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.

• I am not a specialist at cosmology, but if you are interested in relationship between dark energy and iniflation there is model with inflaton coupled to Higgs field, which is a possible candidate for explanation for both phenomena. Maybe you find this references useful - s3.cern.ch/inspire-prod-files-2/…, arxiv.org/pdf/0909.5353.pdf – spiridon_the_sun_rotator Jun 6 '20 at 6:49
• The whole situation could be summarized as: your teacher: where is your last week's homework? you: my dog ate it. your teacher: then where is your this week's homework? you: my dog ate it. your teacher: is that the same dog? you: probably not, I am not sure, and a third mixed-breed dog might be involved too in case you ask about other issues (Hubble tension) with my homework, oh my neighbor's dog called Higgs might have something to do with all the mess as well. – MadMax Aug 3 '20 at 15:50