Can a scalar field grow its energy as the universe expands?

Here is an update to a question we have been looking at this fall. This text is has been copied from a discussion at:

https://www.physicsforums.com/threads/does-a-fields-vacuum-density-violate-conservation-of-energy.977279/

The fields which we know from our everyday experience, for example, the photon field and the electron-positron field, have a constant number of quanta as the universe expands, if we ignore reactions with other particles.

 
            photon
A ~~~~~~~~~~~~~~> B
   <--------------------------------->
      expanding universe

A photon will gain a redshift when it moves in an expanding universe, because if it is emitted by an event A and absorbed in another event B far away, then B "sees" A receding at a great speed - B sees the photon redshifted. If we take the energy of the photon to be what B sees, then the photon has lost energy.

The redshift mechanism reduces the kinetic energy of a electron, too, when it moves in an expanding universe.

What about a scalar field which some people claim, is causing the acceleration of the expansion right now?

The first thing to note is that a field whose energy increases in an expanding universe is "exotic matter", that is, it has a negative pressure and negative gravity. It breaks various energy conditions of General Relativity. We do not know such matter from our everyday experience. It is highly speculative to assume that such matter could exist.

What about the energy content in the hypothetical scalar field? Is that energy contained in quanta of some kind? If yes, does the number of such quanta increase as the universe expands?

Our everyday experience is that energy in a weakly interacting system is, indeed, divided into well-defined quanta which can carry mass and kinetic energy.

The question is harder in a strongly interacting system, say, a crystal. Is the vibrational energy in a crystal divided into quanta of some kind? Certainly not in any unique way.

Suppose that we work in the Minkowski spacetime. The universe is not expanding. We want to excite the Higgs scalar field which has the famous Mexican hat potential. We use a huge particle collider to produce a vast density of Higgs particles. In that way we store a lot of energy in the Higgs field. Can we say that the excited Higgs field has its energy stored in quanta of some kind? It is a strongly interacting system.

We expect the excited Higgs field to release just the same amount of energy as we pumped into it. What if the universe is expanding at the same time? Can the released energy be bigger than the pumped energy?

That would be surprising. Rather, we would expect the Higgs field lose some of the kinetic energy of the Higgs particles as the universe expands.

The hypothesis that a scalar field can grow its energy in an expanding universe is highly speculative. It is at odds with what we know about other fields.