A model for multiproperty galaxy cluster statistics

The massive dark matter haloes that host groups and clusters of galaxies have observable properties that appear to be lognormally distributed about power-law mean scaling relations in halo mass. Coupling this assumption with either quadratic or cubic approximations to the mass function in log space, we derive closed-form expressions for the space density of haloes as a function of multiple observables as well as forms for the low-order moments of properties of observable-selected samples. Using a Tinker mass function in a Λ cold dark matter cosmology, we show that the cubic analytic model reproduces results obtained from direct, numerical convolution at the 10 per cent level or better over nearly the full range of observables covered by current observations and for redshifts extending to z = 1.5. The model provides an efficient framework for estimating effects arising from selection and covariance among observable properties in survey samples.