ZFIRE: Similar Stellar Growth in H$alpha$-emitting Cluster and Field Galaxies at z~2

We compare galaxy scaling relations as a function of environment at $zsim2$ with our ZFIRE survey where we have measured H$alpha$ fluxes for 90 star-forming galaxies selected from a mass-limited [$log(M_{star}/M_{odot})>9$] sample based on ZFOURGE. The cluster galaxies (37) are part of a confirmed system at z=2.095 and the field galaxies (53) are at $1.9<z<2.4$; all are in the COSMOS legacy field. There is no statistical difference between H$alpha$-emitting cluster and field populations when comparing their star formation rate (SFR), stellar mass ($M_{star}$), galaxy size ($r_{eff}$), SFR surface density [$Sigma$(H$alpha_{star}$)], and stellar age distributions. The only difference is that at fixed stellar mass, the H$alpha$-emitting cluster galaxies are $log(r_{eff})sim0.1$ larger than in the field. Approximately 19% of the H$alpha$-emitters in the cluster and 26% in the field are IR-luminous ($L_{IR}>2times10^{11} L_{odot}$). Because the LIRGs in our combined sample are $sim5$ times more massive than the low-IR galaxies, their radii are $sim70$% larger. To track stellar growth, we separate galaxies into those that lie above, on, and below the H$alpha$ star-forming main sequence (SFMS) using $Delta$SFR$(M_{star})=pm0.2$ dex. Galaxies above the SFMS (starbursts) tend to have higher H$alpha$ SFR surface densities and younger light-weighted stellar ages compared to galaxies below the SFMS. Our results indicate that starbursts (+SFMS) in the cluster and field at $zsim2$ are growing their stellar cores. Lastly, we compare to the (SFR-$M_{star}$) relation from RHAPSODY cluster simulations and find the predicted slope is nominally consistent with the observations. However, the predicted cluster SFRs tend to be too low by a factor of $sim2$ which seems to be a common problem for simulations across environment.