The kinematic cosmic dipole beyond Ellis and Baldwin

The cosmic dipole anomaly—currently detected at a significance exceeding 5σ in several independent surveys poses a significant challenge to the standard model of cosmology. The Ellis and 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. Even though for a monochromatic survey fitting a power law on the spectra at the flux limit is always sufficient, it fails for the case of sources with more complicated spectra in photometric surveys, such as galaxies in the visible and near-infrared, which can feature emission lines or breaks. In this work, we demonstrate that the Ellis and Baldwin formula can be generalized to arbitrary luminosity distributions and spectral profiles in particular for photometric surveys. We derived the corresponding expression for the effective spectral index and applied 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 in this sample. These results provide a more general and robust framework to interpret measurements of the cosmic dipole in future photometric large-scale surveys.