This paper presents a comprehensive study of the induced gravitational wave (GW) interpretation of pulsar timing array (PTA) data, considering the unknown equation of state (EoS) of the early universe. The authors perform a Bayesian analysis of NANOGrav data using the PTARCADE code and SIGWFAST for numerical integration of the induced GW spectrum. They focus on two cases: a monochromatic and a log-normal primordial spectrum of fluctuations. For the log-normal spectrum, they find that the results are not very sensitive to the EoS when the GW peak is close to the PTA window, but radiation domination is out of the 2σ contours when only the infra-red power-law tail contributes. For the monochromatic spectrum, the 2σ bounds yield 0.1 ≤ w ≤ 0.9, with radiation domination close to the central value. They also investigate the primordial black hole (PBH) abundance for both spectra, finding that a larger width and stiffer EoS alleviate PBH overproduction. No PBH overproduction requires w ≥ 0.42 up to 2σ for the monochromatic spectrum. Including CMB bounds, they find the PBH mass range is bounded by 10⁻⁵ M☉ ≤ M_PBH ≤ 10⁻¹ M☉. They also find that the PTA signal can explain OGLE microlensing events for 0.42 ≤ w ≤ 0.50. The study shows that induced GWs and PBHs for general w can be analyzed for future data. The paper also discusses the implications of different EoS values on the induced GW spectrum and PBH abundance, and highlights the importance of considering the EoS in the analysis of PTA data. The results suggest that the EoS is constrained to 0.42 ≤ w ≤ 0.87, with a typical PBH mass range of 2.5 × 10⁻⁵ M☉ ≤ M_PBH ≤ 0.09 M☉. The study also notes that the upper bound of O(0.1) M☉ could explain candidate merger events in LVK data. The paper concludes that combining PTA data, CMB constraints, and PBH analysis can significantly narrow down the value of w.This paper presents a comprehensive study of the induced gravitational wave (GW) interpretation of pulsar timing array (PTA) data, considering the unknown equation of state (EoS) of the early universe. The authors perform a Bayesian analysis of NANOGrav data using the PTARCADE code and SIGWFAST for numerical integration of the induced GW spectrum. They focus on two cases: a monochromatic and a log-normal primordial spectrum of fluctuations. For the log-normal spectrum, they find that the results are not very sensitive to the EoS when the GW peak is close to the PTA window, but radiation domination is out of the 2σ contours when only the infra-red power-law tail contributes. For the monochromatic spectrum, the 2σ bounds yield 0.1 ≤ w ≤ 0.9, with radiation domination close to the central value. They also investigate the primordial black hole (PBH) abundance for both spectra, finding that a larger width and stiffer EoS alleviate PBH overproduction. No PBH overproduction requires w ≥ 0.42 up to 2σ for the monochromatic spectrum. Including CMB bounds, they find the PBH mass range is bounded by 10⁻⁵ M☉ ≤ M_PBH ≤ 10⁻¹ M☉. They also find that the PTA signal can explain OGLE microlensing events for 0.42 ≤ w ≤ 0.50. The study shows that induced GWs and PBHs for general w can be analyzed for future data. The paper also discusses the implications of different EoS values on the induced GW spectrum and PBH abundance, and highlights the importance of considering the EoS in the analysis of PTA data. The results suggest that the EoS is constrained to 0.42 ≤ w ≤ 0.87, with a typical PBH mass range of 2.5 × 10⁻⁵ M☉ ≤ M_PBH ≤ 0.09 M☉. The study also notes that the upper bound of O(0.1) M☉ could explain candidate merger events in LVK data. The paper concludes that combining PTA data, CMB constraints, and PBH analysis can significantly narrow down the value of w.