What is cosmological hysteresis?

Last week there were two new papers on the arXiv discussing an effect dubbed “cosmological hysteresis,” which, so the authors argue, would make cyclic cosmological models viable alternatives to inflation

Arrow of time in dissipationless cosmology
Varun Sahni, Yuri Shtanov, Aleksey Toporensky
arXiv:1506.01247 [gr-qc]

Hysteresis in the Sky
Sayantan Choudhury, Shreya Banerjee
arXiv:1506.02260 [hep-th]

Hysteresis is an effect more commonly known from solid state materials, when a material doesn’t return to its original state together with an external control parameter. The textbook example is a ferromagnet’s average magnetization whose orientation can be changed by applying an external magnetic field. Turn up the magnetic field and it drags with it the magnetization, but turn back the magnetic field and the magnetization lags behind. So for the same value of the magnetic field you can have two different values of magnetization, depending on whether you were increasing or decreasing the field.

Hysteresis in ferromagnets. Image credit: Hyperphysics.

This hysteresis is accompanied by the loss of energy into the material in form of heat, because one constantly has to work to turn the magnets, and in this cycle entropy increases. In fact I don’t know any example of hysteresis in which entropy does not increase.

What does this have to do with cosmology? Well, nothing really, except that it’s an analogy that the authors of the mentioned papers are drawing upon. They argue that a simple type of cyclic cosmological model with a scalar field has a similar type of hysteresis, but one which is not accompanied by entropy increase, and that this serves to make cyclic cosmology more appealing.

Cyclic cosmological models have been around since the early days of General Relativity. In such a model, each phase of expansion of the universe ends in a turnaround and subsequent contraction, followed by a bounce and a new phase of expansion. These models are periodic, but note that this doesn’t necessarily mean they are time-reversal invariant. (A sine curve is periodic and has a time-reversal invariance around the maxima and minima. A saw-tooth is periodic but not invariant under time-reversal.)

In any case, that the behavior of a system isn’t time-reversal invariant doesn’t mean its time evolution cannot be inverted. It just means it isn’t symmetric under this inversion. To our best present knowledge the time dependence of all existing systems can be inverted – theoretically. Practically this is normally not possible because such an inversion would require an extremely precise choice of initial conditions. It is easy enough to mix flour, sugar, and eggs to make a dough, but you could mix until we run out of oil (and Roberts) and would never see an egg separate from the sugar again.

Statistical mechanics quantifies the improbability in succeeding to reverse a time-dependence by the increase of entropy. A system is likely to develop into a state of higher entropy, but, except for fluctuations that are normally tiny, entropy doesn’t decrease because this is exceedingly unlikely to happen. That’s the second law of thermodynamics.

This second law of thermodynamics is also the main problem with cyclic cosmologies. Since entropy increases throughout each cycle, the next cycle cannot start from the same initial conditions. Entropy gradually builds up, and this is generally a bad thing if you want conditions in which life can develop because for that you need to maintain some type of order. The major obstacle in making convincing cyclic models is therefore to find a way to indeed reproduce the initial conditions. I don’t really know of a good solution to this. The maybe most appealing idea is that the next cycle isn’t actually initiated by the whole universe but only a small part of it, leading to “baby universe” scenarios. I toyed for some while with the idea to couple two universes that periodically push entropy back and forth, but this ended up in my dead drafts drawer, and ever since I’ve disliked cyclic cosmologies.

In the mentioned papers the authors observe that a cosmology coupled to a scalar field has two different attractors (solutions towards which the field develops) depending on whether the universe is expanding or contracting. In the expanding phase, a scalar field with a potential gets decelerated and slows down, which makes its behavior stable under perturbations because these get damped. In the contracting phase, the field gets accelerated instead, continues to grow, and becomes very sensitive to smallest perturbations because they get blown up. The time-dependence of this system is still reversible in theory, but not in practice for the same reason that you can’t unmix your dough. Since the unstable period tends to be very sensitive to smallest mistakes, you will not be able to reverse it perfectly.

Figure 2 from arXiv:1506.01247. Φ is the scalar field, a dot the time derivative. During expansion of the universe the field evolves along the arrows on the curves in the left figure. Different curves correspond to different initial conditions, but they all converge together. During contraction the field evolves along the curves as shown in the right figure, where the curves diverge with an increasing value of the field.

For the cyclic model this means that basically noise from small fluctuations builds up through each cycle. After the turnaround, the field will not exactly retrace its path but diverge from the time-reversal. That is why they refer to it as hysteresis.

Figure 1 from arXiv:1506.02260. a is the scale factor of the universe - a larger a means a larger universe, and w encodes the equation of state of the scalar field.  In this scenario, the scalar field doesn’t retrace the path it came after turnaround.

It also has the effect that the next cycle starts from a different initial condition (provided there is some mechanism that allows the universe to bounce, which necessitates some quantum gravity theory). In the studies in the paper, the noise is mimicked in a computer simulation by some small random number that is added to the field. More realistically you might think of it as quantum fluctuations.

Now, this all sounds plausible to me. There are two things though I don’t understand about this.

First, I don’t think it’s justified to say that in this case entropy doesn’t increase. The problem of having to finetune initial conditions to reverse the process demonstrates instead that entropy does increase - this is essentially the very definition of entropy increase! Second, and more important, I have no idea why that would makes cyclic cosmological models more interesting because they are just demonstrating exactly that it’s not possible to make these models periodic and one doesn’t return to anywhere close by the initial state.

In summary, cosmological hysteresis seems to exist under quite general circumstances, so there you have another cool word for your next dinner party that will make you sound very sciency. However, I don’t see how that effect makes cyclic cosmologies more appealing. What I learned from the papers though is that this very simple scalar field model already captures the increase in entropy through the cycles, which had not previously been clear to me. In fact this model is so instructive that maybe I should open that drawer with the dead drafts again...