Probing the initial mass function of the first stars with transients
  The emergence of the first, so-called Population III (Pop III), stars shaped
early cosmic history in ways that crucially depends on their initial mass
function (IMF). However, because of the absence of direct observational
constraints, the detailed IMF remains elusive. Nevertheless, numerical
simulations agree in broad terms that the first stars were typically massive
and should often end their lives in violent, explosive deaths. These fates
include extremely luminous pair-instability supernovae (PISNe) and bright
gamma-ray bursts (GRBs), the latter arising from the collapse of rapidly
rotating progenitor stars into black holes. These high-redshift transients are
expected to be within the detection limits of upcoming space telescope
missions, allowing to place effective constraints on the shape of the
primordial IMF that is not easily accessible with other probes. This paper
presents a framework to probe the Pop III IMF, utilizing the cosmological
source densities of high-redshift PISNe and GRBs. Considering these transients
separately could provide useful constraints on the Pop III IMF, but tighter
bounds are obtainable by combining PISN and GRB counts. This combined
diagnostic is more robust as it is independent of the underlying Pop III star
formation rate density, an unknown prior. Future surveys promise to capture
most high-redshift GRBs across the entire sky, but high-redshift PISN searches
with future telescopes, e.g. Roman Space Telescope, will likely be
substantially incomplete. Nevertheless, we demonstrate that even such lower
bounds on the PISN count will be able to provide key constraints on the
primordial IMF, in particular, if it is top-heavy or not.
