Quantum Quadratic Gravity Inflation

• Quantum Quadratic Gravity Inflation is a framework defined by quadratic curvature invariants and quantum corrections that enable UV completion and robust slow-roll dynamics.
• The models incorporate higher-derivative operators, renormalization group improvements, and loop corrections to yield inflationary observables consistent with CMB data.
• Extensions such as non-local form factors, dynamical torsion, and multi-field dynamics produce unique predictions like parity violations and modified tensor-to-scalar ratios.

Quantum Quadratic Gravity Inflation refers to a family of inflationary models in which the early Universe’s dynamics are governed by gravity actions containing quadratic curvature invariants—such as $R^2$ and $$C^{2} \equiv C_{\mu\nu\rho\sigma} C^{\mu\nu\rho\sigma}$$—with quantum corrections taken into account. These frameworks generalize the Starobinsky paradigm by systematically including higher-derivative operators, quantum loop corrections, renormalization group (RG) running, and, in some cases, dynamical fields arising from gauge-theoretic or torsion extensions of gravity. Recent advances provide self-consistent UV completions, nonperturbative control of inflationary observables, and connect Planck-scale quantum gravity to CMB-accessible cosmological signatures.

1. Effective Quadratic Gravity Frameworks and Ultraviolet Completion

Quantum quadratic gravity (QQG) is fundamentally defined by an action of the schematic form

$$S_{\rm QQG} = \int d^4 x\, \sqrt{-g} \left( -\frac{1}{\xi} R^2 - \frac{1}{2\lambda} C^2 + \ldots \right),$$

where $R$ is the Ricci scalar, $C$ the Weyl tensor, $\xi$ and $\lambda$ are dimensionless couplings, and ellipses denote possible matter or higher-order terms. Notably, such pure quadratic gravity actions are perturbatively renormalizable and, when analyzed at one-loop, display asymptotic freedom in the ultraviolet (UV) (Liu et al., 21 Oct 2025). In the extreme UV limit close to the big bang (divergent curvature), gravity is fully described by the quadratic terms; as the Universe expands and cools, quantum corrections (reflected by the running of $\xi$ and $\lambda$) dynamically generate a slow-roll inflationary regime.

When the RG running is computed—including contributions from a large number $\mathcal{N}$ of matter fields—the one-loop $\beta$-functions take the form: $$\beta_\xi = \frac{d\xi}{d\ln\mu} = -\frac{1}{(4\pi)^2} \frac{\xi^2 - 36\lambda \xi - 2520\lambda^2}{36},$$

$$\beta_\lambda = \frac{d\lambda}{d\ln\mu} = -\frac{1}{(4\pi)^2} \frac{(1617 + 90\mathcal{N})\lambda - 20\xi}{90} \lambda,$$

where $\mu$ is an RG scale (often chosen to track the curvature), and $\mathcal{N}$ is large to ensure perturbative control. The RG-improved effective $f(R)$ theory is then $f(R) \sim R^2/\xi(R)$, with logarithmic corrections from the RG flow breaking the classical scaling symmetry and allowing for quasi-de Sitter evolution and reheating (Liu et al., 21 Oct 2025, Rinaldi, 2015). In the deep IR, strong coupling is encountered and GR emerges as an effective description; this is paralleled by the dynamical transition to the familiar post-inflationary hot big bang cosmology.

2. Inflationary Dynamics, Potentials, and Observational Predictions

In QQG inflation, slow-roll dynamics typically arise from the RG-improved potential after mapping to the Einstein frame. For instance, using RG-improvement with a large number of matter fields, the potential for the canonical inflaton field $\varphi$ can be approximated as

$$V(\varphi) \simeq \frac{35 \lambda_0^2 \mu_0^4}{128\pi^2} \left[ 1 - \left( \frac{6\mu_0}{\varphi} \right) \right],$$

with $\lambda_0$, $\mu_0$ constants from the RG flow (Liu et al., 21 Oct 2025). The resulting slow-roll observables for $N$ e-folds before the end of inflation are

$$n_s \simeq 1 - \frac{4}{3N}, \quad r \simeq \frac{8}{3} \left[ \frac{2}{\lambda_{tH}^2 N^4} \right]^{1/3},$$

where $$\lambda_{tH} \equiv (\lambda_0 \mathcal{N})/(4\pi)^2$$ is the effective 't Hooft coupling (Liu et al., 21 Oct 2025). Crucially, generic QQG models predict a minimum tensor-to-scalar ratio $r \gtrsim 0.01$ for consistency with the perturbative regime and correct transition to reheating. For suitable parameter choices and $\mathcal{N} \sim 10^5$–$10^6$, the $(n_s, r)$ predictions fall within currently allowed observational bounds, while the allowed $r$ region is often distinct from the classic Starobinsky plateau $r \sim 0.003$.

