Title: Observational Constraints on Extended Starobinsky and Weyl Gravity Model of Inflation

Abstract: We present constraints on the extended Starobinsky and Weyl gravity model of inflation using updated available observational data. The data includes cosmic microwave background (CMB) anisotropy measurements from Planck and BICEP/Keck 2018 (BK18), as well as large-scale structure data encompassing cosmic shear and galaxy autocorrelation and cross-correlation functions measurements from Dark Energy Survey (DES), baryonic acoustic oscillation (BAO) measurements from 6dF, MGS and BOSS, and distance measurements from supernovae type Ia from Pantheon+ samples. By introducing a single additional parameter, each model extends the Starobinsky model to encompass larger region of parameter space while remaining consistent with all observational data. Our findings demonstrate that the inclusion of higher-order terms loosen the constraint on the upper bound of $e$-folding number $N_{\rm e}$ due to the presence of small additional parameter. The maximum limit on $N_{\rm e}$ could be refined by considering the reheating process to $N_{\rm e}<55-59$ for $k_{*}=0.002-0.05\ {\rm Mpc}^{-1}$. These models extend viable range of tensor-to-scalar ratio ($r$) to very small value $r<0.002$ in contrast to the original $R^2$ Starobinsky model. In addition, our results continue to emphasize the tension in $H_0$ and $S_8$ between early-time CMB measurements and late-time large-scale structure observations.