Weighing Milky Way and Andromeda in an expanding $\Lambda$CDM Universe: Decreasing the Local Group mass

The dynamics of the Local Group (LG), especially the contribution of the Milky Way (MW) and Andromeda (M31) galaxies, is sensitive to the presence of dark energy. This work analyzes the evolution of the LG by considering it as a two-body problem in a homogeneous and isotropic expanding spacetime in a full $\Lambda$ cold dark matter $(\Lambda$CDM) background. Using the timing argument (TA), which links LG dynamics to LG mass, we find that the complete $\Lambda$CDM background predicts a $\sim 10 \%$ lower mass for the LG; while $\Lambda$ alone predicts a $\sim 10 \%$ higher mass. The TA mass is modified by (i) simulations and (ii) the effect of the Large Magellanic Cloud (LMC) to alleviate the poorly constrained internal mass distributions of M31 and the MW, their time evolution, and the unknown distribution of dark matter between them. First, using IllustrisTNG simulations, we accounted for the effects of two extended halos and their environment (rather than point particles) and predicted their mass $\left(3.89 \pm 0.62\right) \cdot 10^{12} M_{\odot}$. Second, the LMC effectively changes the separation and velocities of M31 toward the MW and reduces the predicted mass to $\left(2.33 \pm 0.72\right) \cdot 10^{12} M_\odot$. Despite the uncertainties around dark matter between these galaxies, the overall estimated mass is compatible with the mere sum of the MW and M31 masses. The total mass of the TA is compatible with other estimates, such as the Hubble flow and the Virial Theorem with other dwarf galaxies. The combined result shows, for the first time, that a lower mass estimate can be obtained from the TA, with a consistent embedding and other systematic effects, and without an additional dark matter halo around the galaxies.