Globally stable, ghost-free hyperbolic square-root deformation of the Starobinsky model

We propose an exact, analytic deformation of the Starobinsky model governed by the strictly positive derivative of its Lagrangian, $f'(R) = \alpha R + \sqrt{\alpha^2 R^2 + 1}$, with $\alpha > 0$. This geometric hyperbolic square-root ansatz is designed to eliminate the well-known strong-coupling singularity that arises in quadratic $f(R)$ gravity when $f'(R)=0$. The construction seamlessly recovers general relativity at low curvatures and preserves the successful slow-roll inflationary plateau at extreme positive curvatures. In the limit $R \to -\infty$, the derivative $f'(R)$ asymptotes to zero strictly from above, removing the pathological branch associated with the vanishing of $f'(R)$. This guarantees that the only admissible constant-curvature ($R=A$) solutions correspond to standard Einstein spaces with an effective cosmological constant $\Lambda_{\text{eff}} \equiv A/4$. The first and second derivatives of the action, as well as the scalaron mass squared, remain strictly positive globally, ensuring a perfectly ghost-free and tachyon-free cosmological evolution across the entire spacetime manifold. In the Einstein frame, the dynamics of the scalaron is governed by the globally defined potential $V(\phi) = \frac{1}{8\alpha} [ 1 - (1 + 2\sqrt{2/3}\phi) \exp(-2\sqrt{2/3}\phi) ] + \Lambda \exp(-2\sqrt{2/3}\phi)$, which naturally establishes an impenetrable energetic wall as $\phi \to -\infty$, offering a robust, globally stable mechanism for non-singular bouncing cosmologies. For $N = 60$ inflationary e-folds, the model predicts a scalar spectral index of $n_s \simeq 0.967$ and a strongly suppressed tensor-to-scalar ratio of $r \simeq 0.00083$, which position the proposed theory within the observationally favored parameter space of the Planck and BICEP/Keck Array baseline constraints.