RG flows in QQG have two principal UV completions: (i) asymptotically free flows, where quadratic couplings vanish at high energy, and (ii) asymptotically safe trajectories, where couplings approach finite non-Gaussian fixed-point values (Hoshina, 2022). Both types can accommodate Starobinsky-like inflation at $k_{\rm inf} \sim 10^{22}$ eV, but differ in their extreme UV/Planckian behavior, which potentially induces further differences in high-frequency gravitational phenomena or in the spectrum of primordial fluctuations.

3. Quantum Corrections, Renormalization, and Loop Effects

Quantum gravity corrections in QQG are realized at several levels:

• Perturbative one-loop corrections: These generate Coleman–Weinberg-type potentials, as for the case where the massive spin-2 ghost mode from a $C^2$ term produces the dominant radiative effect, enabling spontaneous breaking of scale invariance and Planck mass generation via one-loop effective potentials

$$U_{\text{eff}}(S) = U_0 + [C_1 + C_2 \ln(S^2/\mu^2)] S^4,$$

with the vacuum expectation value $v_S = \mu \exp[-1/4 - (C_1/2C_2)]$ (Kubo et al., 2022). The resulting radiative symmetry breaking triggers inflation via the gravitational sector itself, without the need for additional scalar fields.

• Nonperturbative RG improvement: Full integration of the quadratic gravity beta functions, with boundary conditions fixed at the Hubble, laboratory, and inflationary scales, reveals a landscape of UV RG trajectories consistent with present-day measurements and Starobinsky inflation (Hoshina, 2022). The running couplings influence early-universe observables and ensure theoretical control over the entire dynamical range.
• Softly broken scale-invariance: Loop resummation and scale-symmetry breaking yield modified $f(R)$ forms, e.g., $$f_{\rm eff}(R) = R^2 / [1 + \gamma \ln(R^2 / \mu^2)]$$, producing attractor predictions $r = (8/3)(1-n_s)$ distinct from $\alpha$-attractor and Starobinsky models (Rinaldi, 2015).
• Non-local form factors: In analytic infinite-derivative gravity, terms such as $R\,\mathcal{F}_R(\Box/M_s^2)\,R$ or $$W_{\mu\nu\rho\sigma}\,\mathcal{F}_W(\Box/M_s^2)\,W^{\mu\nu\rho\sigma}$$ appear, shifting tensor spectra, modifying the consistency relation $r = -8 n_t$, and enabling $r < 0.07$ for appropriate non-locality scales (Koshelev et al., 2020, Koshelev et al., 2017).

4. Extensions: Dynamical Torsion, Multi-Field Dynamics, and Quantum Geometry

Generalizations of quantum quadratic gravity inflation include:

• Dynamical torsion and parity violation: In a quadratic gravity theory with torsion degrees of freedom, the spectrum contains both massless and massive non-ghost states: two spin-0 (0$^\pm$), one spin-1$^+$, and one spin-2$^+$. The minisuperspace action, following a reduction to a flat FLRW background, leads to multi-field inflation on a hyperbolic field-space. The typical potential has a Starobinsky-type $V(\chi)$ plus a periodic ($\theta$-axion-like) sector. Notably, the system generically evolves toward the Starobinsky attractor, even with initial conditions away from the potential summit. Tensor perturbations show mixing between massless and massive spin-2 modes, speed modifications, and parity-violating signatures, which could be probed via B-mode polarization (Aoki et al., 2020).
• Covariant open-system approaches and nonperturbative quantization: Covariant open-system gravity, as formulated in terms of a quantum-corrected metric $\bar{g}_{\alpha\beta}$ and associated geometric field $\hat{\Omega}$, allows for nonperturbative, simultaneous quantization of both inflaton and geometric fluctuations. The normalization of matter field modes becomes directly tied to the quantized spacetime modes, and quadratic fluctuations in geometry yield precise power spectra for geometric perturbations (Morales et al., 23 Sep 2025).
• Multi-field inflation and RG guidance: For models with double inflation (coupling $R + R^2 + C^2$ gravity to a quadratic scalar), cosmic RG invariance can be invoked to identify the correct adiabatic/isocurvature combinations that are RG-invariant at superhorizon scales, with the leading-order predictions for cosmological observables remaining as in pure single-field quantum gravity inflation (Anselmi, 2021).

