Friday, 19 January 2018

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).




No comments:

Post a Comment

Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...