Cosmic Dipole as a Symmetry Response: From the Ellis--Baldwin Formula to Correlation Function Dipoles

Abstract

The cosmic dipole in galaxy number counts is traditionally described by the Ellis--Baldwin (EB) formula under simplifying assumptions of power-law source counts and flux-limited selection. We reformulate the EB dipole as a symmetry response of observed counts to a Lorentz boost, leading to the general expression $D=\beta R$, where $R=\partial\ln N/\partial\ln\beta$ encodes the underlying population and selection effects. The classical EB formula is recovered as a limiting case. We show that this response framework extends beyond one-point statistics: Lorentz boosts induce a dipole component in the two-point correlation function and, more generally, a hierarchy of responses in $n$-point statistics. We further clarify the relation to redshift-space distortions and relativistic galaxy clustering, and provide a unified description in which observer- and source-induced dipoles contribute to the same multipole component. This establishes the cosmic dipole as a symmetry response of finite-sample point-process statistics, offering a new perspective on dipole anisotropies and their observational interpretation.