## The Cosmic Tightrope

Here’s a thought experiment for you.

Imagine you are standing outside a sealed room. The contents of the room are hidden from you, except for a small window covered by a curtain. You are told that you can open the curtain once and only briefly to take a peep at what is inside, and you may do this whenever you feel the urge.

You are told what is in the room. It is bare except for a tightrope suspended across it about two metres in the air. Inside the room is a man who at some time in the past – you’re not told when – began walking along the tightrope. His instructions were to carry on walking backwards and forwards along the tightrope until he falls off, either through fatigue or lack of balance. Once he falls he must lie motionless on the floor.

You are not told whether he is skilled in tightrope-walking or not, so you have no way of telling whether he can stay on the rope for a long time or a short time. Neither are you told when he started his stint as a stuntman.

What do you expect to see when you eventually pull the curtain?

Well, if the man does fall off sometime it will clearly take him a very short time to drop to the floor. Once there he has to stay there.One outcome therefore appears very unlikely: that at the instant you open the curtain, you see him in mid-air between a rope and a hard place.

Whether you expect him to be on the rope or on the floor depends on information you do not have. If he is a trained circus artist, like the great Charles Blondin here, he might well be capable of walking to and fro along the tightrope for days. If not, he would probably only manage a few steps before crashing to the ground. Either way it remains unlikely that you catch a glimpse of him in mid-air during his downward transit. Unless, of course, someone is playing a trick on you and someone has told the guy to jump when he sees the curtain move.

This probably seems to have very little to do with physical cosmology, but now forget about tightropes and think about the behaviour of the mathematical models that describe the Big Bang. To keep things simple, I’m going to ignore the cosmological constant and just consider how things depend on one parameter, the density parameter Ω_{0}. This is basically the ratio between the present density of the matter in the Universe compared to what it would have to be to cause the expansion of the Universe eventually to halt. To put it a slightly different way, it measures the total energy of the Universe. If Ω_{0}>1 then the total energy of the Universe is negative: its (negative) gravitational potential energy dominates over the (positive) kinetic energy. If Ω_{0}<1 then the total energy is positive: kinetic trumps potential. If Ω_{0}=1 exactly then the Universe has zero total energy: energy is precisely balanced, like the man on the tightrope.

A key point, however, is that the trade-off between positive and negative energy contributions changes with time. The result of this is that Ω is not fixed at the same value forever, but changes with cosmic epoch; we use Ω_{0 }to denote the value that it takes now, at cosmic time t_{0}, but it changes with time.

At the beginning, at the Big Bang itself, all the Friedmann models begin with Ω arbitrarily close to unity at arbitrarily early times, i.e. the limit as t tends to zero is Ω=1.

In the case in which the Universe emerges from the Big bang with a value of Ω just a tiny bit greater than one then it expands to a maximum at which point the expansion stops. During this process Ω grows without bound. Gravitational energy wins out over its kinetic opponent.

If, on the other hand, Ω sets out slightly less than unity – and I mean slightly, one part in 10^{60} will do – the Universe evolves to a state where it is very close to zero. In this case kinetic energy is the winner and Ω ends up on the ground, mathematically speaking.

In the compromise situation with total energy zero, this exact balance always applies. The universe is always described by Ω=1. It walks the cosmic tightrope. But any small deviation early on results in runaway expansion or catastrophic recollapse. To get anywhere close to Ω=1 now – I mean even within a factor ten either way – the Universe has to be finely tuned.

A slightly different way of describing this is to think instead about the radius of curvature of the Universe. In general relativity the curvature of space is determined by the energy (and momentum) density. If the Universe has zero total energy it is flat, so it doesn’t have any curvature at all so its curvature radius is infinite. If it has positive total energy the curvature radius is finite and positive, in much the same way that a sphere has positive curvature. In the opposite case it has negative curvature, like a saddle. I’ve blogged about this before.

I hope you can now see how this relates to the curious case of the tightrope walker.

If the case Ω_{0}= 1 applied to our Universe then we can conclude that something trained it to have a fine sense of equilibrium. Without knowing anything about what happened at the initial singularity we might therefore be pre-disposed to assign some degree of probability that this is the case, just as we might be prepared to imagine that our room contained a skilled practitioner of the art of one-dimensional high-level perambulation.

On the other hand, we might equally suspect that the Universe started off slightly over-dense or slightly under-dense, at which point it should either have re-collapsed by now or have expanded so quickly as to be virtually empty.

