The stellar mass - size relation for cluster galaxies at z=1 with high angular resolution from the Gemini/GeMS multi-conjugate adaptive optics system

We present the stellar mass - size relation for 49 galaxies within the $z$ = 1.067 cluster SPT-CL J0546$-$5345, with FWHM $\sim$80-120 mas $K_{\mathrm s}$-band data from the Gemini multi-conjugate adaptive optics system (GeMS/GSAOI). This is the first such measurement in a cluster environment, performed at sub-kpc resolution at rest-frame wavelengths dominated by the light of the underlying old stellar populations. The observed stellar mass - size relation is offset from the local relation by 0.21 dex, corresponding to a size evolution proportional to $(1+z)^{-1.25}$, consistent with the literature. The slope of the stellar mass - size relation $\beta$ = 0.74 $\pm$ 0.06, consistent with the local relation. The absence of slope evolution indicates that the amount of size growth is constant with stellar mass. This suggests that galaxies in massive clusters such as SPT-CL J0546$-$5345 grow via processes that increase the size without significant morphological interference, such as minor mergers and/or adiabatic expansion. The slope of the cluster stellar mass - size relation is significantly shallower if measured in $HST$/ACS imaging at wavelengths blueward of the Balmer break, similar to rest-frame UV relations at $z$ = 1 in the literature. The stellar mass - size relation must be measured at redder wavelengths, which are more sensitive to the old stellar population that dominates the stellar mass of the galaxies. The slope is unchanged when GeMS $K_s$-band imaging is degraded to the resolution of $K$-band HST/NICMOS resolution but dramatically affected when degraded to $K_s$-band Magellan/FourStar resolution. Such measurements must be made with AO in order to accurately characterise the sizes of compact, $z$ = 1 galaxies.