The Relationship between Galaxy and Dark Matter Halo Size from z~3 to the present

We explore empirical constraints on the statistical relationship between the radial size of galaxies and the radius of their host dark matter halos from $z\sim 0.1$--3 using the GAMA and CANDELS surveys. We map dark matter halo mass to galaxy stellar mass using relationships from abundance matching, applied to the Bolshoi-Planck dissipationless N-body simulation. We define SRHR$\equiv r_e/R_h$ as the ratio of galaxy radius to halo virial radius, and SRHR$\lambda \equiv r_e/(\lambda R_h)$ as the ratio of galaxy radius to halo spin parameter times halo radius. At $z\sim 0.1$, we find an average value of SRHR $\simeq 0.018$ and SRHR$\lambda \simeq 0.5$ with very little dependence on stellar mass. SRHR and SRHR$\lambda$ have a weak dependence on cosmic time since $z\sim 3$. SRHR shows a mild decrease over cosmic time for low mass galaxies, but increases slightly or does not evolve for more massive galaxies. We find hints that at high redshift ($z\sim 2$--3), SRHR$\lambda$ is lower for more massive galaxies, while it shows no significant dependence on stellar mass at $z\lesssim 0.5$. We find that for both the GAMA and CANDELS samples, at all redshifts from $z\sim 0.1$--3, the observed conditional size distribution in stellar mass bins is remarkably similar to the conditional distribution of $\lambda R_h$. We discuss the physical interpretation and implications of these results.