Spectrum of Primordial Gravitational Waves in Modified Gravities: A Short Overview

In this work we shall exhaustively study the effects of modified gravity on the energy spectrum of the primordial gravitational waves background. S. Weinberg has also produced significant works related to the primordial gravitational waves with the most important one being the effects of neutrinos on primordial gravitational waves. With this sort review, our main aim is to gather all the necessary information for studying the effects of modified gravity on primordial gravitational waves in a concrete and quantitative way and in a single paper. After reviewing all the necessary techniques for extracting the general relativistic energy spectrum, and how to obtain in a WKB way the modified gravity damping or amplifying factor, we concentrate on specific forms of modified gravity of interest. The most important parameter involved for the calculation of the effects of modified gravity on the energy spectrum is the parameter $a_M$ which we calculate for the cases of $f(R,\phi)$ gravity, Chern-Simons-corrected $f(R,\phi)$ gravity, Einstein-Gauss-Bonnet-corrected $f(R,\phi)$ gravity, and higher derivative extended Einstein-Gauss-Bonnet-corrected $f(R,\phi)$ gravity. The exact forms of $a_M$ is presented explicitly for the first time in the literature. With regard to Einstein-Gauss-Bonnet-corrected $f(R,\phi)$ gravity, and higher derivative extended Einstein-Gauss-Bonnet-corrected $f(R,\phi)$ gravity theories, we focus on the case that the gravitational wave propagating speed is equal to that of light's in vacuum. We provide expressions for $a_M$ expressed in terms of the cosmic time and in terms of the redshift, which can be used directly for the numerical calculation of the effect of modified gravity on the primordial gravitational wave energy spectrum.