Shock breakouts from compact CSM surrounding core-collapse SN progenitors may contribute significantly to the observed $\gtrsim10$ TeV neutrino background
Growing observational evidence suggests that enhanced mass loss from the progenitors of core-collapse supernovae (SNe) is common during $\sim1$ yr preceding the explosion, creating an optically thick circum-stellar medium (CSM) shell at $\sim10^{14.5}$ cm radii. We show that if such mass loss is indeed common, then the breakout of the SN shock through the dense CSM shell produces a neutrino flux that may account for a significant fraction of the observed $\gtrsim10$ TeV neutrino background. The neutrinos are created within a few days from the explosion, during and shortly after the shock breakout, which produces also large UV (and later X-ray) luminosity. The compact size and large UV luminosity imply a pair production optical depth of $\sim10^4$ for $>100$ GeV photons, naturally accounting for the lack of a high-energy gamma-ray background accompanying the neutrino background. SNe producing $>1$ neutrino event in a 1 km$^2$ detector are expected at a rate of $\lesssim0.1$/yr. A quantitative theory describing the evolution of the electromagnetic spectrum during a breakout, as the radiation-mediated shock is transformed into a collisionless one, is required to enable (i) using data from upcoming surveys that will systematically detect large numbers of young, $<1$ d old SNe, to determine the pre-explosion mass loss history of the SN progenitor population, and (ii) a quantitative determination of the neutrino luminosity and spectrum.