Probing the shape of the primordial curvature power spectrum and the energy scale of reheating with pulsar timing arrays

Recent observations by pulsar timing array collaborations have detected a stochastic common-spectrum signal, which may originate from scalar-induced gravitational waves generated by primordial curvature perturbations during inflation. Using the NANOGrav 15-year data set, we explore this hypothesis by constraining the primordial curvature power spectrum and reheating energy scale. We model the primordial power spectrum with a lognormal form and consider reheating with the equation of state parameter w = 1/6. Our Bayesian analysis reveals a narrow peak in the primordial power spectrum (the width of the spectrum Δ < 0.05 at the 95% confidence level) and constrains the reheating temperature to be 0.03 GeV ≲ T_rh ≲ 7.2 GeV. The best-fit SIGW spectrum shows a characteristic turning point near f ∼ 10^−8.1 Hz, marking the transition from reheating to radiation domination, providing a unique probe of the properties of the early Universe.