Reheating and the Horizon Problem 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?
 A: At the end of inflation the inflaton is not in thermal equilibrium. Because the universe expands by a huge amount ($\sim e^{60}$) any thermal bath of particles is diluted away, leaving only the $0$-mode of the inflaton. Historically, it was thought that inflation was preceded by another thermal era (this is why decay of the inflaton was dubbed reheating) but in actual fact we don't really know anything about the pre-inflationary universe. In any case, because the observable universe originates in the a causally connected region the value of the inflaton field is the same everywhere (up to $\sim 10^{-5}$ inhomogeneities generated by inflation) and once the inflaton decays this uniformity is inherited by the decay products, i.e., the temperature is the same everywhere (again up to $\sim 10^{-5}$).
The reheating stage is not very well constrained experimentally. All you need for successful Big Bang cosmology is a thermal soup of relativistic particles with temperature above a few MeV (in order to achieve successful Big Bang Nucleosynthesis). On the other hand, we know from the bound on primordial gravitational waves that the scale of inflation could be as high as $\rho^{1/4} \sim 10^{16}$ GeV ($\sim T_\mathrm{reh}$). The range of allowed temperatures is huge so it's not that difficult to find successful models of reheating.
As far as concrete models go, these are myriad and there are many phenomenological differences, with non-perturbative effects often being important. However, since reheating is poorly constrained most of them are "successful" in the sense of being capable of producing the initial conditions for the Hot Big Bang.
A simple example might be an interaction Lagrangian $\delta \mathcal L = - \sigma \phi \chi^2$ between the inflaton $\phi$ and a boson $\chi$ (this could be the Standard Model Higgs for example) or a Yukawa interaction $\delta\mathcal L = - y\phi\bar\psi\psi$ between the inflaton and a fermion. As the inflaton oscillates at the end of inflation it produces pairs of quanta of the fields it is coupled to. If the couplings are sufficiently small that non-perturbative effects are unimportant then the decay rate of the inflaton is (assuming harmonic inflaton potential)
\begin{equation}
  \Gamma(\phi\rightarrow\chi\chi) = \frac{\sigma^2}{8\pi m}, \qquad \Gamma(\phi\rightarrow \psi\psi) = \frac{y^2m}{8\pi}
\end{equation}
where $m$ is the inflaton mass. Since the inflaton loses energy also to the expansion of the universe, the decay due to this particle production becomes effective when the decay time becomes smaller than the Hubble time ($H\sim \Gamma$). Provided that interactions between decay products and SM are sufficiently strong these thermalize and provide the thermal initial condition for the subsequent Big Bang evolution. 
