The timeless timing argument and the total mass of the Local Group

The timing argument connects the motion of a two-body system to its mass in an expanding Universe with a finite age, under the assumption that it has evolved on a self-gravitating orbit. It is commonly applied to the present-day Milky Way (MW)-M31 system in order to infer its unknown mass from the measured kinematics. We use a set of Local Group analogues from the uchuu simulation to investigate the timing argument over cosmic time. We find that the median inferred mass remains almost constant over the past 12Gyr, even while the haloes themselves grew in mass by more than an order of magnitude. By contrast, we find a closer, and nearly time-invariant agreement between the timing argument value and the mass within a sphere of radius equal to the MW-M31 separation, and we identify this as the total mass of the system. We conclude that the comparatively close present-day agreement between the timing argument and the sum of the halo masses reflects no underlying relation, but merely echoes the fact that the MW and M31 now contain most (but not all) of the mass of the Local Group system.