Milky Way and Andromeda past-encounters in different gravity models: the impact on the estimated Local Group mass

The two-body problem of $M31$ and the Milky Way (MW) galaxies with a cosmological constant background is studied, with emphasis on the possibility that they experienced past encounters. By implementing the initial conditions of the big bang and the last measured relative distance and velocities (i.e. the Timing Argument), it is shown that if $M{31}$ and the MW had more than one encounter then the mass of the Local Group (LG) would be a few times higher than if there had been no encounters. Past encounters are possible only for non-zero transverse velocity, and their viability is subject to observations of the imprints of such near collisions. While it has been previously shown that the presence of the cosmological constant requires a higher mass for the LG, here, using a recent $Gaia$ - based measurement of the transverse velocity the derived LG mass is $3.36^{+1.14}_{-0.7} \cdot 10^{12} M_{\odot}$ with no cosmological constant or $4.54^{+1.2}_{-0.75} \cdot 10^{12} M_{\odot}$ with a cosmological constant background. If the LG has had one past encounter, LG mass is $9.70^{+2.19}_{-1.55}\cdot 10^{12} M_{\odot}$ or $9.99^{+2.22}_{-1.58}\cdot 10^{12} M_{\odot}$ with a cosmological constant background. Modified Newtonian Dynamics (MOND) is also studied, as the accelerations of the Local Group are fully in the deep MOND regime. MOND yields the order of magnitude for the expected baryonic mass only after one past encounter, assuming MOND does not include dark matter. While we only consider the LG as two point masses, our calculations provide a benchmark for future work with simulations to test the effect of the finite size of galaxies and tidal fields due to the neighbouring structures. This model can be also used to test screening mechanisms and alternative theories of gravity.