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.
