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The universality of the halo mass function is investigated in the context of dark energy cosmologies. This widely used approximation assumes that the mass function can be expressed as a function of the matter density  Ω m   and the root‐mean‐square linear density fluctuation  σ  only, with no explicit dependence on the properties of dark energy or redshift. In order to test this hypothesis, we run a series of 15 high‐resolution  N ‐body simulations for different cosmological models. These consist of three  Λ  cold dark matter cosmologies’ best‐fitting  Wilkinson Microwave Anisotropy Probe  1‐, 3‐ and 5‐yr data, which are used for model comparison, and three toy models characterized by a Ratra–Peebles quintessence potential with different slopes and amounts of dark energy density. These toy models have very different evolutionary histories at the background and linear levels, but share the same  σ 8   value. For each of these models, we measure the mass function from catalogues of haloes identified in the simulations using the Friend‐of‐Friend (FoF) algorithm. We find redshift‐dependent deviations from a universal behaviour, well above numerical uncertainties and of non‐stochastic origin, which are correlated with the linear growth factor of the investigated cosmologies. Using the spherical collapse as guidance, we show that such deviations are caused by the cosmology dependence of the non‐linear collapse and virialization process. For practical applications, we provide a fitting formula of the mass function accurate to 5 per cent for all investigated cosmologies. We also derive an empirical relation between the FoF linking parameter and the virial overdensity which can account for most of the deviations from an exact universal behaviour. Overall, these results suggest that measurements of the halo mass function at   z = 0  can provide additional constraints on dark energy since it carries a fossil record of the past cosmic evolution.

Imprints of dark energy on cosmic structure formation – II. Non‐universality of the halo mass function