Natural Metric-Affine Inflation: Reloaded

Abstract: We revisit natural inflation within the framework of metric-affine gravity, considering the impact of a periodic non-minimal coupling between the inflaton and the Nieh-Yan term. Such a term, alone, leads to linear inflation predictions in the strong coupling limit and cannot help to rescue the natural inflation scenario. However, once an analogous non-minimal coupling with the Ricci scalar is added, agreement with data can be easily achieved. Remarkably, the scenario remains viable even with a sub-Planckian periodicity scale and relatively small (order of one) non-minimal couplings.

• The paper introduces a periodic Nieh-Yan coupling that modifies the Einstein-frame potential to improve natural inflation predictions.
• It employs a detailed transition to the Einstein frame and contrasts pure Nieh-Yan with combined Ricci-scalar and Nieh-Yan scenarios.
• Joint couplings generate plateau potentials that restore slow-roll conditions and align inflationary observables with CMB constraints.

Natural Inflation in Metric-Affine Gravity with Nieh-Yan Coupling

Introduction and Motivation

This work revisits the natural inflation paradigm within the context of metric-affine gravity (MAG), focusing explicitly on periodic non-minimal couplings between the inflaton and the Nieh-Yan topological term. While the original natural inflation model, based on a pseudo-Nambu-Goldstone boson (PNGB) potential, elegantly suppresses higher-order corrections, it is tightly constrained (and disfavored) by recent CMB observations. Prior extensions—especially those leveraging non-minimal couplings to the Ricci scalar within both metric and Palatini formulations—have provided partial reconciliation with data, but often require trans-Planckian scales or large coupling constants (Racioppi et al., 2024, Bostan et al., 25 Nov 2025, Antoniadis et al., 2018). The present analysis examines whether the introduction of a periodic coupling to the Nieh-Yan term—available in affine geometric settings with torsion—can rescue natural inflation, particularly in the sub-Planckian periodicity regime and with moderate non-minimal couplings.

Model Formulation: Scalar-Torsion Couplings in MAG

The action is formulated in the Jordan frame with three non-minimal coupling functions: $f(\phi)$ to the Ricci scalar, $\tilde{f}(\phi)$ to the Holst invariant, and $f_{\mathrm{NY}}(\phi)$ to the Nieh-Yan term. The focus is on scalar inflationary models, specifically with the canonical natural inflation potential

$$V(\phi) = \Lambda^4 \left[1 + \cos\left(\phi/M\right)\right]$$

where $M$ is the periodicity scale. Two key coupling scenarios are considered:

1. Nieh-Yan-only coupling: $f(\phi) = 1$, $\tilde{f}(\phi) = 0$, $$f_{\mathrm{NY}}(\phi) = \xi_{\mathrm{NY}} \Omega(\phi)$$,
2. Ricci-scalar plus Nieh-Yan: $f(\phi) = 1 + \xi\Omega(\phi)$, $$f_{\mathrm{NY}}(\phi) = \xi_{\mathrm{NY}} \Omega(\phi)$$, with $\tilde{f}(\phi)$0.

The derivation proceeds with transition to the Einstein frame, following standard manipulation in metric-affine gravity, with specific attention to the non-trivial kinetic normalization induced by the scalar-torsion couplings.

Dynamical Effects of Nieh-Yan Coupling

The non-minimal coupling to the Nieh-Yan term yields a field-dependent kinetic function of the form

$\tilde{f}(\phi)$1

for vanishing Ricci scalar and Holst couplings ($\tilde{f}(\phi)$2, $\tilde{f}(\phi)$3). The key dynamical effect is the flattening and stretching of the Einstein-frame potential, which becomes increasingly linear as $\tilde{f}(\phi)$4 grows, away from stationary points.

Figure 1: $\tilde{f}(\phi)$5 vs. $\tilde{f}(\phi)$6 for various values of $\tilde{f}(\phi)$7 with $\tilde{f}(\phi)$8. Increasing the Nieh-Yan coupling yields a linearized regime.

This linearization has direct implications for resulting scalar spectral index ($\tilde{f}(\phi)$9) and tensor-to-scalar ratio ($f_{\mathrm{NY}}(\phi)$0). In the strong coupling (large $f_{\mathrm{NY}}(\phi)$1) limit, inflationary predictions converge to those of linear inflation, systematically moving away from the parameter region favored by current CMB observations. The $f_{\mathrm{NY}}(\phi)$2–$f_{\mathrm{NY}}(\phi)$3 plane, for both 50 and 60 e-folds, reveals that pure Nieh-Yan natural inflation is incompatible with combined BICEP/Keck and ACT constraints, even as $f_{\mathrm{NY}}(\phi)$4 (the periodicity in units of $f_{\mathrm{NY}}(\phi)$5) is varied.

