  On the frequency-amplitude plane, Gravitational Waves (GWs) from cosmic
strings show a flat plateau at higher frequencies due to the string loop
dynamics in standard radiation dominated post-inflationary epoch. The spectrum
may show an abrupt upward or a downward trend beyond a turning point frequency
$f_*$, if the primordial dark age prior to the Big Bang Nucleosynthesis (BBN),
exhibits non-standard cosmic histories. We argue that such a spectral break
followed by a rising GW amplitude which is a consequence of a post-inflationary
equation of state ($\omega>1/3$) stiffer than the radiation ($\omega=1/3$),
could also be a strong hint of a leptogenesis in the seesaw model of neutrino
masses. Dynamical generation of the right handed (RH) neutrino masses by a
gauged $U(1)$ symmetry breaking leads to the formation of a network of cosmic
strings which emits stochastic GWs. A gravitational interaction of the lepton
current by an operator of the form $\partial_\mu R j^\mu$--which can be
generated in the seesaw model at the two-loop level through RH neutrino
mediation, naturally seeks a stiffer equation of state to efficiently produce
baryon asymmetry proportional to $1-3\omega$. We discuss how GWs with
reasonably strong amplitudes complemented by a neutrino-less double beta decay
signal could probe the onset of the most recent radiation domination and
lightest RH neutrino mass at the intermediate scales.
