Natural inflation in Palatini $F(R)$

Abstract

$F(R)$ Palatini gravity provides a robust framework for constructing viable inflationary potentials. In this study, we examine natural inflation and show that its consistency with observational data can be restored when the model is embedded within $F(R)$ Palatini gravity, specifically for $F(R) = R + αR^n$ with $7/4 \lesssim n \leq 2$. For completeness, we also demonstrate that models with $n > 2$ do not yield comparable improvements, achieving partial agreement with the data only in the limit $n \rightarrow 2$.

Figures (6)

• Figure 1: Reference plots of $G(\zeta)$ (left), generated using $F(R) = R + \alpha R^n$ for $n \leq 2$ (continuous) and $n>2$ (dashed), and the natural inflation potential $V(\phi)$ (right). Note that, for $n>2$ the condition $G(\zeta) = V(\phi)$ can be satisfied for any value of $\phi$, provided that the local maximum of $G$ exceeds the maximum of $V$.
• Figure 2: (a) $r$ vs. $n_s$, (b) $r$ vs. $\log_{10}(\alpha)$, (c) $n_s$ vs. $\log_{10}(\alpha)$, and (d) $\Lambda$ vs. $\log_{10}(\alpha)$ for $n \leq 2$ and $N_e=50$ with $n = 3/2$ (continuous), $n = 7/4$ (dashed), $n = 31/16$ (dot-dashed), and $n = 2$ (dotted), and with $M = 5$ (orange), $M = 6$ (blue), $M = 10$ (green), and $M = 500$ (red). The black line indicates the original prediction for natural inflation, while the dots correspond to predictions for fixed $M$, with larger dots representing increasing $M$. Contours display the 68% and 95% confidence levels based on the latest combinations from BICEP/KeckBICEP:2021xfz (cyan), and ACT collaborations ACT:2025tim (purple).
• Figure 3: Same as Fig. \ref{['fig:n_less_2_N50']}, but for $N_e = 60$.
• Figure 4: The Einstein-frame potential $U(\chi)$ for $\alpha = 0$ (brown), $\alpha = 3.16 \cdot 10^6$ (black, continuous), $\alpha = 10^7$ (black, dashed), and $\alpha = 10^8$ (black, dotted), with $M = 500$ and $n = 7/4$ (left), along with a zoomed-in view of the same potential (right). We indicate $\chi_N$ (stars) and $\chi_{\rm end}$ (dots) for $\alpha = 0$ (blue), $3.16 \cdot 10^6$ (yellow), $10^7$ (orange), and $10^8$ (red). The value $\alpha = 3.16 \cdot 10^6$ corresponds to the maximum of $n_s$, i.e., the "knee" in Fig. \ref{['fig:n_less_2']}. The potentials are normalized by $U_{\alpha}$ (defined in Eq. \ref{['eq:U_max']}) and the Einstein-frame canonical field $\chi$ by $\chi_\alpha$ (defined in Eq. \ref{['eq:chimax']}), so that the potential appears with a period of $2\pi$ in the plot. Further details are given in the text.
• Figure 5: (a) $r$ vs. $n_s$, (b) $r$ vs. $\log_{10}(\alpha)$, (c) $n_s$ vs. $\log_{10}(\alpha)$, and (d) $\Lambda$ vs. $\log_{10}(\alpha)$ for $n > 2$ and $N_e = 50$, with $M = 4$ (blue), $M = 5$ (orange), $M = 6$ (red), and $M = 10$ (green), and $n = 3$ (continuous), $n = 5/2$ (dashed), $n = 9/4$ (dot-dashed), and $n = 33/16$ (dotted). Contours indicate the 68% and 95% confidence levels based on the latest combinations from the BICEP/KeckBICEP:2021xfz (cyan) and ACT ACT:2025tim (purple) collaborations.