  Gravitational-wave detections have revealed a previously unknown population
of stellar mass black holes with masses above $20\, M_{\odot}$. These
observations provide a new way to test models of stellar evolution for massive
stars. By considering the astrophysical processes likely to determine the shape
of the binary black hole mass spectrum, we construct a parameterized model to
capture key spectral features that relate gravitational-wave data to
theoretical stellar astrophysics. In particular, we model the signature of
pulsational pair-instability supernovae, which are expected to cause all stars
with initial mass $100\, M_{\odot}\lesssim M \lesssim 150\, M_{\odot}$ to form
$\sim 40\, M_{\odot}$ black holes. This would cause a cut-off in the black hole
mass spectrum along with an excess of black holes near $40\, M_{\odot}$. We
carry out a simulated data study to illustrate some of the stellar physics that
can be inferred using gravitational-wave measurements of binary black holes and
demonstrate several such inferences that might be made in the near future.
First, we measure the minimum and maximum stellar black hole mass. Second, we
infer the presence of a peak due to pair-instability supernovae. Third, we
measure the black hole mass ratio distribution. Finally, we show how inadequate
models of the black hole mass spectrum lead to biased estimates of the merger
rate and the amplitude of the stochastic gravitational-wave background.
