Merger-associated mass accretion rates as a function of infall time and mass ratio (μ). For each bin, the peak masses of the infalling objects are summed and normalised by the total merger-associated mass accretion for the respective halo (MW or M31). This normalised mass accretion rate (R) is then averaged over simulations, with the resulting values shown by the colours. Left panel: This quantity for the MW. Right panel: Difference between the MW and M31. The histograms at the bottom represent the same quantity but marginalised over mass ratio, showing the 1D distribution as a function of infall time.