Testing the equivalence principle across the Universe: a model-independent approach with galaxy multi-tracing

We present a test of the equivalence principle on cosmological scales. This cornerstone of general relativity has been tested with high precision for standard matter, but its validity for the unknown dark matter remains a crucial open question. We construct a measurable quantity $E_P$ that acts as a null test, i.e. it deviates from unity whenever the weak equivalence principle is violated. This quantity can be directly measured from the cross-correlation of two different galaxy populations, thanks to the inclusion of large-scale relativistic corrections. A key feature of our approach is that it only involves minimal assumptions, without the need to specify the power spectrum shape, the background evolution, the growth rate of cosmic structure, the galaxy bias function or a model for the potential violation of the equivalence principle. We provide forecasts for the Dark Energy Spectroscopic Instrument and the Square Kilometre Array (SKA). While the relativistic corrections can be detected with high significance by both surveys, $E_P$ can be constrained up to an interesting level only by SKA, with a precision around 7-15$\%$ within the redshift range $z < 0.6$.