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

Abstract

We propose an exact, analytic deformation of the Starobinsky model governed by the strictly positive derivative of its Lagrangian, $f'(R) = αR + \sqrt{α^2 R^2 + 1}$, with $α> 0$. 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 $Λ_{\text{eff}} \equiv A/4$. 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 $V(φ) = \frac{1}{8α} [ 1 - (1 + 2\sqrt{2/3}φ) \exp(-2\sqrt{2/3}φ) ] + Λ\exp(-2\sqrt{2/3}φ)$, which naturally establishes an impenetrable energetic wall as $φ\to -\infty$, 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.

Figures (6)

• Figure 1: The first derivative of the modified action, $f'(R) = \alpha R + \sqrt{\alpha^2 R^2 + 1}$, which serves as the foundational geometric ansatz for our deformed Starobinsky model. The function $f'(R)$ remains strictly positive for all real curvature values $R \in (-\infty, +\infty)$. This positivity ensures that the effective gravitational coupling $G_{\mathrm{eff}} = G/f'(R)$ is finite and positive, preventing the spin-2 graviton from becoming a ghost and ensuring the conformal mapping to the Einstein frame remains non-singular.
• Figure 2: The Lagrangian density $f(R)$ for the deformed Starobinsky model, evaluated with a vanishing cosmological constant ($\Lambda=0$). Obtained by integrating the derivative shown in Fig. \ref{['fig:01']}, this function recovers the linear Einstein-Hilbert limit ($f(R) \simeq R$) at low curvatures, while transitioning to the $\mathcal{O}(R^2)$ scaling required for slow-roll inflation at large positive curvatures.
• Figure 3: The second derivative of the modified action, $f"(R)$, plotted as a function of the background Ricci scalar for our exact square-root deformation. The strict positivity $f"(R) > 0$ is maintained across the entire curvature domain $R \in (-\infty, +\infty)$. This is a necessary condition in $f(R)$ gravity to avoid the Dolgov-Kawasaki instability and ensures the conformal mapping to the Einstein frame remains non-singular.
• Figure 4: The effective scalaron mass squared, $m_{\mathrm{s}}^2$, plotted as a continuous function of the background Ricci scalar $R$ for the square-root deformed model. Importantly, the mass squared remains strictly positive ($m_{\mathrm{s}}^2 > 0$) and finite across the entire global curvature domain $R \in (-\infty, +\infty)$. This positivity ensures the classical and perturbative stability of the spacetime, preventing the scalar degree of freedom from becoming tachyonic.
• Figure 5: The scalaron potential expressed as a function of the Jordan-frame Ricci scalar, $V(R)$, evaluated at $\Lambda = 0$ for our exact square-root deformation (solid blue) and the original Starobinsky model (dashed orange). Both potentials exhibit a stable Minkowski vacuum at $R=0$ and behave identically in the high-curvature inflationary regime ($R \to +\infty$). However, their dynamics sharply diverge at negative curvatures. The standard Starobinsky potential suffers a pathological pole at $R_{\mathrm{crit}}=-1/(2\alpha)$, where $f'(R)=0$. At this boundary, the effective gravitational coupling diverges, and the conformal mapping becomes singular. In contrast, the deformed model's potential smoothly bypasses this singularity, remaining finite and globally well-defined for all $R \in (-\infty, +\infty)$.