  In this work we present several characteristic examples of theories of
gravity and particle physics scenarios that may yield an observable energy
spectrum of stochastic primordial gravitational waves, compatible with the 2023
NANOGrav observations. The resulting theories yield a flat or a peak-like
energy spectrum, and we further seek the conditions which if hold true, the
energy spectrum can be compatible with the recent NANOGrav stochastic
gravitational wave detection. As we show, in most cases a blue tilted spectrum
combined with a relatively low reheating temperature is needed, the scale of
which is determined by whether the radiation domination era is ordinary or it
is an abnormal radiation domination era. One intriguing Higgs-axion model,
which predicts short slow-roll eras for the axion field at the post-electroweak
breaking epoch, which eventually change the total equation of state parameter
at the reheating era, can explain the NANOGrav signal, if a blue tilted tensor
spectral index inflationary era precedes the reheating era, and a reheating
temperature of the order $\mathcal{O}(400)\,$GeV. This specific model produces
an energy spectrum of primordial gravitational waves with a characteristic peak
that is detectable from both the NANOGrav and future LISA experiment, but not
from the future Einstein telescope.
