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In our previous lectures we have described what is known as the standard cosmology. This framework is a towering achievement, describing to great accuracy the physical processes leading to the present day universe. However, there remain outstanding issues in cosmology. Many of these come under the heading of initial condition problems and require a more complete description of the sources of energy density in the universe. The most severe of these problems eventually led to a radical new picture of the physics of the early universe - cosmological inflation [163, 164, 165], which is the subject of this lecture.

We will begin by describing some of the problems of the standard cosmology.

5.1. The Flatness Problem

The Friedmann equation may be written as

Equation 172 (172)

where for brevity we are now writing Omega instead of Omegatotal. Differentiating this with respect to the scale factor, this implies

Equation 173 (173)

This equation is easily solved, but its most general properties are all that we shall need and they are qualitatively different depending on the sign of 1 + 3w. There are three fixed points of this differential equation, as given in table 4.

Table 4. Behavior of the density parameter near fixed points.

Fixed Point 1 + 3w > 0 1 + 3w < 0

Omega = 0 attractor repeller
Omega = 1 repeller attractor
Omega = infty attractor repeller

Observationally we know that Omega appeq 1 today - i.e., we are very close to the repeller of this differential equation for a universe dominated by ordinary matter and radiation (w > - 1/3). Even if we only took account of the luminous matter in the universe, we would clearly live in a universe that was far from the attractor points of the equation. It is already quite puzzling that the universe has not reached one of its attractor points, given that the universe has evolved for such a long time. However, we may be more quantitative about this. If the only matter in the universe is radiation and dust, then in order to have Omega in the range observed today requires (conservatively)

Equation 174 (174)

This remarkable degree of fine tuning is the flatness problem. Within the context of the standard cosmology there is no known explanation of this fine-tuning.

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