The field may initially have, say, the value 0 in all space, and it somehow moves slowly to a lower energy value of, say, 1.
What does all this mean? We do have some experience of scalar fields in classical systems, but such fields are not the lowest level primitive fields. Such scalar fields are a high-level abstraction in a system consisting of elementary particles.
For example, if we have a drum skin, its vertical displacement is a scalar value in the 2D space of the skin. The lowest level description of the skin consists of atoms. The vertical displacement is a high-level concept.
If the 2D space expands, then the vertical displacement is affected in various ways.
Does the energy of a primitive field grow as the space expands?
Why would the inflaton field in an expanding space keep its scalar value, so that the energy of the field increases as the space stretches? Such a field certainly is "exotic matter" if its energy grows as the space grows.
The energy of an individual photon decreases in an expanding universe relative to "static" matter. The decrease can be attributed to the redshift that the static matter sees relative to the matter from which the photon originated.
The number of stable elementary particles, like the proton or the electron, probably does not grow in the expansion.
The energy content of the electron field preserves the energy in the rest mass of the electrons, but loses the kinetic energy of the electrons, because an electron keeps moving to a location where its velocity is less relative to static matter.
We face the dilemma why a scalar field would behave differently. Why would its energy grow in the expansion, if the energy of the electron field does not grow?
Is a non-zero value of a field always associated with some quantum?
If we have an electric or a magnetic field which differs from zero, there is a quantum associated with it: a photon, or a quantum of some other field, like the electron field. The electromagnetic field does not have a life of its own independent of these quanta.
If we have a scalar field, why would it have a non-zero value which is not associated with a quantum of some field?
We need to study the logic behind the Higgs field. The Higgs boson is an excitation of the field. The Higgs field is assumed to be non-zero throughout the space. The non-zero value of the field is not born out from some quanta.
The vacuum is often defined as the lowest energy state of fields. There are no quanta present then.
We might then claim that quanta must be present if the fields are not in the lowest energy state. If the inflaton field initially is at a metastable higher energy state, then there must exist quanta which are responsible for that. If space expands, the number or energy of those quanta does not grow. Rather, the quanta are diluted in a larger volume of space. This would imply that the inflation mechanism cannot work. The value of the inflaton field will approach the minimum energy value as the quanta are diluted.
A charge particle of a scalar field?
The reasoning above forbids exotic matter whose energy would grow as space expands. It also predicts a new particle, the charge particle of a scalar field. The charge particle would be responsible when the field has a value which is not the minimum energy value. Can the charge particle decay to other particles, so that the charge is lost? If that is the case, then the scalar charge is not conserved.
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