  Recently, several major pulsar timing array (PTA) collaborations have
assembled strong evidence for the existence of a gravitational-wave background
at frequencies around the nanohertz regime. Assuming that the PTA signal is
attributed to scalar-induced gravitational waves, we jointly employ the PTA
data from the NANOGrav 15-year data set, PPTA DR3, and EPTA DR2 to probe the
conditions of the early Universe. Specifically, we explore the equation of
state parameter ($w$), the reheating temperature ($T_\mathrm{rh}$), and the
sound speed ($c_s$), finding $w = 0.59^{+0.36}_{-0.40}$ (median + $90\%$
credible interval), and $T_\mathrm{rh}\lesssim 0.2\,\mathrm{GeV}$ at the $95\%$
credible interval for a lognormal power spectrum of the curvature perturbation.
Furthermore, we compute Bayes factors to compare different models against the
power-law spectrum model, effectively excluding the pressure-less fluid
domination model. Our study underscores the significance of scalar-induced
gravitational waves as a powerful tool to explore the nature of the early
Universe.
