Time-dependent neutrino emission from Mrk 421 during flares and predictions for IceCube

Blazars are prime candidate sources for the high energy neutrinos recently detected by IceCube. Being one of the brightest sources in the extragalactic X-ray and $\gamma$-ray sky as well as one of the nearest blazars to Earth, Mrk 421 is an excellent source for testing the scenario of the blazar-neutrino connection. Here, we model the spectral energy distribution of Mrk 421 during a 13-day flare in 2010 with unprecedented multi-wavelength coverage, and calculate the respective neutrino flux. We find a correlation between the $>1$ PeV neutrino and photon fluxes, in all energy bands. Using typical IceCube through-going muon event samples with good angular resolution and high statistics, we derive the mean event rate above 100 TeV ($\sim0.57$ evt/yr) and show that it is comparable to that expected from a four-month quiescent period in 2009. Due to the short duration of the flare, an accumulation of similar flares over several years would be necessary to produce a meaningful signal for IceCube. To better assess this, we apply the correlation between the neutrino and $\gamma$-ray fluxes to the 6.9 yr Fermi-LAT light curve of Mrk 421. We find that the mean event count above 1 PeV for the full IceCube detector livetime is $3.59\pm0.60$ ($2.73\pm0.38$) $\nu_\mu+\bar{\nu}_\mu$ with (without) major flares included in our analysis. This estimate exceeds, within the uncertainties, the $95\%$ ($90\%$) threshold value for the detection of one or more muon (anti-)neutrinos. Meanwhile, the most conservative scenario, where no correlation of $\gamma$-rays and neutrinos is assumed, predicts $1.60\pm0.16$ $\nu_\mu+\bar{\nu}_\mu$ events. We conclude that a non-detection of high-energy neutrinos by IceCube would probe the neutrino/$\gamma$-ray flux correlation during major flares or/and the hadronic contribution to the blazar emission.