Constraining the Gravitational-Wave Spectrum from Cosmological
  First-Order Phase Transitions Using Data from LIGO-Virgo First Three
  Observing Runs
  We search for a first-order phase transition (PT) gravitational wave (GW)
signal from Advanced LIGO and Advanced Virgo's first three observing runs. Due
to the large theoretical uncertainties, four GW energy spectral shapes from
bubble and sound wave collisions widely adopted in literature are investigated,
separately. Our results indicate that there is no evidence for the existence of
such GW signals, and therefore we give the upper limits on the amplitude of GW
energy spectrum $\Omega_\text{pt}(f_*)$ in the peak frequency range of $f_*\in
[5,500]$ Hz for these four theoretical models, separately. We find that
$\Omega_\text{pt}(f_*\simeq 40\ \text{Hz})<1.3\times10^{-8}$ at $95\%$ credible
level, and roughly $H_*/\beta\lesssim 0.1$ and $\alpha\lesssim 1$ at $68\%$
credible level in the peak frequency range of $20\lesssim f_*\lesssim 100$ Hz
corresponding to the most sensitive frequency band of Advanced LIGO and
Advanced Virgo's first three observing runs, where $H_*$ is the Hubble
parameter when PT happens, $\beta$ is the bubble nucleation rate and $\alpha$
is the ratio of vacuum and relativistic energy density.
