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, Padé-regulated effective potential, $V_{\text{PadA˜c◯}}(\ddot{I}•)=V_{\text{Staro}}(\ddot{I}•){\left(1+\frac{4{\hat{I}}^2}{\frac{2}{3}\hat{I}{º}^2\ddot{I}{•}^2+1}\right)}^{-1}+\frac{1}{4\hat{I}\pm \hat{I}{º}^2}\left[1-\left(1+\sqrt{\frac{2}{3}}\hat{I}º\ddot{I}•\right)e^{-\sqrt{\frac{2}{3}}\hat{I}º\ddot{I}•}\right]e^{-\sqrt{\frac{2}{3}}\hat{I}º\ddot{I}•}$, which exhibits the inverse-power asymptotic form at high curvatures, $V(\ddot{I}•)\simeq \frac{1}{8\hat{I}\pm \hat{I}{º}^2}\left(1-\frac{6{\hat{I}}^2}{\hat{I}{º}^2\ddot{I}{•}^2}\right)$, with the strength of the deviation from the baseline HSQRT geometry governed by dimensionless parameter ${\hat{I}}^2$. 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 $r\simeq 2\sqrt{3{\hat{I}}^2}/N^{3/2}$. 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 $\hat{I}{\pm }_s\simeq -3/(2N^2)\in [-0.00060,-0.00042]$, establishing a viable target for next-generation CMB observatories.

20 pages, 8 figures