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Context. While the halo mass function is theoretically a very sensitivemeasure of cosmological models, masses of dark-matter halos are poorly defined,global, and unobservable quantities.  Aims. We argue that local, observable quantities such as the X-raytemperatures of galaxy clusters can be directly compared to theoreticalpredictions without invoking masses. We derive the X-ray temperature functiondirectly from the statistics of Gaussian random fluctuations in thegravitational potential.  Methods. We derive the abundance of potential minima constrained by therequirement that they belong to linearly collapsed structures. We then use thespherical-collapse model to relate linear to non-linear perturbations, and thevirial theorem to convert potential depths to temperatures. No reference ismade to mass or other global quantities in the derivation.  Results. Applying a proper high-pass filter that removes large enough modesfrom the gravitational potential, we derive an X-ray temperature function thatagrees very well with the classical Press-Schechter approach on relevanttemperature scales, but avoids the necessity of measuring masses.  Conclusions. TThis first study shows that and how an X-ray temperaturefunction of galaxy clusters can be analytically derived, avoiding theintroduction of poorly defined global quantities such as halo masses. Thisapproach will be useful for reducing scatter in observed cluster distributionsand thus in cosmological conclusions drawn from them.

Statistics of gravitational potential perturbations: A novel approach to deriving the X-ray temperature function