Enhanced HSQRT Deformation in f(R) Gravity

• The paper introduces a quantum logarithmic correction to the baseline HSQRT f(R) model, resolving strong-coupling issues while preserving ghost freedom and stability.
• It derives an exact parametric formulation linking the Jordan and Einstein frames, recovering Einstein gravity at low curvature and modifying the inflationary plateau.
• Inflationary observables, such as the scalar spectral index and tensor-to-scalar ratio, are recalibrated, yielding predictions consistent with current and upcoming cosmological data.

Searching arXiv for the specified papers and closely related context. Enhanced Hyperbolic Square-Root (HSQRT) deformation denotes the logarithmically enhanced extension of the hyperbolic square-root deformation of the Starobinsky model within $f(R)$ gravity. It is built on a baseline HSQRT construction with strictly positive derivative $f'(R)=\alpha R+\sqrt{\alpha^2R^2+1}$, $\alpha>0$, and augments that baseline by a single dimensionless parameter $\beta\ll1$ through a quantum-motivated logarithmic correction. In the stated formulation, the enhancement preserves the recovery of general relativity at low curvature, retains global ghost freedom and tachyon-free stability, regularizes the strong-coupling pathology associated with $f'(R)=0$, and modifies the asymptotic inflationary plateau from purely exponential to inverse-power form in the Einstein frame (Galiautdinov, 16 Mar 2026, Galiautdinov, 10 Mar 2026).

1. Baseline HSQRT deformation of the Starobinsky model

The baseline HSQRT model is defined by the exact ansatz

$$f'(R)=\alpha R+\sqrt{\alpha^2R^2+1},\qquad \alpha>0.$$

Integrating with respect to $R$ gives

$$f(R)=\frac12\,R\Bigl(\alpha R+\sqrt{\alpha^2R^2+1}\Bigr)+\frac{1}{2\alpha}\operatorname{arcsinh}(\alpha R)+C,$$

with the additive constant often chosen as $C=-2\Lambda$. The construction is explicitly designed so that $f'(R)$ never vanishes. This removes the pathological branch associated with quadratic $f'(R)=\alpha R+\sqrt{\alpha^2R^2+1}$0 gravity when $f'(R)=\alpha R+\sqrt{\alpha^2R^2+1}$1, while keeping the model analytic and globally defined (Galiautdinov, 10 Mar 2026).

Its asymptotic behavior interpolates between general relativity and $f'(R)=\alpha R+\sqrt{\alpha^2R^2+1}$2 inflation. For $f'(R)=\alpha R+\sqrt{\alpha^2R^2+1}$3,

$f'(R)=\alpha R+\sqrt{\alpha^2R^2+1}$4

so the theory approaches the Einstein–Hilbert form. For $f'(R)=\alpha R+\sqrt{\alpha^2R^2+1}$5,

$f'(R)=\alpha R+\sqrt{\alpha^2R^2+1}$6

which reproduces the Starobinsky inflationary plateau. In the opposite limit $f'(R)=\alpha R+\sqrt{\alpha^2R^2+1}$7, $f'(R)=\alpha R+\sqrt{\alpha^2R^2+1}$8 asymptotes to zero strictly from above rather than crossing zero. The only admissible constant-curvature solutions are then standard Einstein spaces with effective cosmological constant $f'(R)=\alpha R+\sqrt{\alpha^2R^2+1}$9 when $\alpha>0$0 is constant (Galiautdinov, 10 Mar 2026).

2. Logarithmic enhancement and exact defining form

The enhanced HSQRT deformation keeps the baseline hyperbolic square-root variable

$\alpha>0$1

and introduces a structurally minimal logarithmic correction in parametric form. With $\alpha>0$2, the baseline Lagrangian is written as

$\alpha>0$3

and the correction is

$\alpha>0$4

The full Lagrangian therefore becomes

$\alpha>0$5

In this formulation, the enhancement is single-parameter, quantum-motivated, and explicitly phenomenological (Galiautdinov, 16 Mar 2026).

