# The Two Body Problem in the Presence of Dark Energy and Modified Gravity: Application to the Local Group

## TL;DR

This paper investigates how different dark energy and modified gravity theories affect the mass estimation of the Local Group using the timing argument, revealing inconsistencies and challenges in explaining observed dynamics without dark matter.

## Contribution

It applies the timing argument to various dark energy and modified gravity models, analyzing their implications for the mass of the Local Group and highlighting tensions with observational data.

## Key findings

- MOND predicts a mass about 10% of luminous mass
- Symmetron model suggests a mass too high without dark matter
- H0 measurement uncertainties significantly affect mass estimates

## Abstract

We explore mass estimation of the Local Group via the use of the simple, dynamical `timing argument' in the context of a variety of theories of dark energy and modified gravity: a cosmological constant, a perfect fluid with constant equation of state $w$, quintessence (minimally coupled scalar field), MOND, and symmetrons (coupled scalar field). We explore generic coupled scalar field theories, with the symmetron model as an explicit example. We find that theories which attempt to eliminate dark matter by fitting rotation curves produce mass estimates in the timing argument which are not compatible with the luminous mass of the galaxies alone. Assuming that the galaxies are approaching their first encounter, MOND gives of around $2.7\times 10^{10} M_\odot$, roughly 10\% of the luminous mass of the LG, although a higher mass can be obtained in the case of a previous fly-by event between the MW and M31. The symmetron model suggests a mass too high to be explained without additional dark matter ($\mathcal{O}(10^{12}) M_\odot$), suggesting that there is a missing mass problem in this model. We also demonstrate that tensions in measurements of $H_0$ can produce an uncertainty in the Local Group mass estimate comparable to observational uncertainties on the separation and relative velocity of the galaxies, with values for the mass ranging from $4.5 - 5.4 \times 10^{12} M_{\odot}$ varying $h$ between 0.67 and 0.76.