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  Are we living in a false vacuum? Is there any way to tell?

+ 7 like - 0 dislike

I was thinking of the noted 1980 paper by Sidney Coleman and Frank de Luccia--"Gravitational effects of and on vacuum decay"-- about metastable vacuum states that could tunnel to a lower energy "true vacuum" with catastrophic results. I suspect the answer is that if our vacuum is a false one, we wouldn't "see" the true one til it hit us.


This post imported from StackExchange Physics at 2014-05-14 19:48 (UCT), posted by SE-user Gordon
asked Jan 31, 2011 in Theoretical Physics by Gordon (400 points) [ no revision ]
Please don't post any answers describing how to cause the true vacuum to form!

This post imported from StackExchange Physics at 2014-05-14 19:48 (UCT), posted by SE-user Andrew Grimm
Whoa. Wait. So... Um, what? My brain hurts.

This post imported from StackExchange Physics at 2014-05-14 19:48 (UCT), posted by SE-user user8164

2 Answers

+ 6 like - 0 dislike

Inflation is a rapid stretching which result in cosmic smoothness and uniformity on large scales; as such, inflation is a key component of almost all fundamental cosmological scenarios. Not only does inflation explain the overall uniformity of the universe, but quantum fluctuations during inflation plant the seeds that grow into galaxies and clusters of galaxies that exist today.

The potential for the inflationary potential early in the universe is a de Sitter form. The FLRW equations are $$ \Big(\frac{\dot a}{a}\Big)^2~=~\frac{8\pi G\Lambda}{3}~-~\frac{k}{a^2}, $$ where we assume $k~=~0$ for the generally flat space we appear to observe. The early inflationary universe was driven by a scalar field which generated this vacuum energy where $V(\phi)~=~-a\times\phi$, $a$ a constant. This set the early cosmological constant for the de Sitter expansion with a vacuum energy about 13 orders of magnitude smaller than the Planck energy. The universe had more vacuum energy density than quark-gluon field density in a hadron.

The Lagrangian for a scalar field is $L~=~(1/2)\partial^a\phi\partial_a\phi~–~V(\phi)$ and in QFT we work with the Lagrangian density ${\cal L}~=~L/vol$ so the action $S~=~\int d^3xdt{\cal L}(\phi, \partial\phi)$. We run this into the Euler-Lagrange equation $\partial_a(\partial{\cal L}/\partial(\partial_a\phi))~-~\partial{\cal L}/\partial\phi~=~0$, and keep in mind $vol~\sim~x^3$. This gives a dynamical equation $$ \partial^2\phi ~-~ (3/vol^{4/3})\partial_a\phi~–~ \frac{\partial V(\phi)}{\partial\phi}~=~ 0. $$ If we assume the inflaton field is more or less constant on the space for a given time on the Hubble frame this DE may be simplified to $$ {\ddot\phi}~–~(3/vol^{4/3}){\dot\phi}~–~\frac{\partial V(\phi)}{\partial\phi}~=~0 $$ That middle term is interesting for it is a sort of friction. It indicates the inflaton field, the thing which drives the inflationary expansion, is running down or becoming diffused in the space. The potential function here is complicated and not entirely known, but it is approximately constant, or some small decrease with the value of $\phi$. What then happens, which is not entirely understood, is that the field experiences a phase transition, the potential becomes $V(\phi)~\sim~\phi^2$ with a minimum about 110 orders of magnitude smaller than it was in the unbroken phase. The phase transition has a latent heat of fusion that is released and this is the reheating. If the vacuum is a false vacuum then the $V(\phi)~\sim~\phi^4$

This means the accelerated expansion of the universe should be driven by either of these field and the a force which drives the field: $$ F~=~-\frac{\partial V}{\partial\phi} $$ which is larger for the steep potential, or the quartic. During this period a quantum fluctuation in the field is typically $\delta\phi~=~\pm\sqrt{V(\phi)}$. For the inflationary period the variation in the field due to the force is $\delta\phi_F~=~F/V$ $~sim~\phi^{-1}$ and the quantum fluctuation in the scalar field $\delta\phi_q~=~\pm const\sqrt{\phi}$ The quantum fluctuations can become larger than that classical variation in the field when $$ \delta\phi_F~=~\delta\phi_q~\rightarrow~\phi~\simeq~a^{1/3} $$ For the reheating potential $V(\phi)~=~b\phi^n $, $n~=~2,~4$ the condition for the fluctuation equal the classical field variation is $$ \phi~\simeq~(n^2/a)^{1/{(n+2}} $$ For $n~=~4$ the field may vary far less for the quantum fluctuation to equal the classical variation. If this happens for $n~=~4$ we would expect the universe to tunnel into a lower energy vacuum.

enter image description here

We now turn to some data H. V. Peiris and R. Easther, JCAP 0807, 024 (2008) arXiv:0805.2154 astro-ph. This figure illustrates joint 68% (inner) and 95% (outer) bounds on two variables which characterize the primordial perturbations, derived from a combination of WMAP and SuperNova Legacy Survey data. The predictions for our two inflationary models are superimposed. The numbers refer to the logarithm of the size of universe during the inflationary era. Cosmological perturbations are generated when this quantity is around $60$, so $\phi^4$ inflation is not consistent with the data.

So we are probably out of the danger zone for having one of Coleman-Luccia’s vacuum transitions which destroys everything.

This post imported from StackExchange Physics at 2014-05-14 19:48 (UCT), posted by SE-user Lawrence B. Crowell
answered Jan 31, 2011 by Lawrence B. Crowell (590 points) [ no revision ]
A very succinct explanation of inflation and vacuum decay. +1

This post imported from StackExchange Physics at 2014-05-14 19:48 (UCT), posted by SE-user user346
I must say that looks good, Lawrence +1

This post imported from StackExchange Physics at 2014-05-14 19:48 (UCT), posted by SE-user Gordon
You're making an awfully restrictive assumption here. There are many ways we could be living in a false vacuum; why do you assume it would be related to the potential of a single-field inflaton model?

