V. K. Oikonomou
Abstract 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 m 2 ( 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 m s 2 = κ 2 ρ m (0) /3, with ρ m (0) denoting the energy density of the cold dark matter at the present epoch, that is F ( R ) = F ( R ,Λ, m s 2 ). 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 m 2 ( R ). These viable classes are either deformations of the R 2 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 = RF RRR / F RR 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 and we show that power-law F ( R ) gravities can be viable theories of inflation, for appropriate values of the power-law exponent. Finally we highlight the phenomenological importance of exponential deformations of the R^2 model of the form F ( R ) = R + R 2 / M 2 + λR e ϵ(Λ/ R ) β + λΛ nϵ , which emerge naturally as viable inflationary models which also describe successfully the dark energy era.