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tokens: for extracting information from the phase-space spiral. For example, Li Widrow (2021) infer the gravitational potential of the disk using the bulk stellar density in the ( z , w )-plane and then fit the spiral to the residual of the inferred steady state distribution. The fitted spiral constitutes a consistency check of the inferred potential and is seen as a straight line in the plane of angle and frequency. While our method is accurate in terms of one-dimensional simulations, the actual Milky Way is significantly more complex. To begin with, stars in the Galactic disk do not only oscillate in the vertical direction, but also move in the radial direction as they orbit the Galactic centre. Indeed, the phase-space spiral of the Milky Way has a different form depending on cuts in angular momentum or the galactocentric radius (see for example Fig. 14 in Laporte et al. 2019 , as well as Khanna et al. 2019 ). When applying our method to Gaia data for the first time, we plan to select stars of the solar neighbourhood that have approximately circular orbits, for which the assumptions of our model are most likely to be valid. It would be interesting, although probably quite complicated, to see what kind of information about the global structure of the Milky Way could be extracted from the shape of the spiral in different bins of Galactic angular momentum. Ultimately, we would want to test and refine our method using a high-resolution simulation of a galaxy in three spatial dimensions. Up until now, simulations of Milky Way-like systems (such as Laporte et al. 2018 and Bland-Hawthorn Tepper-García 2021 ) have not had sufficient resolution to produce well-resolved phase-space spirals in stellar density in the ( z , w )-plane. To do so requires billion-particle simulations, which have only recently become feasible ( Asano et al. 2020 , Hunt et al. in prep.). Furthermore, our modelling of the gravitational potential needs to be expanded somewhat; the Poisson equation is no longer one-dimensional but also has a smaller contribution from the radial direction (the so-called rotation curve term). A useful test in terms of systematic uncertainties would be to perform separate analyses of different segments of the phase-space spiral. The assumptions of our analytical model are probably only valid over a certain range in vertical energy; fitting the spiral over a large area in the z , w -plane would give a small statistical uncertainty, but potentially at the price of large systematic errors. It would be possible, at least to some extent, to quantify systematic errors by comparing the results of the smaller, separate spiral segments. Another test in a similar vein would be to abandon the form of the function as defined in Eq. ( 9 ), which might not be completely accurate due to effects of, for example, self-gravity; instead, we could simply assume that the spiral angle is a smooth function with respect to E z and model it as a Gaussian process. Observational