A Tale of Two Couplings: Bayesian Selection in the Interacting Dark Sector

CDM provides a remarkably successful description of the Universe, yet it suffers from persistent discrepancies, with one of the most prominent being the sigma 8 tension between early-Universe Cosmic Microwave Background (CMB) measurements and late-time structure growth probes. To address this mismatch, we investigate generalized interacting dark sector models governed by a quintessence field coupled to dark matter via conformal and disformal transformations. By generically engineering a quintessence potential that ensures the combined evolution of dark matter and dark energy exactly mimics CDM, our models satisfy all background constraints while signaling modified gravity effects within the linear perturbation sector. We perform a statistical analysis using late-time structure growth data anchored to Planck 2018 constraints. Furthermore, we address the initial conditions problem inherent to purely disformal models, demonstrating the necessity of a non-zero initial kinetic energy to initiate dynamical evolution. Our results indicate that purely conformal couplings efficiently transfer energy from dark matter to the quintessence field, suppressing late-time structure formation and successfully accommodating CMB data within late-time modified structure growth. This purely conformal limit yields a strong statistical preference over the CDM baseline according to both the Akaike and Bayesian Information Criteria. Conversely, while non-zero disformal couplings introduce a kinetic friction that dampens the dark sector energy exchange, their expanded parameter space incurs heavy statistical penalties without providing a decisive improvement over the purely conformal scenario.