SEGAL:The  Secular Evolution of GALaxies

1. I.Proposal’s context, positioning and objective(s) 

A         Objectives and research hypothesis

Context:        Galaxies are unique laboratories to test the universal law of gravity, the driving force from their inner cores to their outskirts. Astronomers have recently focused significant amounts of theoretical, computational and observational efforts to understand and explain their cosmic evolution. This rising interest can be explained by the alignment of three transformational shifts:

i) First, new ground-based or space instruments like Muse, Gravity, MaNGA/SAMI and Gaia are collecting an unprecedented wealth of data probing the long-term dynamical state of galaxies on all scales. We now have access to precision astrometric data on the phase-space structure of our Milky Way (literally billions of stars), complemented by statistical samples of kinetic information on large populations of galaxies;

ii) Second, the steady increase in computing power allows us to perform numerical simulations of resolution and complexity greater than ever before in the context of the Cold Dark Matter (CDM) paradigm.  Such a framework is now well-established and successfully describes the formation of structures on large scales, but several challenges are still present on the smaller ones. In this respect, the development of progressively more accurate numerical simulations is essential. This applies not only to isolated and idealised setups, but also to account statistically for the fluctuating environment on different scales, from galactic centers to the outskirts of dark halos. These self-gravitating astrophysical systems can therefore now be considered nested, embedded in their own lively environment, with which they interact throughout their lifetime;

iii) Finally, recent theoretical breakthroughs in our understanding of the kinetic theory of self-gravitating systems allow us for the first time to follow the effects of gravitationally-amplified external and internal perturbation on the orbital structure of galaxies over cosmic time. In particular, we now have self-consistent integro-differential equations describing the quasi-linear evolution of a given system under the effect of self-induced (e.g. through the system’s own graininess) or externally-driven fluctuations.  

        Building upon the seminal works describing the linear response of self-gravitating systems (by Goldreich, Lynden-Bell, Toomre, Kalnajs and others), the development in kinetic theory of a self-consistent framework to characterise their quasi-linear response (Weinberg’01) now offers us a unique opportunity to follow galaxies over cosmic timescales. As for the Earth, galaxies are indeed subject to ‘weather’ and ‘climate’. The former is driven by transients (which reflect recent disturbances which have not yet phase mixed), while the latter encodes distinctive features specifically in orbital space generated over long (secular) timescales. Quasi-linear theory captures the latter. It is, therefore, ideally suited to describe galaxies on large and small scales, because their dynamical times are very short compared to a Hubble time. For galaxies, the most significant (non serendipitous) processes are secular in nature. This motivates the present proposal: through the corresponding stochastic differential equations (SDE), galactic dynamics can now make statistical predictions over cosmic times.

The kinetic developments rely on the inhomogeneous Balescu-Lenard (IBL) equation (Heyvaerts’10, and Figure 1 below), i.e. the master equation describing self-induced fluctuations on the one hand, and a proper accounting of environmentally- driven fluctuations and their gravitational amplification, captured by the dressed Fokker-Planck (dFP) equation on the other hand1. These equations allow us to gauge the respective roles of nature vs. nurture in establishing the long term observed kinetic properties of galaxies, relying on stochastic processes which capture all sources of fluctuations. The corresponding extended kinetic theories (see e.g. Fouvry+’15,’18 and footnote 1 for a mini review) define a computational framework to quantify statistically the long-term evolution of self-gravitating systems, complementing (commonly used) N-body methods. Qualitatively, these kinetic equations all involve diffusion coefficients in orbital (action) space, Dm(J), scaling like the power spectrum of the dressed potential fluctuations projected along the unperturbed trajectories (with m the harmonic number of the resonance, J the action and Ω the frequencies):

In a stellar/collisionless “fluid”, this diffusion coefficient (which drives the secular distortion of its orbital structure, following the dissipation-fluctuation theorem) is amplified by the square of its inverse gravitational susceptibility, ε, evaluated at the natural frequencies of the system. If the perturbation hits these natural frequencies while the system is not far from marginal stability, anisotropic diffusion along that resonance can be extremely efficient and cause rapid changes.

Whether they are external or self-induced, the long-term resonant effects of potential perturbations on galaxies can therefore now be accounted for in detail, by quantifying their spectral properties on small (central cluster), intermediate (disc) and large (halo, globular cluster system) scales. This distinction between processes specific to each scale is made possible by the chosen dichotomy between external and internal fluctuations. It is a requirement to capture secularly the many scales involved.

The distinction also allows us to disentangle their respective roles in sourcing secular evolution, as we quantify the diffusion signatures and the characteristic timescales associated with each source of fluctuations.

For instance, in stellar discs, we must investigate the relative diffusive efficiency of shot noise triggered by Giant Molecular Clouds (GMCs), clumps within the halo, central bars, etc. We must also quantify external sources of fluctuations: flybys, infall etc. We must finally pay attention to the system’s initial reservoir of free energy, which differs greatly between, e.g. spirals and ellipticals, and to the secular transformation of the underlying equilibria through infall. Once all these mechanisms are statistically characterised and quantified, long-term implications (e.g. for the age-dispersion relations within discs, the impact of bars) can be compared to detailed observations, e.g. of the structure of the Milky Way’s distribution function (DF), provided by Gaia (e.g. through structures in the U-V plane). The system’s DF may involve additional parameters (in so-called extended phase space), such as stellar metallicity, which traces the interplay between dynamics and chemistry (abundances, alpha-enhanced elements, etc, which are indirect proxies for stellar age). Hence one also needs to account for cold gas inflow and the birth (and possibly death) of stars, i.e. for the possibility of sources and sinks of particles. We must finally follow multiple masses and describe the corresponding secularly induced mass segregation. This is essential near the Galaxy’s massive black hole (MBH), but more generally of interest in the context of stellar discs, and even galactic halos, where substructures within the halo act as an effective varying number of massive quasi-particles. All these scales have in common that gravity and resonances drive their secular evolution.  

Hence a novel single framework can explain the long-term behaviour of various components of galaxies, seemingly different as they involve nested scales, but in reality, describable through the same approach. SEGAL will quantify the cosmic fate of these self-gravitating systems over a Hubble time and identify asymptotic solutions, specific to each scale. SEGAL will study the statistical impact of e.g. stellar migration, disc thickening via giant molecular clouds and spiral waves, resonant relaxation in galactic centers, bars and globular clusters, or the reshuffling of the orbital structure of dark halo cores. This is both unprecedented and extremely useful.

B Position of the project as it relates to the state of the art

Building up on the practical advantage of quasi-linear theories capturing the long-term effect of self-gravity in an open multi-scale environment, we must proceed via the following computationally-demanding steps:

B.1: quantify the environmental effects induced by the larger scale using ensemble of simulations; compute the relevant power spectra driving externally induced fluctuations at each scale; 


B.2: implement both secular formalisms, solving for the joint kinetic equations and quantifying each characteristic timescale, while accounting for the difference in temperature and degeneracies specific to each scale; identify the generic asymptotic solutions (resp. core halos, exponential discs, cusps...);

B.3: cast our results in terms of observables, tailored to existing and future facilities, and propose new observational diagnostics to test our theoretical predictions. 


