A cosmic Schiff's conjecture test

Schiff’s conjecture posits that any consistent theory of gravity satisfying the Weak Equivalence Principle (WEP) must also satisfy the Einstein Equivalence Principle (EEP), implying a metric theory of gravity. We propose a cosmological test of Schiff’s conjecture, focusing on EEP and its components: WEP, Local Lorentz Invariance (LLI), and Local Position Invariance (LPI). Using galaxy redshift surveys, we analyze the dipole term in the two point correlation function to probe LPI via gravitational redshift, LLI via velocity-dependent effects, and WEP via composition-dependent clustering. The composition difference between galaxy types, such as baryon-to-dark-matter ratio, is parameterized to test WEP. This framework leverages redshift, 2-point correlation functions, power spectra, and cosmological distances to constrain deviations from General Relativity (GR).