Inflation with Nieh-Yan-like terms in metric-affine gravity

We study single-field slow-roll inflation in metric-affine gravity with a scalar field non-minimally coupled to the non-Riemannian Ricci scalar and to the divergences of the torsion and nonmetricity vectors, a structure that generalizes the well-known Nieh-Yan term. By imposing projective coherence of the matter sector and solving the connection field equations, we integrate out torsion and nonmetricity and obtain an equivalent Einstein-frame formulation in which the metric-affine couplings are encoded in a modified kinetic function and potential. For the choice of coupling functions $\mathcal{A}(\phi) = M_P^2 + \xi \phi^2$ to the non-Riemannian Ricci scalar, $\mathcal{C}_i(\phi) = \xi_i \phi$ to the Nieh-Yan-like terms and a monomial Jordan-frame potential $\mathcal{V} \propto \phi^k$, we show that in the limit of a large positive effective Nieh-Yan-like coupling $\bar{\xi}$ the canonical field satisfies $\chi \sim \phi^2$, the Jordan-frame field values during inflation become sub-Planckian, and the Einstein-frame potential reduces to $U \sim \chi^{k/2}$. We compute the slow-roll predictions numerically for quartic and quadratic Jordan-frame potentials and compare them with the current CMB constraints from Planck, BICEP/Keck, ACT, and SPT. We find that intermediate values of $\bar{\xi}$ can restore the compatibility of non-minimally coupled Palatini inflation with observations: in the quartic case, the model predicts a tensor-to-scalar ratio within reach of next-generation CMB experiments for $\bar{\xi}\lesssim10^4$, while in the quadratic case the coupling cures the $\eta$-problem arising for $\xi \gtrsim 10^{-2}$ and yields viable predictions for $10^{-2}\lesssim\bar{\xi}\lesssim 10^2$. In the negative $\bar{\xi}$ regime, the model does not improve upon standard Palatini inflation, though it can still produce distinct, testable predictions.