About fifteen years ago, Guillaume Evrard and I tried to put this argument on firmer mathematical grounds by assigning a sensible prior probability to Ω based on nothing other than the assumption that our Universe is described by a Friedmann model.

The result we got was that it should be of the form

.

I was very pleased with this result, which is based on a principle advanced by physicist Ed Jaynes, but I have no space to go through the mathematics here. Note, however, that this prior has three interesting properties: it is infinite at Ω=0 and Ω=1, and it has a very long “tail” for very large values of Ω. It’s not a very well-behaved measure, in the sense that it can’t be integrated over, but that’s not an unusual state of affairs in this game. In fact it is an *improper **prior*.

I think of this prior as being the probabilistic equivalent of Mark Twain’s description of a horse:

dangerous at both ends, and uncomfortable in the middle.

Of course the prior probability doesn’t tell usall that much. To make further progress we have to make measurements, form a likelihood and then, like good Bayesians, work out the posterior probability . In fields where there is a lot of reliable data the prior becomes irrelevant and the likelihood rules the roost. We weren’t in that situation in 1995 – and we’re arguably still not – so we should still be guided, to some extent by what the prior tells us.

The form we found suggests that we can indeed reasonably assign most of our prior probability to the three special cases I have described. Since we also know that the Universe is neither totally empty nor ready to collapse, it does indicate that, in the absence of compelling evidence to the contrary, it is quite reasonable to have a prior preference for the case Ω=1. Until the late 1980s there was indeed a strong ideological preference for models with Ω=1 exactly, but not because of the rather simple argument given above but because of the idea of cosmic inflation.

From recent observations we now know, or think we know, that Ω is roughly 0.26. To put it another way, this means that the Universe has roughly 26% of the density it would need to have to halt the cosmic expansion at some point in the future. Curiously, this corresponds precisely to the unlikely or “fine-tuned” case where our Universe is in between two states in which we might have expected it to lie.

Even if you accept my argument that Ω=1 is a special case that is in principle possible, it is still the case that it requires the Universe to have been set up with very precisely defined initial conditions. Cosmology can always appeal to special initial conditions to get itself out of trouble because we don’t know how to describe the beginning properly, but it is much more satisfactory if properties of our Universe are explained by understanding the physical processes involved rather than by simply saying that “things are the way they are because they were the way they were.” The latter statement remains true, but it does not enhance our understanding significantly. It’s better to look for a more fundamental explanation because, even if the search is ultimately fruitless, we might turn over a few interesting stones along the way.

The reasoning behind cosmic inflation admits the possibility that, for a very short period in its very early stages, the Universe went through a phase where it was dominated by a third form of energy, vacuum energy. This forces the cosmic expansion to accelerate. This drastically changes the arguments I gave above. Without inflation the case with Ω=1 is unstable: a slight perturbation to the Universe sends it diverging towards a Big Crunch or a Big Freeze. While inflationary dynamics dominate, however, this case has a very different behaviour. Not only stable, it becomes an *attractor* to which all possible universes converge. Whatever the pre-inflationary initial conditions, the Universe will emerge from inflation with Ω very close to unity. Inflation trains our Universe to walk the tightrope.

So how can we reconcile inflation with current observations that suggest a low matter density? The key to this question is that what inflation really does is expand the Universe by such a large factor that the curvature radius becomes infinitesimally small. If there is only “ordinary” matter in the Universe then this requires that the universe have the critical density. However, in Einstein’s theory the curvature is zero only if the total energy is zero. If there are other contributions to the global energy budget besides that associated with familiar material then one can have a low value of the matter density as well as zero curvature. The missing link is dark energy, and the independent evidence we now have for it provides a neat resolution of this problem.

Or does it? Although spatial curvature doesn’t really care about what form of energy causes it, it is surprising to some extent that the dark matter and dark energy densities are similar. To many minds this unexplained coincidence is a blemish on the face of an otherwise rather attractive structure.

It can be argued that there are initial conditions for non-inflationary models that lead to a Universe like ours. This is true. It is not logically necessary to have inflation in order for the Friedmann models to describe a Universe like the one we live in. On the other hand, it does seem to be a reasonable argument that the set of initial data that is consistent with observations is larger in models with inflation than in those without it. It is rational therefore to say that inflation is more probable to have happened than the alternative.

