The Inflationary Dynamics with the Scalar-Tensor Model

TL;DR

This work investigates inflation in a scalar-tensor framework featuring a non-minimal kinetic coupling to the Einstein tensor, aiming to reconcile a dynamical inflation potential with current observational bounds. By performing slow-roll analysis, the authors derive analytical expressions for the observables $n_s$, $r$, and $\alpha_s$ in terms of a dynamical inflationary potential $V(\phi)=V_0[1+(\phi/\mu)^{-p}]$ and a coupling $J(\phi)=\frac{1}{J_0}[\alpha+\beta(\phi/\mu)^{-p}]$. A comprehensive parameter-space mapping identifies viable ranges for the free parameters that yield Planck-consistent $n_s$, $r$, and $\alpha_s$, while highlighting plateaus between asymptotes that imply naturalness with $\mu \sim M_{Pl}$, $\alpha \sim -0.02$, and $\beta \sim 10$. The analysis also shows compatibility with recent ACT DR6 and CMB-S4 forecasts, indicating the model can accommodate tighter future constraints while avoiding extreme fine-tuning. Overall, the study presents a robust scalar-tensor DI scenario that aligns with current data and offers testable predictions for upcoming CMB observations.

Abstract

We investigate the cosmic inflation within a class of the scalar-tensor model with the scalar-dependent non-minimal kinetic couplings. The inflationary dynamical potential will be applied. Using the slow-roll approximation, we compute theoretical predictions for the key observables, like the spectral indexes $n_s$, scalar-to-tensor ratio $r$ and the running of the scalar spectral index $α_s$ in terms of the free parameters of the model. Besides, we find the limitations of these parameters. In addition, these quantities will be compared with the latest observational data from the Planck data. Furthermore, we analyze the sensitivity of $r$, $n_s$ and $α_s$ in terms of the model's free parameters.

Figures (6)

• Figure 1: The tensor-to-scalar ratio $r$ versus the scalar spectral index $n_s$ for $N=50$--$60$ e-folds. The solid lines show the model's predictions for $\alpha=-0.02$, $\beta=10$, $p=3$, in which $\mu$ varies in the range $2.0 \leq \mu \leq 2.4$ (from top to bottom of the $r$ values). The $1\sigma$ and $2\sigma$ confidence contours from the Planck 2018+BICEP/Keck 2018 data have been shown for comparison. For this parameter range, the model's predictions fall well within the observational constraints.
• Figure 2: The tensor-to-scalar ratio $r$ versus the scalar spectral index $n_s$ for $N=50$--$60$ e-folds. The solid lines show the model's predictions for $\beta=10$, $\mu=2$, $p=3$, and $\alpha$ varies in the range $-0.21 \leq \alpha \leq -0.07$ (from top to bottom of the $r$ values). The predictions have been plotted against the Planck 2018+BICEP/Keck 2018 confidence contours.
• Figure 3: The tensor-to-scalar ratio $r$ versus the scalar spectral index $n_s$ for $N=50$--$60$ e-folds. The solid lines show the model's predictions for $\alpha=-0.02$, $\mu=2$, $p=3$, and $\beta$ varies in the range $10.4 \leq \beta \leq 11.1$ (from top to bottom of the $r$ values). The results are consistent with the observational bounds of the Planck 2018+BICEP/Keck 2018.
• Figure 4: The parameter-space analysis of the running of the spectral index $\alpha_s$ shows distinct functional dependencies. Left: $\alpha_s(\alpha)$ exhibits asymptotic divergence at $\alpha = -0.5$ and $-0.1$, which delineates the theoretical boundaries in the coupling parameter-space. Middle: $\alpha_s$ linearly decreases with $\beta$, for the moderate $\beta$ values, it remains within the Planck's $2\sigma$ constraints. Right: $\alpha_s(\mu)$ reveals a resonant structure with the singularities at $\mu = 2.0$ and $2.5$, creating observationally viable plateaus in the intervening parameter regions. The theoretical prediction (the blue curve) has been evaluated against the observational constraints $\alpha_s = -0.0041 \pm 0.0067$ from the Planck collaboration.
• Figure 5: The sensitivity analysis of $n_s$ across the fundamental inflationary parameters. Left: $n_s(\alpha)$ monotonically increases with the coupling strength $\alpha$, which intersects the Planck preferred value $n_s = 0.9663 \pm 0.0041$ at intermediate $\alpha$ values. Middle: $n_s(\beta)$ linearly decreases with the field parameter $\beta$; the central values of $\beta$ yield observationally consistent results. Right: $n_s(\mu)$ features, separated by the asymptotes at $\mu = 2.4$ and $7.0$, indicate the parameter regions where the spectral index remains stable against the mass scale variations. All viable regions (lemon) satisfy the observational constraints within the $2\sigma$ confidence levels.