In the attempt to understand the formation ages of large lunar craters (diameters of 80-100 km), Kirchoff et al.  measured and used the cumulative density of 0.5-10 km diameter craters on the floors of the larger craters. In this work, an effort was made to minimize secondary craters by focusing on young surfaces and mapping crater clusters and chains. Therefore, these data provide new constraints on the shape of the primary impactor size-frequency distribution (SFD) in the time frame ranging from ~1 Ga to ~3 Ga. The characteristics of these crater SFDs were found to change very little in this time frame and indicate that the shape of the impactor population was relatively stable over time. Furthermore, agreement between these data and a cratering production function based on current near-Earth objects models  implies that the shape of the near-Earth objects SFD has remained relatively unchanged for the last ~3 Ga.
Independently, recent sub-kilometer-diameter crater populations measured at Vesta  provide some of the best insights to date on the shape of the small main belt SFD, ranging from ~10 m to ~1 km diameter (or even larger sizes). These objects are the principal reservoirs, through delivery to the NEO population by the size-dependent Yarkovsky effect , of the small lunar impactors responsible for the observed 0.5-10 km lunar craters. Our work suggests that the small main belt asteroids SFD agrees with a collisional model of the main belt  and that its shape was roughly constant for the last ~3 Ga.
Our observations do not constrain (yet) the full temporal behavior of the lunar impactor flux, and it is possible that it may have oscillated by a factor of 2, as suggested by previous work [2,6] as a result of major breakups of large asteroids in the main belt. However, altogether, these results provide strong support for current age determination models [e.g. 2] and that the shape of the small lunar impactor SFD was relatively constant in the past ~3 Ga. This indicates that the formation of main belt families did not drastically alter the flux of the smallest members of the lunar impactor population over long intervals.
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