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We thoroughly study the induced gravitational wave interpretation of the possible gravitational wave background reported by PTA collaborations, considering the unknown equation of state w of the early universe. We perform a Bayesian analysis of the NANOGrav data using the publicly available PTArcade code together with SIGWfast for the numerical integration of the induced gravitational wave spectrum. We focus on two cases: a monochromatic and a log-normal primordial spectrum of fluctuations. For the log-normal spectrum, we show that, while the results are not very sensitive to w when the GW peak is close to the PTA window, radiation domination is out of the 2 contours when only the infra-red power-law tail contributes. For the monochromatic spectrum, the 2 bounds yield 0. 1 w0. 9 so that radiation domination is close to the central value. We also investigate the primordial black hole (PBH) counterpart using the peak formalism. We show that, in general terms, a larger width and stiffer equation of state alleviates the overproduction of PBHs. No PBH overproduction requires w0. 42 up to 2- level for the monochromatic spectrum. Furthermore, including bounds from the cosmic microwave background, we find in general that the mass range of the PBH counterpart is bounded by 10^-5 M_ M ₁₇10^-1 M_. Lastly, we find that the PTA signal can explain the microlensing events reported by OGLE for 0. 42 w 0. 50. Our work showcases a complete treatment of induced gravitational waves and primordial black holes for general w for future data analysis.
Domènech et al. (Thu,) studied this question.