In this paper we investigate the potential of current and upcoming cosmological surveys to constrain the mass and abundance of ultra-light axion (ULA) cosmologies with galaxy cluster number counts. ULAs, sometimes also referred to as Fuzzy Dark Matter, are well-motivated in many theories beyond the Standard Model and could potentially solve the $Lambda$CDM small-scale crisis. Galaxy cluster counts provide a robust probe of the formation of structures in the Universe. Their distribution in mass and redshift is strongly sensitive to the underlying linear matter perturbations. In this forecast paper we explore two scenarios, firstly an exclusion limit on axion mass given a no-axion model and secondly constraints on an axion model. With this we obtain lower limits on the ULA mass on the order of $m_a gtrsim 10^{-24}$ eV. However, this result depends heavily on the mass of the smallest reliably observable clusters for a given survey. Cluster counts, like many other cosmological probes, display an approximate degeneracy in the ULA mass vs. abundance parameter space, which is dependent on the characteristics of the probe. These degeneracies are different for other cosmological probes. Hence galaxy cluster counts might provide a complementary window on the properties of ultra-light axions.
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