This project, based on ideas and insights at the interface between statistical physics and astrophysics, focus on a fundamental problem of classical Newtonian physics: the relaxation towards equilibrium of a system made by many self-gravitating particles. Theoretically, the dynamical evolution of many particles interacting only by Newtonian gravity is a fundamental paradigmatic problem in physics and it remains likewise essential for the modelling and interpretation of astrophysical structures. A distinguishing feature of long-range interacting systems is that, instead of relaxing to thermodynamical equilibria through two-body collisions like short-range interacting ones, they reach, driven by a mean-field collisionless relaxation dynamics, a so-called quasi-stationary state (QSS). This configuration represents a global collective behaviour emerging from the complex dynamics of a large number of elements interacting non-linearly. In most astrophysical systems of interest, two-body relaxation occurs on a time scale longer than the Hubble time. Thus stationary solutions of the collisionless Boltzmann (or Vlasov) equation plus the Poisson equation represent the main analytical framework to describe such QSS; models derived in these approximations represent the key tool to compare stellar dynamic or galactic theory with observations. In particular, the assumption of stationarity is the crucial one for interpreting observations constraining the distribution of mass on the galactic scale: it is under this assumption that the interpretations of the galaxy rotation curves in terms of dark matter or modified Newton dynamics are built. While the assumption of stationarity is usually taken for granted, the time-scale for a complete relaxation from a generic out-of-equilibrium configuration to a QSS is poorly constrained both from a theoretical and numerical point of view.

This theoretical work should be put beside observational studies of galactic velocity fields, a context where there is a growing number of data. We have recently developed a statistical reconstruction method of a Milky Way star’s distance that has allowed us to reach a distance almost three times deeper than the official Gaia maps. In this way we have detected large gradients in all velocity components and we concluded that these data question the most basic hypothesis of stellar dynamics, that of stationarity, and show that the modelling of the galactic disk as a symmetric system with respect to the rotation axis and independent of time is definitely incorrect. The key open question that can be clarified by the next data releases of the Gaia satellite concerns the amplitude of the radial velocities in the outermost part of the disk. Such measurements will allow us to quantify the departure from stationarity allowing thus the corrections to the simple relations, based on the steady state assumption, between mass and velocity usually adopted. Indeed deviations from equilibrium are expected to be relevant especially in the outermost regions of the galaxies, where a star revolution time becomes of the order of the Hubble time.

In addition, analysing in greater details high-resolution two-dimensional maps of the line-of-sight velocity fields of external galaxies would allow determining the possible signatures that are typical of large-scale radial velocities. In this respect it is worth stressing that even for external galaxies the amount of DM is estimated, at first order, by assuming that the observed velocity field corresponds to purely circular motions. The radial velocities are then measured as the residuals between a rotating disk model and the actual data. However, the situation can be in general more complex than that, especially if the galaxy is not axisymmetric. Indeed, we have shown that in such a situation radial velocities can be confused with circular ones, so that the standard methods used for the two-dimensional velocity estimation may be biased by the inconsistent assumption of axisymmetry. A careful study of the velocity fields of external galaxies and the fitting of their properties with template that allows for non-axisymmetric shapes is thus necessary to understand the nature of the kinematics of these galaxies. To this aim we plan to consider two-dimensional velocity fields data from different data sets that map the outermost regions of the galaxies (i.e., by using high resolution HI surveys like Things and Little Things) and to join velocity data to intensity profiles to compute the luminous mass contribution to the velocity field and the possible effect of radial velocities. These analyses will allow determining not only the fraction of DM but also and notably its distribution, i.e. whether or not it is associated with the distribution of visible matter.

In summary this project will aim to (i) understand the basic physical mechanism of collective relaxation in a self-gravitating system and the emerging of a QSS from a complex collective dynamics, (ii) understand the effect of gas dynamics and other dissipational processes in the collapse of an isolated, non spherical and rotating over-density (iii) understand the properties of cosmological initial conditions that are compatible with the occurrence of a monolithic collapse of the type happening for an isolated over-density (iv) obtain the most complete picture of the kinematics of our galaxy (v) constrain the velocity fields of external galaxies estimating the effect of radial velocities for not axisymmetric systems and (vi) obtain a more reliable estimation of the DM fraction and distribution both in our galaxy and in external ones that can provide a crucial information for DM search experiments.

**Contact**: Francesco Sylos Labini

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