# Do the standard cosmology models spontaneously break Lorentz symmetry?

In standard cosmology models (Friedmann equations which your favorite choice of DM and DE), there exists a frame in which the total momenta of any sufficiently large sphere, centered at any point in space, will sum to 0 [1] (this is the reference frame in which the CMB anisotropies are minimal). Is this not a form of spontaneous Lorentz symmetry breaking ? While the underlying laws of nature remain Lorentz invariant, the actual physical system in study (in this case the whole universe) seems to have given special status to a certain frame.

I can understand this sort of symmetry breaking for something like say the Higgs field. In that situation, the field rolls down to one specific position and "settles" in a minima of the Mexican hat potential. While the overall potential $V(\phi)$ remains invariant under a $\phi \rightarrow \phi e^{i \theta}$ rotation, none of its solutions exhibit this invariance. Depending on the Higgs model of choice, one can write down this process of symmetry breaking quite rigorously. Does there exist such a formalism that would help elucidate how the universe can "settle" into one frame ? I have trouble imagining this, because in the case of the Higgs the minima exist along a finite path in $\phi$ space, so the spontaneous symmetry breaking can be intuitively understood as $\phi$ settling randomly into any value of $\phi$ where $V(\phi)$ is minimal. On the other hand, there seems to me to be no clear way of defining a formalism where the underlying physical system will randomly settle into some frame, as opposed to just some value of $\phi$ in a rotationally symmetric potential.

[1] The rigorous way of saying this is : There exists a reference frame S, such that for all points P that are immobile in S (i.e. $\vec{r_P}(t_1) = \vec{r_P}(t_2) \forall (t_1, t_2)$ where $\vec{r_P}(t)$ is the spatial position of P in S at a given time $t$), and any arbitrarily small $\epsilon$, there will exist a sufficiently large radius R such that the sphere of radius R centered on P will have total momenta less than $\epsilon c / E_k$ (where E is the total kinetic energy contained in the sphere).

Ben Crowell gave an interesting response that goes somewhat like this :

Simply put then : Causally disconnected regions of space did not have this same "momentumless frame" (let's call it that unless you have a better idea), inflation brings them into contact, the boost differences result in violent collisions, the whole system eventually thermalises, and so today we have vast swaths of causally connected regions that share this momentumless frame.

Now for my interpretation of what this means. In this view, this seems to indeed be a case of spontaneous symmetry breaking, but only locally speaking, because there should be no reason to expect that a distant causally disconnected volume have this same momentumless frame. In other words the symmetry is spontaneously broken by the random outcome of asking "in what frame is the total momentum of these soon to be causally connected volumes 0?". If I'm understanding you correctly, this answer will be unique to each causally connected volume, which certainly helps explain how volumes can arbitrarily "settle" into one such frame. I'm not sure what the global distribution of boosts would be in this scenario though, and if it would require some sort of fractal distributions to avoid running into the problem again at larger scales (otherwise there would still be some big enough V to satisfy some arbitrarily small total momentum).

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There exists a reference frame S, such that for any point P in S I don't quite follow what you mean here. A frame of reference is basically a timelike, normalized vector, not a set of points. –  Ben Crowell Jul 26 at 15:18
@BenCrowell Fair point, what I should have said was that for any point P that is static in time in S, there exists etc etc. Will edit the question to reflect this. (Is my edit more to your liking ?) –  ticster Jul 26 at 15:20
I think the only possible explanation is that in the early universe, all matter-energy had to be in a well causally-connected and strongly "collisionful" state. A pretty generic development then gives you large-scale isotropy and homogeneity of the equilibrium frame of the matter-energy. Inflation kinda tries to address this long range causality implied by homogeneity. –  Void Jul 26 at 15:25
@Void I understand the intuition behind what you're saying, but it doesn't really address the problem here. "large-scale isotropy and homogeneity of the equilibrium frame of the matter-energy" How does this frame get chosen ? Why this very particular CMB frame and not one that's rushing towards Virgo at $0.5c$ ? And whatever the answer to that, is this a form of spontaneous symmetry breaking ? –  ticster Jul 26 at 15:28

Nice question.

First off, there's a definitional problem because we can't apply a Lorentz boost to the universe as a whole. Lorentz symmetry is a local thing. So when we talk about "Lorentz symmetry" for the universe as a whole, I think we have to keep in mind that we mean something a little different. Basically if the "Lorentz symmetry" has already been broken, I think we mean something like the concept that as various regions of the universe come into causal contact, the process doesn't require any further violent process of collision and equilibration.

There's a logical reason why we expect the universe to have started out in a state the doesn't break the rotational symmetry of the Mexican hat potential. It was very hot, so it hadn't yet undergone a phase transition to a symmetry-breaking state. Here I think the analogy with the broken Lorentz symmetry fails. We don't have any physical principle that constrains what the very early universe should have been like, so it's logically perfectly OK to imagine that the universe just always lacked this symmetry. In fact, many cosmological models assume exactly that, and those models are self-consistent.

However, one could argue that this is unsatisfactory, because there is no physical reason for regions of the universe that were initially out of causal contact to make causal contact with each other "gently" -- there is no causal link between their motions, so we would expect the meeting to be violent, and there would have to be some kind of settling down into a state where the matter was all like a nice, well-mixed perfect fluid whose state of motion we call the Hubble flow. I think this is just a special case of the horizon problem. If inflation, for example, fixes the horizon problem, then probably it also fixes the "Lorentz symmetry" problem.

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Thank you for the interesting response. See my soon to come edit for some issues I have with this idea. –  ticster Jul 26 at 15:55

The special state of motion you're talking about is often called the Hubble flow. (edit: oops, Ben Crowell already mentioned this.)

I think that in modern slow-roll inflation the source of this asymmetry is an asymmetry in the tiny (Planck-scale?) seed of inflation, whatever it was, inflated by a factor of $e^{60}$ or more. The inflaton potential has to be non-constant if it's eventually to reach a minimum and trigger reheating, and the direction of that gradient ends up determining the direction of the Hubble flow. Ben Crowell's idea of violently colliding bubbles sounds more like Guth's original version of inflation.

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