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 $f_*=1/{\rm yr}$, 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.
