The dark matter halo sparsity provides a direct observational proxy of the halo mass profile, characterizing halos in terms of the ratio of masses within radii which enclose two different overdensities. Previous numerical simulation analyses have shown that at a given redshift the halo sparsity carries cosmological information encoded in the halo mass profile. Moreover, its ensemble averaged value can be inferred from prior knowledge of the halo mass function at the overdensities of interest. Here, we present a detailed study of the ensemble average properties of the halo sparsity. In particular, using halo catalogs from high-resolution N-body simulations, we show that its ensemble average value can be estimated from the ratio of the averages of the inverse halo masses as well as the ratio of the averages of the halo masses at the overdensity of interests. This can be relevant for galaxy clusters data analyses. As an example, we have estimated the average sparsity properties of galaxy clusters from the LoCuSS and HIFLUGCS datasets respectively. The results suggest that the expected consistency of the different average sparsity estimates can provide a test of the robustness of mass measurements in galaxy cluster samples.

Large samples of galaxy clusters provide knowledge of both astrophysics in the most massive virialised environments and the properties of the cosmological model that defines our Universe. However, an important issue that affects the interpretation of galaxy cluster samples is the role played by the selection waveband and the potential for this to introduce a bias in the physical properties of clusters thus selected. We aim to investigate waveband-dependent selection effects in the identification of galaxy clusters by comparing the X-ray Multi-Mirror (XMM) Ultimate Extra-galactic Survey (XXL) and Subaru Hyper Suprime-Cam (HSC) CAMIRA cluster samples identified from a common 22.6 deg2 sky area. We compare 150 XXL and 270 CAMIRA clusters in a common parameter space defined by X-ray aperture brightness and optical richness. We find that 71/150 XXL clusters are matched to the location of a CAMIRA cluster, the majority of which (67/71) display richness values N > 15 that exceed the CAMIRA catalogue richness threshold. We find that 67/270 CAMIRA clusters are matched to the location of an XXL cluster (defined within XXL as an extended X-ray source). Of the unmatched CAMIRA clusters, the majority display low X-ray fluxes consistent with the lack of an XXL counterpart. However, a significant fraction (64/107) CAMIRA clusters that display high X-ray fluxes are not asociated with an extended source in the XXL catalogue. We demonstrate that this disparity arises from a variety of effects including the morphological criteria employed to identify X-ray clusters and the properties of the XMM PSF.