Probing the equation of state of the early Universe with pulsar timing arrays

Abstract: The recently released data by pulsar timing array (PTA) collaborations have amassed substantial evidence corroborating the existence of a stochastic signal consistent with a gravitational-wave background at frequencies around the nanohertz regime. We investigate the situation in which the PTA signal originates from scalar-induced gravitational waves~(SIGWs), which serves as a valuable tool to probe the equation of state parameter $w$ during the Universe's early stages. The joint consideration of the PTA data from the NANOGrav 15-year data set, PPTA DR3, and EPTA DR2 yields that $w=0.60^{+0.32}_{-0.39}$ (median + $90\%$ credible interval), indicating a period of condensate domination at the production of SIGWs is allowed by the data. Moreover, the data also supports radiation domination ($w=1/3$) within the $90\%$ credible interval. We also impose an upper bound on the reheating temperature that $T_\mathrm{rh} \lesssim 0.2\,\mathrm{GeV}$ and the constraint on $w$ reveals valuable information on the inflationary potential and the dynamics at the end of inflation.