Impact of merger histories on the timing argument estimate of the Local Group mass

The timing argument (TA) aims to find the total mass of the Local Group (LG) from the relative motions of the Milky Way (MW) and Andromeda galaxy (M31). However, the classical TA always overestimates the LG mass, presumably because it does not account for the hierarchical scenario and other interactions, such as that with the Large Magellanic Cloud (LMC). We focused on the impact of M31's recent major merger by using three merger models to find the peculiar motion of M31 within the simple two-body, point-mass scenario of the TA. We find that the merger correction affects the TA mass by plus or minus 10-15% depending on the M31 tangential motion, which has very large uncertainties. If we consider an M31 merger configuration that reduces the TA mass by 10-15%, to which we add the impact due to the LMC infall into the MW as reported in the literature, the TA mass is consistent with the LG mass from Hubble-Lema & icirc;tre flow. Galaxies are believed to have experienced about 16 major mergers each since z = 11.5. Assuming all these mergers had a similar impact on the TA mass as the most recent M31 merger, the ratio of LG mass to TA mass would be 0.85-0.37+0.65 0 . 85 - 0.37 + 0.65 $ 0.85<^>{+0.65}_{-0.37} $ , and such a TA mass is consistent with all the LG mass estimates. Our result also agrees with findings that used LG analogues in cosmological simulations. We find that the TA mass estimate is limited by the hierarchical scenario, since it is not possible to track the progenitors of both the MW and M31 through so many mergers. We conclude that the MW-M31 dynamical system is far too complex to be modelled as a simple two-body point-mass system.