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

TL;DR

This paper reformulates the cosmic dipole as a symmetry response to Lorentz boosts, unifying various effects in galaxy clustering and providing a new perspective on dipole anisotropies in cosmology.

Contribution

It introduces a general response framework for the cosmic dipole, extending the Ellis--Baldwin formula to higher-order statistics and unifying different sources of dipole anisotropies.

Findings

Derived a general expression for the dipole as a symmetry response.

Extended the response framework to two-point and higher-order correlation functions.

Unified the treatment of observer- and source-induced 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.