B.1 Quantifying the fluctuating environment: We will rely on SEGAL’s computing resources to produce sets of simulations to account statistically for the specific fluctuating environment on the boundary of that scale, from galactic centers to the outskirts of dark halos, so that the inner boundary of each outer-scale corresponds to the environment of its inner-scale. The assumption will be that while the detailed long-term results of N-body or hydrodynamical simulations should be considered with the appropriate level of skepticism, their short-term ensemble-averaged behaviour on the larger scales are more robust. Within SEGAL we will focus only on canonical angular power spectra (involving the Fourier transform of the potential fluctuations w.r.t. the angle coordinates conjugated to the actions – labeling orbits), since the fluctuation-dissipation theorem tells us they are the relevant quantity.  We will use multi-scale, zoomed-in simulations that include massive black holes, stellar and AGN feedback in the spirit of New-Horizon. We pioneered such measurements using over 15 000 dark halos at the Virial radius in Aubert+’07, over 50,000 and 200,000 virtual galaxies, in Pichon+’11 and Dubois+’14, see Figure 0, 5, 7 & 8. The fluctuation dissipation theorem provides us with a shortcut: from the point of view of the underlying orbital structure, only the power spectrum of fluctuations matters. This is a significant compression of the full statistics! We will rely on zoom on multiple resolutions in order to fit and extrapolate these power spectra. These fits will be deliverables of the project and made available to the community. We will also compute explicitly the ensemble average over sets of galaxies of the rate of change of actions which enter the quasi-linear diffusion equations as fluxes.

B.3 Compute observables for upcoming surveys: Today, the synergy of telescopes and spacecrafts at different wavelengths allows us to study galaxies, and embedded MBHs at unprecedented resolution. Gaia, MANGA, SAMI, the VLT with Gravity, are all online and fully working. On longer timescales, following the first detection of gravitational waves by LIGO in 2015, upcoming instruments like LISA will measure the rate at which massive black holes in galactic centers swallow stars and spin up. JWST, DESI, LSST, Euclid, 4MOST, and MOONS will soon provide a comprehensive view of galaxies and their MBH hosts.  All our results (the DFs) must therefore be cast in terms of direct observables, tailored to these existing and future facilities, in order to interpret such datasets (see e.g. Figure 2 bottom panel from Fouvry+’16, or Figure 4 from Fouvry+’18). These observables are typically marginals of the underlying DF(J,τ) and are therefore ‘straightforwardly’ computed. For instance, the so-called Hercules streams in Hipparcos and Gaia data are likely to be kinematic moments of this secular distribution function. Depending on scale, the signatures of our stochastic solutions will be quantified in terms of the evolution of the age-dispersion relations in the Milky Way’s disc, its vertical gradient, and the stellar dynamical evolution of the Galactic center’s cluster (as a probe of e.g. black hole spin). It will also be applied on larger scales to investigate the cusp-core problem of dark matter, the persistence of streams within dark halos.

On each scale, we will quantify the environment, integrate the stochastic differential equations, validate the result using alternative integrators (Nbody, Vlasov, FEM), identify asymptotic solutions and derive observables from the knowledge of the expectation of the phase-averaged time dependent distribution function.

B.4 Scientific themes: SEGAL is focused on three scientific themes, on nested physical scales, embedded in a fourth transverse unifying theme:

• Secular-C: probing the vicinity of supermassive black holes in a Hubble time; 

• Secular-D: the cosmic fate of galactic discs; 

• Secular-H: restructuring life and open halos and their globular cluster population, 

• Secular-T: the secular theory for self-gravitating systems.  

Secular-C: the vicinity of MBHs in a Hubble time(GP,FV,JBF,MV,EV,JM,PDF)

Compelling evidence for massive black holes (MBHs) exists for the nuclei of tens of nearby galaxies, including our own Milky Way (e.g. Genzel+’97). Scaling relations have been identified between MBHs and the large-scale properties of the host galaxy, such as mass, luminosity, and velocity dispersion (e.g. Heckman+’11). The energy pumped by Active Galactic Nuclei (AGN) into the host galaxy can energise the gas, suppressing star formation, thus altering the overall evolution of galactic structures (e.g. Springel+’05).

WPC1 Re-thinking the theory of nuclear star clusters The relationship between host galaxies and MBHs is critical to understanding galaxy evolution. In turn, its central stellar cluster holds the key to probing the MBH properties, such as their spin, their Schwarzschild Barrier or the expected extreme-mass-ratio-in-spirals (EMRI) rates. These stars evolve in a quasi-Keplerian potential hence their orbits take the form of ellipses, which conserve their spatial orientation for many orbital periods. They may therefore be represented as a system of massive Keplerian wires, for which the mass of each star is smeared out along the elliptic path followed by its quasi-Keplerian orbit. Such ideas were first developed in Rauch+’96, which introduced the concept of resonant relaxation by noting that wire-wire interactions greatly enhance the relaxation of the stars’ angular momentum (Alexander’05, Bar-Or+’17, Fouvry+19ab). These Keplerian wires can then undergo various effects on secular timescales: (i) vector resonant relaxation, during which a wire's orbital orientation gets to jitter stochastically through the non-spherical fluctuations in the cluster's potential, (ii) scalar resonant relaxation during which a wire's eccentricity gets to diffuse through resonant coupling between in-plane precessing wires, e.g. precessing because of relativistic effects, (iii) non-resonant relaxation during which a wire's semi-major axis can slowly diffuse under the long-term effects of a series of local Keplerian deflections by nearby stars. A detailed characterisation and implementation of all three dynamical processes will be at the heart of this WP, as will be the link to the existing S cluster at the GC. 
This relaxation process is critical in order to predict the rates of tidal disruptions of stars by MBHs (e.g. Rauch+’96), the merging rates of binary supermassive MBHs (e.g. Yu, 02), or the rate of gravitational wave emissions from star-MBH interactions (e.g. Hopman+’06; Merritt+’11). Resonant relaxation is also the appropriate framework to predict how young stellar populations found in the center of our own Galaxy respond (e.g. Kocsis +’11, Bar-Or+18, and Figure 3) to relativistic effects, in order to quantify the expected rate of infall on the MBH  and its spin. One method to study the secular dynamics of quasi-Keplerian stellar clusters will be to rely on direct N−body simulations. We adopt this approach in WPT2 for validation.

