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 + R 2 8 κ V 0 [ 3 2 κ ln ( R R 0 ) − ϕ 0 ] 2 ( α − 1 ) , while a linear dependence yields, f ( R ) = R + A 0 R 2 ( R R 0 ) 3 α 2 κ . 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.