Inflation from f(R) theories in gravity\u2019s rainbow

Areef Waeming,Phongpichit Channuie

doi:10.1140/epjc/s10052-020-8387-7

Abstract

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.

Highlights

It is expected that the usual dispersion relation will get modified at the energy scale of the order of Planck length in various theories of quantum gravity
It is expected that the usual dispersion relation in the UV limit has to be in principle reformed and captures a modification of the geometry at that limit
The rainbow function is written in the powerlaw form of the Hubble parameter

Summary

Model III

We consider the Hu–Sawicki model in which the f (R) function is of the form n (4.34). We assume that H ≈ H = constant and define a new parameter ≡ (H /M) which is plausible during inflation. For N = 70, we discover that the predictions are consistent with the Planck’ results for TT, 802 Page 18 of 20. +lowE+lensing and with the Planck’ results for TT, TE, EE, +lowE+lensing+BK15+BAO at two sigma confidence level for = 10(100), c2 = 10−11(10−15) only when λ ≤. Using an upper bound on the Hubble parameter during inflation reported by Planck 2018 [58] allows us to determine δ = M/mP: H= mP δ < 2.5 × 10−5 (95% C.L .) → δ < 2.5 × 10−5.

Perturbation equations

Tensor perturbation

Confrontation with the Planck 2018 data

Concluding remarks