This post imported from StackExchange Physics at 2014-05-14 19:48 (UCT), posted by SE-user Matt Reece
Matt, Of course this is not definative. Yet a simple model is most often the best to start with. The false vacuum potential is marked by a $\lambda\phi^4~-~\mu\phi^2$ or quartic potential, whether there is one or there are may fields. The upshot is I don't worry about asteroid impacts much, and I suspect death by mass asteroid impact is a billion billion billion times greater than death by Coleman-Luccia false vacuum transitions.

This post imported from StackExchange Physics at 2014-05-14 19:48 (UCT), posted by SE-user Lawrence B. Crowell

@LawrenceCrowell: The estimate for asteroid impact vs. vacuum decay is simple: we've seen asteroid impacts, and no vacuum decay, so the rate of one is on the order of 1 ~ few million years, and the rate of the other is less than order 1 ~ few billion years. Neither is worrying. The actual question is whether some other situation can lead to vacuum decay.

+ 1 like - 0 dislike

Let me simplify it to this level: you've got a ball moving in 1D. And you observe that the ball is in a stable equilibrium.

Then, you assume that the potential for it has a form of a polynomial of 4th degree in x, with $x=0$ at the position of the ball:
$V(x) = Ax^4+Bx^3+Cx^2$
(No additive constant and no linear term because we are in the equilibrium.)

You can find the coefficients of this polynomial by testing small oscillations around the equilibrium. For this you need to find the dependence of frequency of oscillations on amplitude. And suppose that you have found the following values:
By investigating this function you realise that it has another minimum, which is lower than the current one. So you are in "false vacuum" and you realised it without moving your system into it.

Now, of course this conclution is based on the assumption that the potential has this particular polynomial form and that whole your model works. But, in my opinion, this is the only context, it which the question makes sense.

One can always invent a model where true vacuum is created at some particular energy or even at some particular process. And in order to verify this we must get to the energy either destroying everything, either disproving the theory. Personally, I don't think that this is a sound scientific theory. But if allow yourself this "model freedom", then the answer to your question is: "yes the only way to check if we are in the false vacuum is to create the true one".

This post imported from StackExchange Physics at 2014-05-14 19:48 (UCT), posted by SE-user Kostya
answered Jan 31, 2011 by Kostya (320 points) [ no revision ]
Most voted comments show all comments
Very good points by @Ted and ones that physicists only too often forget about. Sometimes it seem to me that it is enough for physicist that the function is continuous and they expect well-defined Taylor expansion everywhere :) Needless to say, even when the function is smooth, it can converge almost nowhere. I am not sure why this basic point, that expansions are utterly and completely local, is not stressed more vigorously.

This post imported from StackExchange Physics at 2014-05-14 19:48 (UCT), posted by SE-user Marek
These objections fall into two categories. Of course there is no means of probing discontinuities outside the experimental range: that's an absolute limit. All others are contained in the matter of estimating your sensitivity to particular terms and placing consequent bounds on them. My "may not be easy" was intentionally understated. You need big enough perturbations, and sufficiently sensitive measurements. You have the least sensitivity for high degree terms, so this method always has a limited range. You want some reason to say "I think we can rules out terms above $x^8$ because ..."

This post imported from StackExchange Physics at 2014-05-14 19:48 (UCT), posted by SE-user dmckee
The only way a method like this has a hope of working is if you already know that the functional form of the potential is in some very narrow range (e.g., a quartic). Without that a priori knowledge, measurement, no matter how accurate, of the first $N$ terms of a Taylor expansion gives you precisely no reason to believe that the expansion will work well outside the range you've measured. It's not a question of "discontinuities" -- I can give you as many $C^\infty$ functions as you like that match the observations perfectly over the measured range but behave utterly differently outside it.

This post imported from StackExchange Physics at 2014-05-14 19:48 (UCT), posted by SE-user Ted Bunn
I don't see the point of this discussion about continuity. Anyway this gets you into phylosophy and credibility of inductive reasoning. And "there are no right or wrong answers in phylosophy". And, finally, here we are supposed to talk about physics.

This post imported from StackExchange Physics at 2014-05-14 19:48 (UCT), posted by SE-user Kostya

This doesn't work for a vacuum transition involving, say, creation of a brane anti-brane pair over a large volume, which then rearranges a string compactification. The small oscillations around the minimum simply aren't going to be sensitive to the contribution of a pair-creation event at such high energies, it's not a reasonable way to investigate high energy processes. Likely the only answer is going to be theoretical, from finding our string vacuum.

Most recent comments show all comments
But that only worked because you somehow knew the potential was quartic. In reality, if you make measurements only in the neighborhood of the equilibrium, you can measure the first few terms of a Taylor expansion of the potential, but you have no reason to expect those first few terms to be a good approximation over a large enough range of $x$ to justify this conclusion.

This post imported from StackExchange Physics at 2014-05-14 19:48 (UCT), posted by SE-user Ted Bunn
@Ted: With the ball version of the experiment it is possible to evaluate your sensitivity to various terms of the expansion, from which you can put limits on the size of the terms, and from that you can evaluate how "far" your proposed potential is good for. Get enough sensitivity (which may not be easy) and you can feel reasonably confident about the existence of the other minimum. 'Course it is not clear that you can do this with the vacuum.

This post imported from StackExchange Physics at 2014-05-14 19:48 (UCT), posted by SE-user dmckee

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