I am not totally convinced by this reasoning myself, because we still do not know how to put a reasonable measure on the space of possibilities existing prior to inflation. This would have to emerge from a theory of quantum gravity which we don’t have. Nevertheless, inflation is a truly beautiful idea that provides a framework for understanding the early Universe that is both elegant and compelling. So much so, in fact, that I almost believe it.

May 3, 2009 at 2:41 pm

Was your prior in fact 1/{Ω|Ω-1|} so as not to go negative when Ω exceeds 1?

I find inflation to be pragmatic, the very opposite of beautiful. At risk of repeating what I’ve said before, I’ll bet that Ω=1 someday falls out of a more fundamental theory as naturally as spin-half particles fall out of the Dirac equation. No dark energy epicycles.

Anton

May 3, 2009 at 3:14 pm

Good point. I’ve fixed it now. That was a result of my impatience with the limited ability the wordpress stuff has to deal with symbols. I think I’ve done it better now.

I think “neat” would have been more apt that beautiful and I agree that there may be a more fundamental theory that requires large-scale flatness. It does seem odd that our universe is flat when it is supposed to described by a theory that generally requires space-time to be curved.

May 4, 2009 at 11:20 am

I’m not sure whether you are agreeing or disagreeing with what I wrote. The measure in Omega certainly isn’t uniform in the region exceeding unity, but I never said it was.

Perhaps an even simpler way to put this is just to choose the only physical length scale in the problem (the curvature radius) to have a logarithmic prior (i.e. to be invariant under multiplicative rescaling). Plugging this into the Friedman equations gives the same answer for Omega as I mentioned above.

May 4, 2009 at 11:49 am

OK, put it this way that brings time into consideration. There are two scales: the curvature radius, which is fixed by the initial conditions and doesn’t change with time, and the horizon scale which is essentially ct. We can only detect the curvature when it is not very large compared to the horizon.

What I’m saying is that a value of Omega of 0.2, say, requires the curvature scale to be comparable to the horizon scale when it is observed. Too early and the Universe looks flat, too late and it has either recollapsed or gone into free expansion depending on the sign of the curvature.

I contend that values like this are very improbable on minimal information grounds: what tuned the universe to have these two scales comparable to each other?

I think the cases where they are very different are obviously more likely: if the curvature radius is much larger than the horizon then the Universe looks flat. If it is much smaller than Omega is either vanishingly small or infinitely large.

May 4, 2009 at 1:20 pm

I forgot to use the word “comoving” in my comment. The physical curvature radius just changes with the scale factor so is fixed in comoving coordinates, whereas the horizon grows in comoving coordinates.

Converting to time inevitably brings Dicke’s anthropic argument into play. We know that life needs about 1010 years to get going as we need stars to make heavy elements. This means that the Universe can’t have gone into free expansion or have recollapsed before that sort of timescale. This means we can exclude some of the prior space using the observation that life exists.

May 5, 2009 at 2:07 pm

Has anyone seriously tried to find a sensible measure on ‘initial conditions’ or ‘sets of initial data’ in the sense you use it here?

September 14, 2010 at 10:32 am

The most beautiful part of this is that if you look at the function: f(x)=(1/(x-1))/x, then f(phi) = 1. So there you have it, the golden ratio and the universal density constant are related.

September 14, 2010 at 10:59 am

Unfortunately though the probability measure has a normalising factor out front…

February 15, 2012 at 11:20 am

I have looked at the paper but the main line of reasoning is so far not clear… can you summarize the logical structure of the argument?

July 11, 2016 at 1:54 pm

[…] strike me that there’s a similarity with an interesting issue in cosmology that I’ve blogged about before (in different […]

December 12, 2017 at 11:39 am

[…] This paper makes a point that I have wondered about on a number of occasions. One of the problems, in my opinion, is that astrophysicists don’t think enough about their choice of prior. An improper prior is basically a statement of ignorance about the result one expects in advance of incoming data. However, very often we know more than we think we do. I’ve lost track of the number of papers I’ve seen in which the authors blithely assume a flat prior when that makes no sense whatsoever on the basis of what information is available and, indeed, on the structure of the model within which the data are to be interpreted. I discuss a simple example here. […]

March 26, 2018 at 1:56 pm

[…] A comment elsewhere on this blog drew my attention to a paper on the arXiv by Marc Holman with the following abstract: […]