flatness problem

The flatness problem, along with the horizon problem and monopole problem, was one of the motivations leading to the development of the inflationary cosmology. While observations indicate that the Universe has a geometry which is either flat or very close to flat, the standard hot big bang cosmology (without inflation; see ‘Overview’) indicates that the flat geometry is unstable and that only very special choices of initial conditions for the Universe can lead to a present Universe matching our observations.

The problem

Cosmological models are described on average by the Robertson–Walker metric, which features a constant K indicating the spatial curvature of the Universe. Once specified, this constant is unchanged during the evolution of the Universe, with zero value corresponding to the flat geometry. Whether or not the curvature is observable depends on whether it is an important contributor to the expansion rate of the Universe.

It has long been clear that the total density of the Universe is close, say within an order of magnitude, to the critical density which would imply a flat Universe. Recent observations of the cosmic microwave background anisotropies by the WMAP satellite have refined this, placing the density within a percent or so of the critical value.

While the flat geometry is the simplest possibility, and attractive for that reason, the big bang cosmology unfortunately predicts that the Universe can only be so close to flat today if the conditions in the young Universe are very special—the density must be extraordinarily close to the critical density at that time (perhaps within one part in 10³⁰). The reason is that the flat geometry is unstable; as the Universe evolves the geometry is predicted to deviate more and more from the flat case. Arranging the Universe to have negligible curvature by the present epoch is rather like trying to balance a pencil on its tip with such accuracy that after ten billion years it has still not fallen over.

In terms of the Friedmann equation, the flatness problem can be understood from the evolution of the different terms. As a function of the scale factor a(t), the density of matter would evolve in proportion to 1/a³, and radiation as 1/a⁴. Each of these reduces more swiftly with expansion than the curvature term, which falls proportional to 1/a². Accordingly, if the curvature is not to be totally dominant over matter and radiation today, it has to be tiny in comparison to them in the young Universe.

The flatness problem is, on its own, not particularly compelling however, as one could always argue that there is some as-yet-undiscovered principle which means that the Universe must be created with precisely the flat geometry, which would then persevere throughout its existence.

Solution by inflation

In 1981 Alan Guth proposed that the flatness problem could be solved by postulating that the early Universe underwent a period of accelerated expansion, which he named inflation. During such an epoch, the geometry of the Universe is driven towards the spatially flat case, which has become a stable rather than unstable situation. The inflationary period must last for long enough that the geometry finishes extremely close to flatness, since once inflation ends flatness will become unstable again and the geometry will begin to deviate. But provided enough inflation occurs, then even the long epoch between the end of inflation and the present is not enough for the Universe to move significantly away from the nearly flat geometry established by the inflationary epoch.

A more intuitive way to understand the inflationary solution may be to realize that the curvature of the Universe is a characteristic length scale which expands with the Universe. The purpose of the inflationary expansion is to drive the curvature scale to be much larger than the size of our present observable Universe, and hence the portion of the Universe we can see appears to be flat, just as the surface of an ocean appears on average to be flat provided you can only see a small region of it.

To solve the flatness problem, the inflationary expansion would have to increase the size of the Universe by a factor of at least 10²⁷. Such an expansion would simultaneously solve the horizon and monopole problems.