# I Inflation and the false vacuum

1. Apr 27, 2015

### friend

I'm trying to understand more about how our present universe is supposed to be the result of a false vacuum falling to the present vacuum energy.

I've been told (correct this if it's wrong), the universe initially underwent a kind of hyperinflation, expanding exponentially due to a much higher vacuum energy than now. But at some point this high vacuum energy somehow fell to a lower vacuum energy which resulted in a much slower expansion rate. Is this much true, and can someone please supply a few more words about this? For example, was it the inflaton field that decayed into the present particles as the vacuum energy fell? Is this the point where the higgs mechanism gave particles mass? What is meant by vacuum energy? Is this the same concept that the fields had a zero-point-energy. Did the fields of nature during inflation still obey the Heisenberg Uncertainty Principle such that the energy of the field(s) times the duration they existed could be larger than today? Thanks.

Last edited: Apr 27, 2015
2. Apr 27, 2015

### Staff: Mentor

Basically, yes. But the term "vacuum energy" here might be misleading. A better term is "energy in the inflaton field", to keep it distinct from the dark energy we observe in the universe now. See below.

More precisely, the energy in the inflaton field was converted into energy contained in matter and radiation. This process is called "reheating" by cosmologists. At the end of the process, the universe contained a very high density of matter and radiation, plus a very low (by comparison at that time) density of dark energy. The dark energy, at least in the model that AFAIK is currently accepted, was not produced from the energy in the inflaton field; it was there all the time. (This is because "dark energy" in this model is actually just a cosmological constant, which was "built in" to the structure of spacetime from the start, whereas the inflaton field is a separate field.)

No. That came later, during the electroweak phase transition.

Not really. See above.

If they obeyed the HUP, which as far as we know they did, then the energy times the duration was the same as today--at least, heuristically (the actual quantum field theory involved is more complicated than that).

3. Apr 28, 2015

### nikkkom

My understanding of the current situation with our knowledge is as follows:

General Relativity satisfactorily explains "classical" (non-quantum) behavior of space-time, including curvature (gravity). It includes a possibility of empty space having a property of self-expanding (lambda term in GR equations), and observations seem to indicate that lambda in observable Universe has a ery small, but non-zero value.

However, GR doesn't explain *why* lambda has this value.

Standard Model, the quantum theory of particles and forces, satisfactorily explains behavior of particles. Their interactions. Their decays.

However, these two theories don't mesh fully with each other. For example, in SM one can calculate the energy of empty space. It's non-zero because of vacuum fluctuations. And the result is *vastly* larger than tiny observed value of lambda. Basically, if we use SM'd predictions on energy density of Universe, then GM predicts that such Universe must have collapsed very soon after creation.

SM and/or GR needs modifying so that their predictions match observations. We know that this must be done.

Inflaton field is one way to extend SM. Roughly, it can work as follows: in "unbroken", symmetric state, vacuum energy is high and Universe expands very rapidly. This solves a few difficulties in early Big Bang scenarios.

But this state is not a state with lowest energy. "Eventually" inflaton field finds a state with energy minimum (similar to how "symmetric" Higgs field is not stable and finds its own minimum), in which vacuum energy is very low (matches observed lambda).

Since currently all "inflaton" theories are very young, they can't yet give predictions testable with experiment (e.g. they can't predict current lambda). Giving time, we can start making predictions, comparing them to observations and see how well it goes.

4. Apr 28, 2015

### Staff: Mentor

I'm not sure this is an accurate description of all inflation models. In at least some of them, as I understand it, the dark energy (or lambda) that we currently observe is a different field (or cosmological constant) from the inflaton field.

5. Apr 28, 2015

### friend

Was a field (e.g. inflaton field) necessary to cause inflation? Or could inflation occur without any fields being involved? It does seem necessary, don't you think, that something with energy must exist everywhere (thus a "field") to cause change (expansion) everywhere all at once? If a field was involved, was it necessary to be a quantum field, by which I mean did it have a conjugate momentum and a non-zero commutator between the field and its conjugate momentum? If it had a non-zero commutator, then doesn't this imply a non-zero uncertainty principle between the expectation value of the field times the expectation of the conjugate momentum, and wouldn't that result in a vacuum energy? Then doesn't this all imply that there are virtual particles (of inflatons to begin with)? And if these virtual inflatons are not charged, then how would these virtual particles recombine and what would they recombine into? (This last question might be asked about higgs bosons and gluons, if such particles can exist as virtual particles) Is the inflaton field the same thing as the unified field that is supposed to exist from which all the other fields came from?

I appreciate any answers. You don't have to answer all the questions in order to answer one. Thanks.

Last edited: Apr 28, 2015
6. Apr 28, 2015

### Staff: Mentor

Yes. More precisely, from a GR point of view, inflation requires a nonzero stress-energy tensor with particular properties. The only ways we know of to produce such a stress-energy tensor are a cosmological constant or a scalar field. But a cosmological constant is, well, constant, so if that had caused inflation it would never have stopped. So the only possibility left is a scalar field.

