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    The origin of the radio synchrotron background (RSB) is currently unknown. Its understanding might have profound implications in fundamental physics or might reveal a new class of radio emitters. In this work, we consider the scenario in which the RSB is due to extragalactic radio sources and measure the angular cross-correlation of Low-Frequency Array (LOFAR) images of the diffuse radio sky with matter tracers at different redshifts, provided by galaxy catalogues and cosmic microwave background lensing. We compare these measured cross-correlations to those expected for models of RSB sources. We find that low-redshift populations of discrete sources are excluded by the data, while higher redshift explanations are compatible with available observations. We also conclude that at least 20 per cent of the RSB surface brightness level must originate from populations tracing the large-scale distribution of matter in the Universe, indicating that at least this fraction of the RSB is of extragalactic origin. Future measurements of the correlation between the RSB and tracers of high-redshift sources will be crucial to constraining the source population of the RSB.

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