Observational Constraints on Asymptotic Safety Inflation in Gravity Rainbow

Using suitable Renormalization Group (RG) based re-summation of quantum corrections to $R^2$ term, a re-summed version of the effective Lagrangian can be obtained \cite{Demmel:2015oqa}. In the context of gravity as an Asymptotically Safe (AS) theory, authors of Refs.\cite{Liu:2018hno,Koshelev:2022olc} proposed a refined Starobinsky model, $L_{\rm AS} = M^{2}_{p} R/2+(\alpha/2)R^{2}/[1+\beta \ln(R/\mu^{2})]$, where $R$ is the Ricci scalar, $\alpha$ and $\beta$ are constants and $\mu$ is an energy scale. In the present work, we embed this underlying effective Lagrangian within the framework of gravity's rainbow. By implementing the COBE normalization and the Planck constraint on the scalar spectrum, we demonstrate that the power spectrum of curvature perturbation relies on $\alpha$ and $\beta$, as well as on a rainbow parameter. Similarly, the scalar spectral index $n_s$ is influenced by $\beta$ and the rainbow parameter, yet remains unaffected by $\alpha$. Additionally, the tensor-to-scalar ratio $r$ solely depends on the rainbow parameter. Remarkably, when requiring $n_s$ to be consistent with the Planck collaboration at $1\sigma$ confidence level, the upper limit on the tensor-to-scalar ratio $r < 0.036$ can be naturally satisfied. This value potentially holds promise for potential measurement by Stage IV CMB ground experiments and is certainly within reach of future dedicated space missions.