5. Canonical Quantization and Ghost Sectors

Quadratic gravity models generically introduce higher-derivative data and thereby extra degrees of freedom, notably a massive spin-2 state with a ghost signature (negative kinetic energy). The analysis of primordial gravitational wave spectra in these models must carefully treat the ghost sector, particularly in canonical quantization:

• By introducing an auxiliary tensor field, the theory is reformulated as a second-order system: the massless graviton and its massive spin-2 companion decouple at quadratic order. Canonical quantization, with commutation relations carefully imposed, allows for the computation of the tensor power spectrum. Both amplitude and tilt are suppressed by a universal factor $(1+2H_*^2/m_{\mathrm{gh}}^2)^{-1}$, where $m_{\mathrm{gh}}$ is the ghost mass. In the limit $H_*^2/m_{\mathrm{gh}}^2 \ll 1$, the system reduces to standard inflationary predictions, with the classic slow-roll relation $r = -8n_t$ restored. The ghost is rendered virtual and does not propagate as a physical degree of freedom (Kubo et al., 5 Feb 2025).
• When the ghost mass is near the Hubble scale during inflation, observable modifications to the primordial gravitational wave spectrum may appear (e.g., a further $\sim$11% suppression of $r$ beyond Starobinsky), but for higher masses or safe parameter ranges, unitarity and predictivity are maintained (Kubo et al., 2022, Kubo et al., 5 Feb 2025).

6. Comparison to Starobinsky and Other Inflationary Models

QQG inflation provides both extensions to and, in some regimes, alternatives to the standard Starobinsky $R+R^2$ scenario. Distinguishing features include:

• Modified phenomenology: With loop corrections and RG-improvement, the slow-roll observables $(n_s, r)$ depart from the Starobinsky attractor. For instance, $n_s \simeq 1 - 4/(3N)$ and $r \gtrsim 0.01$ set by the UV completion and matter content, compared to $n_s \simeq 1 - 2/N$ and $r \simeq 12/N^2$ for Starobinsky (Liu et al., 21 Oct 2025).
• Boundaries for inflation model-building: New CMB and BAO data suggest mild tension ($>2\sigma$) with pure Starobinsky predictions; consistent QQG models avoid this, particularly for large effective matter content $\mathcal{N}$. Moreover, certain non-local extensions and softly broken scale-invariant models predict a universal relation $r = (8/3)(1-n_s)$ that generically sits above the Starobinsky plateau (Rinaldi, 2015, Koshelev et al., 2017).
• Unifying inflation with late-time acceleration: Quantum quadratic gravity models naturally accommodate late-time cosmic acceleration via exponential $F(R)$ terms and renormalization-group-improved couplings, with oscillations stabilized by “curing” terms that tame singularities in the dark energy sector (Elizalde et al., 2017).

7. Observational Tests and Future Directions

Quantum quadratic gravity inflation frameworks yield predictions that are sharply testable with CMB B-mode observations (e.g., tensor-to-scalar ratio $r$ and its scale dependence), measurements of non-Gaussianity (e.g., $f_{\mathrm{NL}}$ in squeezed configurations), and potentially parity-violating signals or modified speed of gravitational waves for models with torsion. Upcoming and next-generation CMB experiments such as LiteBIRD, the Simons Observatory, and CMB-S4 will further probe the viability of these models. Additionally, their consistent UV completion and robust predictions connect early Universe cosmology to foundational questions in quantum gravity, extending inflationary tests to the Planck scale and beyond.

A summary of representative features across key settings is provided below:

Aspect	Main QQG Mechanism	Observable Consequence
RG-improved quadratic gravity	One-loop running of $R^2$, $C^2$ terms	$n_s \simeq 1 - 4/(3N)$, $r \gtrsim 0.01$ and transition to GR at reheating
Non-local modifications	AID form factors (e.g., $e^{\gamma \Box}$)	Modified $r, n_t$, altered inflationary consistency relation
Dynamical torsion extensions	Massive spin-0/1/2 fields in Cartan geometry	Multi-field inflation, parity-violating GW, attractor behavior in field space
Scale-invariant scenarios	Loop/resummation, softly broken SI	Universal relation $r = (8/3)(1-n_s)$, distinct from $\alpha$- or Starobinsky attractors
Canonical quantization & ghosts	Auxiliary tensor field, virtual ghost	Tensor spectrum suppressed by $(1+2H_*^2/m_\text{gh}^2)^{-1}$, $r = -8n_t$

Quantum quadratic gravity inflation thus constitutes a comprehensive, theoretically robust, and observationally distinguishable framework for primordial cosmology, allowing for the systematic probing of quantum gravitational corrections via the fossil light from the early Universe.