The kinematic cosmic dipole beyond Ellis and Baldwin

The cosmic dipole anomaly, currently detected at a significance exceeding 5$\sigma$ in several independent survey poses a significant challenge to the standard model of cosmology. The Ellis & Baldwin formula provides a theoretical link between the intrinsic dipole anisotropy in the sky distribution of extragalactic light sources and the observer's velocity relative to the cosmic rest frame, under the assumptions that the sources follow a power-law luminosity function and exhibit power-law spectral energy distributions. In this work, we demonstrate that this relation can be generalized to arbitrary luminosity distributions and spectral profiles. We derive the corresponding expression for the effective spectral index and apply it to a sample of quasars observed in the W1 band of the CatWISE survey. We show that the anomalous cosmic dipole persists beyond the power-law assumption. These results provide a more general and robust framework to interpret measurements of the cosmic dipole in future large-scale surveys.