Complementary consistency test of the Copernican principle via Noether's theorem and machine learning forecasts

The Copernican principle (CP), i.e. the assumption that we are not privileged observers of the Universe, is a fundamental tenet of the standard cosmological model. A violation of this postulate implies the possibility that the apparent cosmic acceleration could be explained without the need of a cosmological constant, dark energy or covariant modifications of gravity. In this paper we present a new test of the CP relating the distance and the expansion rate, derived via Noether's theorem, which is complementary to other tests found in the literature. We also simulate fiducial data based on upcoming stage IV galaxy surveys and use them to reconstruct the Hubble rate H(z) and the angular diameter distance d(A) (z) in order to forecast how well our null test can constrain deviations from the cosmological constant model. We find that our new test can easily rule out several scenarios based on the Lemaitre-Tolman-Bondi void model at confidence of greater than or similar to 3 sigma at middle to high redshifts (z > 0.5).