Implications of NANOGrav results and UV freeze-in in a fast-expanding
  Universe
  Recent pulsar timing data reported by the NANOGrav collaboration indicates
the existence of a stochastic gravitational wave (GW) background at a frequency
$f\sim 10^{-8}~\rm Hz$. We show that a dark sector consisting of a Standard
Model (SM) gauge singlet fermion $\chi$ and a singlet scalar $\phi$, both
charged under a $Z_4$ symmetry, is capable of generating such a low frequency
GW via strong first order phase transition (SFOPT) through the modification of
the standard cosmological history, where we assume faster-than-usual expansion
at pre-BBN times driven by a new cosmological species $\varphi$ whose energy
density red-shifts with the scale factor as $\rho_\varphi\propto
a^{-\left(4+n\right)}$. Depending on the choice of the fast expansion
parameters, reheat temperature and effective scale of the theory, it is also
possible to address correct dark matter (DM) relic abundance via freeze-in. We
show that a successful first order phase transition explaining NANOGrav results
together with PLANCK observed DM abundance put bound on the fast expansion
parameters requiring $n\lesssim 4$ to explain both.
