If by "inflaton field" you mean the scalar field that drove the period of primordial accelerated expansion, then that field is not generally considered to be relevant to the late-time expansion. While indeed both primordial and late-time accelerated expansion could be referred to as "inflation", the term is usually, when used without qualification, meant to refer to primordial inflation.

It is, however, possible to model the dark energy-fueled late-time expansion as being due to an evolving scalar field as well; this idea is called quintessence. The only difference between the primordial and late-time accelerated expansion is its rate, which is in proportion to the energy density of field.

There was an early model of primordial inflation, called quintessential inflation, that posited that the same field is responsible for both epochs, though the shape of this field's potential energy function was arguably fine-tuned in order to accomplish this. See: https://arxiv.org/abs/astro-ph/9810509

Also, slight modification to your description of inflation. When inflation was underway, the inflaton field was evolving in a very flat portion of its potential (the energy density was nearly constant). As the inflaton rolled down toward the false vacuum, the energy density began to decay and when it reached the vacuum it underwent oscillations which resulted in the decay of the inflaton field into thermal degrees of freedom.

If by "inflaton field" you mean the scalar field that drove the period of primordial accelerated expansion, then that field is not generally considered to be relevant to the late-time expansion. While indeed both primordial and late-time accelerated expansion could be referred to as "inflation", the term is usually, when used without qualification, meant to refer to primordial inflation.

It is, however, possible to model the dark energy-fueled late-time expansion as being due to an evolving scalar field as well; this idea is called quintessence. The only difference between the primordial and late-time accelerated expansion is its rate, which is in proportion to the energy density of the field.

There was an early model of primordial inflation, called quintessential inflation, that posited that the same field is responsible for both epochs, though the shape of this field's potential energy function was arguably fine-tuned in order to accomplish this. See: https://arxiv.org/abs/astro-ph/9810509

Also, slight modification to your description of inflation. When inflation was underway, the inflaton field was evolving in a very flat portion of its potential (the energy density was nearly constant). As the inflaton rolled down toward the true vacuum, the energy density began to decay and when it reached the vacuum it underwent oscillations which resulted in the decay of the inflaton field into thermal degrees of freedom.

If by "inflaton field" you mean the scalar field that drove the period of primordial accelerated expansion, then that field is not generally considered to be relevant to the late-time expansion. While indeed both primordial and late-time accelerated expansion could be referred to as "inflation", the term is usually, when used without qualification, meant to refer to primordial inflation.

It is, however, possible to model the dark energy-fueled late-time expansion as being due to an evolving scalar field as well; this idea is called quintessence. The only difference between the primordial and late-time accelerated expansion is its rate, which is in proportion to the energy density of field.

There was an early model of primordial inflation, called quintessential inflation, that posited that the same field is responsible for both epochs, though the shape of this field's potential energy function was arguably fine-tuned in order to accomplish this.

Also, slight modification to your description of inflation. When inflation was underway, the inflaton field was evolving in a very flat portion of its potential (the energy density was nearly constant). As the inflaton rolled down toward the false vacuum, the energy density began to decay and when it reached the vacuum it underwent oscillations which resulted in the decay of the inflaton field into thermal degrees of freedom.

If by "inflaton field" you mean the scalar field that drove the period of primordial accelerated expansion, then that field is not generally considered to be relevant to the late-time expansion. While indeed both primordial and late-time accelerated expansion could be referred to as "inflation", the term is usually, when used without qualification, meant to refer to primordial inflation.

It is, however, possible to model the dark energy-fueled late-time expansion as being due to an evolving scalar field as well; this idea is called quintessence. The only difference between the primordial and late-time accelerated expansion is its rate, which is in proportion to the energy density of field.

There was an early model of primordial inflation, called quintessential inflation, that posited that the same field is responsible for both epochs, though the shape of this field's potential energy function was arguably fine-tuned in order to accomplish this. See: https://arxiv.org/abs/astro-ph/9810509

Also, slight modification to your description of inflation. When inflation was underway, the inflaton field was evolving in a very flat portion of its potential (the energy density was nearly constant). As the inflaton rolled down toward the false vacuum, the energy density began to decay and when it reached the vacuum it underwent oscillations which resulted in the decay of the inflaton field into thermal degrees of freedom.