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

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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 Omega in Eq. 2 leads to the relation

Equation 10 10.

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

Equation 11 11.

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

Equation 12 12.

The present astronomical observations place Omega within a range of 0.1 to 2. Thus |Omega - 1| is of order unity. At the GUTs epoch, however, with a propto t1/2, we have adot2 propto t-1. Putting in numbers, we find that adot02 / adot2 leq 10-48. In other words, |Omega (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 = infty (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 pi / 45) T 3. Since a propto T-1, the total entropy in a proper volume, v propto 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.

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