  In this work we shall study a possible pre-inflationary scenario for our
Universe and how this might be realized by $f(R)$ gravity. Specifically, we
shall introduce a scenario in which the Universe in the pre-inflationary era
contracts until it reaches a minimum magnitude, and subsequently expands,
slowly entering a slow-roll quasi-de Sitter inflationary era. This
pre-inflationary bounce avoids the cosmic singularity, and for the eras before
and after the quasi-de Sitter inflationary stage, approximately satisfies the
string theory motivated scale factor duality $a(t)=a^{-1}(-t)$. We investigate
which approximate forms of $f(R)$ can realize such a non-singular
pre-inflationary scenario, the quasi-de Sitter patch of which is described by
an $R^2$ gravity, thus the exit from inflation is guaranteed. Furthermore,
since in string theory pre-Big Bang scenarios lead to an overall amplification
of the gravitational wave energy spectrum, we examine in detail this
perspective for the $f(R)$ gravity generating this pre-inflationary
non-singular bounce. As we show, in the $f(R)$ gravity case, the energy
spectrum of the primordial gravitational waves background is also amplified,
however the drawback is that the amplification is too small to be detected by
future high frequency interferometers. Thus we conclude that, as in the case of
single scalar field theories, $f(R)$ gravity cannot produce detectable
stochastic gravitational waves and a synergistic theory of scalars and higher
order curvature terms might be needed.
