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 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 .