Complementarity of Stacking and Multiplet Constraints on the Blazar Contribution to the Cumulative High-Energy Neutrino Intensity

We investigate the blazar contribution to the cumulative neutrino intensity assuming a generic relationship between neutrino and gamma-ray luminosities, $L_{\nu}\propto (L_{\rm ph})^{\gamma_{\rm lw}}$. Using the gamma-ray luminosity functions for blazars including flat spectrum radio quasars (FSRQs) and BL Lac objects, as well as the $Fermi$-LAT detection efficiency, we estimate contributions from {blazars resolved by $Fermi$-LAT as well as the unresolved counterpart.} Combining the existing upper limits from stacking analyses, the cumulative neutrino flux from all blazars ({including $Fermi$-LAT resolved and unresolved ones}) are constrained { in the range $0\lesssim\gamma_{\rm lw}\lesssim2.5$}. We also evaluate the effects of the redshift evolution and the effective local number densities for each class of FSRQs, BL Lacs, and all blazars, by which we place another type of constraints on the blazar contribution using the non-detection of high-energy neutrino multiplets. We demonstrate that these two upper limits are complementary, and that the joint consideration of the stacking and multiplet analyses { not only supports the argument that blazars are disfavored as the dominant sources of the 100-TeV neutrino background, but it extends this argument by including also $Fermi$-LAT-unresolved blazars as well, for a more generic luminosity correlation $L_{\nu}\propto (L_{\rm ph})^{\gamma_{\rm lw}}$}.