From geometry to cosmology: a pedagogical review of inflation in curvature, torsion, and extended gravity theories

Abstract We present a comprehensive and pedagogical review of inflationary cosmology in the context of a wide spectrum of modified theories of gravity. This includes curvature-based frameworks such as f (R) gravity, Gauss–Bonnet gravity f (G), and mixed curvature invariants like f (R, G) ; torsion-based models including teleparallel gravity f (T) and Einstein–Cartan theory; and non-metricity-inspired theories such as symmetric teleparallel gravity f (Q). Scalar–tensor extensions are systematically explored where scalar fields couple not only to the Ricci scalar R, but also to torsion T, non-metricity Q, the Gauss–Bonnet invariant G, and the Ricci tensor R R μ ν. For each class of models, we derive and analyze inflationary dynamics under slow-roll approximations, highlighting the effects of geometry-induced modifications to the inflaton field equations. We examine background evolution, reheating mechanisms, and perturbation spectra, with explicit focus on observables including the spectral index nₛ n s, the tensor-to-scalar ratio r, running ₛ α s, and non-Gaussianity f₍₋ f NL. Numerical predictions are compared with Planck 2018, BICEP/Keck, and upcoming CMB-S4 forecasts. The review includes auxiliary formulations such as conformal transformations, dynamical system reconstructions, and effective field theory embeddings. We explore both analytical solutions and numerical phase-space portraits, employing Bayesian model selection, machine learning classifiers, and Hubble-flow reconstruction techniques to compare the viability of models across observational parameter space. Special attention is paid to exotic inflationary scenarios motivated by string theory (e. g. , brane inflation, axion monodromy), extra-dimensional constructions, mimetic gravity, non-local infinite-derivative models, and theories with varying fundamental constants à la Bekenstein. We also assess the formation of primordial black holes and stochastic gravitational waves as potential discriminants among models. This work aims to serve as both a detailed theoretical resource and a phenomenological guide, offering an integrated, high-precision roadmap for navigating the inflationary landscape beyond general relativity. By connecting modified gravity frameworks with cosmological data and forward-looking missions (e. g. , LiteBIRD, LISA), we seek to elucidate the next generation of viable inflationary paradigms.