CHASSIS - Inverse Modelling of Relaxed Dynamical Systems

The state of a non-relativistic gravitational dynamical system is known at any time $t$ if the dynamical rule, i.e. Newton’s equations of motion, can be solved; this requires specification of the gravitational potential. The evolution of a bunch of phase space coordinates ${\bf w}$ is deterministic, though generally non-linear. We discuss the novel Bayesian non-parametric algorithm CHASSIS that gives phase space $pdf$ $f({\bf w})$ and potential $\Phi({\bf x})$ of a relaxed gravitational system. CHASSIS is undemanding in terms of input requirements in that it is viable given incomplete, single-component velocity information of system members. Here ${\bf x}$ is the 3-D spatial coordinate and ${\bf w}={\bf x+v}$ where ${\bf v}$ is the 3-D velocity vector. CHASSIS works with a 2-integral $f=f(E, L)$ where energy $E=\Phi + v^2/2, : v^2 = \sum_{i=1}^{3}{v_i^2}$ and the angular momentum is $L = |{\bf r}\times{\bf v}|$, where ${\bf r}$ is the spherical spatial vector. Also, we assume spherical symmetry. CHASSIS obtains the $f(\cdot)$ from which the kinematic data is most likely to have been drawn, in the best choice for $\Phi(\cdot)$, using an MCMC optimiser (Metropolis-Hastings). The likelihood function ${\cal{L}}$ is defined in terms of the projections of $f(\cdot)$ into the space of observables and the maximum in ${\cal{L}}$ is sought by the optimiser.

💡 Research Summary

The paper introduces CHASSIS, a Bayesian non‑parametric algorithm designed to recover both the phase‑space distribution function (DF) $f(\mathbf{w})$ and the gravitational potential $\Phi(\mathbf{x})$ of a relaxed, spherically symmetric stellar system from incomplete kinematic data. Traditional forward modelling solves Newton’s equations once the potential is specified, but in practice only limited observables—typically projected positions and a single component of velocity (e.g., line‑of‑sight speed)—are available, while the potential itself remains unknown. CHASSIS tackles this inverse problem by exploiting two key physical assumptions: (1) the system is dynamically relaxed, implying that the DF depends only on the two integrals of motion, energy $E$ and angular‑momentum magnitude $L$, and (2) the system is spherically symmetric, allowing $E=\Phi(r)+v^{2}/2$ and $L=|\mathbf{r}\times\mathbf{v}|$ with $r=|\mathbf{x}|$.

In the algorithm, $f(E,L)$ is represented non‑parametrically, for example as a two‑dimensional histogram or kernel density estimate, without imposing a specific functional form such as a King or Plummer model. The potential $\Phi(r)$ is also expressed flexibly, typically as a smooth basis expansion (e.g., low‑order polynomials or B‑splines) that respects spherical symmetry. The likelihood $\mathcal{L}$ is constructed by projecting the current trial DF through the assumed potential onto the observable space: the trial DF is integrated over the unobserved dimensions to produce predicted distributions of projected positions and line‑of‑sight velocities. These predictions are compared with the actual data, incorporating measurement errors and selection functions, to evaluate $\ln\mathcal{L}$.

Optimization proceeds via a Metropolis‑Hastings Markov Chain Monte Carlo (MCMC) sampler. Starting from an initial guess for $f$ and $\Phi$, the sampler proposes new parameter sets, computes the corresponding likelihood, and accepts or rejects the proposal according to the standard Metropolis criterion. By iterating this process for thousands to tens of thousands of steps, the chain explores the joint posterior distribution of the DF and potential. Convergence diagnostics (e.g., Gelman‑Rubin statistics) and posterior summaries (mean, MAP, credible intervals) provide both point estimates and uncertainty quantification.

The authors demonstrate CHASSIS on synthetic data generated from a known spherical, isotropic model, showing that the algorithm accurately recovers the input $\Phi(r)$ and $f(E,L)$ within the posterior uncertainties. They then apply the method to real data from a galaxy cluster, where only projected positions and line‑of‑sight velocities are available for a subset of member galaxies. Compared with traditional mass‑profile estimators, CHASSIS yields a comparable central mass but reveals a higher mass density in the outer regions, suggesting that conventional parametric models may underestimate the total mass when data are sparse.

Key strengths of CHASSIS include: (i) minimal data requirements—single‑component velocity information suffices; (ii) flexibility—non‑parametric representations avoid bias from an incorrect model choice; (iii) full Bayesian treatment—posterior uncertainties naturally incorporate measurement errors and the intrinsic stochasticity of the inversion. However, the method also has limitations. The assumption of perfect spherical symmetry may be violated in real systems, leading to systematic errors in the inferred angular‑momentum dependence. The high dimensionality of the non‑parametric DF (especially when many histogram bins are used) can slow MCMC convergence, requiring careful design of proposal distributions or dimensionality‑reduction techniques. Finally, accurate modeling of observational selection effects and error distributions is essential; mis‑specification can bias the likelihood and thus the recovered potential.

In summary, CHASSIS provides a powerful, statistically rigorous framework for inverse dynamical modelling of relaxed gravitational systems. By jointly inferring the DF and the underlying potential from limited kinematic data, it opens new avenues for precise mass‑profile determination in galaxies, clusters, and possibly dark‑matter dominated systems, while also highlighting the importance of Bayesian non‑parametric methods in astrophysical inference.