Natural inflation in Palatini $F(R,X)$

TL;DR

The paper analyzes natural inflation within Palatini $F(R,X)$ gravity with $F(R,X)=F(R+X)$, focusing on a positive bounded Jordan-frame potential $V(oldsymbol{φ})=oldsymbol{Λ}^4[1+ ext{cos}(oldsymbol{φ}/oldsymbol{M})]$. By introducing an auxiliary field $oldsymbol{ζ}$ and performing a Weyl transformation to the Einstein frame, the inflaton dynamics reduce to a potential $U(oldsymbol{ζ})=rac{1}{4}rac{oldsymbol{ζ}}{F'(oldsymbol{ζ})}$ subject to $G(oldsymbol{ζ})=V(oldsymbol{φ})$, with $G(oldsymbol{ζ})=rac{1}{4}[2F(oldsymbol{ζ})-oldsymbol{ζ}F'(oldsymbol{ζ})]$. The analysis splits into $n\le2$ and $n>2$ regimes for $F(oldsymbol{ζ})=oldsymbol{ζ}+oldsymbol{α}oldsymbol{ζ}^n$, deriving slow-roll observables and showing that for $n\le2$ the model can be made viable (often improving standard natural inflation predictions) as $oldsymbol{α}$ grows, while for $n>2$ the construction is limited by a maximal $oldsymbol{α}$ and cannot rescue the model from observational exclusion. The normalization from the scalar amplitude $A_s$ ties $oldsymbol{α}$ to $oldsymbol{Λ}^4$, yielding testable predictions for upcoming CMB surveys, especially in the $n\le2$ regime.

Abstract

In the context of Palatini gravity, $F(R+X)$ models, with X the inflaton kinetic term, are characterized by the appealing property of generating asymptotically flat inflaton potentials, exactly like the more commonly studied Palatini $F(R)$ models, but without the complication of non-canonical inflaton kinetic terms in the Einstein frame. In this paper, we study the case of a Jordan frame potential which is positive and bounded, specifically, natural inflation. We compute the CMB observables and show that for a wide class of $F(R + X)$ theories, including the quadratic one, natural inflation is still viable.

Figures (5)

• Figure 1: Einstein frame potential for the quadratic $F(R_X) = R_X +\alpha R_X^2$ with $M=5$, and $\alpha=2\cdot 10^8$ (red), $\alpha= 10^9$ (blue), $\Lambda \sim 5\cdot 10^{-3}$ is fixed by imposing the constraint $A_s \simeq 2.1\cdot 10^{-9}$. We also show for reference the field values $\phi_N$ (star) and $\phi_{end}$ (dot). As $\alpha$ increases, the width of the plateaus becomes much larger than the width of the minima of the potential where the field will perform reheating.
• 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 the model presented in section \ref{['sec:n<2']}, with $n=3/2$ (blue) $n = 7/4$ (green) and $n = 2$ (red) with $M = 3$ (thick), $M = 4$ (dashed), $M = 5$ (dash-dotted), and $M = 10$ (dotted). The black dots show the predictions of the original natural inflation model. All observables are computed at $N_e = 60$. The amplitude of the power spectrum $A_s$ is fixed to its observed value. The gray regions indicate the 95% (dark-gray) and 68% (light-gray) confidence levels (CL), respectively, based on the latest combination of Planck, BICEP/Keck, and BAO data BICEP:2021xfz.
• Figure 3: Reference plots of a $G(\zeta)$ (left) generated by a $F_{>2}(R_X)$ and natural inflation $V(\phi)$ (right). Notice that $G(\zeta)=V(\phi)$ is satisfied easily for any $\phi$ when $\zeta \leq \zeta_0$, provided that the local maximum of $G \geq$ is higher than the maximum of $V$.
• Figure 4: Einstein frame potential for the model $F(R_X)=R_X+\alpha R_X^3$ with $M=5$, and $\alpha=0$ (i.e. the original natural inflation potential) (blue), and for $\alpha_{max}=1.1\cdot 10^{16}$ (red). We also show for reference the field values $\phi_N$ (star) and $\phi_{end}$ (dot). We notice that even for $\alpha=\alpha_{max}$ the potential change is minor. As a consequence, the CMB observables do not change enough, giving $r= 0.02, n_s =0.957$, still outside the experimental region.
• 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 the model presented in section \ref{['sec:n>2']}, with $M = 4$ (thick), $M = 5$ (dashed), $M = 6$ (dashed-dotted), and $M = 10$ (dotted) and $n=3$. The black dots show the prediction for the original natural inflation potential. The blue line in (d) is $\qty(\frac{G(\zeta_\text{max})}{2})^{1/4}$ as a function of $\alpha$ to make manifest that the model is only defined for $\alpha<\alpha_{max}$. The amplitude of the power spectrum $A_s$ is fixed to its observed value. The gray regions indicate the 95% (dark-gray) and 68% (light-gray) confidence levels (CL), respectively, based on the latest combination of Planck, BICEP/Keck, and BAO data BICEP:2021xfz.