Essay

FLATNESS PROBLEM

Adapted from P. Coles, 1999, The Routledge Critical Dictionary of the New Cosmology, Routledge Inc., New York. Reprinted with the author's permission. To order this book click here: http://www.routledge-ny.com/books.cfm?isbn=0415923549

The problem, left unresolved in the standard version of the Big Bang theory, stemming from the impossibility of predicting a priori the value of the density parameter Omega which determines whether the Universe will expand for ever or will ultimately recollapse. This shortcoming is ultimately a result of the breakdown of the laws of physics at the initial singularity in the Big Bang model.

To understand the nature of the mystery of cosmological flatness, imagine you are in the following situation. You are standing outside a sealed room. The contents are hidden from you, except for a small window covered by a small door. You are told that you can open the door at any time you wish, but only once, and only briefly. You are told that the room is bare, except for a tightrope suspended in the middle about two metres in the air, and a man who, at some indeterminate time in the past, began to walk the tightrope. You know also that if the man falls, he will stay on the floor until you open the door. If he does not fall, he will continue walking the tightrope until you look in.

What do you expect to see when you open the door? One thing is obvious: if the man falls, it will take him a very short time to fall from the rope to the floor. You would be very surprised, therefore, if your peep through the window happened to catch the man in transit from rope to floor. Whether you expect the man to be on the rope depends on information you do not have. If he is a circus artist, he might well be able to walk to and fro along the rope for hours on end without falling. If, on the other hand, he is (like most of us) not a specialist in this area, his time on the rope would be relatively brief. Either way, we would not expect to catch him in mid-air. It is reasonable, on the grounds of what we know about this situation, to expect the man to be either on the rope or on the floor when we look.

This may not seem to have much to do with Omega , but the analogy can be recognised when we realise that Omega does not have a constant value as time goes by in the Big Bang theory. In fact, in the standard Friedmann models Omega evolves in a very peculiar way. At times arbitrarily close to the Big Bang, these models are all described by a value of Omega arbitrarily close to 1. To put this another way, consider the Figure under density parameter. Regardless of the behaviour at later times, all three curves shown get closer and closer near the beginning, and in particular they approach the flat universe line. As time goes by, models with Omega just a little greater than 1 in the early stages develop larger and larger values of Omega , reaching values far greater than 1 when recollapse begins. Universes that start out with values of Omega just less than 1 eventually expand much faster than the flat model, and reach values of Omega very close to 0. In the latter case, which is probably more relevant given the contemporary estimates of Omega < 1, the transition from Omega near 1 to a value near 0 is very rapid.

Now we can see the analogy. If Omega is, say, 0.3. then in the very early stages of cosmic history it was very close to 1, but less than this value by a tiny amount. In fact, it really is a tiny amount indeed! At the Planck time, for example, Omega has to differ from 1 only in the sixtieth decimal place. As time went by, Omega hovered close to the critical density value for most of the expansion, beginning to diverge rapidly only in the recent past. In the very near future it will be extremely close to 0. But now, it is as if we had caught the tightrope walker right in the middle of his fall. This seems very surprising, to put it mildly, and is the essence of the flatness problem.

The value of Omega determines the curvature of spacetime. It is helpful to think about the radius of spatial curvature - the characteristic scale over which the geometry appears to be non-Euclidean, like the radius of a balloon or of the Earth. The Earth looks flat if we make measurements on its surface over distances significantly less than its radius (about 6400 km). But on scales larger than this the effect of curvature appears. The curvature radius is inversely proportional to 1 - Omega in such a way that the closer Omega is to unity, the larger is the radius. (A flat universe has a radius of infinite curvature.) If Omega is not too different from 1, the scale of curvature is similar to the scale of our cosmological horizon, something that again appears to be a coincidence.

There is another way of looking at this problem by focusing on the Planck time. At this epoch, where our knowledge of the relevant physical laws is scant, there seems to be only one natural timescale for evolution, and that is the Planck time itself. Likewise, there is only one relevant length scale: the Planck length. The characteristic scale of its spatial curvature would have been the Planck length. If spacetime was not flat, then it should either have recollapsed (if it were positively curved) or entered a phase of rapid undecelerated expansion (if it were negatively curved) on a timescale of order the Planek time. But the Universe has avoided going to either of these extremes for around 10⁶⁰ Planek times.

These paradoxes are different ways of looking at what has become known as the cosmological flatness problem (or sometimes, because of the arguments that are set out in the preceding paragraph, the age problem or the curvature problem), and it arises from the incompleteness of the standard Big Bang theory. That it is such a big problem has convinced many scientists that it needs a big solution. The only thing that seemed likely to resolve the conundrum was that our Universe really is a professional circus artist, to stretch the above metaphor to breaking point. Obviously, Omega is not close to zero, as we have strong evidence of a lower limit to its value of around 0.1. This rules out the man-on-the-floor alternative. The argument then goes that Omega must be extremely close to 1, and that something must have happened in primordial times to single out this value very accurately.

The happening that did this is now believed to be cosmological inflation, a speculation by Alan Guth in 1981 about the very early stages of the Big Bang model. The inflationary Universe involves a curious change in the properties of matter at very high energies resulting from a phase transition involving a quantum phenomenon called a scallar field. Under certain conditions, the Universe begins to expand much more rapidly than it does in standard Friedmann models, which are based on properties of low-energy matter with which we are more familiar. This extravagant expansion - the inflation - actually reverses the kind of behaviour expected for Omega in the standard models. Omega is driven hard towards 1 when inflation starts, rather than drifting away from it as in the cases described above.

A clear way of thinking about this is to consider the connection between the value of Omega and the curvature of spacetime. If we take a highly curved balloon and blows it up to an enormous size, say the size of the Earth, then its surface will appear to be flat. In inflationary cosmology, the balloon starts off a tiny fraction of a centimetre across and ends up larger than the entire observable Universe. If the theory of inflation is correct, we should expect to be living in a Universe which is very flat indeed, with an enormous radius of curvature and in which Omega differs from 1 by no more than one part in a hundred thousand.

The reason why Omega cannot be assigned a value closer to 1 is that inflation generates a spectrum of primordial density fluctuations on all scales, from the microscopic to the scale of our observable Universe and beyond. The density fluctuations on the scale of our horizon correspond to an uncertainty in the mean density of matter, and hence to an uncertainty in the value of Omega .

One of the problems with inflation as a solution to the flatness problem is that, despite the evidence for the existence of dark matter, there is no really compelling evidence of enough such material to make the Universe closed. The question then is that if, as seems likely, Omega is significantly smaller than 1, do we have to abandon inflation? The answer is not necessarily, because some models of inflation have been constructed that can produce an open universe. We should also remember that inflation predicts a flat universe, and the flatness could be achieved with a low matter density if there were a cosmological constant or, in the language of particle physics, a nonzero vacuum energy density.

On the other hand, even if Omega were to turn out to be very close to 1, that would not necessarily prove that inflation happened either. Some other mechanism, perhaps associated with the epoch of quantum gravity, might have trained our Universe to walk the tightrope. It maybe, for example, that for some reason quantum gravity favours a flat spatial geometry. Perhaps, then, we should not regard the flatness `problem' as a problem: the real problem is that we do not have a theory of the very beginning in the Big Bang cosmology.