Constraining $β$-Exponential Inflation with the latest ACT observations

Abstract

Recent observations from the Atacama Cosmology Telescope (ACT), especially when combined with DESI baryon acoustic oscillation data, indicate a scalar spectral index $n_s$ higher than the value reported by \textit{Planck} 2018, placing tension on universal inflationary attractor models. Motivated by this discrepancy, we investigate the inflationary predictions of the $β$-exponential potential, $V(φ)=V_0\left(1-λβφ/M_p\right)^{1/β}$ considering both minimally and non-minimally coupled realizations. This potential generalizes standard exponential inflation and naturally arises in braneworld scenarios. We derive analytical expressions for the slow-roll parameters and inflationary observables using a perturbative expansion in the non-minimal coupling $ξ$, and validate these results through numerical calculations. In the minimally coupled case, the model predicts $n_s \simeq 0.976$ and $r \simeq 0.035$ for $N=50$ and moderate values of β, remaining compatible with ACT+DESI constraints at the 1σlevel while yielding a spectral tilt larger than the universal attractor prediction. Introducing a small non-minimal coupling significantly improves agreement with observations by suppressing the tensor-to-scalar ratio while preserving the enhanced scalar tilt. For $N=60, λ\sim 0.3-0.5$, and $β\sim O(1-5)$, the non-minimally coupled model yields $n_s \simeq 0.974-0.976$ and $r \lesssim 0.03$, comfortably consistent with ACT, DESI, and BICEP/Keck bounds. Our results show that the $β$-exponential potential, especially when implemented with a non-minimal coupling, exhibits good agreement with the latest CMB observations. Our inflationary predictions of the non-minimal model of $n_s$ and $r$ confirming the leading-order contributions in $ξ$ are sufficient to capture the essential features of both $r$ and $n_s$ in observationally relevant regimes.

Figures (5)

• Figure 1: Constraints on the scalar and tensor primordial power spectra, shown in the $r-n_{s}$ parameter space. The bounds on $r$ are primarily determined by the BK18 observations, whereas the limits on $n_s$ are set by Planck (red) and P-ACT (green) data. We fix $N=50,\,60$ and vary a parameter $\beta$ for a minimally-coupled model.
• Figure 2: The variation of the normalized potential $V_0^{1/4}/M_p$ as a function of $\beta$ for different values of the coupling parameter $\lambda$, with the number of e-folds fixed at $N = 60$. The black curves correspond to $\lambda = 0.1$, where the dashed line denotes the minimally coupled case ($\xi = 0$) and the solid line represents the nonminimally coupled case ($\xi = 0.0005$). The purple curves show the results for $\lambda = 0.3$, while the orange curves correspond to $\lambda = 0.5$; in both cases, dashed lines indicate minimal coupling ($\xi = 0$) and solid lines indicate nonminimal coupling ($\xi = 0.0005$).
• Figure 3: We present a comparison between the analytical predictions obtained from the perturbative expansion in the non-minimal coupling parameter $\xi$ and the exact numerical results for the tensor-to-scalar ratio $r$ and the scalar spectral index $n_s$, evaluated at $N=60$ for two representative values of the self-coupling parameter, $\lambda = 0.3$ and $\lambda = 0.5$, with $\xi = 5 \times 10^{-4}$.
• Figure 4: Constraints on the scalar and tensor primordial power spectra, shown in the $r-n_{s}$ parameter space predicted by the nonminimally-coupled model (color) compared with a minimally-coupled one (black). The bounds on $r$ are primarily determined by the BK18 observations, whereas the limits on $n_s$ are set by Planck (red) and P-ACT (green) data. We fix $N=60$, $\lambda=0.1$ (purple), $\lambda=0.5$ (orange) and vary a parameter $\beta$ from $1$ to $5$.
• Figure 5: The running of the spectral index predicted by our model (solid line) is shown alongside the ACT 95% confidence limits (dashed lines) over the interval $\beta\in[1,\,5]$ and using $N=60,\,\lambda=0.5,\,\xi=0.0005$.