# FeaturedI Inflation, horizon-problem and thermal equilibrium

1. Feb 28, 2017

### haushofer

Dear all,

for the topic

I came across the following paper: https://arxiv.org/abs/1505.01456. In this paper the following arguments are given why the horizon problem is a fake problem. It raised by me the following question: Why is it assumed in the formulation of the horizon problem that the observed thermal equilibrium from the CMB has to be explained by thermalization right after t=0?

Or: Why did the energy content in the universe have to start out of equilibrium, while equilibrium itself is statistically much more common/probable than an out-of-equilibrium state?

Maybe 'there is something about the combination of statistical mechanics and gravity I don't grasp. Can anyone elaborate? Many thanks!

2. Feb 28, 2017

### Staff: Mentor

I'm not sure I buy the author's assertion that equilibrium is the "natural" state. His argument for that makes an implicit assumption of causal contact between all parts of the system. But that is precisely the assumption that is violated in non-inflationary cosmological models, and the lack of causal contact is precisely the issue that gives rise to the horizon problem.

3. Feb 28, 2017

### Chalnoth

Yup. Or to put it another way: how did location A know to be at the same temperature as location B if there has never been any contact between the two locations?

4. Feb 28, 2017

### Staff: Mentor

I was actually thinking of a more specific argument, because it looks to me like the author of the paper has an answer to this one: that location A being at the same temperature as location B is the "natural" state anyway, so it doesn't need to be explained; what would need to be explained would be if they were not at the same temperature.

The more specific argument I was thinking of goes like this: suppose I have a system S that is split into two subsystems, A and B, each of which contains an equal portion of S (equal volume, equal numbers of particles, etc.). (We make no assumptions at this point about the relationship between the two subsystems.) The total energy of S is fixed at E. The phase space of S includes a huge region, much huger than any other region, corresponding to thermal equilibrium.

Now suppose we pick, at random, a particular microstate in the phase space of S. The author's argument is that we are overwhelmingly likely to pick a microstate that is in the thermal equilibrium region of the phase space; and on that basis, he claims that it is overwhelmingly likely that the temperatures of the subsystems, A and B, are the same. However, that argument leaves out a crucial step. To see what it is, ask the question: in the particular microstate we picked, how is the total energy E split up between the energies of the two subsystems, which we can call EA and EB? The answer is that it is overwhelmingly likely that EA and EB are not equal.

Now, if the subsystems A and B are causally connected, the fact that EA and EB are not equal in a particular randomly chosen microstate doesn't affect the author's claim about the temperature, because that microstate will evolve to another microstate in which EA and EB are different than the first one--and then to another where they are different again, etc., etc. And on average, we will find that all of those differences cancel out, so that on average, the energy E is partitioned equally between EA and EB. In other words, the causal connection allows energy to be exchanged between the subsystems A and B, and this energy exchange is what makes their temperatures equal, on average.

But if the subsystems A and B are not causally connected, they cannot exchange energy, and so if we start in a microstate where EA and EB are different (which is overwhelmingly likely), they will stay different (because they will each be constant, since the subsystems are isolated). And because EA and EB are different, the temperatures TA and TB will be different as well (because everything else about the two subsystems is the same).

Another way of stating this argument is that the author has assumed a particular Hamiltonian for the total system S, one that induces an evolution on phase space that exchanges energy between the subsystems A and B. But there is another possible Hamiltonian which does not exchange energy between A and B. (In fact there will be many possible Hamiltonians of each type.) So the real question is, which Hamiltonian is the right one? And that depends on the causal connection, or lack thereof, between the two subsystems.

5. Feb 28, 2017

### Chalnoth

The problem with that, as I see it, is that in no location is the temperature close to constant. On the contrary, the temperature at a given point in the universe varies by many orders of magnitude as the universe evolves.

You might be able to say that, "If physical process X occurs, it results in a system with temperature Y," but then why would this same physical process happen at exactly the same time in different regions if there is no communication?

