We report on the first measurement of the astrophysical neutrino flux using particle showers (cascades) in IceCube data from 2010 -- 2015. Assuming standard oscillations, the astrophysical neutrinos in this dedicated cascade sample are dominated ($sim 90 %$) by electron and tau flavors. The flux, observed in the sensitive energy range from $16,mathrm{TeV}$ to $2.6,mathrm{PeV}$, is consistent with a single power-law model as expected from Fermi-type acceleration of high energy particles at astrophysical sources. We find the flux spectral index to be $gamma=2.53pm0.07$ and a flux normalization for each neutrino flavor of $phi_{astro} = 1.66^{+0.25}_{-0.27}$ at $E_{0} = 100, mathrm{TeV}$, in agreement with IceCubes complementary muon neutrino results and with all-neutrino flavor fit results. In the measured energy range we reject spectral indices $gammaleq2.28$ at $ge3sigma$ significance level. Due to high neutrino energy resolution and low atmospheric neutrino backgrounds, this analysis provides the most detailed characterization of the neutrino flux at energies below $sim100,{rm{TeV}}$ compared to previous IceCube results. Results from fits assuming more complex neutrino flux models suggest a flux softening at high energies and a flux hardening at low energies (p-value $ge 0.06$). The sizable and smooth flux measured below $sim 100,{rm{TeV}}$ remains a puzzle. In order to not violate the isotropic diffuse gamma-ray background as measured by the Fermi-LAT, it suggests the existence of astrophysical neutrino sources characterized by dense environments which are opaque to gamma-rays.
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