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

2.2.2. THE FLATNESS PROBLEM We have ignored the curvature term kc² / a² in Eq. 2 in considering the very early universe. The use of the density parameter in Eq. 2 leads to the relation

10.

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

11.

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

12.

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 ₀² / 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 a³, in an expanding universe is conserved. Its present value in the observable region is about 10⁸⁵. Such a large value for a dimensionless conserved quantity is hard to explain.