However, gaining physical insights from these simulations is challenging, as various complex dynamical processes are intimately entangled there. In addition, because of the significant breadth of timescales between the fast Keplerian motion and cosmic times, the computational costs are such that one can typically only run a few realisations, limited to a relatively small number N of particles. Moreover, this cannot be scaled up easily to astrophysical systems, as different dynamical mechanisms scale differently with N (Heggie+’03). When focusing on resonant relaxation,

We will specifically consider flat and spherical multi-mass clusters and answer the following questions i) Can we model the stellar mass accretion rate on the MBH? ii) Can the long timescale phenomenology in the vicinity of the so-called Schwarzschild barrier be reproduced by SDE diffusion based on the first principle Balescu-Lenard kinetic equation? iii) Is resonant relaxation between stellar orbits only effective in a restricted region of phase-space away from the last stable orbit lines? iv) Can stochastic solutions produce numerical results consistent with direct N-body results for plunge rates? v) What should be the observed distribution of GR-detected mergers? vi) What drives the interplay between the MBH and its nuclear cluster? vii) what role does the stellar cluster play in spinning Kerr black holes?

Results & Deliverables: Characteristic timescale for each diffusion process; Detailed computations of the relaxation timescales in the whole orbital space surrounding SgrA* for scalar resonant relaxation, vector resonant relaxation, non-resonant relaxation. All shared as a publicly available code. Prediction of the EMRI rate as a function of a stellar cluster's properties (e.g. density power law index and mass spectrum). Computation of the MBH mass and spin growth as sourced by the relaxation of the stellar cluster. Comparison of the statistical distribution of observed stellar properties in SgrA* (e.g. eccentricity or density distribution) w.r.t. the expectations from the relaxation processes. Quantitative estimation of the dissolution time of stellar discs in the vicinity of SgrA* and comparison with the observed discs. Constraints on the efficiency of mass segregation (e.g. between stars and intermediate mass black holes) from observations and theoretical expectations.

Secular-D: the cosmic fate of galactic discs 
(JBF,CP,BF,YD,AH,MW,AS,DA,GM,SP,JD,AS,PDF)

WPD1:Radial migration via churning and blurring Most stars, perhaps all, are born in stellar discs. Major mergers destroyed some of these discs quite early in the history of the universe, but some discs have survived up to the present day, including the Milky Way. Understanding the secular dynamics of stellar discs therefore appears as an essential ingredient of cosmology, as the discs’ cosmological environments are now firmly established in the ΛCDM model. Self-gravitating stellar discs are cold responsive dynamical systems in which rotation provides an important reservoir of free energy and where orbital resonances play a key role. The availability of free energy leads to some stimuli being strongly amplified, while resonances tend to localise their dissipation, with the net result that even a very small perturbation can lead to discs evolving to significantly distinct equilibria. These discs are embedded in various sources of gravitational noise, from shot noise arising from the finite number of giant molecular clouds in the interstellar medium to globular clusters and substructures orbiting around the galaxy.

 Spiral arms in the gas distribution also provide another source of fluctuations, while the central bar of the disc offers yet another coherent stimuli. The history of galactic discs likely comprises the joint responses to all these various stimuli (internal and external). 
One can find in the solar neighbourhood at least three illustrations of such effects. First, the random velocity of each coeval cohort of stars increases with the cohort’s age (Wielen,’77; Aumer+’09). In addition, the velocity distribution around the Sun exhibits several ‘streams’ of stars (Dehnen,’98). Each of these streams contains stars of various ages and chemistries, which are all responding to some stimulus in a similar fashion (Famaey+’05). Finally, in the two-dimensional action-space, resonant ridges form and play an important role in the secular dynamics of razor-thin stellar discs, as argued in Sellwood+’14. Indeed, Gaia’DR2 has revealed a very rich structure in local velocity space, which is related to resonances with multiple non-axisymmetric patterns (including the bar) and possibly to incomplete phase-mixing. In terms of in-plane motions, this rich structure is also seen as ridges in the actions of the axisymmetric background potential of the Galaxy. We have  preliminary shown (Monari+’19) that a second prominent ridge in action space then corresponds to the 4:1 outer resonance of the m = 4 mode of such a bar, and that the velocity structure seen as an arch at high azimuthal velocities in Gaia data can be related to its 2:1 outer Lindblad resonance. Direct numerical simulations of thin stellar discs over secular timescales are very challenging because their two dimensional geometry combined with their responsiveness causes discreteness noise to be important unless very large number of particles is employed. It is only recently that it became possible to simulate a disc with a sufficient number of particles for Poisson shot noise to be dynamically unimportant for many orbital times. A complementary approach to understand their dynamics is therefore to rely on extended kinetic theory via a stochastic framework. This will be the topic of this WP, relying on in WPT2 for validation. SEGAL will follow up on Fouvry+’15 and evolve both kinetic equations in this context, accounting for mass and angular momentum inflow within the disc, and modeling self-consistently churning (drift in guiding center) and blurring (heating) over a Hubble time, while accounting for secular cold gas infall from the large-scale-structure environment (Pichon+’11, Kimm+’11), see WPH2. A realistic radial profile of accreted cold gas is a requirement for the chemo-dynamical modelling of the Milky Way.

WPD2: Galactic disc thickening/settling The problem of explaining the origin of thick discs in our Galaxy has been around for some time (e.g. Gilmore+’83). The interest for this dynamical question has been revived recently in the light of the APOGEE survey (Eisenstein+’11) and Gaia data. Star formation within stellar discs typically occurs on the circular orbits of the gas, so that young stars should form a very thin disc (Wielen’77). However, chemo-kinematic observations of old stars within our Milky Way (Jurić+’08; Bovy+’12) or in other galactic discs (Burstein’79; Comerón+’11) have all shown that thick components are very common. Yet the formation of thickened stellar discs remains a significant puzzle for galactic formation theory. Various dynamical mechanisms, either internal or external, have been proposed to explain the observed thickening, but their respective impacts and roles remain to be quantified. First, some violent major events could be at the origin of the vertically extended distribution of stars in disc galaxies: accretion of galaxy satellites (Meza+’05; Abadi+’03), major mergers of gas-rich systems (Brook+’04), or even gravitational instabilities in gas-rich turbulent clumpy discs (Noguchi’98). Violent mergers definitely have a strong impact on galactic structure, but thickened stellar discs could also originate from the slow and continuous heating of pre-existing thin discs, via for example galactic infall leading to multiple minor mergers (Toth+’92; Villalobos+’08). Spiral density waves (Sellwood+’84; Monari+’16) are also possible candidates to increase the disc's velocity dispersion, which can be converted into vertical motion through deflections from giant molecular clouds (GMCs) (Spitzer+’53; Hänninen+’02). In addition, radial migration could also play an important rol­e in the secular evolution of stellar discs. This migration could be induced by spiral-bar coupling (Minchev+’10), transient spiral structures (Barbanis+’67; Solway+’12), or perturbations induced by minor mergers (Bird+’12). Finally, recent large numerical simulations are now attempted in a self-consistent cosmological setup (Grand+’16), to probe the interplay between these competing mechanisms (Fig. 5). All investigations can be broadly characterised as induced by an external or internal source of fluctuations to trigger a vertical orbital reshuffling in the disc. SEGAL will quantify in WPD2 the expected cosmic evolution of the vertical structure of the disc and its population, relying on the multi-component generality of the kinetic formalism. This will involve building perturbatively thickened equilibria with a mapping in action-space from an integrable to a non-integrable model via fits of generating functions (Kaasalainen+’94, see WPT1 below). We will then solve the exact field equations, construct an appropriate basis of potentials, and deal with the full response matrix in order to solve for the corresponding Balescu-Lenard and Fokker Planck equations. Validation will be done in WPT2. We will finally derive the corresponding global kinetic observables: age-vertical dispersion gradients and 3D velocity ellipsoid as a function of position and cosmic age and population, etc.

