The impact cratering record of the Moon and the terrestrial planets provides important clues about the formation and evolution of the Solar System. Especially intriguing is the epoch ≃3.8-3.9 Gyr ago (Ga), known as the Late Heavy Bombardment (LHB), when the youngest lunar basins such as Imbrium and Orientale formed. The LHB was suggested to originate from a slowly declining impactor flux or from a late dynamical instability. Here we develop a model for the historical flux of large asteroid impacts and discuss how it depends on various parameters, including the time and nature of the planetary migration/instability. We find that the asteroid impact flux dropped by 1 to 2 orders of magnitude during the first 1 Gyr and remained relatively unchanged over the last 3 Gyr. The early impacts were produced by asteroids whose orbits became excited during the planetary migration/instability, and by those originating from the inner extension of the main belt (E-belt; semimajor axis 1.6 < a < 2.1 au). The inner main belt dominated the asteroid impact record after ∼3-4 Ga. The profiles obtained for the early and late versions of the planetary instability initially differ, but end up being similar after ∼3 Ga. Thus, the time of the instability can only be determined by considering the cratering and other constraints during the first ≃1.5 Gyr of the Solar System history. Our absolute calibration of the impact flux indicates that asteroids were probably not responsible for the LHB, independently of whether the instability happened early or late, because the calibrated flux is not large enough to explain Imbrium/Orientale and a significant share of large lunar craters. Comets and leftovers of the terrestrial planet formation provided additional, and probably dominant source of impacts during early epochs. The cometary impacts occur during a narrow interval after the instability and would not be recorded on surfaces if the planetary instability happened early.-2 -

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