RETRACTED ARTICLE: From geometry to cosmology: a pedagogical review of inflation in curvature, torsion, and extended gravity theories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 $$R_{\mu \nu }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>R</mml:mi> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mi>ν</mml:mi> </mml:mrow> </mml:msub> </mml:math> . 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 $$n_s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:math> , the tensor-to-scalar ratio r , running $$\alpha _s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:math> , and non-Gaussianity $$f_{\textrm{NL}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>f</mml:mi> <mml:mtext>NL</mml:mtext> </mml:msub> </mml:math> . 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.