Mixed constraints to inflationary models
  We show how to constrain inflationary models and reheating by using mixed
constraints. In particular we study the physics of the reheating phase after
inflation from observational constraints to the inflationary stage. We show
that it is possible to determine $\omega$, the equation of state during
reheating, by using the reported values of the spectral index and the {\it
full} number of $e$-folds $N(n_s,\omega)= N_H(n_s)+N_{re}(n_s,w)\approx 60$,
which includes the accelerated expansion and the reheating phase. We show that
the reheating number of $e$-folds $N_{re}$ is quite sensitive to this equation
of state. Requiring $N_{re}>0$ and a sensible value for the thermalization
scale $T_{re}$, demands in general a reheating phase with $\omega\neq 0$. We
exemplify the constraints with two particular examples: We show how the
Starobinsky model allows only large values of $T_{re}$ if the reheating phase
is dominated by dust ($w =0$), and if Primordial Black Hole production is
subdominant. For the case of $N=1$ Supergravity inflation, the extra parameter
of the potential provides the necessary freedom to afford lower-scale
thermalization in a dust-like reheating phase and yet our method serves to
determine the rest of the observable parameters.