Results & Deliverables: Computations of the timescale for disc thickening, while accounting for self-gravity, as induced by, e.g. DM halo fluctuations, GMCs, passing-by satellites. Computations of the diffusion coefficients in orbital space for churning (diffusion in angular momentum) and blurring (diffusion in eccentricity), while accounting for self-gravity, as induced by different heating mechanisms, e.g. bars, spiral arms, GMCs, passing-by satellites, and DM halo fluctuations. Characterisation of diffusion in extended phase space, i.e. accounting for metallicity and age, and the respective signatures associated with the different heating sources. Global asymptotic and steady-state solution for exponential discs.  Characterisation of the orbital diffusion signatures in the vicinity of the Sun, as traced by GAIA DR2 via marginals of DF(J,τ) and the corresponding kinetic estimators. Theory for bar dissolution and buckling.  Self-consistent model for cosmic disc settling.

Secular-H: Secular evolution: restructuring live and open halos 
(JBF,CP,ALV,SR,PB,MW,PDF)

WPH1 The cusp-core problem of galactic halos Dark matter (DM) only simulations favour the formation of a cusp in the inner region of DM haloes (Dubinski+’91; Navarro+’97), following what appears to be a universal cuspy profile, the NFW profile. However, observations tend to recover profiles more consistent with a shallower core (Moore’94; de Blok+’02). This discrepancy between the cuspy profile predicted by direct DM-only simulations and the core profile inferred from observations is an important challenge in cosmology, coined the cusp-core problem. Various solutions have been proposed to resolve this discrepancy. A first set of solutions involves modifying the dynamical properties of the collisionless DM, preventing it from collapsing into cuspy profile in the first place. Examples include the possibility of warm dark matter (Kuzio de Naray+’10) or of self-interacting dark matter (Spergel+’00). Another set of investigations rely on the idea that not accounting self-consistently for the baryonic physics and its back-reactions on the DM may also be at the origin of the discrepancy. These mechanisms can be divided into three broad categories. The first one relies on dynamical friction from infalling baryonic clumps and disc instabilities (El-Zant+’01; Weinberg+’02; Goerdt+’10; Del Popolo+’16). A second mechanism is associated with AGN-driven feedback (Peirani+’08; Martizzi+’12; Dubois+’16). Finally, a third option involves the long-term fluctuations associated with supernova-driven feedback (Binney+’01; Penarrubia+ '12). The collisionless diffusion equation is the ideal framework to investigate in detail their role on the secular evolution of DM haloes. 
In SEGAL, the first step will be to characterise these fluctuations. To do so, we will rely on hydrodynamical simulations, see Fig. 6.

In order to decouple the source of perturbations, i.e. the disc, from the perturbed system, i.e. the halo, these simulations are performed while using a static and inert halo. Such a setup allows us to measure and characterise the statistical properties of the fluctuations induced by the disc directly from simulations. Because the DM halo is analytical, this also prevents any shot noise associated with using a finite number of DM particles. Once these fluctuations are estimated, their effects on the DM halo may then be quantified using the secular collisionless diffusion equation. In order to characterise these fluctuations, we will consider an analytic NFW halo profile, and embed within it a gaseous and stellar disc, after preparing the system in a quasi-stationary state. In addition, we will implement supernova feedback allowing for the release of energy from the supernova into the interstellar medium. Figure 5 illustrates two successive snapshots of such a hydrodynamical simulation. In this figure, one can note that because of supernova feedback, the gas density fluctuates. These fluctuations in the potential due to the gas will be felt by the DM halo and may therefore drive resonant secular diffusion in the DM halo. We will ensemble-average various realisations of this same physical setup. The same approach will allow us to investigate how much this diffusion depends on the strength of the feedback, while changing the recipes (e.g. cooling/heating, etc.) used in the hydrodynamical simulations. We will quantify the typical fluctuation power spectrum and find quantitative bounds on feedback strengths sufficient to induce a softening of the DM halo’s profile. Similarly, the diffusion efficiency w.r.t. the disc and halo masses will also be investigated. Finally, the efficiency of AGN feedback to induce secular diffusion on larger scales in more massive DM halo will also be quantified with the same toolbox.


WPH2: The cosmic fate of open halos and galaxies It now appears clearly that the dynamical (azimuthal instabilities, warps, accretion), physical (heating, cooling) and secular (radial migration) evolution of galaxies are processes which are in part driven by the nature of their live halo, in particular by the boundary conditions imposed by their cosmic environment (e.g. Stewart+’16).It is therefore of prime importance to quantify the secular response of a galaxy or a halo induced by its interaction with this near environment. Interaction should be understood in a general sense and involve tidal potential interactions (like that corresponding to a satellite orbiting around the galaxy), shot noise (e.g. the population of globular clusters within the halo), and infall, where external components (virialized or not) are advected into the halo (Weinberg’93).

Halo transmission and amplification can then foster communication between spatially separated regions through gravitational wakes (see e.g. Murali’99) and continuously excites the galactic structure. For example, spirals can be induced by encounters with satellites and/or by mass injection (e.g. Toomre +’72; Howard+’90), while warps result from torque interactions with the surrounding matter (Jiang+’99).

 The statistical link between the inner properties of galactic haloes, and their cosmic boundary can be reversed to attempt and constrain the strength of infall while investigating their secular evolution. Following Pichon+’06, the dressed Fokker Planck equation will be extended in SEGAL and applied to systems open to their cosmic environment. We will derive the kinetic equation which governs the quasi-linear evolution of DM profiles induced by cosmic infall. Under the assumption of ergodicity, we will relate the corresponding source, drift and diffusion coefficients of the ensemble-average distribution to the underlying cosmic two-point statistics of the infall. We will also account for the slow evolution of the underlying equilibrium over half a Hubble time (see also Aubert+’07), and quantify cold gas infall towards the disc (cf WPD1). Finally SEGAL will revisit the Balescu equation in a context where the system’s number of particles gets to evolve during secular evolution, to describe for example the dissolution of over-densities via phase mixing and tidal stripping. A connection with chemical potentials and (later stages of) violent relaxation will be addressed (see also WPT3).

