Comparing different realizations of modified Newtonian dynamics: virial theorem and elliptical shells

TL;DR

This paper compares different modified gravity theories, specifically BIMOND/QUMOND and TeVeS, analyzing their force fields and virial theorem predictions in non-spherical systems, revealing conditions under which they produce identical virial results despite different force fields.

Contribution

It provides analytical derivations of the virial theorem in two modified gravity theories and identifies conditions where their predictions coincide even outside spherical symmetry.

Findings

Certain free function choices lead to identical virial theorems in both theories.

Force fields differ outside spherical symmetry, but the virial theorem can be the same.

Analytical results have implications for two-body force calculations in modified gravity.

Abstract

There exists several modified gravity theories designed to reproduce the empirical Milgrom's formula (MOND). Here we derive analytical results in the context of the static weak-field limit of two of them (BIMOND, leading for a given set of parameters to QUMOND, and TeVeS). In this limit, these theories are constructed to give the same force field for spherical symmetry, but their predictions generally differ out of it. However, for certain realizations of these theories (characterized by specific choices for their free functions), the binding potential-energy of a system is increased, compared to its Newtonian counterpart, by a constant amount independent of the shape and size of the system. In that case, the virial theorem is exactly the same in these two theories, for the whole gravity regime and even outside of spherical symmetry, although the exact force fields are different. We explicitly show this for the force field generated by the two theories inside an elliptical shell. For more general free functions, the virial theorems are however not identical in these two theories. We finally explore the consequences of these analytical results for the two-body force.