The abundances of dark matter haloes in the universe are described by the halo mass function (HMF). It enters most cosmological analyses and parametrizes how the linear growth of primordial perturbations is connected to these abundances. Interestingly, this connection can be made approximately cosmology independent. This made it possible to map in detail its near-universal behaviour through large-scale simulations. However, such simulations may suffer from systematic effects, especially if baryonic physics is included. In this paper, we ask how well observations can constrain directly the HMF. The observables we consider are galaxy cluster number counts, galaxy cluster power spectrum and lensing of Type Ia supernovae. Our results show that Dark Energy Survey is capable of putting the first meaningful constraints on the HMF, while both Euclid and J-PAS (Javalambre-Physics of the Accelerated Universe Astrophysical Survey) can give stronger constraints, comparable to the ones from state-of-the-art simulations. We also find that an independent measurement of cluster masses is even more important for measuring the HMF than for constraining the cosmological parameters, and can vastly improve the determination of the HMF. Measuring the HMF could thus be used to cross-check simulations and their implementation of baryon physics. It could even, if deviations cannot be accounted for, hint at new physics.
