We use the sky-average spectrum measured by EDGES High-Band ($90-190$ MHz) to constrain parameters of early galaxies independent of the absorption feature at $78$~MHz reported by Bowman et al. (2018). These parameters represent traditional models of cosmic dawn and the epoch of reionization produced with the 21cmFAST simulation code (Mesinger & Furlanetto 2007, Mesinger et al. 2011). The parameters considered are: (1) the UV ionizing efficiency ($zeta$), (2) minimum halo virial temperature hosting efficient star-forming galaxies ($T^{rm min}_{rm vir}$), (3) integrated soft-band X-ray luminosity ($L_{rm X,<,2,keV}/{rm SFR}$), and (4) minimum X-ray energy escaping the first galaxies ($E_{0}$), corresponding to a typical H${rm scriptstyle I}$ column density for attenuation through the interstellar medium. The High-Band spectrum disfavors high values of $T^{rm min}_{rm vir}$ and $zeta$, which correspond to signals with late absorption troughs and sharp reionization transitions. It also disfavors intermediate values of $L_{rm X,<,2,keV}/{rm SFR}$, which produce relatively deep and narrow troughs within the band. Specifically, we rule out $39.4<log_{10}left(L_{rm X,<,2,keV}/{rm SFR}right)<39.8$ ($95%$ C.L.). We then combine the EDGES High-Band data with constraints on the electron scattering optical depth from Planck and the hydrogen neutral fraction from high-$z$ quasars. This produces a lower degeneracy between $zeta$ and $T^{rm min}_{rm vir}$ than that reported in Greig & Mesinger (2017a) using the Planck and quasar constraints alone. Our main result in this combined analysis is the estimate $4.5$~$leq log_{10}left(T^{rm min}_{rm vir}/rm Kright)leq$~$5.7$ ($95%$ C.L.). We leave for future work the evaluation of $21$~cm models using simultaneously data from EDGES Low- and High-Band.
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