Starobinsky-inflation in asymptotically safe shift-symmetric scalar-tensor theory

Abstract: We investigate the constraining power of scalaron-driven inflation on asymptotically safe scalar-tensor theories. Starting from a Horndeski-type theory and applying a renormalization group improvement procedure generates higher-derivative couplings which are fixed in terms of the microscopic parameters - a structure that is expected to occur also within first principle computations based on the asymptotic safety mechanism. The latter are taken to be the free parameters appearing at the Gaussian fixed point. We find that the free parameter initially associated with the non-minimal gravity-matter coupling is not confined to the gravity-matter sector of the theory and also enters the effective higher-derivative couplings in the gravitational sector. We review the setting of multi-field inflationary models which is appropriate to analyze the inflationary dynamics in this context and illustrate their applicability by working out the explicit bounds on the non-minimal gravity-matter coupling resulting from cosmological observations. Given the fixed point structure of asymptotically safe scalar-tensor theories, the results indicate that UV-completions by two of the three available non-Gaussian fixed points can be ruled out while pinpointing phenomenologically viable RG trajectories emanating from the third fixed point.

• The paper demonstrates that RG improvements in asymptotically safe, shift-symmetric scalar-tensor theory yield a Starobinsky-like inflationary model.
• It derives an improved action with R² and R³ corrections that constrain the gravity-matter coupling D0 using observational data.
• The two-field Einstein frame analysis provides predictions for the spectral index and tensor-to-scalar ratio consistent with Planck results, guiding UV completion viability.

Starobinsky Inflation in Asymptotically Safe Shift-Symmetric Scalar-Tensor Theory

Introduction and Theoretical Motivation

The paper "Starobinsky-inflation in asymptotically safe shift-symmetric scalar-tensor theory" (2606.22408) systematically addresses the interplay between scalaron-driven inflation and asymptotically safe scalar-tensor models endowed with shift symmetry. The analysis begins with Horndeski-type scalar-tensor actions, which extend the gravitational sector by a scalar field without introducing higher-order equations of motion, preserving second-order dynamics. The central focus is on renormalization group (RG) improvement methods applied to these models to investigate which inflationary scenarios are allowed under asymptotic safety constraints.

The asymptotic safety paradigm in quantum gravity posits that a non-Gaussian fixed point (NGFP) exists for the RG flow of gravitational couplings, ensuring ultraviolet (UV) completeness. When matter and gravity are coupled, the RG trajectories interpolate between an NGFP in the UV and a Gaussian fixed point (GFP) in the infrared (IR). The shift symmetry of the scalar sector severely constrains the allowed interactions, excluding scalar potentials generated radiatively and tightly structuring the non-minimal gravity-matter couplings.

Figure 1: Schematic RG flow for the couplings $\tilde{g}_k$ and $\tilde{\lambda}_k$ showing interpolation from a NGFP in the UV (blue circle) to a GFP in the IR (red circle); inflationary energy scales reside in the intermediate region (orange circle).

Renormalization Group Improvement and Action Construction

The RG improvement procedure is pivotal to quantifying quantum corrections without requiring explicit computation of all effective couplings. Two expansion strategies are contrasted: the traditional top-down approach expanding at the NGFP and a novel bottom-up approach expanding at the GFP, justified by the inflationary scale residing well below the Planck mass.

The improved action derived via the IR expansion exhibits only integer powers in curvature invariants, primarily $R^2$ and $R^3$, with non-minimal $RX$ couplings. The RG-improved Lagrangian in this regime is:

$$\mathcal{L}^{IR}_{RG} = \frac{1}{2 \kappa^2} \left( R -2 \Lambda_{0} + \frac{1}{6m^2}R^2 + \frac{\lambda}{3m^4} R^3 \right) - X \left(\frac{1}{2} + D_0 R - \alpha R^2 \right)$$

where $D_0$ encodes the gravity-matter coupling determined by RG trajectory initial conditions rather than arbitrary IR tuning. The shift symmetry ensures technical naturalness, preventing destabilization of $D_0$ via radiative corrections.

