Constraint on the early Universe by relic gravitational waves: From pulsar timing observations

Abstract: Recent pulsar timing observations by the Parkers Pulsar Timing Array and European Pulsar Timing Array teams obtained the constraint on the relic gravitational waves at the frequency $f_=1/{\rm yr}$, which provides the opportunity to constrain $H_$, the Hubble parameter when these waves crossed the horizon during inflation. In this paper, we investigate this constraint by considering the general scenario for the early Universe: we assume that the effective (average) equation-of-state $w$ before the big bang nucleosynthesis stage is a free parameter. In the standard hot big-bang scenario with $w=1/3$, we find that the current PPTA result follows a bound $H_\leq 1.15\times10^{-1}\mpl$, and the EPTA result follows $H_\leq 6.92\times10^{-2}\mpl$. We also find that these bounds become much tighter in the nonstandard scenarios with $w>1/3$. When $w=1$, the bounds become $H_\leq5.89\times10^{-3}\mpl$ for the current PPTA and $H_\leq3.39\times10^{-3}\mpl$ for the current EPTA. In contrast, in the nonstandard scenario with $w=0$, the bound becomes $H_*\leq7.76\mpl$ for the current PPTA.