Stellar Initial Mass Function (IMF) Probed with Supernova Rates and Neutrino Background: Cosmic Average IMF Slope is $\simeq 2-3$ Similar to the Salpeter IMF

The stellar initial mass function (IMF) is expressed by $\phi(m) \propto m^{-\alpha}$ with the slope $\alpha$, and known as the poorly-constrained but very important function in studies of star and galaxy formation. There are no sensible observational constraints on the IMF slopes beyond Milky Way and nearby galaxies. Here we combine two sets of observational results, 1) cosmic densities of core-collapse supernova explosion (CCSNe) rates and 2) cosmic far ultraviolet radiation (and infrared re-radiation) densities, which are sensitive to massive ($\simeq 8-50 \,{\rm M}_\odot$) and moderately massive ($\simeq 2.5-7 \,{\rm M}_\odot$) stars, respectively, and constrain the IMF slope at $m>1\,{\rm M}_\odot$ with a freedom of redshift evolution. Although no redshift evolution is identified beyond the uncertainties, we find that the cosmic average IMF slope at $z=0$ is $\alpha=1.8-3.2$ at the 95\% confidence level that is comparable with the Salpeter IMF, $\alpha=2.35$, which marks the first constraint on the cosmic average IMF. We show a forecast for the Nancy Grace Roman Space Telescope supernova survey that will provide significantly strong constraints on the IMF slope with $\delta \alpha\simeq 0.5$ over $z=0-2$. Moreover, as for an independent IMF probe instead of 1), we suggest to use diffuse supernovae neutrino background (DSNB), relic neutrinos from CCSNe. We expect that the Hyper-Kamiokande neutrino observations over 20 years will improve the constraints on the IMF slope and the redshift evolution significantly better than those obtained today, if the systematic uncertainties of DSNB production physics are reduced in the future numerical simulations.