Induced gravitational waves as a cosmological probe of the sound speed
  during the QCD phase transition
  The standard model of particle physics is known to be intriguingly
successful. However their rich phenomena represented by the phase transitions
(PTs) have not been completely understood yet, including the possibility of the
existence of unknown dark sectors. In this Letter, we investigate the
measurement of the equation of state parameter $w$ and the sound speed $c_{\rm
s}$ of the PT plasma with use of the gravitational waves (GWs) of the universe.
Though the propagation of GW is insensitive to $c_{\rm s}$ in itself, the sound
speed value affects the dynamics of primordial density (or scalar curvature)
perturbations and the induced GW by their horizon reentry can then be an
indirect probe both $w$ and $c_{\rm s}$. We numerically reveal the concrete
spectrum of the predicted induced GW with two simple examples of the scalar
perturbation spectrum: the monochromatic and scale-invariant spectra. In the
monochromatic case, we see that the resonant amplification and cancellation
scales of the induced GW depend on the $c_{\rm s}$ values at different time
respectively. The scale-invariant case gives a more realistic spectrum and its
specific shape will be compared with observations. In particular, the QCD phase
transition corresponds with the frequency range of the pulsar timing array
(PTA) observations. If the amplitude of primordial scalar power is in the range
of $10^{-4}\lesssim A_\zeta\lesssim10^{-2}$, the induced GW is consistent with
current observational constraints and detectable in the future observation in
Square Kilometer Array. Futhermore the recent possible detection of stochastic
GWs by NANOGrav 12.5 yr analysis~[1] can be explained by the induced GW if
$A_\zeta\sim\sqrt{7}\times10^{-3}$.
