Mass-Richness relations for X-ray and SZE-selected clusters at 0.4 < z <2.0 as seen by Spitzer at 4.5μm

We study the mass-richness relation of 116 spectroscopically-confirmed massive clusters at $0.4 < z < 2$ by mining the $Spitzer$ archive. We homogeneously measure the richness at 4.5$\mu$m for our cluster sample within a fixed aperture of $2^{\prime}$ radius and above a fixed brightness threshold, making appropriate corrections for both background galaxies and foreground stars. We have two subsamples, those which have a) literature X-ray luminosities and b) literature Sunyaev-Zeldovich effect masses. For the X-ray subsample we re-derive masses adopting the most recent calibrations. We then calibrate an empirical mass-richness relation for the combined sample spanning more than one decade in cluster mass and find the associated uncertainties in mass at fixed richness to be $\pm 0.25$ dex. We study the dependance of the scatter of this relation with galaxy concentration, defined as the ratio between richness measured within an aperture radius of 1 and 2 arcminutes. We find that at fixed aperture radius the scatter increases for clusters with higher concentrations. We study the dependance of our richness estimates with depth of the [4.5]$\mu$m imaging data and find that reaching a depth of at least [4.5]= 21 AB mag is sufficient to derive reasonable mass estimates. We discuss the possible extension of our method to the mid-infrared $WISE$ all-sky survey data, and the application of our results to the $Euclid$ mission. This technique makes richness-based cluster mass estimates available for large samples of clusters at very low observational cost.