Dark matter (DM) plays a central role in modern physics. It was first introduced to explain the motion of galaxies in a cluster and then to explain the speed of stars in individual galaxies. In both cases the measured speeds were too high to be balanced by the mass estimated by the light emission. The cosmological picture provides a different, complementary, albeit indirect, evidence of the need to introduce DM: in this case DM is necessary to relate the tiny temperature fluctuations in the cosmic background radiation with the distribution of visible mass in the universe. Cosmological DM must be of a non-baryon nature as its interaction with photons must occur only through the force of gravity otherwise fluctuations of the cosmic background would be too large compared to observations. In fact, to have non-linear perturbations today (which correspond to galaxies and clusters of galaxies) it is necessary that when decoupling between matter and radiation occurred (at redshift 1,000) fluctuations in the matter density field were 1/1,000. However, if the DM were baryonic these fluctuations would correspond to fluctuations of the same order in the radiation field, while in the latter the observed fluctuations are 100 times smaller, that is 1/100,000. With non-baryon DM this important problem is solved at the price of introducing a large quantity (in the current model called LCDM it represents about 25% of the matter of the universe and about 5 times more than the baryon) of which at the moment there is no experimental direct trace.

The major effort, both theoretically and experimentally, concerns the search for galactic DM which does not necessarily have the same non-baryon nature as cosmological DM. To this aim, large collaborations have been developed between particle physicists and astrophysicists: since the mid-1980s, dozens of projects have sought the rare interactions envisaged by different theoretical approaches, between DM particles and normal matter. However, the most recent DM research just concluded, like all other previous DM detection experiments, did not report evidence of the existence of DM particles. Furthermore, neither the Large Hadron Collider nor the large experiments to detect the DM (such as those performed at the Gran Sasso) have observed any particle beyond the Standard Model, of the type of particles that should constitute the cosmological DM. Of course, these negative results do not exclude the existence of DM and indeed the theories of DM particles have become more and more sophisticated: to evade the conflict with experimental null results, theorists now assume that particles interact with normal matter even less than initially thought. This proliferation of invisible particles has become so common in the literature that it has been given a collective name: the “hidden sector”. An alternative idea to solve the hidden mass problem has been proposed in the literature since the 1980s: an ad-hoc modification of Newton’s gravity. In particular, in this approach, instead of invoking more mass in the form of unknown particles, the gravitational force is increased in intensity for the same distance by a law of decay less rapid than the inverse of the radius squared, that is, as the inverse distance.

*Figure 1: The initial condition is represented on the left and the result of self-gravitating evolution on the right. The color code is proportional to the density. D. Benhaiem, F. Sylos Labini M. Joyce, Physical Review E 99, 022125 (2019)*

**Relaxation to equilibrium of a self-gravitating system**

This project, based on ideas and insights at the interface between statistical physics and astrophysics, proposes a new attempt to understand the problem of the galactic DM which is motivated by the recent observational results of the Gaia mission (which is still ongoing). This has just produced the largest and most accurate census of stellar positions, velocities and other properties for over a billion stars in our galaxy. Maps published by the Gaia collaboration show that the velocity field of stars in the galactic disk has an unexpected complexity: the collective motions of stars are observed in all three velocity components and show structures with a variety of morphologies whose nature implies that the galactic disk is in a state of imbalance. The extent of the deviation from equilibrium is now one of the main observational issues that will be clarified in the near future in the forthcoming publications of Gaia’s data. These observations suggest considering a third theoretical possibility to explain the relationship between the speed of stars and their mass.

This concerns a fundamental problem involving classical Newtonian physics and ordinary matter (i.e. stars and gases) and which has been neglected in the literature: the relaxation towards equilibrium of a system consisting of many self-gravitating particles, which is a problem which is part of the physics of systems with long-range interaction. In other words, in a given system, it is possible to simply correlate the rotation speed to the mass only if this is in a stationary situation in which, for example, the centrifugal force is balanced by the centripetal force due to gravity, i.e. basic assumption used to estimate DM. Our goal is instead to study under what conditions a stable equilibrium state is reached in a self-gravitating system, how long it takes to relax in such a configuration from generic out-of-equilibrium initial conditions and, from an observational point of view, whether the velocity fields of both our galaxies and the outer ones are compatible with such a situation.

