Exploring the Equation of State of the Early Universe: Insights from BBN, CMB, and PTA Observations

Strong evidence for a gravitational-wave background (GWB) has been reported in the nano-Hertz band. Interpreting the origin of this background to be scalar-induced gravitational waves (SIGWs), we explore the equation of state (EoS) of the early universe by performing Bayes parameter inferences across the big-bang nucleosynthesis (BBN), cosmic microwave background (CMB), and pulsar timing array (PTA) joint observations for the first time. Assuming a monochromatic power spectrum for primordial curvature perturbations, we obtain the spectral amplitude $A\sim10^{-3}-10^{-1}$ and spectral peak frequency $f_\ast\sim10^{-7}-10^{-6}$ Hz. We find that the radiation domination with EoS $w=1/3$ is compatible with the current observational data, the kination domination with EoS $w=1$ is not forbidden, while the early matter domination with EoS $w=0$ is excluded at more than $2\sigma$ confidence level. These results can be tested with future observations.