Characterizing the post-inflationary reheating history, Part I: single
  daughter field with quadratic-quadratic interaction
  We study the evolution of the energy distribution and equation of state of
the Universe from the end of inflation until the onset of either radiation
domination (RD) or a transient period of matter domination (MD). We use both
analytical techniques and lattice simulations. We consider two-field models
where the inflaton $\Phi$ has a monomial potential after inflation $V(\Phi)
\propto |\Phi - v|^p$ ($p\geq2$), and is coupled to a daughter field $X$
through a quadratic-quadratic interaction $g^2\Phi^2 X^2$. We consider two
situations, depending on whether the potential has a minimum at $i)$ $v = 0$,
or $ii)$ $v > 0$. In the scenario $i)$, the final energy transferred to $X$ is
independent of $g^2$ and entirely determined by $p$: it is negligible for $p <
4$, and of order $\sim 50\%$ for $p \geq 4$. The system goes to MD at late
times for $p = 2$, while it goes to RD for $p > 2$. In the later case, we can
calculate exactly the number of e-folds until RD as a function of $g^2$, and
hence predict accurately inflationary observables like the scalar tilt $n_s$
and the tensor-to-scalar ratio $r$. In the scenario $ii)$, the energy is always
transferred completely to $X$ for $p>2$, as long as its effective mass $m_X^2 =
g^2(\Phi-v)^2$ is not negligible. For $p=2$, the final ratio between the energy
densities of $X$ and $\Phi$ depends strongly on $g^2$. For all $p \ge 2$, the
system always goes to MD at late times.
