Quantitative Predictions for $f(R)$ Gravity Primordial Gravitational Waves

In this work we shall develop a quantitative approach for extracting predictions on the primordial gravitational waves energy spectrum for $f(R)$ gravity. We shall consider two distinct models which yield different phenomenology, one pure $f(R)$ gravity model and one Chern-Simons corrected potential-less $k$-essence $f(R)$ gravity model in the presence of radiation and non-relativistic perfect matter fluids. The two $f(R)$ gravity models were carefully chosen in order for them to describe in a unified way inflation and the dark energy era, in both cases viable and compatible with the latest Planck data. Also both models mimic the $\Lambda$-Cold-Dark-Matter model and specifically the pure $f(R)$ model only at late times, but the Chern-Simons $k$-essence model during the whole evolution of the model up to the radiation domination era. In addition they guarantee a smooth transition from the inflationary era to the radiation, matter domination and subsequently to the dark energy era. Using a WKB approach introduced in the relevant literature by Nishizawa, we derive formulas depending on the redshift that yield the modified gravity effect, quantified by a multiplicative factor, a ``damping'' in front of the General Relativistic waveform. In order to calculate the effect of the modified gravity, which is the ``damping'' factor, we solve numerically the Friedmann equations using appropriate initial conditions and by introducing specific statefinder quantities. As we show, the pure $f(R)$ gravity gravitational wave energy spectrum is slightly enhanced, but it remains well below the sensitivity curves of future gravitational waves experiments. In contrast, the Chern-Simons $k$-essence $f(R)$ gravity model gravitational wave energy spectrum is significantly enhanced and two signals are predicted which can be verified by future gravitational wave experiments.