Quadratic, Higgs and hilltop potentials in the Palatini gravity

In this work, we study inflation with the non-minimally coupled quadratic, Standard Model (SM) Higgs and hilltop potentials through $\xi \phi^2R$ term in the Palatini gravity. We first analyze observational parameters of Palatini quadratic potential as functions of $\xi$ for high-$N$ scenario and low-$N$ scenario. In addition to this, taking into account inflaton $\phi$ has a non-zero vacuum expectation value $v$ afterwards inflation, we display observational parameters of well-known symmetry-breaking potentials type of Higgs potential and its generalizations which are hilltop potentials in the Palatini formalism for high-$N$ scenario and low-$N$ scenario. We calculate inflationary parameters of Palatini Higgs potential as functions of $v$ for different $\xi$ values where inflaton values both $\phi>v$ and $\phi<v$ during inflation as well as we show that observational parameters of Palatini Higgs potential in the induced gravity limit for high-$N$ scenario. On the other hand, we illustrate different from the Higgs potential the effect of $\xi$ on hilltop potentials which can agree with the observations for inflaton value solely $\phi<v$ and $\xi$, $v\ll1$ for both two scenarios, which we mentioned above. For each considered potentials, we also display $n_s-r$ values fit the current data given by the Keck Array/BICEP2 and Planck collaborations.