Joint Ricci Scalar and Nieh-Yan Coupling: Viable Sub-Planckian Natural Inflation

When a non-minimal coupling to the Ricci scalar is added $f_{\mathrm{NY}}(\phi)$6, the model's phenomenology changes significantly. Notably, for $f_{\mathrm{NY}}(\phi)$7 and sub-Planckian periodicity $f_{\mathrm{NY}}(\phi)$8, the Einstein-frame potential develops a plateau structure, with the Nieh-Yan coupling extending the flat region and lowering the peak.

Figure 3: $f_{\mathrm{NY}}(\phi)$9 vs. $$V(\phi) = \Lambda^4 \left[1 + \cos\left(\phi/M\right)\right]$$0 illustrating the comparative flattening for standard natural inflation, Palatini-type non-minimal coupling, and combined Palatini plus Nieh-Yan couplings for $$V(\phi) = \Lambda^4 \left[1 + \cos\left(\phi/M\right)\right]$$1.

For this parameter regime, inflationary observables such as $$V(\phi) = \Lambda^4 \left[1 + \cos\left(\phi/M\right)\right]$$2, $$V(\phi) = \Lambda^4 \left[1 + \cos\left(\phi/M\right)\right]$$3, and running $$V(\phi) = \Lambda^4 \left[1 + \cos\left(\phi/M\right)\right]$$4 can fall within the observed ranges, provided $$V(\phi) = \Lambda^4 \left[1 + \cos\left(\phi/M\right)\right]$$5 is moderately large ($$V(\phi) = \Lambda^4 \left[1 + \cos\left(\phi/M\right)\right]$$6) and the energy scale $$V(\phi) = \Lambda^4 \left[1 + \cos\left(\phi/M\right)\right]$$7. Numerical analysis establishes that for vanishing or very small $$V(\phi) = \Lambda^4 \left[1 + \cos\left(\phi/M\right)\right]$$8, slow-roll is violated due to large $$V(\phi) = \Lambda^4 \left[1 + \cos\left(\phi/M\right)\right]$$9, but increasing $M$0 rapidly restores compatibility with slow-roll and hence with CMB constraints.

Figure 5: $M$1 vs. $M$2, $M$3 vs. $M$4, and other relevant observable correlations for $M$5, $M$6, and $M$7, showing entry into the favored CMB region as $M$8 increases.

Compatibility with CMB anisotropy measurements (Planck, BICEP/Keck, ACT) is thus achieved without recourse to trans-Planckian $M$9, large couplings, or higher-derivative operators. The combined model robustly interpolates between standard Palatini natural inflation and the linear inflation regime, with the latter recovered for very large $f(\phi) = 1$0.

Implications and Future Directions

The findings demonstrate that a periodic Nieh-Yan coupling, in synergy with a periodic Ricci-scalar coupling, can both flatten the inflaton potential and bring natural inflation into agreement with data for sub-Planckian periodicities and order-unity non-minimal couplings. This mechanism provides an alternative to previous approaches that relied on trans-Planckian scales or an uncomfortably large non-minimal coupling hierarchy.

The capability to engineer viable potentials with only two degrees of freedom (graviton and PNGB inflaton), moderate $f(\phi) = 1$1 and $f(\phi) = 1$2, and without higher-derivative actions is theoretically appealing. It also leverages the geometric latitude of MAG frameworks, where topological and torsion-induced invariants such as Nieh-Yan can have direct cosmological phenomenology. This opens further avenues to explore inflationary dynamics via other affine, torsional, and parity-odd invariants—potentially constraining torsion signatures via CMB or seeking unique predictions in primordial tensor spectra.

Conclusion

By incorporating a periodic non-minimal coupling to the Nieh-Yan term into the natural inflation scenario in metric-affine gravity, and in conjunction with a Ricci-scalar periodic coupling, it is possible to obtain observationally viable inflationary predictions for sub-Planckian decay constants and moderate coupling strengths. The scenario does not alleviate the tension for Nieh-Yan coupling alone, but achieves consistency with CMB data for a physically motivated set of parameters once the Palatini-type scalar-curvature coupling is included. These results establish the phenomenological relevance of topological invariants in early-universe cosmology within extended geometric frameworks.

References: (2605.23827, Racioppi et al., 2024, Bostan et al., 25 Nov 2025, Antoniadis et al., 2018), and related works.