Neutral hydrogen pervades the infant Universe, and its redshifted 21-cm signal allows one to chart the Universe. This signal allows one to probe astrophysical processes such as the formation of the first stars, galaxies, (super)massive black holes and enrichment of the pristine gas from z~6 to z~30, as well as fundamental physics related to gravity, dark matter, dark energy and particle physics at redshifts beyond that. As one enters the Dark Ages (z>30), the Universe becomes pristine. Ground-based low-frequency radio telescopes aim to detect the spatial fluctuations of the 21-cm signal. Complementary, global 21-cm experiments aim to measure the sky-averaged 21-cm signal. Escaping RFI and the ionosphere has motivated space-based missions, such as the Dutch-Chinese NCLE instrument (currently in lunar L2), the proposed US-driven lunar or space-based instruments DAPPER and FARSIDE, the lunar-orbit interferometer DSL (China), and PRATUSH (India). To push beyond the current z~25 frontier, though, and measure both the global and spatial fluctuations (power-spectra/tomography) of the 21-cm signal, low-frequency (1-100MHz; BW~50MHz; z>13) space-based interferometers with vast scalable collecting areas (1-10-100 km2), large filling factors (~1) and large fields-of-view (4pi sr.) are needed over a mission lifetime of >5 years. In this ESA White Paper, we argue for the development of new technologies enabling interferometers to be deployed, in space (e.g. Earth-Sun L2) or in the lunar vicinity (e.g. surface, orbit or Earth-Moon L2), to target this 21-cm signal. This places them in a stable environment beyond the reach of most RFI from Earth and its ionospheric corruptions, enabling them to probe the Dark Ages as well as the Cosmic Dawn, and allowing one to investigate new (astro)physics that is inaccessible in any other way in the coming decades. [Abridged]
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