f(R) Cosmology, analytical inversion method

Inflation, the rapid expansion of spacetime in the early universe, provides an elegant resolution to several fundamental cosmological problems. Among various theoretical approaches, f ( R ) gravity offers a minimal yet powerful extension of General Relativity capable of generating inflationary dynamics. In this work, we aim to provide a detailed and systematic derivation of a bottom-up analytical reconstruction method for f ( R ) gravity, given the number of e-folds and the scalar power spectrum under the slow-roll approximation. Through explicit calculation, we demonstrate that an exponential dependence of the scalar spectrum on the e-fold number leads to the theory f(R)=R+R²8κV₀[32κln(RR0)−ϕ0]^2(α−1), while a linear dependence yields, f(R)=R+A₀R²(RR0)^3α2κ. Our analysis provides a transparent derivation of these functional forms and highlights how specific choices of the scalar spectrum naturally reconstruct distinct inflationary f ( R ) models.