Quantifying the Rarity of the Local Super-Volume

We investigate the extent to which the number of clusters of mass exceeding ${10}^{15}{M}_{\odot }{h}^{-1}$ within the local super-volume ($<135\mathrm{M}\mathrm{p}\mathrm{c}{h}^{-1}$) is compatible with the standard $\mathrm{\Lambda }$CDM cosmological model. Depending on the mass estimator used, we find that the observed number $N$ of such massive structures can vary between $0$ and $5$. Adopting $N=5$ yields $\mathrm{\Lambda }$CDM likelihoods as low as $2.4\times {10}^{-3}$ (with ${\sigma }_8=0.81$) or $3.8\times {10}^{-5}$ (with ${\sigma }_8=0.74$). However, at the other extreme ($N=0$), the likelihood is of order unity. Thus, while potentially very powerful, this method is currently limited by systematic uncertainties in cluster mass estimates. This motivates efforts to reduce these systematics with additional observations and improved modelling.