We use ZFIRE and ZFOURGE observations with the Spectral Energy Distribution (SED) fitting tool Prospector to reconstruct the star formation histories (SFHs) of proto-cluster and field galaxies at $zsim 2 $ and compare our results to the TNG100 run of the IllustrisTNG cosmological simulation suite. In the observations, we find that massive proto-cluster galaxies ($log[{rm M}_{ast}/{rm M}_{odot}]>$10.5) form $45 pm 8 %$ of their total stellar mass in the first $2$ Gyr of the Universe compared to $31 pm 2 %$ formed in the field galaxies. In both observations and simulations, massive proto-cluster galaxies have a flat/declining SFH with decreasing redshift compared to rising SFH in their field counterparts. Using IllustrisTNG, we find that massive galaxies ($log[{rm M}_{ast}/{rm M}_{odot}] geq 10.5$) in both environments are on average $approx190$ Myr older than low mass galaxies ($log[{rm M}_{ast}/{rm M}_{odot}]= 9-9.5$). However, the difference in mean stellar ages of cluster and field galaxies is minimal when considering the full range in stellar mass ($log[{rm M}_{ast}/{rm M}_{odot}] geq 9$). We explore the role of mergers in driving the SFH in IllustrisTNG and find that massive cluster galaxies consistently experience mergers with low gas fraction compared to other galaxies after 1 Gyr from the Big Bang. We hypothesize that the low gas fraction in the progenitors of massive cluster galaxies is responsible for the reduced star formation.
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