  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
\textsc{PTArcade} code together with \textsc{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 $2\sigma$ contours when only the infra-red power-law tail contributes. For
the monochromatic spectrum, the $2\sigma$ bounds yield $0.1\lesssim
w\lesssim0.9$ 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
$w\gtrsim0.42$ up to 2-$\sigma$ 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 $10^{-5}
M_\odot\lesssim M_{\rm PBH}\lesssim10^{-1} M_\odot$. Lastly, we find that the
PTA signal can explain the microlensing events reported by OGLE for
$0.42\lesssim w\lesssim 0.50$. Our work showcases a complete treatment of
induced gravitational waves and primordial black holes for general $w$ for
future data analysis.
