Obviating PBH overproduction for SIGWs generated by Pulsar Timing Arrays in loop corrected EFT of bounce

In order to unravel the present situation of the PBH overproduction problem, our study emphasizes the critical role played by the equation of state (EoS) parameter $w$ within the framework of effective field theory (EFT) of non-singular bounce. Our analysis focuses on a wide range of EoS parameter values that are still optimal for explaining the latest data from the pulsar timing array (PTA). As a result of our study, the most advantageous window, $0.31 \leq w \leq 1/3$, is identified as the location of a substantial PBH abundance, $f_{\rm PBH} \in (10^{-3},1)$ with large mass PBHs, $M_{\rm PBH}\sim {\cal O}(10^{-7}-10^{-3})M_{\odot}$, in the SIGW interpretation of the PTA signal. When confronted with PTA, we find that the overproduction avoiding circumstances are between $1\sigma-2\sigma$, while the EoS parameter lies inside the narrow window, $0.31<w\leq 1/3$. We propose a regularized-renormalized-resummed (RRR) scalar power spectrum that is large enough to produce EoS dependent scalar generated gravitational waves compatible with PTA evidence, while satisfying the perturbativity, causality, and unitarity criteria, within the range of $0.88 \leq c_{s} \leq 1$.