Bayesian Leave-One-Out Cross-Validation for Astrophysical Model Comparison Using Gravitational-Wave Background Observations

Prior investigations demonstrated that ultralight dark matter solitons are capable of generating dynamical friction acting on supermassive black-hole binaries, thereby attenuating low-frequency power in the pulsar-timing-array gravitational-wave background and placing constraints on both the particle mass and the effective ultralight-dark-matter abundance. The present study extends that framework by evaluating the predictive performance of four distinct models: a simplified and a physically detailed ultralight-dark-matter formulation, a phenomenological environmental-hardening description, and a gravitational-wave-driven-only model. Predictive performance is assessed through Bayesian leave-one-out cross-validation applied to the five lowest pulsar-timing-array frequency bins. The phenomenological model yields the highest expected log predictive density; however, its margin over the remaining models is modest relative to the associated standard errors. Consequently, the available data do not decisively favor any single model. The most unambiguous pairwise outcome arises within the ultralight-dark-matter framework itself: the simplified formulation surpasses the physically detailed implementation across all five frequency bins. Present pulsar-timing-array observations are therefore consistent with ultralight-dark-matter-driven low-frequency suppression, yet they do not sufficiently discriminate ultralight dark matter from broader environmental characterizations of supermassive-black-hole-binary evolution.