Observational constraints on asymptotic safety inflation in gravity's 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 (Demmel et al., 2015). In the context of gravity as an Asymptotically Safe (AS) theory, authors of Refs. Liu et al. (2018), Koshelev et al. (2023) proposed a refined Starobinsky model, LAS=(MpR)-R-2/2+(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 ns 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 ns 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.