Charting galactic accelerations: when and how to extract a unique potential from the distribution function

Abstract

The advent of datasets of stars in the Milky Way with six-dimensional phase-space information makes it possible to construct empirically the distribution function (DF). Here, we show that the accelerations can be uniquely determined from the DF using the collisionless Boltzmann equation, providing the Hessian determinant of the DF with respect to the velocities is non-vanishing. We illustrate this procedure and requirement with some analytic examples. Methods to extract the potential from datasets of discrete positions and velocities of stars are then discussed. Following Green & Ting, we advocate the use of normalizing flows on a sample of observed phase-space positions to obtain a differentiable approximation of the DF. To then derive gravitational accelerations, we outline a semi-analytic method involving direct solutions of the over-constrained linear equations provided by the collisionless Boltzmann equation. Testing our algorithm on mock datasets derived from isotropic and anisotropic Hernquist models, we obtain excellent accuracies even with added noise. Our method represents a new, flexible and robust means of extracting the underlying gravitational accelerations from snapshots of six-dimensional stellar kinematics of an equilibrium system.

Snowmass2021 Cosmic Frontier White Paper: Dark Matter Physics from Halo Measurements

“…There is progress in relaxing these assumptions of a steady state and Galactic symmetries, which will be necessary to make further progress in extracting information about DM microphysics from galactic dynamics using Gaia (noting again that Gaia data is precisely what has allowed us to uncover the deviations from these assumptions). For example, the full three-dimensional gravitational potential can analysed with a method using "normalising flows" [355], which is model independent in terms of symmetry assumptions. It is also possible to relax the assumption of a steady state and extract information from the shape of time-varying dynamical structures of the Milky Way; this is exemplified by a new method [354,356] that infers the gravitational potential using the recently discovered phase-space spiral [357].…”

Charting galactic accelerations II: how to 'learn' accelerations in the solar neighbourhood

“…A novel procedure for calculating gravitational accelerations from stellar kinematical data was introduced in An et al (2021). In this article, we test the methodology in the context of the neighbourhood of the Sun, with a view towards an upcoming paper in which we apply our technique to the Gaia data.…”

Charting galactic accelerations – II. How to ‘learn’ accelerations in the solar neighbourhood

“…A no v el procedure for calculating gravitational accelerations from stellar kinematical data was introduced in An et al ( 2021 ). In this article, we test the methodology in the context of the neighbourhood of the Sun, with a view towards an upcoming paper in which we apply our technique to the Gaia data.…”