Cosmological Inflation in $f(Q,\mathcal{L}_{m})$ Gravity

Abstract: Cosmological inflation remains a key paradigm for explaining the earliest stages of the Universe, yet the theoretical limitations of General Relativity (GR) motivate the development of alternative formulations capable of addressing both early and late cosmic acceleration. In this work, we investigate cosmological inflation within the $f(Q,\mathcal{L}{m})$ gravity framework based on symmetric teleparallel geometry, where the non-metricity scalar $Q$ couples directly to the matter Lagrangian. We formulate the slow-roll dynamics and derive analytical predictions for the scalar spectral index $n{s}$ and tensor-to-scalar ratio $r$ in both linear and nonlinear non-minimal coupling models, assuming a power-law inflaton potential. Our findings show that the linear case, $f(Q,\mathcal{L}{m})=-αQ + 2\mathcal{L}{m}+β$, becomes compatible with Planck+BK15+BAO constraints for positive $α$ and $β$, producing narrow viable contours in parameter space. In contrast, the nonlinear model, $f(Q,\mathcal{L}{m})=-αQ+(2\mathcal{L}{m})^{2}+β$, achieves observational viability only for negative $α$ and $β$, and its predictions predominantly fall inside the $68\%$ confidence region of joint data. These results demonstrate that $f(Q,\mathcal{L}_{m})$ gravity produces distinct inflationary regimes, providing a highly competitive alternative to GR.