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Pre-Inflationary Bounce Effects on Primordial Gravitational Waves of $f(R)$ Gravity

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.