differently, had this fine-tuning not occurred, the universe
would have contracted back to *a* = 0 (for *k* = 1) or
diffused to *a* =
(for *k* = -1) long before the present epoch. In the absence of any
physical mechanism, this fine-tuning has to be imposed ad hoc at the
GUTs epoch in the standard model.

The flatness problem can be posed alternatively as follows. The
entropy density of photons at thermal equilibrium at temperature *T* is
given by (4 / 45)
*T* ^{3}. Since *a* *T*^{-1}, the total entropy in a proper
volume, *v*
*a*^{3}, in an expanding universe is conserved. Its present
value in the observable region is about 10^{85}. Such a large
value for a dimensionless conserved quantity is hard to explain.