Einstein Frame and Inflationary Dynamics

By employing a conformal transformation to the Einstein frame, the $F(R, \phi)$ theory is recast as a two-field system with a canonical scalaron $\rho$ and an additional shift-symmetric scalar $\tilde{\lambda}_k$0. The scalaron potential governs inflation, and $\tilde{\lambda}_k$1 remains a flat direction, protected by symmetry. The resulting inflaton potential reproduces the Starobinsky form when higher-order terms are negligible:

$\tilde{\lambda}_k$2

With the $\tilde{\lambda}_k$3 correction, the plateau of the potential attains finite extension, enabling stringent constraints on $\tilde{\lambda}_k$4.

Figure 2: Potential $\tilde{\lambda}_k$5 as a function of $\tilde{\lambda}_k$6 for varying $\tilde{\lambda}_k$7, showing plateau shrinkage as higher-order corrections become significant.

Covariant Multi-Field Formalism and Slow-Roll Observables

The dynamics adhere to a covariant two-field formalism, but inflation predominantly follows the $\tilde{\lambda}_k$8 direction. Slow-roll parameters $\tilde{\lambda}_k$9 and $R^2$0 are derived analytically in terms of the number of e-folds $R^2$1, the plateau parameter $R^2$2, and underlying couplings. Their explicit expressions enable computation of the spectral index $R^2$3 and tensor-to-scalar ratio $R^2$4, directly connecting theoretical predictions with CMB observations:

$R^2$5

The model yields $R^2$6 and $R^2$7 predictions in excellent agreement with Planck 2018 results for suitable $R^2$8 and $R^2$9 values, confirming the viability of Starobinsky-type inflation with subleading $R^3$0 corrections.

Figure 3: Predictions for the spectral index ($R^3$1) and tensor-to-scalar ratio ($R^3$2) in the two-field model for $R^3$3, overlaid with Planck 2018 confidence contours.

Constraints on Gravity-Matter Coupling and Implications

The observational requirement of a sufficiently flat inflationary plateau translates into a non-trivial constraint on $R^3$4. Explicitly,

$R^3$5

for plateau parameters in the regime $R^3$6. This tight constraint acts as a selection criterion for RG trajectories in the UV-complete theory and is preserved by shift symmetry, guaranteeing its radiative stability. The model’s predictions for $R^3$7—$R^3$8—fall well within current Planck limits and are testable by upcoming missions (e.g., LiteBIRD), offering a direct probe of quantum gravitational effects in cosmology.

Further, the analysis enables falsification or selection of UV completions among NGFPs. Specifically, two of three NGFPs are excluded due to their critical exponent structure failing to support RG trajectories passing through phenomenologically viable $R^3$9 values, with only NGFP$RX$0 accommodating observational constraints.

Theoretical and Phenomenological Outlook

The approach establishes a direct bridge between UV quantum gravity physics and inflationary cosmology, utilizing the asymptotic safety paradigm and shift symmetry. It identifies $RX$1 as a phenomenologically relevant parameter encoding quantum gravitational signatures, with forthcoming data poised to further constrain its magnitude.

The formalism is scalable to scenarios involving generation of non-trivial scalar potentials (via inclusion of Yukawa couplings or kinetial terms), at which point multi-field effects become important. Current reduction to a single-field inflation is a consequence of the fixed-point structure, not an ad hoc assumption, and may evolve as the matter sector is extended.

Conclusion

This paper develops a rigorous connection between Starobinsky-type inflation and the landscape of asymptotically safe scalar-tensor gravity with shift symmetry (2606.22408). By leveraging RG improvements anchored at the Gaussian fixed point, the analysis constrains the non-minimal gravity-matter coupling $RX$2 through inflationary observables. The approach excludes certain UV completions and delineates viable RG trajectories, yielding model predictions compatible with Planck constraints and amenable to future experimental falsification. The results emphasize the pivotal role of technical naturalness and symmetry protection in retaining phenomenological consistency, and establish a template for integrating quantum gravitational corrections into inflationary cosmology.