Gravity's Rainbow Deformed Starobinsky Model

• The model integrates energy-dependent spacetime deformations with Starobinsky R² inflation, yielding a novel mechanism for UV regularization without ad hoc cutoffs.
• It employs rainbow functions to modify dispersion relations and embeds quantum and RG-induced corrections into f(R) actions, affecting key inflationary dynamics.
• Observable predictions, such as modified spectral indices and tensor-to-scalar ratios, provide practical tests for distinguishing these deformed models in future CMB experiments.

The Gravity’s Rainbow Deformed Starobinsky Model constitutes an overview of two quantum-gravity-motivated extensions of gravitational physics: the energy-dependent metric structure of Gravity’s Rainbow and the higher-curvature $R^2$-based inflationary dynamics of the Starobinsky model. The framework systematically integrates modifications to both the gravitational Lagrangian (including quantum and RG-induced corrections) and the spacetime metric itself (through rainbow functions), with far-reaching implications for ultraviolet (UV) regularization, inflationary predictions, and observable cosmological signatures.

1. Fundamental Structure: Modified Dispersion Relations and Rainbow-Deformed Gravity

The model’s underlying principle is the replacement of the standard (energy-independent) spacetime metric and particle dispersion relations with an explicitly energy-dependent (“rainbow”) structure. In Gravity’s Rainbow the square of the energy-momentum relation is deformed via rainbow functions $g_1(E/E_P)$ and $g_2(E/E_P)$, such that

$E^2 \, g_1^2(E/E_P) - p^2 \, g_2^2(E/E_P) = m^2,$

with $g_1, g_2 \to 1$ as $E/E_P \to 0$ to recover standard relativity in the infrared (Garattini et al., 2011, Garattini et al., 2013). These functions also “rainbow deform” the gravitational metric, so that the lapse, extrinsic curvature, and scalar curvatures pick up explicit energy dependence.

Concrete cosmological and astrophysical models are constructed by embedding these rainbow functions directly into cosmological metrics, for example modifying FLRW as

$$ds^2 = -\frac{dt^2}{\tilde{f}^2(\varepsilon)} + \frac{a^2(t)}{\tilde{g}^2(\varepsilon)} \delta_{ij} dx^i dx^j,$$

with typical parameterizations $\tilde{f}(\varepsilon) \simeq (H/M)^\lambda$ and $\tilde{g} = 1$ during inflation, where $H$ is the Hubble rate, $M$ the characteristic $R^2$ scale, and $\lambda$ the “rainbow parameter” (Chatrabhuti et al., 2015, Channuie, 23 Apr 2024).

2. Regularization of Divergences and Zero-Point Energy

A key outcome is that rainbow deformation acts as an intrinsic UV regulator for divergences arising in gravitational quantum field theory. In the computation of zero-point gravitational energy (ZPE), the Wheeler–DeWitt (WDW) equation is recast as a Sturm-Liouville eigenproblem, with the cosmological constant interpreted as an eigenvalue:

$$\frac{1}{V} \frac{\langle\Psi|\int_{\Sigma} d^3x\, \hat{\Lambda}_\Sigma|\Psi\rangle}{\langle\Psi|\Psi\rangle} = -\frac{\Lambda}{\kappa},$$

where $\hat{\Lambda}_\Sigma$ contains quadratic kinetic and curvature terms depending on the rainbow-altered canonical variables (Garattini et al., 2011, Garattini et al., 2013). The graviton fluctuation spectrum and the density of states are modified by $g_1,g_2$, such that divergent mode sums are suppressed for choices like $g_1/g_2 \sim \exp(-E/E_P)$. The induced cosmological constant becomes finite without ad hoc cutoff prescriptions, aligning the mechanism of inflationary vacuum energy generation with the scale set by rainbow functions rather than arbitrary regularization (Garattini et al., 2011, Garattini et al., 2013).

3. Quantum-Corrected $f(R)$ Actions and Deformation Parameters

The Starobinsky model originally utilizes an $R^2$ term to drive inflation:

$f(R) = R + \frac{R^2}{6M^2}.$

Quantum and RG-improved treatments motivate a more general class of actions, such as:

$$f(R) = R^{2(1-\alpha)} \quad \text{or} \quad f(R) = R + \frac{R^{2(1-\alpha)}}{6\tilde{M}^2}.$$

Here, $\alpha$ quantifies the “marginal” quantum deformation arising from one-loop or matter-field corrections (Codello et al., 2014, Channuie, 2019). More elaborate proposals include RG-resummed effective Lagrangians with running or logarithmic corrections:

$$L_{\rm eff} = \frac{M_p^2 R}{2} + \frac{\alpha}{2}\frac{R^2}{1+\beta \ln(R/\mu^2)}$$

with RG constants $\alpha, \beta$ and reference scale $\mu$ (Channuie, 23 Apr 2024). In string-inspired settings, higher-curvature terms such as the Bel–Robinson tensor squared (Ketov et al., 2022, Toyama et al., 31 Jul 2024) and quartic Grisaru–Zanon terms can additionally deform the potential, with quantum-coupling constants tightly constrained by causality and ghost-absence criteria.

The rainbow deformation’s effect is embedded in both the $f(R)$ background dynamics and the perturbation equations, formally through the presence of $g_1$/$g_2$ or equivalently via the rainbow parameter $\lambda$.

