Inflationary Scenarios in $f(Q,\phi)$ Gravity with Scalar Field Coupling

In this work, we investigated several inflationary scenarios within the framework of modified $f(Q,\phi)$ gravity with a nonminimal coupling between the scalar field and the nonmetricity scalar. We focused on the impact of the coupling parameter $\xi$ on the inflationary observables, namely the scalar spectral index $n_s$ and the tensor-to-scalar ratio $r$. In the case of De Sitter inflation, we showed that the model can reproduce observationally viable predictions only within a restricted range of the coupling parameter. Specifically, we found that $n_s$ increases with $\xi$, while $r$ decreases, leading to a narrow allowed region $10^{-3} \lesssim \xi \lesssim 10^{-2}$ compatible with Planck data. Outside this range, the model either predicts excessively large tensor modes or an unphysical blue-tilted spectrum. We also derived theoretical constraints on $\xi$ from the consistency of the model, leading to an upper bound $\xi < \frac{\kappa}{2p}$. For $\kappa = 1$ and $p = 60$, this implies $\xi < 0.00833$, with a preferred region around $\xi \sim \mathcal{O}(10^{-3})$. Furthermore, we analyzed the Cosh-type inflationary model and showed that it provides a robust and consistent description of inflation. In this case, the tensor-to-scalar ratio decreases while the scalar spectral index increases with the number of e-folds $N$. For $N = 60$, the model predicts $n_s \approx 0.965 - 0.967, \qquad r \approx 0.017 - 0.018$, in excellent agreement with current observational constraints. Overall, our results highlight the crucial role of the nonminimal coupling in shaping the inflationary dynamics and ensuring compatibility with cosmological observations.