Statistics of structures in the gravitational potential - a possible way to constrain halo populations without reference to mass

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The distribution of dark-matter halos with mass is one of the central predictions of the standard cosmological model. The Press-Schechter mass function or variants thereof predict the mass function of dark-matter halos. Although based on questionably simple assumptions, numerical simulations confirm it with astonishing accuracy. The mass function of real dark-matter halos is in principle simple and in practice very difficult to measure. The main problem is that mass is not an astronomical observable. Mass is inferred from kinematic or dynamical observables such as the motion of tracers, the temperature of hot gas, or from light deflection by gravitational lensing. None of these observables measures mass, but rather the depth of the gravitational potential or derivatives thereof. Mass is obtained only after integrating local information within some volume which is typically arbitrarily defined, and whose centre is typically uncertain. Proxies to the mass, such as X-ray temperature or luminosity, help practically to some degree, but do not conceptually improve the situation. Besides, they introduce calibration uncertainties. The question thus arises whether distribution functions of observable quantities can directly be predicted without reference to mass. Having completed a successful pilot study, we propose to investigate distribution functions of structures in the gravitational potential, which are directly related to local observables such as temperatures, velocity dispersions or gravitational-lensing effects. The statistics of potential sinks of sufficient depth and size can be derived from the theory of Gaussian random fields, allowing the direct prediction of, e.g., X-ray temperature functions which agree very well with the observations at least at the high-mass end. Within the proposed project, we wish to extend and improve our prototype method such that distribution functions of several observables can directly be predicted on all scales.