It is my understanding that inflation solves the horizon problem associated with an FLRW description of the universe by introducing a period during which $\varrho + 3p = \varrho(1 + 3w) < 0$, so that the co-moving Hubble radius (for $w \neq -1$) $$ (aH)^{-1} = H_0^{-1}a^{(1+3w)/2} $$ shrinks, thus allowing the widely separated parts of the universe to once have been in thermal contact.
I also believe one usually considers all the energy content of the universe to be bound up in the inflaton field. During reheating, the field oscillates about the minimum of its potential, and decays into matter and radiation fields.
I have two questions, and I do hope my understanding is not too flawed:
At the end of the inflationary epoch, the inflaton field remains in thermal equilibrium, from the period of causal contact. The decay ought to be a non-equilibrium process, however. I suppose that the argument is that the fields then themselves thermalize, and because the different regions were in equilibrium they remain so. Although this seems very plausible, I wonder if there is any quantitative argument that shows this?
The decay of the inflaton must be described by an interaction Lagrangian that couples the inflaton to the fields it decays into. Which, if any, are the concrete models that succeed at this, yielding an acceptable quantitative description of the fields of the radiation era? Do different models differ in predictions in any fundamental way?