Logarithmically enhanced hyperbolic square-root deformation of Starobinsky inflation

Abstract

We propose and analyze 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 characteristic exponential slow-roll plateau at large positive curvatures. Motivated by recent precision cosmological observations (such as ACT DR6 and DESI) indicating an upward shift in the scalar spectral index and a preference for deviations from pure exponential plateau behavior, we introduce a structurally minimal, quantum-motivated logarithmic enhancement. This phenomenological enhancement modifies the deep ultraviolet asymptotic regime while maintaining global tachyon-free stability, ghost freedom, and the recovery of general relativity at low curvatures to preserve standard reheating. By developing an exact parametric formulation of the Einstein frame, we demonstrate that the scalar potential of the enhanced model transitions from an exponential to an inverse-power asymptotic form, $V(φ) \simeq V_0 \left[ 1 - 6β/ (κ^2φ^2) \right]$, where the strength of this deviation is governed by a single dimensionless parameter $β$. We derive exact analytic expansions for the slow-roll observables, yielding a parameter-free leading-order spectral index $n_s \simeq 1 - 3/(2N)$ and a tunable tensor-to-scalar ratio $r \simeq 2(3β)^{1/2}/N^{3/2}$. For standard inflationary 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 running $α_s \simeq -3/(2N^2) \in [-0.00060, -0.00042]$, while providing a viable, parameter-controlled target for next-generation cosmic microwave background observatories.

Figures (6)

• Figure 1: The Lagrangian densities, $f(R)$, of the base HSQRT (blue), enhanced HSQRT (green), and standard Starobinsky (orange) models, at $\beta = 0.1$ with $\Lambda=0$.
• Figure 2: The first derivative of the Lagrangian densities, $f'(R)$, in the base HSQRT (blue), enhanced HSQRT (green), and standard Starobinsky (orange) models, at $\beta = 0.1$.
• Figure 3: The second derivative of the Lagrangians, $f"(R)$, in the base HSQRT (blue), enhanced HSQRT (green), and standard Starobinsky (orange) models, at $\beta = 0.1$.
• Figure 4: The effective scalaron mass squared, $m_{\mathrm{s}}^2$, in the base HSQRT (blue), enhanced HSQRT (green), and standard Starobinsky (orange) models, at $\beta = 0.1$.
• Figure 5: The scalaron potential expressed as a function of the Jordan-frame Ricci scalar, $V(R)$, in the base HSQRT (blue), enhanced HSQRT (green), and standard Starobinsky (orange) models, at $\beta = 0.1$.