Relations Between the Sizes of Galaxies and their Dark Matter Halos at Redshifts $0 < z < 3$

We derive relations between the effective radii $R_{\rm{eff}}$ of galaxies and the virial radii $R_{200c}$ of their dark matter halos over the redshift range $0 < z < 3$. For galaxies, we use the measured sizes from deep images taken with \emph{Hubble Space Telescope} for the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey; for halos, we use the inferred sizes from abundance matching to cosmological dark matter simulations via a stellar mass--halo mass (SMHM) relation. For this purpose, we derive a new SMHM relation based on the same selection criteria and other assumptions as for our sample of galaxies with size measurements. As a check on the robustness of our results, we also derive $R_{\rm{eff}}$--$R_{200c}$ relations for three independent SMHM relations from the literature. We find that galaxy $R_{\rm{eff}}$ is proportional on average to halo $R_{200c}$, confirming and extending to high redshifts the $z=0$ results of Kravtsov. Late-type galaxies (with low S\'ersic index and high specific star formation rate [sSFR]) follow a linear $R_{\rm{eff}}$--$R_{200c}$ relation, with effective radii at $0.5 < z < 3$ close to those predicted by simple models of disk formation; at $z < 0.5$, the sizes of late-type galaxies appear to be slightly below this prediction. Early-type galaxies (with high S\'ersic index and low sSFR) follow a roughly parallel $R_{\rm{eff}}$--$R_{200c}$ relation, $\sim$ 0.2--0.3 dex below the one for late-type galaxies. Our observational results, reinforced by recent hydrodynamical simulations, indicate that galaxies grow quasi-homologously with their dark matter halos.