The ultraviolet and infrared limits are central to the construction. At large positive curvature, $\alpha>0$6 and $\alpha>0$7, yielding

$\alpha>0$8

The enhancement thus preserves $\alpha>0$9-type growth while introducing a slowly running logarithmic correction. For $\beta\ll1$0, corresponding to $\beta\ll1$1, the correction decouples: $\beta\ll1$2 For small curvature, the expansion

$\beta\ll1$3

is used to state that $\beta\ll1$4, $\beta\ll1$5, and standard Einstein–Hilbert gravity is recovered (Galiautdinov, 16 Mar 2026).

3. Stability structure and exact Einstein-frame formulation

The baseline HSQRT theory is globally stable in the standard $\beta\ll1$6 sense because both first and second derivatives remain strictly positive: $\beta\ll1$7 These conditions guarantee no ghost, $\beta\ll1$8, and no Dolgov–Kawasaki instability, $\beta\ll1$9. The Jordan-frame scalaron mass squared,

$f'(R)=0$0

is strictly positive for all $f'(R)=0$1, excluding tachyonic instability (Galiautdinov, 10 Mar 2026).

The enhanced model is constructed so that these global properties survive. In the Jordan frame one starts from

$f'(R)=0$2

defines

$f'(R)=0$3

and performs the conformal transformation $f'(R)=0$4. The Einstein-frame action takes the canonical form

$f'(R)=0$5

with

$f'(R)=0$6

Because $f'(R)=0$7 cannot be inverted algebraically once $f'(R)=0$8, the enhanced theory is kept in an exact parametric form based on $f'(R)=0$9. The canonical field and potential are then represented parametrically and are stated to remain exact and real for all $$f'(R)=\alpha R+\sqrt{\alpha^2R^2+1},\qquad \alpha>0.$$0. Near small curvature the enhanced theory has

$$f'(R)=\alpha R+\sqrt{\alpha^2R^2+1},\qquad \alpha>0.$$1

which is used to recover standard reheating dynamics (Galiautdinov, 16 Mar 2026).

4. Einstein-frame potential and asymptotic geometry

For the baseline HSQRT model, the Einstein-frame scalaron potential is globally defined as

$$f'(R)=\alpha R+\sqrt{\alpha^2R^2+1},\qquad \alpha>0.$$2

Its defining geometric feature is an impenetrable energetic wall at $$f'(R)=\alpha R+\sqrt{\alpha^2R^2+1},\qquad \alpha>0.$$3. In that limit the dominant term rises without bound, so the field cannot run off to $$f'(R)=\alpha R+\sqrt{\alpha^2R^2+1},\qquad \alpha>0.$$4; the stated dynamical interpretation is a non-singular bounce rather than a Big-Crunch singularity (Galiautdinov, 10 Mar 2026).

The enhanced HSQRT deformation preserves that negative-curvature barrier while altering the large-field plateau. For $$f'(R)=\alpha R+\sqrt{\alpha^2R^2+1},\qquad \alpha>0.$$5,

$$f'(R)=\alpha R+\sqrt{\alpha^2R^2+1},\qquad \alpha>0.$$6

and the field map implies the asymptotic Einstein-frame form

$$f'(R)=\alpha R+\sqrt{\alpha^2R^2+1},\qquad \alpha>0.$$7

This is the characteristic transition from an exponential plateau to an inverse-power plateau. By contrast, in the deep negative-curvature regime $$f'(R)=\alpha R+\sqrt{\alpha^2R^2+1},\qquad \alpha>0.$$8, one has $$f'(R)=\alpha R+\sqrt{\alpha^2R^2+1},\qquad \alpha>0.$$9, $R$0, and $R$1, so the infinite barrier is preserved. Around $R$2, the expansion $R$3 and the quadratic form $R$4 recover the standard low-curvature sector (Galiautdinov, 16 Mar 2026).

5. Inflationary observables

The baseline HSQRT model retains the standard large-$R$5 predictions of Starobinsky-type inflation. Using

$R$6

together with the slow-roll approximation at $R$7, the leading-order observables are

$R$8

For $R$9, the explicit values quoted are $$f(R)=\frac12\,R\Bigl(\alpha R+\sqrt{\alpha^2R^2+1}\Bigr)+\frac{1}{2\alpha}\operatorname{arcsinh}(\alpha R)+C,$$0 and $$f(R)=\frac12\,R\Bigl(\alpha R+\sqrt{\alpha^2R^2+1}\Bigr)+\frac{1}{2\alpha}\operatorname{arcsinh}(\alpha R)+C,$$1, placing the model within the observationally favored parameter space of the Planck and BICEP/Keck Array baseline constraints (Galiautdinov, 10 Mar 2026).

