An Inconsistency in standard Maximum Likelihood Estimation of Bulk Flows

TL;DR

This paper identifies an inconsistency in the standard maximum likelihood method for estimating galaxy bulk flows, showing it assumes a constant velocity field which conflicts with the bulk flow's definition, and proposes a correction.

Contribution

It reveals a fundamental inconsistency in the standard MLE approach for bulk flows and introduces a simple correction method to address it.

Findings

The standard MLE assumes a constant velocity field, conflicting with the bulk flow definition.

The inconsistency persists even with ideal data and full spatial coverage.

A proposed modification corrects the inconsistency in bulk flow estimation.

Abstract

Maximum Likelihood estimation of the bulk flow from radial peculiar motions of galaxies, generally assumes a constant velocity field inside the survey volume. The assumption is inconsistent with the definition of the bulk flow as the average of the peculiar velocity field over the relevant volume. This follows from a straightforward mathematical relation between the bulk flow of a sphere and the velocity potential on its surface. The inconsistency exists also for ideal data with exact radial velocities and full spatial coverage. Based on the same relation we propose a simple modification to correct for this inconsistency.