  The chirality imbalance in QCD is spontaneously induced by a repulsive
axial-vector interaction from the instanton anti-instanton pairing at high
temperature above the chiral phase transition, and vanishes at low temperature.
The chiral chemical potential $\mu_5$ is in the same magnitude as estimated
from the sphaleron transition. Phase transition of the chirality imbalance is
always a first-order one in the early universe with $\mathcal {P}$ and
$\mathcal {CP}$ violation. The spectra of gravitational waves and the formation
of the primordial black holes from this first-order phase transition is
investigated in this work, and the effect of a strong magnetic field is also
analyzed. The gravitational waves produced by chirality imbalance can be
detected by LISA, Taiji and DECIGO, with the peak energy density locating in
the range of $10^{-11}$ to $10^{-9}$ and the peak frequencies lying in the
range of $10^{-5}$ Hz to $10^{-2}$ Hz. The spectrum with larger axial vector
coupling strength and stronger magnetic field has higher peak energy density
and lower peak frequency. According to this trend, the gravitational waves
spectra might also be able to be detected by SKA, IPTA and EPTA. The Phase
transition inverse duration $\beta/H_*$ calculated from the bounce solution is
in the order of $10^4$, which is much higher than typical value $10-100$ from
electroweak phase transitions. Based on the mechanism of postponement of the
false vacuum decay, it is found that the formation of the primordial black hole
is not favored because the phase transition completes in an extreme short time
due to the large value of $\beta/H_*$ and thus the false vacuum energy density
decays sharply.
