MaxEnt 2016: Norton’s Dome and the Cosmological Density Parameter

The second in my sequence of posts tangentially related to talks at this meeting on Maximum Entropy and Bayesian Methods in Science and Engineering is inspired by a presentation this morning by Sylvia Wenmackers. The talk featured an example which was quite new to me called Norton’s Dome. There’s a full discussion of the implications of this example at John D. Norton’s own website, from which I have taken the following picture:


This is basically a problem in Newtonian mechanics, in which a particle rolls down from the apex of a dome with a particular shape in response to a vertical gravitational field. The solution is well-determined and shown in the diagram.

An issue arises, however, when you consider the case where the particle starts at the apex of the dome with zero velocity. One solution in this case is that the particle stays put forever. However it can be shown that there are other solutions in which the particle sits at the top for an arbitrary (finite) time before rolling down. An example could be for example if the particle were launched up the dome from some point with just enough kinetic energy to reach the top where it is momentarily at rest, but then rolls down again.

Norton argues that this problem demonstrates a certain kind of indeterminism in Newtonian Mechanics. The mathematical problem with the specified initial conditions clearly has a solution in which the ball stays at the top forever. This solution is unstable, which is a familiar situation in mechanics, but this equilibrium has an unusual property related to the absence of Lipschitz continuity. One might expect that an infinitesimal asymmetric perturbation of the particle or the shape of the surface would be needed to send the particle rolling down the slope, but in this case it doesn’t. This is because there isn’t just one solution that has zero velocity at the equilibrium, but an entirely family as described above. This is both curious and interesting, and it does raise the question of how to define a probability measure that describes these solutions.

I don’t really want to go into the philosophical implications of this cute example, but it did strike me that there’s a similarity with an interesting issue in cosmology that I’ve blogged about before (in different terms).

This probably seems to have very little to do with physical cosmology, but now forget about domes and think instead 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 Ω0. 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 Ω0>1 then the total energy of the Universe is negative: its (negative) gravitational potential energy dominates over the (positive) kinetic energy. If Ω0<1 then the total energy is positive: kinetic trumps potential. If Ω0=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 Ω0 to denote the value that it takes now, at cosmic time t0, but it changes with time.

At the beginning, i.e. 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 1060 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.

The evolution of Ω  is neatly illustrated by the following phase-plane diagram (taken from an old paper by Madsen & Ellis) describing a cosmological model involving a perflect fluid with an equation of state p=(γ-1)ρc2. This is what happens for γ>2/3 (which includes dust, relativistic particles, etc):


The top panel shows how the density parameter evolves with scale factor S; the bottom panel shows a completion of this portrait obtained using a transformation that allows the point at infinity to be plotted on a finite piece of paper (or computer screen).

As discussed above this picture shows that all these Friedmann models begin at S=0 with Ω arbitrarily close to unity and that the value of Ω=1 is an unstable fixed point, just like the situation of the particle at the top of the dome. If the universe has Ω=1 exactly at some time then it will stay that way forever. If it is perturbed, however, then it will eventually diverge and end up collapsing (Ω>1) or going into free expansion (Ω<1).  The smaller the initial perturbation,  the longer the system stays close to Ω=1.

The fact that all trajectories start at Ω(S=0)=1 means that one has to be very careful in assigning some sort of probability measure on this parameter, ust as is the case with the Norton’s Dome problem I started with. About twenty 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

P(\Omega) \propto \Omega^{-1}(\Omega-1)^{-1}.

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 what is called 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 means basically that the equation of state of the contents of the universe is described by γ<2/3 rather than the case γ>2/3 described above. 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. Here’s what the phase plane looks like in this case:



Whatever the pre-inflationary initial conditions, the Universe will emerge from inflation with Ω very close to unity.

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.


9 Responses to “MaxEnt 2016: Norton’s Dome and the Cosmological Density Parameter”

  1. Anton Garrett Says:

    The dome is cute. But physically, I question whether a particle sent directly up the dome towards the apex with exactly enough kinetic energy to reach it (but no more) is any different, once it has got there, from a particle that is simply placed motionless at the apex; why should the latter particle stay where it is but the former kick off again after a while? Perhaps a small calculation is worthwhile to see how long the former particle actually takes to get to the apex – might this be infinite?

    The equation of motion of the particle reduces (we are assured) to

    r” = k sqrt(r)

    where r is the length of the trajectory along the dome, from the apex, and a prime ‘ denotes d/dt. The initial conditions are r = r’ = 0. There is the trivial and entirely correct solution r = 0 for all time, although this is unstable. The surprise is that there is a further family of solutions (or a generalisation) in which r = 0 up to an arbitrary time T and is thereafter proportional to (t-T)^4. This solution obviously solves the equation of motion and the boundary conditions.

    I do wonder if this family of solutions is an artifact of the chosen shape of the dome, and whether it goes away if the dome is only infinitesimally different. This will be the case, in particular, if the perturbation of the dome shape is asymmetrical – for then the particle has a preferred direction to slide down. I also wonder whether the T-family of solutions satisfies the equation of motion at the actual instant t=T. Differentiating the equation of motion twice more with respect to time and investigating that question is amusing…

    The presentation of this problem at Ghent advocated a tool called nonstandard analysis (NSA) to tackle it, which imagines different sorts of infinitesimals, rather as Cantor imagined different sorts of infinities. But, after some controversy, it has now been settled that NSA offers nothing that the usual careful “epsilon-delta” techniques of approaching a limit do not encompass.