WPH3: The phase space structure of tidal streams and their progenitor clusters.The stellar streams within our MW’s dark halo as seen by GAIA will be analysed in terms of an impulsive diffusion process which will allow us to constrain the dark halo’s shape, its flattening and its clumpiness, (since each stream provides an estimate of the local diffusion tensor, which in turn scales like the local power-spectrum of the (dark+visible) potential). All quantities are of interest to constrain  the ΛCDM paradigm on Galactic scales. One additional goal will be investigate the subtle connection between the structural and kinematic properties of the stars in the tidal tails and the phase space properties of the progenitor star clusters (Varri+ '18). It has important implications on the morphology and dynamics of the streams, as recently demonstrated by the complex outer structures of the globular cluster omega Centari, for which the modelling of the internal rotation has been crucial (Ibata+ '19, Nature). A key aspect will be the continuation of our current investigation of the fundamental role played by  'kinematic complexity' in the evolution of collisional stellar systems (Breen+ '17, 19, Rozier+ '19, Hamilton+ '19.

 Results & Deliverables: Measurement of the power-spectrum of the potential fluctuations induced by the supernova feedback from a galactic disc, as a function of a galaxy's properties (e.g. SFR or radial profile). Implementation of the diffusion equation to constrain the timescale for the cusp-core transformation from inner baryonic perturbations, as a function of a halo's properties (e.g. mass or concentration). Measurement of the power-spectrum of fluctuations induced by a DM halo, as a function of a halo's properties (e.g. cold DM vs. fuzzy DM, role of the large-scale structures). Computation of the galactic disc heating induced by the DM halo perturbations. Formulation of a theory of angular momentum diffusion in collisional stellar systems.  Characterization of the phase space properties of tidal streams resulting from progenitor clusters modelled with direct summation N-body simulations, including rotation and pressure anisotropy.   Computation of the dissolution time of stellar streams from DM halo perturbations, and comparisons with constraints from GAIA measurements. Construction of a theory of inflow-driven diffusion to account for the statistics of inflows on galactic scales.


Secular-T: Secular theory for self-gravitating systems  (PHC,BM,FB,SP,CS, PDF, JBF,CP)

The kinetic theory of gravitating systems is a very active and fruitful field of research, as illustrated by Villani’s’10 Fields medal, or Mouhot’s’16 Adams prize for their “link between kinetic theory and molecular dynamics... and the relaxation towards equilibrium for collisional equations of Boltzmann type”. One of the purposes of SEGAL will also be to continue stimulating theoretical physics with novel kinetic theories arising in the context of galactic dynamics and 'gravitational plasma physics'. In this work-package, we will address some of the limitations of the existing kinetic theories, validate our findings using alternative methods, and investigate how our extended kinetic equations can be generalised to describe wider classes of dynamical processes, on both shorter and longer timescales.

WPT1: At first order, in the previous themes, we assumed integrability, i.e. the existence of global angle-action coordinates. It can be either guaranteed by the system’s symmetry (spherical halo, razor-thin discs) or by additional assumptions, such as the epicyclic approximation. When it does not hold, the system’s dynamics may become partially chaotic and its secular evolution may need to be described via explicit stochastic diffusions, which we will need to calibrate from simulations. For example, for stellar discs, chaos is likely to play a role in the central bar. In SEGAL, we will rely on perturbation theory (Goldstein’50) to build new (fast and slow) angle-actions customized to each resonance, using as a perturbation the departure from symmetry of the sought system: this strategy, known as Torus Mapping (Binney+’16), yields new actions with which secular diffusion will be reformulated. We will in particular build secular diffusion equations for thickened discs models and triaxial cusps. Eventually, it would be of significant interest to account for the anisotropy of the environment as well (following e.g. Aubert+’04).

WPT2: When implementing complex kinetic theories with loopback via self-gravity, validation is essential. Since we aim to make SDE solution a competitive tool to N-body simulations, we must demonstrate that the kinetic formulation accurately describes the secular response of modelled systems. Relying on ColDICE, a state-of-the-art public Vlasov (waterbag) solver developed at IAP (via our Vlasix project), we will compare the effect of secular evolution accurately described by such codes to the prediction of N-body on the one hand, and kinetic theory using SDE on the other hand. We will also rely on FEM for comparison, relying on the (hired) expertise of the PDF, and the PI and CoI upcoming investment in this technology (using the discontinuous Galerkin method). This comparison will also be used to study and understand the process of entropy production, having access via kinetic theory to an explicit analytic expression for its rate. It will allow us to study secular dynamical phase transitions (Fouvry+’15), which capture the slow and irreversible build-up of collisional effects leading to a destabilisation of secularly metastable states. At the technical level, the IBL equation can be used to quantify numerical shot-noise errors occurring in N-body codes, and partially controled in Vlasov codes such as ColDICE. Conversely, ColDICE can be used to study coherent transients in phase space, as seen by GAIA. 

WPT3 The derivation of the existing resonant orbital kinetic equations assumes full phase-average. They do not in their present form capture commensurability in the orbital frequencies (such as the 2:3 Pluto–Neptune resonances in our solar system) and are only concerned with orbit-orbit interactions. We will investigate in SEGAL how the secular formalism can be generalized to include such mean-motion resonances, following the procedure implemented for the Keplerian problem (i.e. integration over the fast angle before closure). SEGAL will also investigate how kinetic theory can be generalized to systems with a small or fluctuating number of effective particles, for example as a result of the progressive dissolution of over-densities. This will allow us to explore intermediate timescales, when the system has not fully phase mixed nor violently relaxed. We will attempt to develop a theory for the rate of entropy  production, while distinguishing the rapidly rising phase from the asymptotic one.

WPT4 In some regimes, the resonant orbital diffusion described by the IBL equation may vanish or weaken significantly. This can for example occur in galactic centers, as illustrated by the Schwarzschild barrier, driven by the divergence of the relativistic precession frequencies as stars move closer to the BH. This vanishing can also be imposed by symmetry, e.g. the collision operator vanishes for 1D homogeneous systems. Finally, it may happen on longer secular timescales if steady states for the IBL equation have been reached by the system. In such regimes, additional collisional effects involving three-body resonances will be accounted for within SEGAL. In addition to strong collisions, resonant effects associated with 1/N2 correlations can also drive the dynamics. This requires to consider the third equation of the BBGKY hierarchy and focus on slower effects associated with 3−body correlations. SEGAL will generalise the functional derivation of the inhomogeneous Landau equation (Fouvry+’16a) to account for collective effects and re-derive the IBL equation. This derivation will be challenging, as it involves a Fredholm integral equation for closure, but it is potentially very useful, since path integrals can easily be tailored to variational principles, which should help us formalise optimal FEM solutions. SEGAL will also apply this method to derive a closed kinetic equation involving higher order correlations. This could for example allow us to describe the dynamics of 1D homogeneous systems, for which the 1/N IBL collision term vanishes by symmetry (Eldridge+’63). It is also relevant for the Hamiltonian Mean Field toy model (HMF) (Bouchet +’05), the archetype of bar instability in galaxies. Finally, kinetic equations such as the Balescu equation describe the most probable evolution of the one particle distribution function through the law of large numbers. Following recent developments in non-equilibrium statistical mechanics, we will attempt to describe rare-event fluctuations away from this most likely PDF using Large Deviation Theory (e.g. Saint-Raymond’09, Codis+’15).

