Nanohertz gravitational waves from NEC violation in the early universe
  We study nanohertz gravitational waves relevant to pulsar timing array
experiments from quantum fluctuations in the early universe with null energy
condition (NEC) violation. The NEC violation admits accelerated expansion with
the scale factor $a\propto (-t)^{-p}$ ($p>0$), which gives the tensor spectral
index $n_t=2/(p+1)>0$. To evade the constraint from Big Bang nucleosynthesis
(BBN), we connect the NEC-violating phase to a subsequent short slow-roll
inflationary phase which ends with standard reheating, and thereby reduce the
high frequency part of the spectrum. An explicit model is constructed within
the cubic Horndeski theory which allows for stable violation of the NEC. We
present numerical examples of the background evolution having the different
maximal Hubble parameters (which determine the peak amplitude of gravitational
waves), the different inflationary Hubble parameters (which determine the
amplitudes of high frequency gravitational waves), and different durations of
the inflationary phase (which essentially determine the peak frequency of the
spectrum). We display the spectra with $n_t=0.8$, $0.9$, and $0.95$ for
$f\lesssim 1/{\rm yr}$, which are consistent with the recent NANOGrav result.
We also check that they do not contradict the BBN constraint. We discuss how
the nearly scale-invariant spectrum of curvature perturbations is produced in
the NEC-violating phase.