I don't know that the quantum aspects of the field would come into play in predicting inflation itself; a simple classical scalar field can do that. But quantum fluctuations in the field are required to explain the "wrinkles" that are being found in, for example, the CMB, by the WMAP satellite and now by the Planck satellite.

No. Inflation is not a perturbative phenomenon, i.e., the perturbation theory formalism is not used to describe it. So the virtual particle idea isn't useful in trying to visualize how the inflaton field works.

No. The "unified field" idea applies to the fields in the Standard Model (plus, in some versions, gravity). The inflaton field is a separate field, not included in any of that.

7. Apr 28, 2015

### friend

What about reheating? Isn't this where the inflation field decays to ordinary particles (perhaps without mass yet)? That would suggest that the inflaton field is the unified field, right?

8. Apr 28, 2015

### Staff: Mentor

No. In reheating, the potential energy stored in the inflaton field gets transferred to the fields corresponding to the ordinary Standard Model particles (and yes, they are all massless at this temperature--they don't gain mass until the electroweak phase transition, which comes later, when the temperature has fallen far enough for the Higgs field to condense out and take on a vacuum expectation value). It's just an ordinary transfer of energy from one field to another.

9. Apr 28, 2015

### friend

How is this different than what I was thinking? Was it my use of the term "decay"? Of course I meant the energy of the inflaton field was "transfered" to the SM fields, which is what you said. The same thing happens in the unified field, right, the energy of the unified field gets transfered to the SM fields? I fail to see the distinction.

10. Apr 28, 2015

### Staff: Mentor

You asked if the inflaton field "is" the unified field. It isn't.

That's not the same as saying the inflation field "is" the unified field. "Is" means "is", not "can transfer energy to".

No, the unified field is the SM fields. More precisely, what we call "the SM fields" are all just different low-energy states of the unified field. They're not different fields that the energy of the unified field gets transferred to. (Note that this all assumes that we have a grand unification theory that incorporates all the Standard Model fields, which we don't; we have candidates for such a theory, but no good experimental data to test them.)

11. Apr 28, 2015

### friend

Is the unified field the same as when energies are so high that the coupling constants for charge and mass become equal so that we can not tell the difference between the forces and particles?

12. Apr 28, 2015

### nikkkom

13. Apr 28, 2015

### nikkkom

Unified field is a field which contains (in case of electroweak and strong unification) leptons and quarks as its components, showing that they are not unrelated things. Components should change into each other under a suitable symmetry group, such as SU(5).

14. Apr 28, 2015

### Staff: Mentor

What's called the "grand unification energy" is the energy at which the coupling constants for the three Standard Model interactions (strong, weak, and electromagnetic) become equal. Charge is one of these (the electromagnetic coupling constant). Mass is not; the Higgs mechanism for giving the SM particles mass is separate from all this.

The "unified field" is a hypothesis for why the coupling constants become equal at the grand unification energy--because all three SM interactions are really just aspects of the same interaction of the unified field.

You can still tell the difference between "forces" (interactions, i.e., gauge bosons) and "particles" (fermions) even at energies at or above the grand unification energy.

15. Apr 29, 2015

### friend

So what is supposed to cause the forces to distinguish themselves? Is this where symmetry breaking comes in? Is that accomplished by the higgs mechanism? Or is it something else? Thanks.

16. Apr 29, 2015

### Staff: Mentor

Yes. At least, that is the hypothesis on which all of the current candidates for a grand unified theory are based. But as I said, we don't really have a way to experimentally test any of them. The Standard Model does not include any of this.

No. The Higgs mechanism is associated with electroweak symmetry breaking, which is included in the Standard Model, and which is a different symmetry breaking process than the ones hypothesized in connection with grand unification. Electroweak symmetry breaking happens at a much lower energy than the hypothesized grand unified symmetry breaking.

Last edited: Apr 29, 2015
17. Apr 29, 2015

### friend

What is supposed to break then the unified field symmetry if not the higgs mechanism? Is there supposed to be a different field like the higgs, but not the higgs, that breaks this greater symmetry? Or perhaps more basic, what generally breaks a symmetry? Is this where something turns on or off or changes the value of the structure constants of the group associated with that symmetry? Thanks again.

18. Apr 29, 2015

### Staff: Mentor

I believe that's one of the candidate hypotheses, yes.

A good general way to look at symmetry breaking--or more precisely, "spontaneous symmetry breaking", which is what the various mechanisms we've been talking about are, is that it happens in systems which have two dynamic regimes. In the "high energy" regime, solutions to the equations describing the system have the same symmetry as the equations themselves. But in the "low energy" regime, solutions no longer have that same symmetry; they may have a lesser symmetry or they may have none at all. Instead, there will be families of solutions that, taken all together, manifest the symmetry of the equations.

So what breaks the symmetry is simply a lowering of energy of the system; for example, the average temperature of the universe decreases as it expands, and when the temperature of the early universe became lower than the threshold for electroweak symmetry breaking, the electroweak phase transition happened, and the symmetry of the underlying electroweak equations was no longer manifested directly in the solution describing the particles in the universe. Instead, you would have to look at a whole family of possible solutions (only one of which was actually realized in our universe) to see the underlying symmetry of the equations.