4. Inflationary Dynamics and Observable Predictions

4.1 Background Evolution

In rainbow-deformed Starobinsky inflation, the effective slow-roll dynamics are subtly altered. For a rainbow function of the form $\tilde{f} \propto (H/M)^\lambda$, the background Hubble evolution and the number of e-folds get rescaled (Chatrabhuti et al., 2015, Channuie, 2019):

$$N \simeq \frac{1}{2\epsilon_1},\qquad \epsilon_1 \simeq \frac{\tilde{M}^{2\lambda+2}}{6(\lambda+1)} H^{-2(\lambda+1)}$$

for the deformed quadratic model.

4.2 Scalar and Tensor Perturbations

Perturbation analysis in this context yields the following leading-order expressions for the scalar spectral index $n_s$ and the tensor-to-scalar ratio $r$:

$$n_s - 1 = -\frac{2}{N} - \frac{4\alpha(\lambda+1)}{N}, \qquad r = \frac{12(\lambda+1)^2}{N^2}$$

for the marginally-deformed $f(R)\sim R^{2(1-\alpha)}$ case (Channuie, 2019).

For the standard rainbow-deformed Starobinsky scenario without marginal deformation ($\alpha\to 0$), $n_s-1=-2/N$, as in the classic Starobinsky prediction; however, $r$ is enhanced by $(\lambda+1)^2$ compared to the canonical $r=12/N^2$ (Chatrabhuti et al., 2015, Channuie, 23 Apr 2024).

Further quantum and higher-curvature corrections introduce additional (typically subleading) shifts in $n_s$ and $r$. For instance, in string-inspired scenarios with quartic curvature corrections, the shift in spectral tilt due to a coupling $\gamma$ is of order $\Delta n_s \sim +2.5 \times 10^{-4}$, while $\Delta r \sim -4.9 \times 10^{-5}$ (Toyama et al., 31 Jul 2024).

4.3 Compatibility with Observational Data

Parameters are constrained by Planck and advanced CMB observations. For $N=60$, compatibility with $n_s = 0.965 \pm 0.004$ and $r < 0.036$ imposes upper limits on $\lambda$—typically, $\lambda < 3.4$ for marginally-deformed models with $\alpha\sim 0.01$, and up to $\lambda \lesssim 6.0$ in pure rainbow-deformed ($\alpha\to 0$) models (Chatrabhuti et al., 2015, Channuie, 2019, Channuie, 23 Apr 2024).

Notably, the tensor-to-scalar ratio $r$ is highly sensitive to $\lambda$, providing a distinctive observational signature. Models also admit further tuning via $\alpha$ and $\beta$ in RG-improved Lagrangians. Current and forthcoming data from Planck, BICEP/Keck, ACT, and anticipated Stage IV CMB experiments probe the allowed parameter space, particularly in the sub-$r\lesssim 0.036$ regime (Channuie, 23 Apr 2024, Odintsov et al., 24 Aug 2025).

5. Quantum Effects, UV Completion, and Gravitational Baryogenesis

Quantum-induced corrections, both in the Lagrangian (via RG, superstring, or effective field theory inputs) and in the rainbow-deformed metric, act as mechanisms for UV completion. The deformed models provide a natural regularization of the ZPE, eliminating the need for arbitrary cutoffs and enabling a finite induced cosmological constant without conventional renormalization (Garattini et al., 2011, Garattini et al., 2013).

Gravity’s Rainbow-deformed f(R) gravity also allows for alternative baryogenesis scenarios. The baryon-number asymmetry is generated via CPT-violating couplings of $(\partial_\mu R) J^\mu$, with the baryon-to-entropy ratio $Y_B$ acutely dependent on the time-derivative of the Ricci scalar, which itself is deformed by the rainbow function (e.g., $R \sim t^{-2(\alpha+1)}$) (Goodarzi, 23 Aug 2025). The strength and timing of baryogenesis are thus controlled by the rainbow parameter $\alpha$, the f(R) power index, and the dynamics of the scale factor.

6. Astrophysical and Thermodynamic Signatures

Beyond cosmic inflation and baryogenesis, rainbow-deformed Starobinsky models have been applied to non-static, collapsing spacetimes and black hole physics (Rudra, 2019). The alteration of the causal structure in gravitational collapse is found to increase the likelihood of naked singularity formation, challenging cosmic censorship. Thermodynamic properties (event horizon, temperature, entropy, and specific heat) are also directly affected by rainbow deformations, leading to quantum-gravitational deviations from classical results.

7. Future Directions: Observational Discriminants and Quantum Gravity Links

Ongoing refinement of rainbow-deformed $R^2$ inflation models includes embedding RG-resummed Lagrangians to systematically encode asymptotically safe quantum gravity (Channuie, 23 Apr 2024). Superstring-inspired higher-curvature corrections (e.g., Bel–Robinson tensor squared, Grisaru–Zanon quartics) are constrained by ghost-avoidance and unitarity bounds on their coupling constants, yielding quantum gravity corrections that are comparable in size to higher-order classical “$1/N^3$” corrections in CMB observables (Ketov et al., 2022, Toyama et al., 31 Jul 2024).

The “kination” and post-inflationary expansion histories are shown to interplay nontrivially with rainbow-induced slow-roll modifications, affecting observable predictions. Power-law f(R) deformations around the Starobinsky point ($n \simeq 2$) serve as additional tuning possibilities for model compatibility with high-precision ACT and Planck datasets (Odintsov et al., 24 Aug 2025).

A plausible implication is that the combined analysis of $n_s$, $r$, non-Gaussianity, and baryogenesis outcomes provides a testbed for discriminating quantum gravity imprints on inflationary cosmology. The ultimate test of the Gravity’s Rainbow Deformed Starobinsky scenario lies in measurable deviations from Starobinsky’s predictions at high-precision CMB experiments and in the discovery or exclusion of quantum gravity–modulated features in the primordial universe.