Theoretically, the dynamic evolution of many particles that interact only with Newtonian gravity is a fundamental paradigmatic problem in physics that remains equally essential for the modeling and interpretation of astrophysical structures. A distinctive feature of long-range interacting systems (such as gravity) is that, instead of relaxing to a state of thermodynamic equilibrium through two-body collisions such as those with short-range interacting, they reach, guided by a dynamic of non-relaxation. mean field collision, a so-called quasi steady state (QSS). This configuration represents a collective and global behavior that emerges from the complex dynamics of a large number of elements interacting in a non-linear way. In most systems of astrophysical interest, relaxation of two bodies occurs on a time scale longer than the Hubble time. Therefore, the stationary solutions of the Boltzmann (or Vlasov) equation without collisions plus the Poisson equation represent the main analytical framework to describe such QSS; models derived in these approximations represent the key tool for comparing stellar dynamics or galactic theory with observations. In particular, the assumption of stationarity is crucial for the interpretation of the observations from which we want to estimate the distribution of mass on a galactic scale: it is under this assumption that the interpretations of the rotation curves of the galaxy in terms of DM or dynamics are constructed. Newton’s modified. While the assumption of stationarity is generally taken for granted, the time scale for complete relaxation from a generic configuration out of equilibrium to a QSS is scarcely limited both from a theoretical and numerical point of view.

To study these issues, controlled numerical experiments are generally considered, in which a system is initially prepared in a certain relatively simple initial condition, and then evolves numerically through gravitational dynamics with an N-body code that solves the equations of motion for a large number of particles. In this way we have recently [2,3] studied the gravitational collapse of isolated over-densities of self-gravitating particles with a small initial rotation speed. We have shown that collective relaxation brings the system closer to virial equilibrium, but also generates quite generically, when the initial condition breaks the spherical symmetry, long-lasting non-stationary structures with a rich morphological variety and characterized by spiral arms, bars and even ring structures in special cases, qualitatively similar to spiral galaxies. In these systems the particles do not follow circular and stationary orbits but instead form long-lasting transients, which have the shape of spiral arms with bars or rings, dominated by radial movements that prevent relaxation towards an equilibrium configuration. Therefore, a central objective of the present project is to obtain a systematic understanding of the relaxation times of a QSS starting from an initial generic out-of-equilibrium condition; furthermore, we wish to consider non-gravitational physics (such as gas dynamics, star formation, etc.) and study the impact of such dissipative effects on purely gravitational dynamics in order to link our results to a more realistic and complete theory of formation of galaxies.

**The cosmological galaxy formation**

We also aim to develop complete cosmological simulations in which such systems can form in a complex environment (i.e. when non-isolated systems are considered). In particular, our goal is to modify the properties of the correlation of density fluctuations in standard cosmological models so that monolithic “top-down” collapses, of the same type that occur in the case of isolated over-densities, can occur: this it is in fact the key dynamic feature that involves a collective relaxation process, giving rise to the variety of structures we have observed in isolated collapses. Such simulations must therefore have initial conditions that are qualitatively different from the typical ones used in cosmological literature (i.e. those with cold DM, etc.) in which the aggregation proceeds in a “bottom-up” manner and the galaxies are formed through the aggregation of smaller systems. In this type of dynamic process there is no collective relaxation and, as long as only non-dissipative effects are considered, no spherical symmetry breaking mechanisms are active: in fact, spherical-like systems are formed (the so-called DM halos) which believed to surround spiral galaxies characterized by the presence of a disk. A careful study of the numerical resolution effects is necessary to correctly simulate these systems [4].

**The speed field of the Milky Way**

This theoretical work will have to be accompanied by observational studies of galactic velocity fields, a context in which there is an increasing number of data. In this context, we have recently developed a method of statistical reconstruction of the distances of the stars of the Milky Way that has allowed us to reach a distance almost three times deeper than the official maps of Gaia [5]. In this way we have detected large gradients in all the velocity components and we have concluded that these data question the most elementary hypothesis of stellar dynamics, that of stationarity: in fact, they show that the modeling of the galactic disk as a symmetrical system with respect to the rotation and time independent is definitely wrong. The key question that remains open and that can be clarified by forthcoming publications of the Gaia satellite data concerns the amplitude of the radial velocities in the outermost part of the disk. These measurements will allow us to quantify the deviation from stationarity thus allowing corrections to simple relationships, based on the assumption that the system is stationary, between mass and velocity normally adopted. In fact, the deviations from equilibrium are expected to be relevant especially in the ultra-peripheral regions of galaxies, where a stellar revolution time becomes of the order of Hubble time [6].