No. The structure constants are just that: constants. They never change. The underlying group structure is still there; it just manifests differently.

19. Apr 29, 2015

### friend

I was thinking more mathematically. As I understand it, symmetry means that the fields involved have a non-zero commutator that defines the Lie algebra for those fields, and there are certain structure constants associated to those commutators. So the only way for symmetry to break or change would be to change the structure constants of that commutator that defines the Lie algebra of the Lie group involved. This makes me think that somehow the structure constants have to change in order to break the symmetry. Does any of this sound right?

20. Apr 29, 2015

### Staff: Mentor

No, that's not what symmetry means. Symmetry in general means that there is some group of transformations that leaves something unchanged. In physics, what is left unchanged is the laws or equations that apply to something. Some examples:

(1) Lorentz invariance is a symmetry in SR that applies to physical laws: all valid physical laws in SR must remain unchanged under the group of Lorentz transformations.

(2) Gauge invariance is a symmetry that applies to the laws of electromagnetism; those laws remain unchanged under the group of gauge transformations. For electromagnetism, that group is the group U(1).

(3) Electromagnetism is just one example of a gauge theory; there is a more general class of similar theories where the group of gauge transformations is some other group besides U(1). Since those other groups are not Abelian (i.e., the group operation does not commute), these theories are often called non-Abelian gauge theories. Another name for them is Yang-Mills theories, after the two physicists who first discovered them. The Standard Model is an example of such a theory; the gauge group in the SM is SU(3) x SU(2) x U(1). (Actually, that's not precisely true, but it's close enough for this discussion.)

The connection with Lie algebras, commutators, and structure constants is simply that all of the groups I mentioned above are Lie groups, and any Lie group has an associated Lie algebra, which can be thought of as the algebraic structure of the group in an infinitesimal neighborhood of the identity. For an Abelian group such as U(1), since the group operation commutes, all commutators of the generators of the Lie algebra (the "generators" are the set of linearly independent operators that form the basis of the Lie algebra) are zero. (Note that this shows that a symmetry does not need to be associated with nonzero commutators of anything.)

For a non-Abelian group, however, such as SU(2) or SU(3) (or the tensor product group that is the gauge group for the SM), the commutators of the generators of the Lie algebra are nonzero. In such cases, it turns out that each commutator of a pair of generators can be written as a linear combination of the generators; the structure constants are simply the coefficients in the linear combinations. (Note that these are commutators of operators, not fields.)

The structure constants of the Lie algebra of any Lie group, as I said before, cannot change; they are inherent properties of the group. When spontaneous symmetry breaking occurs, what happens, as I said before, is that, instead of the full symmetry of the equations being manifest in a solution, only a reduced symmetry (or no symmetry at all) is manifest in a solution--the full symmetry is only manifest in a family of solutions. If a reduced symmetry is still present, the reduced symmetry group must be a subgroup of the full symmetry group.

For example, if we just look at the electroweak sector of the Standard Model, the full symmetry group is SU(2) x U(1). At energies above the electroweak phase transition energy, this full symmetry group is manifest in the actual physical solution. This is usually described as the symmetry transformations--i.e., the SU(2) x U(1) gauge transformations--leaving invariant the vacuum state--the state of lowest energy. When electroweak symmetry breaking occurs, the full set of SU(2) x U(1) gauge transformations no longer leave the vacuum state invariant; instead, the vacuum state is left invariant only by a reduced set of U(1) gauge transformations. The reduced symmetry, U(1), is a subgroup of the full symmetry group SU(2) x U(1). (These U(1) gauge transformations are in fact just the gauge transformations of electromagnetism.)

21. Apr 29, 2015

### friend

Thank you. That was a very well written answer. You've given me a lot to think about. I can not think of any more questions.

22. May 1, 2015

### friend

Thanks for your responses. But I guess what I was really after, then, is HOW the inflaton field cause spacetime to inflate. Is there a word description of how this occurred?

23. May 1, 2015

### Staff: Mentor

The inflaton field, dynamically, works just like dark energy; it behaves like a cosmological fluid with an equation of state of $p = - \rho$, so that it causes accelerating expansion. Since the energy density of the inflaton field was very high, much higher than the density of dark energy in our current universe, the rate of acceleration of the expansion was correspondingly much larger.

24. May 1, 2015

### friend

OK, so you seem to be describing inflation in terms of global variables, $p = - \rho$ . But I'm wondering about it in microphysical terms. Since it's called an inflaton, I'm assuming it is a quantum field. Does this suggest particles or virtual particles of inflatons? I know we don't have a quantum theory of gravity yet. I don't know if we don't have a quantum theory of inflation, though.

25. May 1, 2015

### Staff: Mentor

Strictly speaking, yes, but quantum fluctuations in the field don't really play a role in the dynamics. A classical scalar field can produce inflation by the mechanism I described. As I understand it, quantum fluctuations in the field are necessary to trigger the phase transition that ends inflation, though.

Not necessarily. Not all states of a quantum field have particle interpretation.