6. Mar 1, 2017

### Staff: Mentor

This is true, but I don't think it affects the relative temperatures of different parts of the universe. If they started out the same, evolution of the universe would not by itself make them different.

I don't think his argument even considers specific physical processes; he's just making a very general assumption about the phase space of the system. In the terms I used at the end of my last post, I don't think he assumes any specific Hamiltonian (which is what would describe the actual physical processes going on), but he does appear to assume that the Hamiltonian will induce an evolution that allows energy exchange between all parts of the system.

I suppose he could be making an alternate assumption along the lines you suggest, where somehow the same physical process "should" happen in different regions of the universe that are causally disconnected. But I'm not sure how that would translate into the statements he makes about phase space and distribution functions.

7. Mar 1, 2017

### haushofer

Maybe this is the point I'm missing: why exactly is this statement true?

8. Mar 1, 2017

### Staff: Mentor

Because, heuristically, the set of points in the phase space in which the total energy E is exactly equally distributed between the two subsystems should be a set of measure zero; almost all microstates of the total system S should be microstates in which the energy distribution among the individual particles is not perfectly uniform. Thermal equilibrium when the subsystems are in causal contact is achieved not by always keeping the total energies of the two subsystems exactly equal, but by constantly exchanging energy in the subsystems, so if we have $EA > EB$ at one instant, we will most likely have $EA < EB$ at the next instant, and over time the fluctuations will average out and the temperatures of both subsystems will be equal. But that only works if the subsystems are in causal contact.

9. Mar 1, 2017

### Chalnoth

But why would they start out the same at the same time? Presumably some sort of event had to precipitate the start of the universe. That event might always happen in exactly the same way, but why would it happen at exactly the same time in locations that are causally-disconnected?

That doesn't seem to me to be worthwhile, for the simple reason that the entropy in the universe was extraordinarily low compared to today. If you're just going to make a phase-space argument, the natural conclusion is not a uniform universe, but no dynamical universe at all.

I think my problem with his whole line of argument is that he's trying to discuss this matter without reference to any specific early-universe physics, but in doing so he's implicitly assuming certain properties of the early-universe physics. By making those assumptions implicit, it hides their impact on the conclusions.

10. Mar 1, 2017

### Chronos

The absence of causality has unpredictable consequences. The common term for this phenomenon is magic.

11. Mar 1, 2017

### Staff: Mentor

I think this is a fair criticism.

12. Mar 6, 2017

### haushofer

I'm busy with work (developing some new courses at my university), but I'm still intrigued by this topic, as I want to include something about it in my own "soon to come" (that was sarcastic) book on quantum gravity in Dutch.

What would those assumptions exactly be, according to you? I still have the intention to read up a a little about the ekpyrotic scenario by Steinhardt and others; as I understand, such a scenario doesn't need inflation and hence I expect that thermal equilibrium is achieved somehow.

13. Mar 6, 2017

### Chalnoth

I think I laid them out about as well as I can above.

14. Mar 7, 2017

### haushofer

Ah, overlooked it, I see it now. Clear enough, thanks!

15. Mar 8, 2017

### Khashishi

If you believe the universe is infinite, then at some far enough distance there hasn't been time to equilibrate even with inflation. If you assume the universe is globally isotropic and homogeneous, then you are making assumptions that the universe started out with similar conditions everywhere, in which case, it's only natural to assume it had the same temperature everywhere. On the other hand, inflation is consistent with universes which are not globally isotropic and homogeneous, where it only appears so because our observable universe is too small.

16. Mar 8, 2017

### Chronos

17. Mar 31, 2017

### Chronos

18. Apr 3, 2017

### Chalnoth

I don't understand. This is an article by Tim Maudlin, and I don't see that it's a very reasonable argument against locality. The main thing that grates me about it is it relies mostly upon quotes from famous scientists, rather than any sort of serious analysis.

19. Apr 3, 2017

### Chronos

Maes authored the paper cited by the OP. I assume Maudlin's purpose was to review the significance of Bell's work, not validate it. It appeared both authors were making a case for relaxing the notion of locality.