Hubble's Law Implies Benford's Law for Distances to Galaxies

TL;DR

This paper provides a theoretical explanation linking Hubble's law to Benford's law, suggesting galaxy distances should follow Benford's distribution, and supports this with empirical evidence and robustness analysis.

Contribution

It derives a new galaxy-distance law based on Hubble's law that explains why galaxy distances follow Benford's law, with robustness to errors and variability.

Findings

Galaxy distances follow Benford's law empirically.

Theoretical derivation links Hubble's law to Benford's distribution.

Conformity to Benford's law improves with more data.

Abstract

A recent article by Alexopoulos and Leontsinis presented empirical evidence that the first digits of the distances to galaxies are a reasonably good fit to the probabilities predicted by Benford's law, the well known logarithmic statistical distribution of significant digits. The purpose of the present article is to give a theoretical explanation, based on Hubble's law and mathematical properties of Benford's law, why galaxy distances might be expected to follow Benford's law. The new galaxy-distance law derived here, which is robust with respect to change of scale and base, to additive and multiplicative computational or observational errors, and to variability of the Hubble constant in both time and space, predicts that conformity to Benford's law will improve as more data on distances to galaxies becomes available. Conversely, with the logical derivation of this law presented here, the recent empirical observations may be viewed as independent evidence of the validity of Hubble's law.