Constant-roll $β$-exponential inflation: Palatini formalism

TL;DR

The paper addresses inflation in a Palatini gravity setting with an $R^2$ term and a non-minimally coupled $\phi$ featuring a $\beta$-exponential potential. It derives the Einstein-frame generalized $k$-inflation Lagrangian $\\mathcal{L}(\\phi,X)=A(\\phi)X+B(\\phi)X^2-U(\\phi)$ by introducing and eliminating an auxiliary field, and imposes the constant-roll condition $\\ddot{\\phi}=\\kappa H\\dot{\\phi}$. Through this framework, it computes $n_s$ and $r$ and demonstrates that, for a broad range of parameters $\\beta,\\lambda,\\kappa,\\xi,\\alpha$, the predictions lie within current ACT DR6 and Planck constraints. The results highlight that Palatini dynamics with non-minimal coupling and higher-curvature terms can yield viable early-Universe inflation with the flexibility to fit observations. Overall, the work extends constant-roll $k$-inflation to a richer gravitational sector and confirms its compatibility with modern CMB data.

Abstract

In this paper, we explore the inflationary dynamics of the $β$-exponential potential model, where a scalar field couples to quadratic $(R + R^2)$ gravity. In this model, the inflaton is the field that determines the size of the extra dimension. We employ the Palatini formalism to derive the resulting Einstein-frame generalized $k$-inflation effective theory, which we analyze under the assumption that the constant-roll condition is satisfied. We scan the parameter space for inflationary predictions, specifically the spectral index $n_s$ and the tensor-to-scalar ratio $r$, ensuring consistency with the results from ACT DR6. The compliant regions are depicted accordingly. For a suitable range of the model parameters, the values obtained for the inflationary observables align with the most recent observations by the Atacama Cosmology Telescope (ACT) collaboration and/or the Planck mission.

Figures (5)

• Figure 1: The predictions for the $n_s - r$ parameter space corresponding to selected values of the $\beta$ and $\lambda$ parameters for constant-roll $\beta$-exponential inflation within the Palatini formalism. The constant-roll parameter, $\kappa$, is fixed at 0.005, and the non-minimal coupling, $\xi$, is set at 0.0011. Meanwhile, the parameter $\alpha$ varies within the range of $[1 \times 10^5 - 8 \times 10^{10}]$. The light and dark contours represent the $95\%$ and $68\%$ confidence levels, respectively. The constraints on $r$ are driven by the BK18 data BICEP:2021xfz, while the constraints on $n_s$ are driven by Planck (orange) Akrami2018, ACT (blue), or P-ACT (purple) ACTcollab2. The combined dataset also includes CMB lensing and BAO in all cases.
• Figure 2: The impact of the constant-roll parameter $\kappa$ on $n_s - r$ for fixed $\beta$, $\lambda$, and $\xi$ values in the context of constant-roll $\beta$-exponential inflation, with $\alpha$ in the range $[1 \times 10^5-8 \times 10^{10}]$.
• Figure 3: Constant-roll parameter $\kappa$ versus $n_s-r$ for fixed $\beta$, $\lambda$ and $\xi$ parameters of constant-roll $\beta$-exponential inflation and selected values of $\alpha$. For each curve $\kappa$ continuously varies in the range $[1 \times 10^{-5}-11 \times 10^{-3}]$.
• Figure 4: $\beta$-exponential parameter $\lambda$ versus $n_s-r$ for fixed $\beta$, $\kappa$ and $\xi$ parameters of constant-roll $\beta$-exponential inflation and selected values of $\alpha$. For each curve, the value of $\lambda$ varies continuously within the range of $0.001$ to $1.0$.
• Figure 5: The non-minimal coupling parameter $\xi$ vs $n_s - r$ for selected $\beta$, $\lambda$ values and fixed $\kappa$ in the context of constant-roll $\beta$-exponential inflation, with $\alpha$ in the range $[1 \times 10^5-8 \times 10^{10}]$.