How far are the sources of IceCube neutrinos? Constraints from the diffuse TeV gamma-ray background

The nearly isotropic distribution of the TeV-PeV neutrinos recently detected by IceCube suggests that they come from sources at distance beyond our Galaxy, but how far they are is largely unknown due to lack of any associations with known sources. In this paper, we propose that the cumulative TeV gamma-ray emission accompanying the production of neutrinos can be used to constrain the distance of these neutrino sources, since the opacity of TeV gamma rays due to absorption by the extragalactic background light (EBL) depends on the distance that these TeV gamma rays have travelled. As the diffuse extragalactic TeV background measured by \emph{Fermi} is much weaker than the expected cumulative flux associated with IceCube neutrinos, the majority of IceCube neutrinos, if their sources are transparent to TeV gamma rays, must come from distances larger than the horizon of TeV gamma rays. We find that above 80\% of the IceCube neutrinos should come from sources at redshift $z>0.5$. Thus, the chance for finding nearby sources correlated with IceCube neutrinos would be small. We also find that, to explain the flux of neutrinos under the TeV gamma-ray emission constraint, the redshift evolution of neutrino source density must be at least as fast as the the cosmic star-formation rate.