We study the energy distribution and equation of state of the universe between the end of inflation and the onset of radiation domination (RD), considering observationally consistent single-field inflationary scenarios, with a potential ‘flattening’ at large field values, and a monomial shape V(ϕ)∝|ϕ|p around the origin. As a proxy for (p)reheating, we include a quadratic interaction g2ϕ2X2 between the inflaton ϕ and a light scalar ‘daughter’ field X, with g2>0. We capture the non-perturbative and non-linear nature of the system dynamics with lattice simulations, obtaining that: i) the final energy transferred to X depends only on p, not on g2; ii) the final transfer of energy is always negligible for 2≤p<4, and of order ∼50% for p≥4; iii) the system goes at late times to matter-domination for p=2, and always to RD for p>2. In the latter case we calculate the number of e-folds until RD, significantly reducing the uncertainty in the inflationary observables ns and r.