Galaxy clusters are luminous tracers of the most massive dark matter haloes in the Universe. To use them as a cosmological probe, a detailed description of the properties of dark matter haloes is required. We characterize how the dynamical state of haloes impacts the halo mass function at the high-mass end. We used the dark matter-only MultiDark suite of simulations and the high-mass objects M > 2.7e13 M/h therein. We measured mean relations of concentration, offset, and spin as a function of halo mass and redshift. We investigated the distributions around the mean relations. We measured the halo mass function as a function of offset, spin, and redshift. We formulated a generalized mass function framework that accounts for the dynamical state of the dark matter haloes. We confirm the discovery of the concentration upturn at high masses and provide a model that predicts the concentration for different values of mass and redshift with one single equation. We model the distributions around the mean concentration, offset, and spin with modified Schechter functions. The concentration of low-mass haloes shows a faster redshift evolution compared to high-mass haloes, especially in the high-concentration regime. The offset parameter is smaller at low redshift, in agreement with the relaxation of structures at recent times. The peak of its distribution shifts by a factor of 1.5 from z = 1.4 to z = 0. The individual models are combined into a comprehensive mass function model, as a function of spin and offset. Our model recovers the fiducial mass function with 3% accuracy at redshift 0 and accounts for redshift evolution up to z = 1.5. This approach accounts for the dynamical state of the halo when measuring the halo mass function. It offers a connection with dynamical selection effects in galaxy cluster observations. This is key toward precision cosmology using cluster counts as a probe.
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