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3.1. The flatness problem

Taking advantage of the definition of the density parameter, and ignoring a possible cosmological constant contribution, the Friedmann equation can be written in the form

Equation 20 (20)

During standard big bang evolution, a2H2 is decreasing, and so Omega moves away from one, for example

Equation 21 (21)

Equation 22 (22)

where the solutions apply provided Omega is close to one. So Omega = 1 is an unstable critical point. Since we know that today Omega is certainly within an order of magnitude of one, it must have been much closer in the past. Inserting the appropriate behaviours for the matter and radiation eras (or if you like just assuming radiation domination all the way to the present) gives

Equation 23 (23)

Equation 24 (24)

That is, hardly any choices of the initial density lead to a Universe like our own. Typically, the Universe will either swiftly recollapse, or will rapidly expand and cool below 3K within its first second of existence.

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