Effective Field Theory for a Baryon-Correlated Dark Matter Profile

While the standard $\Lambda$CDM model succeeds on large cosmological scales, it faces persistent small-scale challenges, including the core-cusp problem, the diversity of galaxy rotation curves, and the tight correlation between dark matter and baryons observed in the Tully-Fisher relation. To address these issues, we recently proposed an empirical law where the effective dark matter energy density is directly correlated with the baryonic gravitational potential, $\rho_{\rm DM} \propto \Phi_b^2$, which reproduces observed rotation curves and resolves the core-cusp and diversity problems. To provide a theoretical foundation for this empirical law, we construct an effective field theory (EFT) introducing massive scalar, vector, and tensor mediators between baryons and a dark sector field $\chi$. We demonstrate that aligning the mediator couplings to a specific ratio (4:6:3) with degenerate masses cancels the additional fifth forces acting on baryons up to $\mathcal{O}(v^2)$. We then show that this theoretical framework originates from a 5-dimensional (5D) spacetime. Treating the baryonic source as a 5D null fluid reveals that the three mediators emerge from a single 5D symmetric tensor field. By confining the field $\chi$ to a 4D brane, we show that its interaction with these mediators generates a pressureless energy density ($\rho_{\rm int} \propto \Phi_b^2$) that yields the empirically required baryon-correlated profile. Consequently, the field $\chi$ exhibits a scale-dependent transition: on cosmological scales, its mass energy acts as standard Cold Dark Matter (CDM), whereas on galactic scales, its interaction energy governs local dynamics. Finally, by evaluating the dynamical boundary of this localized interaction region, we provide a physical interpretation that yields the relation $\mu = K M_b^{-3/2}$, offering a theoretical basis for the Tully-Fisher relation.