The enhanced deformation changes these asymptotics in a specific way. Starting from

$$f(R)=\frac12\,R\Bigl(\alpha R+\sqrt{\alpha^2R^2+1}\Bigr)+\frac{1}{2\alpha}\operatorname{arcsinh}(\alpha R)+C,$$2

the slow-roll analysis yields

$$f(R)=\frac12\,R\Bigl(\alpha R+\sqrt{\alpha^2R^2+1}\Bigr)+\frac{1}{2\alpha}\operatorname{arcsinh}(\alpha R)+C,$$3

For $$f(R)=\frac12\,R\Bigl(\alpha R+\sqrt{\alpha^2R^2+1}\Bigr)+\frac{1}{2\alpha}\operatorname{arcsinh}(\alpha R)+C,$$4, the resulting spectral index lies in the range

$$f(R)=\frac12\,R\Bigl(\alpha R+\sqrt{\alpha^2R^2+1}\Bigr)+\frac{1}{2\alpha}\operatorname{arcsinh}(\alpha R)+C,$$5

and the running in the range

$$f(R)=\frac12\,R\Bigl(\alpha R+\sqrt{\alpha^2R^2+1}\Bigr)+\frac{1}{2\alpha}\operatorname{arcsinh}(\alpha R)+C,$$6

The tensor-to-scalar ratio remains tunable through $$f(R)=\frac12\,R\Bigl(\alpha R+\sqrt{\alpha^2R^2+1}\Bigr)+\frac{1}{2\alpha}\operatorname{arcsinh}(\alpha R)+C,$$7, with the text emphasizing that it is easily below current upper bounds $$f(R)=\frac12\,R\Bigl(\alpha R+\sqrt{\alpha^2R^2+1}\Bigr)+\frac{1}{2\alpha}\operatorname{arcsinh}(\alpha R)+C,$$8 while remaining a target for next-generation $$f(R)=\frac12\,R\Bigl(\alpha R+\sqrt{\alpha^2R^2+1}\Bigr)+\frac{1}{2\alpha}\operatorname{arcsinh}(\alpha R)+C,$$9-mode searches. The observational motivation is explicitly connected to ACT DR6 and DESI, which are described as indicating an upward shift in the scalar spectral index and a preference for deviations from a pure exponential plateau (Galiautdinov, 16 Mar 2026).

6. Physical interpretation, scope, and common misunderstandings

A central point of the enhanced HSQRT deformation is that the logarithmic correction does not replace the original HSQRT regularization mechanism. In the negative-curvature limit $C=-2\Lambda$0, the $C=-2\Lambda$1-correction decouples and $C=-2\Lambda$2, so the ghost-singularity is regularized exactly as in the baseline model. The enhancement therefore leaves intact the feature that $C=-2\Lambda$3 approaches zero strictly from above rather than vanishing or becoming negative (Galiautdinov, 16 Mar 2026).

It is also inaccurate to read the enhancement as a wholesale abandonment of Starobinsky inflation. The theory continues to recover general relativity at low curvatures, preserves standard reheating through scalaron oscillations around a quadratic minimum, and retains an $C=-2\Lambda$4-type inflationary sector. What changes is the deep ultraviolet asymptotic regime, where the Einstein-frame potential crosses over from the original exponential plateau to an inverse-power plateau controlled by $C=-2\Lambda$5. This suggests that the enhanced model is best understood as a precision-oriented deformation of the HSQRT baseline rather than as a distinct infrared theory.

The construction is presented as phenomenological, quantum-motivated, and exact in parametric form, with no piecewise approximations. Its stated significance lies in combining global regularity, ghost freedom, and tachyon-free evolution with a modified large-$C=-2\Lambda$6 inflationary prediction,

$C=-2\Lambda$7

that is intended to move the scalar tilt into the observational window favored by 2025–2026 data while preserving the infinite barrier and global stability inherited from the baseline HSQRT framework (Galiautdinov, 16 Mar 2026).