Logarithmically enhanced hyperbolic square-root deformation of Starobinsky inflation

Abstract

We propose an enhanced hyperbolic square-root (HSQRT) deformation of the Starobinsky model in the context of $f(R)$ gravity. The original HSQRT construction provided a globally regular modification of $R^2$ inflation, curing the strong-coupling singularity at negative curvatures while preserving the exponential slow-roll plateau at large positive curvatures. Motivated by recent cosmological data (ACT DR6 and DESI), we introduce a structurally minimal, rational-logarithmic enhancement. This enhancement modifies the deep UV asymptotic regime while preserving global tachyon-free stability, ghost freedom, and the recovery of general relativity at low curvatures. In the Einstein frame, the scalaron dynamics is described by a globally defined, Pad\'{e}-regulated effective potential, $V_{\text{Pad\'{e}}}(\phi) = V_{\text{Staro}}(\phi) \left(1+ \frac{4\beta }{\frac{2}{3}\kappa^2\phi^2 + 1}\right)^{-1} + \frac{1}{4\alpha\kappa^2} \left[1 - \left( 1 + \sqrt{\frac{2}{3}}\kappa\phi\right) e^{-\sqrt{\frac{2}{3}}\kappa\phi} \right] e^{-\sqrt{\frac{2}{3}}\kappa\phi}$, which exhibits the inverse-power asymptotic form at high curvatures, $V(\phi) \simeq \frac{1}{8\alpha\kappa^2} \left( 1 - \frac{6\beta}{\kappa^2\phi^2} \right)$, with the strength of the deviation from the baseline HSQRT geometry governed by dimensionless parameter $\beta$. We derive analytic expansions for the slow-roll observables, yielding a leading-order spectral index $n_s \simeq 1 - 3/(2N)$ and a tensor-to-scalar ratio $r \simeq 2\sqrt{3\beta}/N^{3/2}$. For standard e-fold durations ($N \in [50, 60]$), this model drives the spectral index directly into the newly favored observational window ($n_s \simeq 0.970\text{--}0.975$) and predicts an exceptionally small spectral running $\alpha_s \simeq -3/(2N^2) \in [-0.00060, -0.00042]$, establishing a viable target for next-generation CMB observatories.