Timing argument take on the Milky Way and Andromeda past encounter

The two-body problem of the M31 and Milky Way (MW) galaxies with a cosmological constant background is studied, with an emphasis on the possibility that they experienced a past encounter (PE). Past encounters are possible only for a nonzero transverse velocity, and their viability is subject to observations of the imprints of such near collisions. By implementing the timing argument (TA) for two isolated point bodies, it is shown that if M31 and the MW had experienced a PE, then the predicted mass of the Local Group (LG) would be twice larger. This predicted mass is too large and implies that the MW and M31 galaxies would have collided at similar to 8 Gys. Therefore, the TA analysis shows that a PE is not possible for the LG system.