Gravitational wave signatures of non-singular matter bouncing cosmology in NANOGrav and beyond

Non-singular matter bouncing cosmological setups are of particular interest since apart from adressing the initial singularity problem they can give rise as well to a nearly scale-invariant curvature power spectrum on scales $k<10^{4}\mathrm{Mpc}^{-1}$, favored by Cosmic Microwave Background (CMB) experiments. Interestingly enough, one can find that within such non-singular bouncing cosmological setups, curvature perturbations grow on super-horizon scales during the matter contracting phase. In this work, we account for the evolution of cosmological perturbations during the transition to the Hot Big Bang expanding Universe, finding at the end naturally enhanced curvature perturbations on very small scales at horizon-crossing time during the expanding phase. These enhanced cosmological perturbations can induce gravitational waves (GWs) due to second order gravitational interactions and collapse as well to form primordial black holes (PBHs), with the latter acting as one of the most viable dark matter candidates. Remarkably, we find an induced GW background with a universal infrared (IR) frequency scaling of $f^2$, in excellent agreement with the recently released $\mathrm{nHz}$ GW data by the NANOGrav collaboration, being potentially detectable as well by other GW observatories such as LISA and ET, depending on the values of the bouncing cosmological parameters at hand.