We exploit EAGLE, a cosmological hydrodynamical simulation, to reproduce the selection of the observed sub-millimeter (submm) galaxy population by selecting the model galaxies at $z geq 1$ with mock submm fluxes $S_{850} geq 1$ mJy. There is a reasonable agreement between the galaxies within this sample and the properties of the observed submm population, such as their star formation rates (SFRs) at $z<3$, redshift distribution and many integrated galaxy properties. We find that the bulk of the $S_{850} geq 1$ mJy model population is at $z = 2.5$, and that they are massive galaxies ($M_* sim 10^{11}$ Msol) with high dust masses ($M_{mathrm{dust}} sim 10^{8}$ Msol), gas fractions ($f_{mathrm{gas}} approx 50$%) and SFRs ($dot M_* approx 100$ Msol/yr). They have major and minor merger fractions similar to the general population, suggesting that mergers are not the primary driver of the model submm galaxies. Instead, the $S_{850} geq 1$ mJy model galaxies yield high SFRs primarily because they maintain a significant gas reservoir as a result of hosting an undermassive black hole. In addition, we find that not all highly star-forming EAGLE galaxies have submm fluxes $S_{850} > 1$ mJy. Thus, we investigate the nature of $z geq 1$ highly star-forming Submm-Faint galaxies (i.e., $dot M_* geq 80$ Msol/yr but $S_{850}< 1$ mJy). We find they are similar to the model submm galaxies; being gas rich and hosting undermassive black holes, however they are typically lower mass ($M_* sim 10^{10}$ Msol) and are at higher redshifts ($z>4$). These typically higher-$z$ galaxies show stronger evidence for having been triggered by major mergers, and critically, they are likely missed by current submm surveys due to their higher dust temperatures. This suggests a potentially even larger contribution to the SFR density at $z > 3$ from dust-obscured systems than implied by current observations.
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