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

Abstract: We present a simplified review of inflationary cosmology across various modified gravity theories. These include models based on curvature, torsion, and non-metricity. We explore how scalar fields interact with different geometric quantities and how these interactions affect inflationary dynamics. Key cosmological features such as background evolution, reheating, and observable parameters are discussed. We also examine exotic scenarios inspired by string theory, extra dimensions, and non-local models. This work aims to connect theoretical models with observational data and future missions, offering guidance for exploring inflation beyond general relativity.

• The paper synthesizes modified gravity models for inflation, showcasing curvature-, torsion-, and non-metricity-based frameworks with detailed observational benchmarks.
• It systematically compares theoretical predictions such as nₛ and r, highlighting models like Starobinsky inflation that match precise CMB data.
• The review discusses diverse reheating mechanisms and stability issues, paving the way for future tests with missions like LiteBIRD, CMB-S4, and LISA.

Inflationary Cosmology in Curvature, Torsion, and Extended Gravity Theories

Introduction and Motivation

This review provides a comprehensive synthesis of inflationary cosmology within a broad spectrum of modified gravity frameworks, including curvature-based ($f(R)$, $f(G)$, $f(R,G)$), torsion-based ($f(T)$, Einstein–Cartan), non-metricity-inspired ($f(Q)$), and scalar–tensor extensions. The motivation stems from both theoretical and observational challenges in canonical inflation, such as fine-tuning, UV sensitivity, and embedding in high-energy physics. The increasing precision of CMB and large-scale structure data, notably from Planck, BICEP/Keck, and upcoming missions (LiteBIRD, CMB-S4), necessitates models that go beyond standard scalar field inflation and can be robustly confronted with data.

Observational Constraints and Model Viability

The review systematically analyzes inflationary observables: scalar spectral index $n_s$, tensor-to-scalar ratio $r$, running $\alpha_s$, and non-Gaussianity $f_{\mathrm{NL}}$. Current constraints ($n_s \approx 0.965$, $r < 0.056$, $f_{\mathrm{NL}}^{\mathrm{local}} \approx 0$) strongly favor plateau-like potentials and geometric inflationary mechanisms. Starobinsky $f(R)$ inflation ($n_s \sim 0.965$, $r \sim 0.003$) is highlighted as a benchmark, with $f(T)$, $f(Q)$, and scalar–Gauss–Bonnet models remaining viable under parameter tuning and stability conditions. Models predicting high $r$ (e.g., chaotic inflation) are disfavored.

Curvature-Based Inflation: $f(R)$, $f(G)$, and $f(R,G)$

$f(R)$ Gravity and Starobinsky Inflation

The $f(R)$ framework, particularly the Starobinsky model ($f(R) = R + \alpha R^2$), provides a geometric origin for inflation, eliminating the need for an explicit inflaton. The scalaron field, arising from higher-order curvature corrections, drives slow-roll inflation with robust predictions for $n_s$ and $r$. Reheating is achieved via scalaron decay, yielding $T_{\rm rh} \sim 10^9$–$10^{10}$ GeV, compatible with leptogenesis. Generalizations (power-law, exponential, quantum-corrected $f(R)$) are discussed, with observational viability contingent on avoiding ghosts and matching CMB constraints.

Gauss–Bonnet and Mixed Curvature Models

$f(G)$ and $f(R,G)$ models introduce higher-order invariants motivated by string theory and quantum gravity. Scalar–Gauss–Bonnet couplings ($\xi(\phi) G$) enable inflation on steeper potentials and can suppress tensor modes. Stability requires $c_T = c$ (GW170817 constraint) and absence of ghosts. Mixed models ($f(R,G)$) offer unified descriptions of inflation and late-time acceleration but require careful functional choices to avoid instabilities.

Torsion and Non-Metricity-Based Inflation: $f(T)$, $f(Q)$, Einstein–Cartan

Teleparallel Gravity and $f(T)$ Inflation

Teleparallel gravity replaces curvature with torsion as the source of gravitational dynamics. $f(T)$ models admit second-order field equations and can realize inflation without scalar fields. Viable forms (power-law, exponential, logarithmic $f(T)$) yield $n_s \sim 0.965$, $r < 0.01$, with reheating via geometric mechanisms. Covariant formulations restore local Lorentz invariance, addressing tetrad ambiguities.

