  The stochastic gravitational wave background (SGWB) provides a unique
opportunity to probe the early Universe, potentially encoding information about
the primordial curvature power spectrum and the energy scale of reheating.
Recent observations by collaborations such as NANOGrav, PPTA, EPTA+InPTA, and
CPTA have detected a stochastic common-spectrum signal, which may originate
from scalar-induced gravitational waves (SIGWs) generated by primordial
curvature perturbations during inflation. In this study, we explore the
hypothesis that the NANOGrav signal is sourced by SIGWs and aim to constrain
the shape of the primordial curvature power spectrum and the reheating energy
scale using the NANOGrav 15-year data set. We model the primordial curvature
power spectrum with a lognormal form and focus on the case where the equation
of state during reheating is $w=1/6$, corresponding to an inflaton potential
$V(\phi) \sim \phi^{14/5}$. Employing Bayesian inference, we obtain posterior
distributions for the lognormal power spectrum parameters and the reheating
temperature. Our results indicate a narrow peak in the primordial power
spectrum ($\Delta < 0.001$ at 90\% confidence) and a lower bound on the
reheating temperature ($T_{\rm rh} \geq 0.1 {\rm GeV}$), consistent with Big
Bang Nucleosynthesis constraints. The best-fit SIGW energy density spectrum
exhibits a distinct turning point around $f \sim 10^{-8.1}\,{\rm Hz}$,
corresponding to the transition from reheating to the radiation-dominated era.
This feature, combined with the sharp high-frequency decrease due to the narrow
primordial power spectrum peak, offers a unique signature for probing early
Universe properties.
