  In the non-standard model of relic gravitational waves (RGWs) generated in
the early universe, the theoretical spectrum of is mainly described by an
amplitude $r$ and a spectral index $\beta$, the latter usually being determined
by the slope of the inflation potential. Pulsar timing arrays (PTAs) data have
imposed constraints on the amplitude of strain spectrum for a power-law form as
a phenomenological model. Applying these constraints to a generic, theoretical
spectrum with $r$ and $\beta$ as independent parameters, we convert the PTAs
constraint into an upper bound on the index $\beta$, which turns out to be less
stringent than those upper bounds from BBN, CMB, and LIGO/VIRGO, respectively.
Moreover, it is found that PTAs constrain the non-standard RGWs more stringent
than the standard RGWs. If the condition of the quantum normalization is
imposed upon a theoretical spectrum of RGWs, $r$ and $\beta$ become related.
With this condition, a minimum requirement of the horizon size during inflation
is greater than the Planck length results in an upper bound on $\beta$, which
is comparable in magnitude to that by PTAs. When both PTAs and the quantum
normalization are applied to a theoretical spectrum of RGWs, constraints can be
obtained for other cosmic processes of the early universe, such as the
reheating, a process less understood observationally so far. The resulting
constraint is consistent with the slow-roll, massive scalar inflation model.
The future SKA will be able to constrain RGWs further and might even detect
RGWs, rendering an important probe to the very early universe.
