  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.
