Recent long wavelength observations on the thermal dust continuum suggest that the Rayleigh-Jeans (RJ) tail can be used as a time-efficient quantitative probe of the dust and ISM mass in high-$z$ galaxies. We use high-resolution cosmological simulations from the Feedback in Realistic Environment (FIRE) project to analyze the dust emission of $M_*>10^{10};M_{odot}$ galaxies at $z=2-4$. Our simulations (MassiveFIRE) explicitly include various forms of stellar feedback, and they produce the stellar masses and star formation rates of high-$z$ galaxies in agreement with observations. Using radiative transfer modelling, we show that sub-millimeter (sub-mm) luminosity and molecular ISM mass are tightly correlated and that the overall normalization is in quantitative agreement with observations. Notably, sub-mm luminosity traces molecular ISM mass even during starburst episodes as dust mass and mass-weighted temperature evolve only moderately between $z=4$ and $z=2$, including during starbursts. Our finding supports the empirical approach of using broadband sub-mm flux as a proxy for molecular gas content in high-$z$ galaxies. We thus expect single-band sub-mm observations with ALMA to dramatically increase the sample size of high-$z$ galaxies with reliable ISM masses in the near future.
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