Why does the Jeans Swindle work?

TL;DR

This paper provides a formal justification for the 'Jeans Swindle' by demonstrating that including the universe's expansion in the Jeans equation cancels the divergent background density term, legitimizing this common but unjustified approximation.

Contribution

It shows that the expansion of the universe naturally cancels the divergent background term, providing a theoretical basis for the Jeans Swindle.

Findings

The expansion term cancels the divergent background density term.

The Jeans Swindle is formally justified when including cosmic expansion.

Provides a theoretical foundation for a common astrophysical approximation.

Abstract

When measuring the mass profile of any given cosmological structure through internal kinematics, the distant background density is always ignored. This trick is often refereed to as the "Jeans Swindle". Without this trick a divergent term from the background density renders the mass profile undefined, however, this trick has no formal justification. We show that when one includes the expansion of the Universe in the Jeans equation, a term appears which exactly cancels the divergent term from the background. We thereby establish a formal justification for using the Jeans Swindle.