Strong-coupling scale and frame-dependence of the initial conditions for chaotic inflation in models with modified (coupling to) gravity

A classical evolution in chaotic inflationary models starts at high energy densities with semi-classical initial conditions presumably consistent with universal quantum nature of all the fundamental forces. That is each quantum contributes the same amount to the energy density. We point out the upper limit on this amount inherent in this approach, so that all the quanta are inside the weak-coupling domain. We discuss this issue in realistic models with modified gravity, $R^2$- and Higgs-inflations, emphasizing the specific change of the initial conditions with metric frame, while all the quanta still contribute equal parts. The analysis can be performed straightforwardly in any model with modified gravity ($F(R)$-gravity, scalars with non-minimal couplings to gravity, etc).