Eur. Phys. J. C (2024) 84: 766
https://doi.org/10.1140/epjc/s10052-024-13155-0

Regular Article - Theoretical Physics

Gravity’s rainbow effects on higher curvature modification of $R^2$ inflation

¹ Faculty of Science and Technology, Nakhon Si Thammarat Rajabhat University, 80280, Nakhon Si Thammarat, Thailand
² School of Science, Walailak University, 80160, Nakhon Si Thammarat, Thailand
³ College of Graduate Studies, Walailak University, 80160, Nakhon Si Thammarat, Thailand
⁴ Northern Virginia Community College, 8333 Little River Turnpike, 22003, Annandale, VA, USA
⁵ Centre for Space Research, North-West University, 2520, Potchefstroom, South Africa

^b phongpichit.ch@mail.wu.ac.th

Received: 23 May 2024
Accepted: 21 July 2024
Published online: 1 August 2024

Abstract

In this work, we study several extensions of the higher curvature modification of $R^2$ inflation in the context of gravity’s rainbow. We modify the $(R+R^2)$ model by adding an $f_1{R}^3$-term, an $f_2{R}^4$-term, and an $f_3{R}^{3/2}$-term to the original model. We calculate the inflationary observables and confront them using the latest observational bounds from Planck 2018 data. We assume the rainbow function of the form $\tilde{f}=1+{\left(\frac{H}{M}\right)}^{\lambda }$ with $\lambda$ being a rainbow parameter and M a mass-dimensional parameter. We demonstrate that the power spectrum of curvature perturbation relies on the dimensionless coefficient $f_i,i=1,2,3$, a rainbow parameter $\lambda$ and a ratio H/M. Likewise, the scalar spectral index $n_s$ is affected by both $f_i$ and the rainbow parameter. Moreover, the tensor-to-scalar ratio r is solely determined by the rainbow parameter. Interestingly, by ensuring that $n_s$ aligns with the Planck collaboration’s findings at the $1\sigma$ confidence level, the tensor-to-scalar ratio could reach up to $r\sim 0.01$, which is possibly measurable for detection in forthcoming Stage IV CMB ground experiments and is certainly feasible for future dedicated space missions.