Towards Model Agnostic $F(R)$ Gravity Inflation

In this work we construct a formalism that can reveal the general characteristics of classes of viable $F(R)$ inflationary theories. The assumptions we make is that the slow-roll era occurs, and that the de Sitter scalaron mass m²(R) of the $F(R)$ gravity is positive or zero, for both the inflationary and late-time quasi de Sitter eras, a necessary condition for the stability of the de Sitter spacetime. In addition, we require that the de Sitter scalaron mass is also a monotonically increasing function of the Ricci scalar, or it has an extremum. Also the $F(R)$ gravity function is required to depend on the two known fundamental scales in cosmology, the cosmological constant $Λ$ and the mass scale mₛ²=κ² ρₘ⁽⁰⁾3, with ρₘ⁽⁰⁾ denoting the energy density of the cold dark matter at the present epoch, that is F(R)=F(R,Λ,mₛ²). Using these general assumptions we provide the general features of viable classes of $F(R)$ gravity inflationary theories which remarkably can also simultaneously describe successfully the dark energy era. This unique feature of a unified description of the dark energy and inflationary eras stems from the requirement of the monotonicity of the de Sitter scalaron mass m²(R). These viable classes are either deformations of the R² model or $α$-attractors type theories. The analysis of the viability of a general $F(R)$ gravity inflationary theory is reduced in evaluating the parameter x=R FRRRFRR and the first slow-roll index of the theory, either numerically or approximately. We also disentangle the power-law $F(R)$ gravities from power-law evolution.