  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.