Results & Deliverables: Extension of the kinetic theory to account for higher-order correlations, e.g. as needed in the homogeneous HMF model. Extension of the kinetic theory to account for mean-motion resonances in quasi-Keplerian systems (in galactic nuclei and proto-planetary systems), i.e. going beyond the uncorrelated double Keplerian orbit-average. Extension of the kinetic theory to account for non-resonant effects, e.g. as in the diffusion à la Chandrasekhar through local Keplerian deflections. Exploration of non-integrable geometries (e.g. thickened discs, flattened spheres, and barred discs) to account for the richness of orbits and possible chaotic diffusion. Development of a numerical solver and integrator for the linear response theory of self-gravitating systems and the associated Balescu-Lenard kinetic equation. Validation of the kinetic theory through detailed comparisons with other approaches, in particular using Vlasov solver, direct N-body methods, and FEM methods.

Methodology and risk management

Quantitative assessment of success: A measure of the impact of this programme – beyond our detailed secular understanding of galactic evolution, black-hole and dark-halo physics – will be i) a statistical understanding of multi-scale galactic evolution over half a Hubble time  ii) the computation of the typical diffusion time of each secular process and the identification of the asymptotic solutions, and iii) the promotion of quasi-linear theories as a general universal and powerful means to rival/complement brute force Monte Carlo (N-body) methods. While N-body simulations have been extremely successful in experimentally improving our understanding of galaxy dynamics, they would still definitely benefit from validation through independent methods. This programme aims to do this via analytical and intensive numerical work of an intrinsically novel nature, by solving the corresponding SDEs and seeking finite element methods solutions. At the technical level, SEGAL’s deliverables will include new public routines for solving extended kinetic equations as stochastic Langevin processes, or using finite element methods (following our online distribution of codes such as MatrixMethod or MPgrafic and simulations outputs such as horizon-skymap). SEGAL will also deliver asymptotic solutions on all three scales, sets of estimators for standard observables, a better understanding of the effect of orbital resonances, and an extension of kinetic theory to closures involving three body encounters or more.

Feasibility: SEGAL is an exploratory, innovative and challenging project: it involves novel theoretical and numerical developments, addressing key questions in long term galactic evolution which have proven to be difficult. We are nevertheless confident that stochastic methods will allow us to understand the long term evolution of galaxies which will complement – or indeed in some context rival the classical N-body methods. Indeed we have already demonstrated via a series of forerunner papers that we control all the expertise required to carry out the programme presented in this proposal. This includes i) full scientific leadership on these truly original, unexplored and timely techniques; ii) managerial skills in supervising and training half a dozen PhD students around this topic. Several of them (Aubert, Ocvick, Dubois) are indeed now world-leaders in the production of massive state-of-the-art numerical simulations; others have received numerous prizes for the secular work; iii) having pioneered the statistical description of the cosmic environment of halos, first as dark matter flows (e.g. Aubert+’04), then as baryonic flows (e.g. Kimm+’11, Welker et al 15a,b); iv) having computed for the first time the exact drift and diffusion coefficient of the Balescu-Lenard equation and the corresponding diffusion time (Fouvry+‘15,17,Hamilton+’18); and v) having convinced the highly competitive (with pressure factors of 12) French ANR research council to fund at a level of 0.5 M€ the precursor (cosmic scale) project, ‘The origin of the Hubble sequence’, which has proven very successful (> 80 rank A publications) and has led to the intensive numerical (Dubois+‘14,17) and analytical (Fouvry+‘15a-c,16,17) work that makes this proposal possible (see the end of project report).

Risk assessment and mitigation: Since most work-packages are independent, we do not expect any significant stumbling blocks. Extended kinetic theory has already proven effective in explaining the origin of ridges in disc, the MW and clusters, and the behaviour of stars near our Galaxy’s massive black hole.  A transverse work-package Secular-T is devoted to quality assessment using a comparison with existing Vlasov/N-body/FEM solvers. Having matured this research over the last ten years, and successfully implemented three existing case of flux estimation, we envisage further improvement (e.g. going beyond the assumed integrability of the underlying system, or accounting for the effect of possible overlapping resonances). Each of these upgrades will be addressed explicitly within the unifying theme Secular-T. In the most unlikely event that secular kinetic diffusion evolution (via both SDE and FEM) would turn out to be too time demanding within one of the work-packages, we would resort to the other packages, to direct estimates of diffusion fluxes using Vlasov and N-body approach. Finally, within Secular-T we will also carry out more exploratory research which will only impact the rest of the project as a bonus. We therefore anticipate that the resources requested will allow us to carry out SEGAL’s most of this programme within four years, and possibly exceed our target.

1. II.Organisation and implementation of the project 

 SEGAL involves four complementary nodes led respectively by C. Pichon (Paris), P.H. Chavanis (Toulouse), B. Marcos (Nice), and B. Famaey (Strasbourg). Each node’s personnel is supplemented by external collaborators (mostly from the UK) which will contribute to the project as a whole, and to the scientific aims of that group. This structure provides the consortium with effective theoretical and observational expertise rooted in astrophysics and kinetic theory, as demonstrated in our recent (>20) joint publications which pioneered this field. Marcos’ group will bring his expertise in kinetic equations in the context of statistical physics and exactly solvable models, as well as expertise in modeling numerically progentiors of the MW, Famaey’s on modeling the chemo-dynamical evolution of the Milky Way with the Gaia data. Chavanis’ group will continue to provide a theoretical understanding of kinetic theory, while Pichon’s will focus on data in the vicinity of Sgr A*, and acquiring knowledge in SDE and FEM methods. The expected synergies between the four nodes will be key to reaching the milestones of the project. The PDF will allow us to bridge the four institutes through joint work, spending two years in Paris and one in Strasbourg. (S)he will also provide new expertise in FEM and relativistic simulations. As a group, we aim to carry out the programme presented in this proposal via re-enforced collaboration funded by the ANR, and shared computed resources hosted by IAP.

Given the expected close collaboration between the various nodes, we now describe our requested resources globally. The main emphasis of the budget of this ANR project is indeed on promoting such collaboration between the nodes, which has proven very effective for our past ANR (during which we organized 3 international meeting and 6 internal collaborative meetings). Hence a significant fraction of the budget will once again be devoted to travel/visitor costs.

