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It is generally believed that $10^{-35}$ seconds after the Big Bang, the symmetry of a GUT was broken and after $10^{-12}$ seconds the electroweak force was broken:

\begin{equation} \mathrm{SU(2)} \times \mathrm{U(1)} \rightarrow \mathrm{U(1)} \end{equation}

This symmetry breaking is a result of the universe cooling down and undergoing a phase transition. I'm aware that the temperature of the universe it about $2.7$ Kelvin, so the temperature of the universe cannot decrease much more, but I was wondering if there is a chance that another phase transition might happen again in the future?

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In vacuum and with only the particles we know about the answer is no. Let's look at the symmetries we know exist in nature:

  1. $SU(3)$ colour: confined, only colourless states exist below the QCD phase transition
  2. $SU(2)\times U(1)_Y$ electroweak: Higgsed to $U(1)_{EM}$ electromagnetism
  3. $U(1)_{EM}$: Here we have opportunity. See below...
  4. $U(1)_{B-L}$: Global symmetry in SM, possible gauged symmetry of GUT or not a true symmetry at all. If gauged it's broken at a high scale. Note that we don't see any massless Goldstone bosons for this symmetry, so it can't be a spontaneously broken global symmetry. It must be Higgsed or not a symmetry at all. In either case breaking has already happened or never will. Breaking of this is relevant to baryogenesis but not to much else.
  5. QCD chiral flavour symmetry: only an approximate symmetry. Broken by chiral condensates, the pseudo-Nambu-Goldstones are the mesons. (I'll stop listing approximate symmetries otherwise this list will get very long, but this one is rather important.)
  6. Local Poincare invariance: Exact and "gauged" in general relativity. (Note: there is an ongoing debate about the semantics of whether gravity is a gauge theory. There are important similarities and differences between gravity and the standard Yang-Mills gauge theories. Hence the scare quotes on "gauged.")
  7. Global Poincare invariance ($SO(1,3)\ltimes \mathbb{R}_4$): Spontaneously broken by the fact that the universe is expanding and there is stuff in it. This is a symmetry of Minkowski spacetime, so it is often used in particle physics, but it is not a symmetry of our actual universe because it is expanding. In general there are no global symmetries or conservation laws in GR, but the usual spacetime symmetries hold to a very good approximation on galactic cluster and smaller scales. My previous language meant to convey this but was sloppy and inaccurate.

As far as I can see there are two options for spontaneous symmetry breaking in the current universe: either attack #3 or #6. What do you need to have to break either of these groups? You need an order parameter that transforms nontrivially under the symmetries to take a nonzero expecation value.

For electromagnetism that means you need a charged condensate, but we don't know of any charged scalars and the chiral condensate is necessarily neutral (why is that? good question ;)). In principle one of the $W^\pm$ could serve (spontaneously breaking Lorentz invariance as well), but this can't happen because they have large positive mass squared through the Higgs mechanism. You would have to generate a negative effective mass squared using some fancy new mechanism that definitely doesn't exist at the low energies we can see. So you can't break EM in vacuum, but you can in a medium where collective motions of many particles serve as the condensate. These exotic materials are called superconductors, and a few people think they are mildly interesting. ;)

That leaves local Poincare invariance. This can be broken by a vector or tensor field developing a condensate. People have looked at these sorts of models, but needless to say there is nothing like this in known physics. Experiments have demonstrated Poincare invariance to an incredible accuracy. Given the accuracy of the experiments and the cosmological scale of the transition temperature we're talking about you would need a vector or tensor with a cosmologically small negative mass squared. Needless to say this is problematic, especially if you want to identify these with the known gauge bosons or graviton.

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  • $\begingroup$ Interesting answer. Some points: 1) Global Poincare invariance is not spontaneously (where are the Goldstone bosons?). 2) Gauged Poincare invariance doesn't give GR in 4 dimensions. 3) I think you have forgotten (perhaps you want to add it) the most interesting symmetry for this answer: the Peccei–Quinn symmetry to solve the CP strong problem. $\endgroup$ – Diego Mazón Aug 15 '13 at 6:21
  • $\begingroup$ @drake Good points. 1) What are the global Poincare transformations for an FRW universe? I guess the answer is that it never was a symmetry to begin with... I was sloppy. :) I'll edit accordingly. 2) There is a semantic question over whether gravity is a gauge theory or not. For the purposes of this question I landed on the "yes" side, but it is a legitimate point, hence the scare quotes I put on "gauged". 3) I hadn't forgotten, but I left it out for two reasons: it's not in the SM and I'm focusing on known physics, and I don't feel competent enough to write about it at any length. $\endgroup$ – Michael Brown Aug 15 '13 at 6:29
  • $\begingroup$ Fair enough +1. Many people unjustifiably say that Lorentz invariance is spontaneously broken by the matter content of the universe. Please read physics.stackexchange.com/questions/71476/… to see in which precise senses general relativity is a gauge theory. Non of them includes the gauge theory obtained after gauging the Poincare invariance of special relativistic field theories. $\endgroup$ – Diego Mazón Aug 15 '13 at 21:51
  • $\begingroup$ @drake Interesting post, but I don't really get the issue. Doesn't local Poincare include diffeomorphims? I know there is no such thing as tetrads or torsion in Yang-Mills, but I'm not claiming gravity is Yang-Mills. So is the issue this: " Note that a gauge theory is a gauge invariant theory, but a gauge invariant theory doesn't has to be a gauge theory (for example, the Standard Model is gauge invariant, but it's not a gauge theory since the scalar self-interaction doesn't enlarge the gauge symmetry of the model)."? $\endgroup$ – Michael Brown Aug 16 '13 at 1:09
  • $\begingroup$ Because I would disagree with that, but it's a semantic question. I've never found it useful to draw a distinction between "gauge theories" and "gauge invariant theories," but maybe that's just me! Thanks for your vote anyway. :) $\endgroup$ – Michael Brown Aug 16 '13 at 1:10
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As Michael says, there don't seem to be any serious possibilities for a symmetry breaking, however there are some possibilities for tunneling into a different vacuum state. Ironically this would probably be a symmetry restoration rather than a symmetry breaking.

You may have heard of the String Landscape. This is the idea that string theory allows many metastable solutions for the universe and we may be in one of those metastable states. It's possible that the universe could tunnel into a state where supersymmetry is unbroken and all the compact dimensions are fully extended.

Slightly less speculative is the suggestion that the electroweak vacuum may be unstable. This has been briefly discussed in Measured Higgs mass and vacuum stability, or see the paper by Alekhin, Djouadi and Moch.

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