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Initial-Final Mass Relation for 3 to 4 M$_\odot$ Progenitors of White Dwarfs from the Single Cluster NGC 2099

We have expanded the sample of observed white dwarfs in the rich open cluster NGC 2099 (M37) with the Keck Low-Resolution Imaging Spectrometer. Of 20 white dwarf candidates, the spectroscopy shows 19 to be true white dwarfs with 14 of these having high S/N. We find 11 of these 14 to be consistent with singly evolved cluster members. They span a mass range of $\sim$0.7 to 0.95 M$_\odot$, excluding a low-mass outlier, corresponding to progenitor masses of $\sim$3 to 4 M$_\odot$. This region of the initial final mass relation (IFMR) has large scatter and a slope that remains to be precisely determined. With this large sample of white dwarfs that belong to a single age and metallicity population, we find an initial-final mass relation of (0.171$\pm$0.057)M$_{\rm initial}$+0.219$\pm$0.187 M$_\odot$, significantly steeper than the linear relation adopted over the full observed white dwarf mass range in many previous studies. Comparison of this new relation from the solar metallicity NGC 2099 to 18 white dwarfs in the metal-rich Hyades and Praesepe shows that their IFMR also has a consistently steep slope. This strong consistency also suggests that there is no significant metallicity dependence of the IFMR at this mass and metallicity range. As a result, the IFMR can be more reliably determined with this broad sample of 29 total white dwarfs giving M$_{\rm final}$=(0.163$\pm$0.022)M$_{\rm initial}$+0.238$\pm$0.071 M$_\odot$ from M$_{\rm initial}$ of 3 to 4 M$_\odot$. A steep IFMR in this mass range indicates that the full IFMR is nonlinear.