Partner 1: IAP

The PI’s group activity has focused on gravitational dynamics, with a special emphasis on the statistical characterisation of matter. They have in particular analysed the instability and secular mechanisms driving the evolution of galaxies embedded in their cosmic environment. This investigation has led them to explore the dynamics of the large-scale structure, the intergalactic and interstellar media and the secular dynamics of galaxies and black holes. They have promoted novel tools and theories to trace and understand the impact of the cosmic web in simulations and observations. They used state-of-the-art “full-physics” cosmological simulations (Marenostrum’07, Horizon’08, Horizon-AGN’14, New Horizon’18), theoretical priors (tidal torque, excursion set…), and surveys (COSMOS, VIPERS, SDSS) to disentangle the relative effects of all these interconnected influences, to understand the acquisition of angular momentum driving the morphology of galaxies. One of the highlights of their work is a complete theory for conditional spin alignment (see Codis+’12,15, Laigle+’15), conditional assembly bias (Musso+17’, Kraljic+18’) and cosmic web connectivity (Codis+18’).        

They have also worked on smaller scales on the secular evolution of stellar systems driven by discreteness and their cosmic environment. They investigated analytically the secular transformation of cusps into cores, radial migration, Galactic center and globular cluster dynamics (including rotation) and disc thickening driven by GMCs, relying on breakthroughs in kinetic theory.  It reflects their recurrent focus on statistical approaches – which have been the successful backbone of large-scale structure cosmology – promoted to galactic scales, in order to follow galactic evolution over cosmic time. The astrophysical motivation is Galactic archeology/near field cosmology: to use the chemo-dynamical information encoded in extended phase space to time-reverse the history of galaxies and their host black holes and constrain dark matter. It is also of physical and formal interest, as the archetype of a stochastic anisotropic process with positive feedback loops. Two of their outstanding contributions have been i) accounting for the secular evolution of open systems (Pichon+’06) and ii) implementing the Balescu-Lenard equation to stellar systems for the first time (Fouvry+’15), and in the process demonstrating why swing-amplified sequences need not be correlated to account for the formation of ridges in action space, now observed in details by GAIA. The PI has a proven track record of leading large scientific projects – such as ANR spin(e), involving training and supervising many PhDs and postdocs, and successfully promoting collaborations amongst scientists across different institutions (Oxford, Marseille, Lyon, Seoul, Edinburgh). Halle brings numerical expertise in secular galactic processes while Volonteri on BH physics. Weinberg pioneered this field of research. Perrin’s group first hand knowledge of Sgr A* will contribute to the project with new astrometric points for S2 and new constraints on orbital elements. They will fit the data with the GYOTO ray tracing code and perform a new test of GR: the pericenter shift. The test will also comprise the search for a possible extended mass around the black hole. Bernardeau and Prunet’s in depth expertise in gravitational theory will complement  the recent hiring of Fouvry at CNRS, which will prove critical to the success of this ANR.

Partner 2: Strasbourg

The Strasbourg’s group activity has focused on gravitational dynamics and its application to the Milky Way galaxy. Their main focus is to constrain the Milky Way gravitational potential by making use of large surveys – such as RAVE, and now Gaia and WEAVE – and by devising new creative and beyond-the-state-of-the-art theoretical tools to extract all the relevant dynamical information. At the observational level, they have detected early-on a stellar radial velocity gradient in the Galactic disk with the RAVE survey (Siebert+’11), which has been recently confirmed with the Gaia DR2 (Katz+18), and whose origin due to the effect of the Galactic bar and/or spiral arms is still heavily debated. They have also put forward the effect of bar-spiral resonance overlaps in enhancing the radial migration of stars (Minchev+’10). Recently, they made a crucial new theoretical step in developing the most efficient analytical approach to the effects of the Galactic bar and spirals, by computing explicit distribution functions through perturbation theory, both away from resonances (Monari+’16) and in the resonant trapping regions of phase-space (Monari+’17). They will combine all the relevant dynamical information on field stars from Gaia and complementary spectroscopic surveys such as WEAVE, in order to build a state-of-the-art Galactic model, and to understand the effect of the disk's secular evolution. The model will include the effects of non-asymmetries (the bar and spiral arms) and vertical perturbations of the disk, such as those recently revealed with the Gaia DR2 (Antoja+’18). One crucial aspect is to take into account properly the self-gravity of the disk, which is one of the prime focuses of the WP-D part of the project. Beyond the specific resources requested for this project, the team has also access to fat node servers available at the Strasbourg HPC center.  The supplementary storage space as part of the material costs associated to the project will prove critical. The group’s active involvement in observational Galactic data will complement the IAP’s on the GC. 

Partner 3: Toulouse

The Toulouse node is focusing on the statistical mechanics and kinetic theory of Hamiltonian and Brownian systems with long-range interactions, using analytical tools. This has led to the derivation of sophisticated kinetic equations such as the inhomogeneous Balescu-Lenard equation (Chavanis‘12), self-gravitating Brownian particles (Chavanis+’02), and toy-models such as the Hamiltonian Mean Field models. Another activity of this group concerns the structure of dark matter halos as a possible solution to the cusp-core problem, and proposing as alternatives to the supermassive black holes that are purported to exist at the centers of galaxies.  Phase transitions and instabilities such as the gravothermal catastrophe have also been studied both in Newtonian and General Relativity. The Scotts bring established expertise on the dynamics of collisional stellar systems and related applications to the astrophysics of globular clusters and high-performance computing (direct-summation N-body simulations on GPUs). Novel themes explored in a collaborative fashion, include the theoretical exploration of 'kinematic complexity' in stellar and fluid systems, with attention to linear stability and long-term evolution.    More recent collaborations on the kinetic theory of self-gravitating discs, spheres and nuclear clusters (vector resonant relaxation) will continue to lead to fruitful collaborations with IAP.  The theoretical expertise of this group will prove crucial to the SEGAL proposal, both to highlight the physical processes captured by WP1-3 but also to reach the milestones of WP4.

Partner 4: Nice

The Nice node is split between Laboratory J.A. Dieudonné – devoted to the research on Mathematics and Theoretical Physics, and Observatoire de Nice – focused on astronomy. The former brings an expertise on kinetic theory, statistical physics and applied mathematics in general, for example in the context of the HMF model (Benetti+17), which will be crucial to tackle the ambitious theoretical goals of the collaboration. The latter brings expertise on numerical methods, hydrodynamical simulations of the Galaxy, and direct expertise in GAIA data. They work on different aspects of large scale structure formation in cosmology, formation and evolution of galaxies, self-gravitating systems, within a statistical physics perspective of simple and / or exactly solvable models. Peirani’s group at the Observatoire (together with Dubois at IAP) are key players in modelling numerically MilkyWay-like galaxies in a cosmological context (they were granted ~40 Mhrs of CPU in this context in France, the UK and Korea). Some of their simulations involve 100s of such galaxies over which statistical averaging will be possible. In addition to his theoretical skills, Marcos also brings expertise in GPUs (together with Aubert in Strasbourg, Rozier in Paris and Breen in Toulouse) which will be available via the cluster@IAP in order to use e.g. self-consistent field methods as gravity solver. 

