Senjaya D.Mahidol University2026-02-062026-02-062026-01-01Nuclear Physics B Vol.1022 (2026)05503213https://repository.li.mahidol.ac.th/handle/123456789/114797Inflation, 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+R²8κV<inf>0</inf>[32κln(RR0)−ϕ0]^2(α−1), while a linear dependence yields, f(R)=R+A<inf>0</inf>R²(RR0)^{^{^3α^2κ}}. 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.Physics and AstronomyArticleSCOPUS10.1016/j.nuclphysb.2025.1172142-s2.0-105023496790