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

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 $f_{\ast }=1\mathrm{/}\mathrm{y}\mathrm{r}$, which provides the opportunity to constrain $H_{\ast }$, 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 $w=1\mathrm{/}3$, we find that the current PPTA result follows a bound $H_\leq 1.15\times10^{-1}\mpl$,andtheEPTAresultfollows$H_\leq 6.92\times10^{-2}\mpl$. We also find that these bounds become much tighter in the nonstandard scenarios with $w>1\mathrm{/}3$. When $w=1$, the bounds become $H_{\ast }\le 5.89\times {10}^{-3}\text{mpl}$ for the current PPTA and $H_{\ast }\le 3.39\times {10}^{-3}\text{mpl}$ for the current EPTA. In contrast, in the nonstandard scenario with $w=0$, the bound becomes $H_{\ast }\le 7.76\text{mpl}$ for the current PPTA.