Implications of a Scalar Field Interacting with the Dark Matter Fluid on the Primordial Gravitational Waves

We consider the implications of a scalar field interacting with the dark matter fluid on the energy spectrum of the primordial gravitational waves. We choose an interaction type which before a critical matter density $\rho_m^c$ during the reheating era or early radiation domination era, the scalar field loses energy transferring it to the dark matter fluid, while after the critical matter density $\rho_m^c$ the dark matter fluid loses energy transferring it to the scalar field. The scalar field is assumed to have an exponential potential and at the critical matter density with $\rho_m\sim \rho_m^c$, at which point, the interaction between the scalar and the dark matter fluid is switched off, we demand that the effective equation of state of scalar field is described by a matter dominated era. This is crucial since it affects the behavior of the trajectories and the fixed points of the two dimensional dynamical system composed by the dark matter fluid and the scalar field. Specifically the phase space contains two stable dark matter dominated final attractors and two unstable stiff era dominated fixed points. Thus, there exists the remarkable possibility that the Universe might feel the passing of the scalar field through the unstable kinetic dominated fixed points, during the reheating era, with the total equation of state parameter of the Universe being deformed to be larger than $w=1/3$. This deformation of the total equation of state parameter during the reheating era can potentially have significant effects on the energy spectrum of the primordial gravitational waves. Also the model we use also contains an $F(R)$ gravity which controls in a dominant way inflation and the late-time acceleration, in a phenomenologically viable way.