Characterizing the postinflationary reheating history: 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 $\mathrm{\Phi }$ has a monomial potential after inflation $V(\mathrm{\Phi })\propto |\mathrm{\Phi }-v{|}^p$ ($p\ge 2$) and is coupled to a daughter field $X$ through a quadratic-quadratic interaction $g^2{\mathrm{\Phi }}^2{X}^2$. 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 $p\ge 4$. 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 $m_X^2=g^2(\mathrm{\Phi }-v{)}^2$ is not negligible. For $p=2$, the final ratio between the energy densities of $X$ and $\mathrm{\Phi }$ depends strongly on $g^2$. For all $p\ge 2$, the system always goes to MD at late times.