Symmetric Teleparallel and $f(Q)$ Gravity

$f(Q)$ gravity, based on non-metricity, offers a novel geometric platform for inflation. Theories remain second-order and ghost-free, with inflation driven by geometric terms or scalar–non-metricity couplings. Starobinsky-type $f(Q)$ models reproduce $n_s \sim 0.965$, $r \sim 0.004$. Perturbation theory and reheating mechanisms are under active development.

Einstein–Cartan Theory

Incorporating spin and torsion, Einstein–Cartan gravity provides singularity avoidance and bounce cosmologies. Spinor fluids or scalar–torsion couplings can drive inflation, with predictions $n_s \sim 0.96$, $r < 0.01$. The algebraic nature of torsion ensures ghost-freedom, but perturbative analyses and quantum embeddings require further work.

Scalar–Tensor and Exotic Couplings

Non-minimal couplings to curvature ($\xi \phi^2 R$), torsion ($\phi^2 T$), Ricci tensor ($R_{\mu\nu} \partial^\mu \phi \partial^\nu \phi$), and Gauss–Bonnet ($\xi(\phi) G$) are systematically explored. Higgs inflation and G-inflation are notable examples, with large $\xi$ flattening potentials and suppressing $r$. Stability and unitarity bounds are critical, especially for large couplings.

String-Inspired, Braneworld, and Non-Local Models

String theory and extra-dimensional scenarios (brane inflation, axion monodromy, DBI inflation) provide UV-complete frameworks with distinctive kinetic structures and non-Gaussian signatures. Non-local gravity (e.g., $R \mathcal{F}(\Box) R$) offers ghost-free, UV-finite inflationary dynamics, with de Sitter phases and suppressed tensor modes.

Mimetic Gravity, Carmeli's Cosmological Relativity, and Varying Constants

Mimetic gravity isolates the conformal degree of freedom, enabling inflation and dark matter phenomenology via constraint-enforced scalar fields. Carmeli's 5D cosmological relativity geometrizes cosmic acceleration without scalar fields, though perturbative and quantum aspects remain underdeveloped. Bekenstein-type varying constant models embed inflation in the dynamics of fundamental couplings, linking cosmology with quantum field theory.

Dynamical Systems, Bayesian Model Selection, and Machine Learning

Dynamical systems techniques classify inflationary attractors, stability, and phase-space trajectories across models. Bayesian inference (evidence, Bayes factors) and machine learning (emulators, classifiers, reconstruction) are essential for navigating high-dimensional parameter spaces and confronting theory with data.

Reheating, Preheating, PBHs, and Gravitational Waves

Reheating mechanisms vary across models: scalaron decay ($f(R)$), geometric dissipation ($f(Q)$), tachyonic preheating (scalar–GB), and spin-induced transitions (Einstein–Cartan). PBH formation and stochastic GW backgrounds serve as discriminants, with scalar-induced GWs and PBH mass spectra providing observational tests.

Comparative Assessment and Future Prospects

A taxonomy of models is presented, organized by geometric origin (curvature, torsion, non-metricity), coupling structure, and UV embedding. Starobinsky $f(R)$, scalar–GB, non-local, and $f(Q)$ models are currently favored. Upcoming missions (LiteBIRD, CMB-S4, LISA) will constrain $r$ to $10^{-3}$, probe GW backgrounds, and test reheating and PBH predictions.

Open Problems and Directions

Key unresolved issues include:

• Ghost and instability avoidance in higher-derivative models
• Reheating mechanisms in torsion and non-metricity theories
• Embedding in quantum gravity (string, LQG, asymptotic safety)
• Perturbation theory and observational signatures (isocurvature, non-Gaussianity)
• Model selection frameworks integrating Bayesian and ML approaches

Conclusion

Modified gravity inflationary models offer a diverse and technically rich landscape, with geometric, string-inspired, and effective field theory approaches providing viable alternatives to canonical inflation. Theoretical consistency, observational compatibility, and computational advances are converging to enable rigorous discrimination among models. The next generation of cosmological and gravitational wave experiments will be decisive in determining the geometric and physical origin of inflation, with modified gravity frameworks poised as leading contenders.