*Figure 3 The azimuth velocity in the disk of our galaxy is represented in the left panel while the radial velocity in the right panel. The color code is proportional to the speed module. M. López-Corredoira, F. Sylos Labini*

**Velocity fields of external galaxies**

The more detailed analysis of the two-dimensional high-resolution maps of the velocity fields of the line of sight of external galaxies, would allow to determine the possible “footprints” typical of large-scale radial velocities. In this regard it is worth underlining that also for external galaxies the quantity of DM is estimated, in the first order, assuming that the observed velocity field corresponds to purely circular movements. The radial velocities are then measured as residuals between a spinning disk model and the actual data. However, the situation in general can be more complex than that, especially if the galaxy has no axial symmetry [7]. In this regard, we have shown that in this situation the radial speeds can be confused with the circular ones, so that the standard methods used for the estimation of the two-dimensional speed can be distorted by the inconsistent assumption of axial symmetry. A careful study of the velocity fields of external galaxies and the adaptation of their properties to a model that allows non-axis-symmetrical shapes is therefore necessary to understand the nature of the kinematics of these galaxies. For this purpose, we intend to consider two-dimensional velocity field data from different datasets that map the ultra-peripheral regions of galaxies (i.e., using high-resolution HI observations such as Things and Little Things) and merge the data. of velocity to intensity profiles to calculate the contribution of light mass to the velocity field and the possible effect of radial velocities. These analyzes will allow to determine not only the fraction of DM but also and in particular its distribution, or whether it is associated or not with the distribution of visible matter.

*Figure 4 Velocity field along the line of sight of NGC 628 (F. Sylos Labini, D. Benhaiem, S. Comeron, M. López-Corredoira, Astron.Astrophys. 622, A58 (2019))
*

**Large Scale Structure of the Universe**

In the past decades we have assisted to an exponential growth of the galaxy redshift surveys data which have revealed that galaxies are organized in a large scale network of filaments and voids. Statistical analyses that we have performed of these surveys have shown that the galaxy distribution is characterized by power-law correlations in the range of scales [0.1-20] Mpc/h with a correlation exponent 1 corresponding to a fractal dimension D = 2. Further we found that the density depends, for 20 < r < 80 Mpc/h, only weakly on the system size, i.e. D =2.7 but density fluctuations are not self-averaging which implies that the size of structures is of the same order of the samples and thus it is not possible to average out fluctuations. Correspondingly, we have found that density fluctuations follow the Gumbel distribution of extreme-value statistics, different from a Gaussian distribution which would arise for a homogeneous spatial galaxy configuration.

Whether or not on scales r > 80 Mpc/h correlations decay and the distribution crossovers to uniformity, is still matter of considerable debate. This debate was originated by the use of different statistical methods to measure two-point correlations, to estimate statistical and systematic errors and to control the selection effects that maybe present in the data. In particular, the critical points concern the a priori assumptions which are usually used, without being directly tested, in the statistical analysis of the data and the a posteriori hypotheses that are invoked to interpret the results. Ongoing galaxy surveys, such as the Dark Energy Survey, will create in the next few years the largest three-dimensional map of galaxies to date that, covering a contiguous large spatial volume and controlling luminosity selection effects, will allow to study galaxy correlations on scales larger than 100 Mpc/h. Analyses of such sample represent one key-objective for our activities.

**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.

**Publications (selected)**

[1] M. López-Corredoira, Foundations of Physics, 47, 711 (2017),

[2] D. Benhaiem, M. Joyce, F. Sylos Labini, Astrophysical Journal, 851, 19 (2017)

[3] D. Benhaiem, F. Sylos Labini M. Joyce, Physical Review E 99, 022125 (2019); http://physics.aps.org/synopsis-for/10.1103/ PhysRevE.99.022125

[4] D. Benhaiem, M. Joyce, F. Sylos Labini, T. Worrakitpoonpon, Mon.Not.R.Acad.Soc, 473, 2348, (2018)

[5] M. López-Corredoira, F. Sylos Labini, Astron.Astrophys., 621, A48 (2019)

[6] M. López-Corredoira, F. Sylos Labini, P. M. W. Kalberla, C. Allende Prieto Astron.J., 157, 26 (2019), M. Lopez-Corredoira, F. Garzon, H.-F. Wang, F. Sylos Labini, R. Nagy, Z. Chrobakova, J. Chang, B. Villarroel, Astronomy & Astrophysics, Volume 634, id.A66, 14 pp

[7] F. Sylos Labini, D. Benhaiem, S. Comeron, M. López-Corredoira, Astron.Astrophys. 622, A58 (2019)

**Collaborations**

- Canary Islands Institute of Astrophysics (La Laguna, Tenerife, Spain)
- Pierre et Marie Curie University (Paris, France)
- Department of Physics, University of Rome Sapienza
- Department of Physics University of Florence
- Institute of Complex Systems CNR (Florence)
- National Institute of Astrophysics (Florence)
- Department of Physics University of St. Petersburg

**Projects**

- DYNamics and non-equilibrium states of complex SYStems: MATHematical methods and physical concepts ”INFN Research Network. Specific initiatives of the CSN4 / INFN National Institute of Nuclear Physics.
- HPC resources of The Institute for Scientific Computing and Simulation, project Equip @ Meso (University Pierre et Marie Curie, Paris, France)