Probing the stellar initial mass function with high-$z$ supernovae
  The first supernovae will soon be visible at the edge of the observable
universe, revealing the birthplaces of Population III stars. With upcoming
near-infrared missions, a broad analysis of the detectability of high-$z$
supernovae is paramount. We combine cosmological and radiation transport
simulations, instrument specifications, and survey strategies to create
synthetic observations of primeval core-collapse, Type IIn and pair-instability
supernovae with the James Webb Space Telescope ($JWST$). We show that a
dedicated observational campaign with the $JWST$ can detect up to $\sim 15$
pair-instability explosions, $\sim 300$ core-collapse supernovae, but less than
one Type IIn explosion per year, depending on the Population III star formation
history. Our synthetic survey also shows that $\approx 1-2 \times10^2$
supernovae detections, depending on the accuracy of the classification, are
sufficient to discriminate between a Salpeter and flat mass distribution for
high redshift stars with a confidence level greater than 99.5 per cent. We
discuss how the purity of the sample affects our results and how supervised
learning methods may help to discriminate between CC and PI SNe.
