We illustrate the extraordinary discovery potential for extragalactic astrophysics of a far-IR/submm all-sky spectroscopic survey with a 3m-class space telescope. Spectroscopy provides both a 3D view of the Universe and allows us to take full advantage of the sensitivity of present-day instrumentation, overcoming the spatial confusion that affects broadband far-IR/submm surveys. Emission lines powered by star formation will be detected in galaxies out to $z simeq 8$. It will provide measurements of spectroscopic redshifts, SFRs, dust masses, and metal content for millions of galaxies at the peak epoch of cosmic star formation and of hundreds of them at the epoch of reionization. Many of these galaxies will be strongly lensed; the brightness amplification and stretching of their sizes will make it possible to investigate (by means of follow-up with high-resolution instruments) their internal structure and dynamics on the scales of giant molecular clouds. This will provide direct information on the physics driving the evolution. Furthermore, the arc-min resolution of the telescope at submm wavelengths is ideal for detecting the cores of galaxy proto-clusters, out to the epoch of reionization. Tens of millions of these galaxy-clusters-in-formation will be detected at $z simeq 2$-3, with a tail out to $z simeq 7$, and thousands of detections at 6 < z < 7. Their study will allow us to track the growth of the most massive halos well beyond what is possible with classical cluster surveys (mostly limited to $z < 1.5$-2), tracing the history of star formation in dense environments and teaching us how star formation and galaxy-cluster formation are related across all epochs. Such a survey will overcome the current lack of spectroscopic redshifts of dusty star-forming galaxies and galaxy proto-clusters, representing a quantum leap in far-IR/submm extragalactic astrophysics.
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