  The recent observation of $^4$He favors a large lepton asymmetry at the big
bang nucleosynthesis. If Q-balls with a lepton charge decay after the
electroweak phase transition, such a large lepton asymmetry can be generated
without producing too large baryon asymmetry. In this scenario, Q-balls
dominate the universe before the decay and induces the sharp transition from
the early matter-dominated era to the radiation-dominated era. In this
transition, the gravitational waves (GWs) are enhanced through a second-order
effect of the scalar perturbations. We evaluate the density of the produced GWs
and show that pulsar timing array observations can probe this scenario
depending on the amplitude of the scalar perturbations.
