Constraints on the Richness-Mass Relation and the Optical-SZE Positional Offset Distribution for SZE-Selected Clusters

We cross-match galaxy cluster candidates selected via their Sunyaev-Zeldovich effect (SZE) signatures in 129.1 deg$^2$ of the South Pole Telescope 2500d SPT-SZ survey with optically identified clusters selected from the Dark Energy Survey (DES) science verification data. We identify 25 clusters between $0.1lesssim zlesssim 0.8$ in the union of the SPT-SZ and redMaPPer (RM) samples. RM is an optical cluster finding algorithm that also returns a richness estimate for each cluster. We model the richness $lambda$-mass relation with the following function $langlelnlambda|M_{500}ranglepropto B_lambdaln M_{500}+C_lambdaln E(z)$ and use SPT-SZ cluster masses and RM richnesses $lambda$ to constrain the parameters. We find $B_lambda= 1.14^{+0.21}_{-0.18}$ and $C_lambda=0.73^{+0.77}_{-0.75}$. The associated scatter in mass at fixed richness is $sigma_{ln M|lambda} = 0.18^{+0.08}_{-0.05}$ at a characteristic richness $lambda=70$. We demonstrate that our model provides an adequate description of the matched sample, showing that the fraction of SPT-SZ selected clusters with RM counterparts is consistent with expectations and that the fraction of RM selected clusters with SPT-SZ counterparts is in mild tension with expectation. We model the optical-SZE cluster positional offset distribution with the sum of two Gaussians, showing that it is consistent with a dominant, centrally peaked population and a sub-dominant population characterized by larger offsets. We also cross-match the RM catalog with SPT-SZ candidates below the official catalog threshold significance $xi=4.5$, using the RM catalog to provide optical confirmation and redshifts for additional low-$xi$ SPT-SZ candidates.In this way, we identify 15 additional clusters with $xiin [4,4.5]$ over the redshift regime explored by RM in the overlapping region between DES science verification data and the SPT-SZ survey.