1. III Impact and benefits of the project .

Understanding the long-term evolution of the components of galaxies such as stellar discs and galactic centers is now a subject of intensive research. In this context, the purpose of SEGAL is to establish quasi-linear theory as an ideal tool and enlightening addition to N-body simulations to probe statistically the cosmic fate of galaxies on multiple scales over a good fraction of a Hubble time. With the present public release of the GAIA data, a detailed theoretical modelling of the-long term evolution of the Milky Way’s internal structure is in order. Quasi-linear theory is for instance ideally suited to explain the novel observed features in the phase-space of our Milky Way. The cross-validation of N-body simulations via kinetic theory on secular timescales is now also of prime importance for upcoming projects like DESI, LSST, Euclid, 4MOST, MOONS etc, which rely heavily on modeling the dynamics of galaxies to mock their surveys. Eventually we will be able to gauge the roles of nature vs. nurture in establishing the observed properties of galaxy population on small and large scales, something currently out of reach of standard N-body techniques. Finally, the universal nature of gravitational kinetic theory could eventually allow us to address related problems, such as the secular evolution of proto-planetary discs. More generally, gravity, with its rich phenomenology, is ideally suited to help us understand in detail the secular implications of collective modes, shot noise and resonances. As such, it will continue to enlighten our understanding of the underlying mathematics, capturing fundamental physical processes such as entropy production, anisotropic resonant diffusion, secular phase transition etc. Hence gravitational kinetic theory will also stimulate progress and innovative research in theoretical physics and beyond. Following our past distribution of existing codes (e.g. MatrixMethod, MPgrafic, or simulations results like horizon-simulation.org and projet-horizon.fr) all our deliverables will be made available online on the public page of SEGAL for dissemination and validation to maximize impact. In essence, this area of research will, therefore, thrive in the coming years, as the communities active in cosmology and formation and dynamics of galaxies and star clusters are already converging towards a common set of open questions and challenges, and have much to learn from one another. SEGAL has the ambition to tackle exactly this - mostly unexplored - scientific intersection. Such a spirit, coupled with the novelty of the proposed methodology, generates great potential for discovery. The ultimate outcome of this unique synergy will be a deeper understanding of the fundamental physical processes in the assembly of cosmic structures in our universe and the proposed programme will contribute significantly to this endeavour.

Summary table of persons involved in the project: the colour code reflect WP

1. IV.References related to the project 

1 In some sense, the last 70+ years theoretical effort in explaining secular evolution of galaxies has been partially misled, because the source of fluctuation was assumed to be local Keplerian deflections.

Externally driven secular evolution is addressed in the context of kinetic theory via the so-called Fokker-Planck collisionless diffusion equation, where the source of evolution are long distance potential fluctuations from an external bath. Binney+’88 computed these diffusion coefficients in action-space assuming correlated noise driven fluctuations in the gravitational potential. This first approach did not account for gravitational amplification. Weinberg’93 emphasised the importance of self-gravity for the non-local and collective relaxation of stellar systems, but for a somewhat contrived uniform equilibrium. Weinberg’01 considered the secular evolution of a stellar sphere in a bath while accounting for self-gravity. Pichon+’06  presented a time-decoupling approach to solve the collisionless Boltzmann equation in the presence of external perturbations and infall to quantify the fluctuating effects of dynamical flows on secular timescales.

 Conversely, internally driven secular evolution is described via truncations of the BBGKY hierarchy, valid at order 1/N. While self-gravitating systems are spatially inhomogeneous, the first internally driven kinetic theories (Jeans’29) were also based on the assumption that close encounters between stars are treated with a local approximation, as if the system were infinite and homogeneous. Accounting for a large number of weak deflections, Chandrasekhar’49 developed an analogy with Brownian motion, relying on binary-collision theory. This led to the gravitational equivalent of Landau’s equation from plasmas, which does not account for long-range induced recurrences, nor collective effects (which increase effective gravitational masses and reduce collisional relaxation times). The kinetic theory of gravitating systems was recently generalised to fully inhomogeneous systems, either when collective effects are neglected (Chavanis’12) with the inhomogeneous Landau equation, or when they are accounted for, leading to the inhomogeneous Balescu-Lenard equation (Heyvaerts’10, Figure 1 below). These novel theories have now proven worthy in explaining secular processes on Galactic, Globular and nuclear cluster scales, illustrating the shortcomings of the local approximations.

2 e.g. the radial distribution of cosmic cold gas accretion directly impacts the metallicity of the new stars, which are the chemical clocks within the disc. This is a prime example of scale coupling that SEGAL will address.

3 Shadowing (Quinlan+’92), the fact that numerical shot noise statistically populates the relevant tori in phase space is of little help in this regime, unless it preserves orbits, since kinetic theory shows that resonances drive secular evolution in relaxed systems, as exemplified by the structure of the linear response operator.

4 Indeed, gravitational energy dominates on most scales. The impact of gas will be addressed here in setting up e.g. disc galaxies in low entropy states, as a drift in the mass of the components (following e.g. Pichon+’07), as a cold contribution to the gravitational susceptibility, and as extra sources of ‘external’ potential fluctuations (e.g. driven via AGN and stellar feedback see WPH below). Other baryonic processes relevant to galactic scales were investigated in the previous spin(e) ANR (which in particular showed that cooling led the gas to follow closely the dark-matter-imposed cosmic web). This stance restricts our analysis to the quieter lower redshift universe, where the acquisition of baryons can be treated perturbatively, or as an extra source of drift in the secular DF. Jointly with spin(e), SEGAL will allow us to quantify where and when secular processes dominate over environmentally driven ones.

5 The effect of general relativity is to induce a prograde pericenter shift, i.e. a shift of the orbit at the closest distance to the black hole. The warping of space-time due to frame dragging by a spinning black hole has a more subtle effect: a shift of the orbit at apocenter. Compared to a Schwarzschild black hole, the position of the apocenter of the orbit of the star S2 (which has a period of 16 years) around a maximum spinning black hole shifts by 10 μas, 24 μas et 40 μas, at the 1st, 2nd and 3rd orbit (Grould et al., 2017). A central goal of the proposal is to disentangle this relativistic effect from the Newtonian shift that an extended mass component would imply. Note that dozens of stellar mass black holes and invisible remnants are likely to perturb the trajectories of the S cluster. Their effect is partially degenerate with constrains on the relativistic effects of Sgr A*.