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: $$f(R)=R+\alpha (R/M)^{n},\,f(R)=R+\alpha R^{2}+\beta R^{2}\log (R/M^{2})$$ and the Einstein–Hu–Sawicki model with $$n,\,\alpha ,\,\beta $$ 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.