A non-parametric approach to the relation between the halo mass function and internal dark matter structure of haloes

Context. Galaxy cluster masses are usually defined as the mass within a spherical region enclosing a given matter overdensity (in units of the critical density). Converting masses from one overdensity definition to another can have several useful applications. Aims. In this article we present a generic non-parametric formalism that allows one to accurately map the halo mass function between two different mass overdensity definitions using the distribution of halo sparsities defined as the ratio of the two masses. We show that changing mass definitions reduces to modelling the distribution of halo sparsities. Methods. Using standard transformation rules of random variates, we derive relations between the halo mass function at different overdensities and the distribution of halo sparsities. Results. We show that these relations reproduce the N-body halo mass functions from the Uchuu simulation within the statistical errors at a few percent level. Furthermore, these relations allow the halo mass functions at different overdensities to be related to parametric descriptions of the halo density profile. In particular, we discuss the case of the concentration-mass relation of the Navarro-Frenk-White profile. Finally, we show that the use of such relations allows us to predict the distribution of sparsities of a sample of haloes of a given mass, thus opening the way to inferring cosmological constraints from individual galaxy cluster sparsity measurements