How are Dark Energy and the Inflaton Field related? My understanding of Inflation Theory:
Before $10^{-35}$ seconds the universe began to cool and the Inflaton Field approached a false vacuum.  When it reached this false vacuum, there was a constant energy density in the universe which cause the universe to expand (for some reason? I don't really understand why a constant energy density would cause repulsive gravity. If you could briefly explain this too that would be appreciated).  As it expanded the field became supercooled until it quantum tunneled to its true vacuum around $10^{-34}$ seconds , ending inflation in our pocket of space.
My main question:
Is Dark Energy caused by or related to the Inflaton Field. Could today's expansion be a result of the Inflaton Field but just a much slower, weaker version than inflation because the Inflaton Field is in a much lower energy state now than it was during inflation.
 A: 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.