    • energiesombre Says:

      As I understand it, the non-uniqueness of the solution is entirely due to the shape of the dome, that is how the problem is constructed. There does not exist a unique solution of the differential equation because the square root is not a Lipschitz function.

      • telescoper Says:

        That’s correct.

      • Anton Garrett Says:

        It’s good to have the mathematical criterion for what dome forms give non-unique solutions, but the issues raised are still worth discussing in the case of non-uniqueness.

  2. The dome is very cute! I’d never seen that before! The mathematical underpinnings (Lipschitz continuity) is probably something with a ton of depth to it that I’ve also never seen.

    Anton, Norton remarks that for some domes (such as a hemisphere) the time taken for the ball to reach the apex is indeed infinite. But in the case of this construction, the time taken is finite.

    I’m not entirely sure what to make of it. It seems like, from the time-reversal case, that were one to imagine each T-solution as a video tape recording the ball rising to the apex and sitting there for a time T (where the video tape is switched off), the strange time-reversed solutions only look strange because of the fact they are…well, time reversed! Is that right?

    • Anton Garrett Says:

      Another way to get insight is to find the solution of

      r” = sqrt(r)

      by two consecutive quadratures, which can be done by writing

      r” = (1/2) d (r’^2)/dr

      The first integral (ie, the energy equation) is easy; the next quadrature, to get t(r), is where the fun comes, and things depend on the order in which various limits get taken.

    • “The dome is very cute! I’d never seen that before! The mathematical underpinnings (Lipschitz continuity) is probably something with a ton of depth to it that I’ve also never seen.”

      If you are interested in this, you will probably enjoy the wonderful book by Thanu Padmanabhan, which manages to be both fun and deep.

  3. Yes, inflation can solve the flatness problem. However, it is important to remember that, when it was originally formulated, the question was not why Ω (the “total Ω” here, including the cosmological constant) is quite close to 1, as we now know it is, but rather why it is “of order 1”, i.e. between, say, 0.01 and 100, but not, say, 0.0000003 or 1,000,323,232,111.

    Let’s look at the original flatness problem.

    Yes, if Ω is slightly greater than 1, it will become arbitrarily large, but such large values happen only near the point of maximum expansion, before the universe collapses. So, as I have pointed out, a typical observer will not measure such large values.

    Adler and Overduin have also questioned the idea that the typical flatness scenario is improbable, illustrating it with a wonderful example from classical mechanics (which, via Newtonian cosmology, is clearly applicable here).

    If it is slightly less than 1, and the universe expands forever, then there are weak-anthropic (the dash is crucial) arguments for it not being arbitrarily small. In any universe which lasts forever, in some sense we are “near the beginning”. The flatness problem for Ω less than 1 is nothing stranger than this.

    We know today that the cosmological constant is positive. As pointed out in brilliant paper by Kayll Lake, in this case fine tuning is required to get a large value of Ω, not the other way around. Think about it: Ω is essentially the density divided by the square of the Hubble constant. The matter density decreases as the universe expands, that due to the cosmological constant is constant (which, in contrast to the Hubble constant, which owes its name to a constant factor in fitting a straight line, is why it is called the cosmological constant). So, the only way for Ω to become large is for the Hubble constant to become small, which happens only if the universe almost stops expanding and “coasts” for a while. This is possible only if the matter and cosmological constant are balanced—fine tuned. Ironically, Dicke, who, along with Peebles, is responsible for the classic formulation of the flatness problem, argued that such a coasting universe (which for a time was a hypothesis to explain why the universe can be much older than the Hubble time, back when one believed the Hubble constant to be much larger than we do today and the Hubble time correspondingly much less than the age of the Earth) is improbable on fine-tuning grounds, apparently not realizing that this solves the flatness problem (at least for a positive cosmological constant) which he would later formulate with Peebles (as far as I know, though, they never published anything on this in a refereed journal).

    All of the papers mentioned above (except the formulation of the flatness problem by Dicke and Peebles, which appeared in General Relativity: an Einstein Centenary Survey) are in leading journals in the field. As far as I know, and correct me if I am wrong, no-one has shown that any of the arguments in any of these papers are invalid. Usually, when one puts something wrong on arXiv, it is usually shot down, usually by more than one (team of) author(s), within a few weeks, at least if it is on an interesting topic (wrong orbital elements for a binary system might go unnoticed). This hasn’t happened, but people still talk about the flatness problem as if it were some big mystery. (OK, I might not be worth shooting down, but Lake is a well respected member of the GR community.) Maybe young people are scared of being on the weak side with so many luminaries expounding on the flatness problem, and the luminaries themselves are on record stating it is a problem, so would rather not address the issue.

    Coles and Ellis in their risk-your-career-with-a-low-Ω book and Evrard and Coles (mentioned above) come close, but sort of like Playboy before the pubic wars*, don’t go all the way. 🙂

    I don’t know. So, if you agree with Lake, Adler, Overduin, and me, cite our papers. If not, debunk them in the same or similar journals.


    * “What else is there? Sex and physics.”

    —Dennis Overbye

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