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Inflation from $f(R)$ theories in gravity's rainbow

In this work, we study the $f(R)$ models of inflation in the context of gravity's rainbow theory. We choose three types of $f(R)$ models: $f(R)=R+α(R/M)^{n},\,f(R)=R+αR^{2}+βR^{2}\log(R/M^{2})$ and the Einstein-Hu-Sawicki model with $n,\,α,\,β$ being arbitrary real constants. Here $R$ and $M$ are the Ricci scalar and mass scale, respectively. For all models, the rainbow function is written in the power-law form of the Hubble parameter. We present a detailed derivation of the spectral index of curvature perturbation and the tensor-to-scalar ratio and compare the predictions of our results with the latest Planck 2018 data. With the sizeable number of e-foldings and proper choices of parameters, we discover that the predictions of all $f(R)$ models present in this work are in excellent agreement with the Planck analysis.