  A population of binary black hole mergers has now been observed in
gravitational waves by Advanced LIGO and Virgo. The masses of these black holes
appear to show evidence for a pile-up between $30$--$45$ $M_\odot$ and a
cut-off above $\sim 45$ $M_\odot$. One possible explanation for such a pile-up
and subsequent cut-off are pulsational pair-instability supernovae (PPISNe) and
pair-instability supernovae (PISNe) in massive stars. We investigate the
plausibility of this explanation in the context of isolated massive binaries.
We study a population of massive binaries using the rapid population synthesis
software COMPAS, incorporating models for PPISNe and PISNe. Our models predict
a maximum black hole mass of $40$ $M_\odot$. We expect $\sim 10$\% of all
binary black hole mergers at redshift z = 0 will include at least one component
that went through a PPISN (with mass $30$--$40$ $M_\odot$), constituting $\sim
20$--$50$\% of binary black hole mergers observed during the first two
observing runs of Advanced LIGO and Virgo. Empirical models based on fitting
the gravitational-wave mass measurements to a combination of a power law and a
Gaussian find a fraction too large to be associated with PPISNe in our models.
The rates of PPISNe and PISNe track the low metallicity star formation rate,
increasing out to redshift $z = 2$. These predictions may be tested both with
future gravitational-wave observations and with observations of superluminous
supernovae.
