tokens: Thus, while large-scale structure for different types of DM can appear identical, careful attention to structure on subgalactic scales can be a powerful tool to distinguish DM models. A powerful probe of the gravitationally bound structures of dark matter is strong gravitational lensing. Given a matter over or under density, the deflection angle along the line of sight is given by an integral over the induced gravitational potential (Narayan Bartelmann 1997 ),where χ is the distance along the line of sight, χ s is the distance to the source, and is the gravitational potential. In the thin lens approximation (Narayan Bartelmann 1997 ), valid in the limit that the thickness of the lensing galaxy is small compared to the distance to the lens, this takes a simplified form,where D LS , D L , and D S are the angular diameter distances from the lens to the source, from the observer to the lens, and from the observer to the source, respectively, and z is the distance along the line of sight. From this expression one can straightforwardly compute the lensing due to any gravitational potential Ψ. The gravitational potential is in turn determined by matter density via the Poisson equation, ∇ 2 Ψ ∝ ρ . The linearity of this equation implies that the total lensing due to the separate contributions, e.g., of a halo and halo substructure, is simply the sum of the individual contributions. That is,where is the external shear due to large-scale structure, and are the lensing due to the halo and halo substructure respectively. The well studied case is the lensing due to the spherical substructures expected from hierarchical structure formation in the context of noninteracting particle dark matter. However, as mentioned in the 1 , other types of substructure can exist in models of dark matter outside the WIMP paradigm. As a prototypical example, we will consider dark matter condensates, namely superfluids, which exhibit substructure in the form of vortices. We now proceed to develop this in detail. 3. Case Study: Dark Matter Superfluidity The canonical example of a condensate dark matter model is axion dark matter. Axions were introduced as a solution to the strong-CP problem of the SM (Peccei Quinn 1977 ; Weinberg 1978 ; Wilczek 1978 ), and soon thereafter proposed as a dark matter candidate (Abbott Sikivie 1983 ; Dine Fischler 1983 ; Preskill et al. 1983 ). It was later argued that axions could form a Bose–Einstein condensate and exhibit superfluidity (Sin 1994 ; Hu et al. 2000 ; Silverman Mallett 2002 ; Hui et al. 2017 ; Sikivie Yang 2009 ; Berezhiani Khoury 2015 ; Ferreira et al. 2019 ). To emphasize the superfluid nature of axions, we can rewrite the field equations in terms of fluid equations. Being comprised of extremely light particles at an incredibly high number density, axion dark matter is well described by a coherent scalar field. Moreover, because dark matter as we observe it is cold, the system is well described by a nonrelativistic
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