Constraints on primordial curvature power spectrum with kination era: Insights from NANOGrav 15-year data set

The stochastic signal detected by the NANOGrav, PPTA, EPTA, and CPTA collaborations can be attributed to gravitational waves induced by the primordial curvature perturbations generated during inflation. These scalar-induced gravitational waves provide valuable insights into the small-scale inflation and reheating epochs. In this paper, we assume an equation of state of $w=1$ during reheating and adopt a log-normal form for the primordial curvature power spectrum to elucidate the observed stochastic signal. The inflation and reheating scenarios are rigorously constrained utilizing Bayesian methods applied to the NANOGrav 15-year data set. The analysis yields constraints on the reheating temperature, indicating $T_{\mathrm{rh}}\gtrsim 0.01\mathrm{Gev}$, a result consistent with constraints derived from Big Bang nucleosynthesis. Furthermore, the NANOGrav 15-year data set necessitates the primordial curvature power spectrum's amplitude and width to satisfy $A\sim 0.01$ and $\Delta \lesssim 0.1$, respectively. Due to the change in the equation of state w, there exists a turning point in the energy density spectrum of scalar-induced gravitational waves. This suggests that if more data about the scalar-induced gravitational waves is observed, it could potentially provide constraints on the time when the reheating epoch transitions to radiation domination.