Pseudo Phase-space Density Research Articles

Structural and dynamical modeling of WINGS clusters

Numerical simulations indicate that cosmological halos display power-law radial profiles of pseudo phase-space density (PPSD), Q ≡ ρ/σ3, where ρ is the mass density and σ is the velocity dispersion. We tested these predictions for Q(r) using the parameters derived from the Markov chain Monte Carlo (MCMC) analysis performed with the MAMPOSSt mass-orbit modeling code on the observed kinematics of a velocity dispersion based stack (σv) of 54 nearby regular clusters of galaxies from the WINGS data set. In the definition of PPSD, the density is either in total mass ρ (Qρ) or in galaxy number density ν (Qν) of three morphological classes of galaxies (ellipticals, lenticulars, and spirals), while the velocity dispersion (obtained by inversion of the Jeans equation using the MCMC parameters) is either the total (Qρ and Qν) or its radial component (Qr, ρ and Qr, ν). We find that the PPSD profiles are indeed power-law relations for nearly all MCMC parameters. The logarithmic slopes of our observed Qρ(r) and Qr, ρ(r) for ellipticals and spirals are in excellent agreement with the predictions for particles in simulations, but slightly shallower for S0s. For Qν(r) and Qr, ν(r), only the ellipticals have a PPSD slope matching that of particles in simulations, while the slope for spirals is much shallower, similar to that of subhalos. However, for cluster stacks based on the richness or gas temperature, the fraction of power-law PPSDs is lower (esp. Qν) and the Qρ slopes are shallower, except for S0s. The observed PPSD profiles, defined using ρ rather than ν, appear to be a fundamental property of galaxy clusters. They would be imprinted during an early phase of violent relaxation for dark matter and ellipticals, and later for spirals as they move toward dynamical equilibrium in the cluster gravitational potential, while S0s are either intermediate (richness and temperature-based stacks) or a mixed class (σv stack).

The GOGREEN survey: Internal dynamics of clusters of galaxies at redshift 0.9–1.4

Context.The study of galaxy cluster mass profiles (M(r)) provides constraints on the nature of dark matter and on physical processes affecting the mass distribution. The study of galaxy cluster velocity anisotropy profiles (β(r)) informs the orbits of galaxies in clusters, which are related to their evolution. The combination of mass profiles and velocity anisotropy profiles allows us to determine the pseudo phase-space density profiles (Q(r)); numerical simulations predict that these profiles follow a simple power law in cluster-centric distance.Aims.We determine the mass, velocity anisotropy, and pseudo phase-space density profiles of clusters of galaxies at the highest redshifts investigated in detail to date.Methods.We exploited the combination of the GOGREEN and GCLASS spectroscopic data-sets for 14 clusters with massM200 ≥ 1014 M⊙at redshifts 0.9 ≤ z ≤ 1.4. We constructed anensemblecluster by stacking 581 spectroscopically identified cluster members with stellar massM⋆ ≥ 109.5 M⊙. We used the MAMPOSSt method to constrain severalM(r) andβ(r) models, and we then inverted the Jeans equation to determine theensembleclusterβ(r) in a non-parametric way. Finally, we combined the results of theM(r) andβ(r) analysis to determineQ(r) for theensemblecluster.Results.The concentrationc200of theensemblecluster mass profile is in excellent agreement with predictions from Λ cold dark matter (ΛCDM) cosmological numerical simulations, and with previous determinations for clusters of similar mass and at similar redshifts, obtained from gravitational lensing and X-ray data. We see no significant difference between the total mass density and either the galaxy number density distributions or the stellar mass distribution. Star-forming galaxies are spatially significantly less concentrated than quiescent galaxies. The orbits of cluster galaxies are isotropic near the center and more radial outside. Star-forming galaxies and galaxies of low stellar mass tend to move on more radially elongated orbits than quiescent galaxies and galaxies of high stellar mass. The profileQ(r), determined using either the total mass or the number density profile, is very close to the power-law behavior predicted by numerical simulations.Conclusions.The internal dynamics of clusters at the highest redshift probed in detail to date are very similar to those of lower-redshift clusters, and in excellent agreement with predictions of numerical simulations. The clusters in our sample have already reached a high degree of dynamical relaxation.

Phase-space structure of protohalos: Vlasov versus particle-mesh

The phase-space structure of primordial dark matter halos is revisited using cosmological simulations with three sine waves and cold dark matter (CDM) initial conditions. The simulations are performed with the tessellation based Vlasov solver ColDICE and a particle-mesh (PM) N-body code. The analyses include projected density, phase-space diagrams, radial density ρ(r), and pseudo-phase space density: Q(r) = ρ(r)/σv(r)3 with σv the local velocity dispersion. Particular attention is paid to force and mass resolution. Because the phase-space sheet complexity, estimated in terms of total volume and simplex (tetrahedron) count, increases very quickly, ColDICE can follow only the early violent relaxation phase of halo formation. During the violent relaxation phase, agreement between ColDICE and PM simulations having one particle per cell or more is excellent and halos have a power-law density profile, ρ(r) ∝ r−α, α ∈ [1.5, 1.8]. This slope, measured prior to any merger, is slightly larger than in the literature. The phase-space diagrams evidence complex but coherent patterns with clear signatures of self-similarity in the sine wave simulations, while the CDM halos are somewhat scribbly. After additional mass resolution tests, the PM simulations are used to follow the next stages of evolution. The power law progressively breaks down with a convergence of the density profile to the well-known Navarro–Frenk–White universal attractor, irrespective of initial conditions, that is even in the three-sine-wave simulations. This demonstrates again that mergers do not represent a necessary condition for convergence to the dynamical attractor. Not surprisingly, the measured pseudo phase-space density is a power law Q(r) ∝ r−αQ, with αQ close to the prediction of secondary spherical infall model, αQ ≃ 1.875. However this property is also verified during the early relaxation phase, which is non-trivial.

Power-law Pseudo-phase-space Density Profiles of Dark Matter Halos: A Fluke of Physics?

It has been known for nearly 20 years that the pseudo phase-space density profile of equilibrium simulated dark matter halos, $\rho(r)/\sigma^3(r)$, is well described by a power law over 3 decades in radius, even though both the density $\rho(r)$, and the velocity dispersion $\sigma(r)$ deviate significantly from power laws. The origin of this scale-free behavior is not understood. It could be an inherent property of self-gravitating collisionless systems, or it could be a mere coincidence. To address the question we work with equilibrium halos, and more specifically, the second derivative of the Jeans equation, which, under the assumptions of (i) Einasto density profile, (ii) linear velocity anisotropy - density slope relation, and (iii) $\rho/\sigma^3\propto r^{-\alpha}$, can be transformed from a differential equation to a cubic algebraic equation. Relations (i)-(iii) are all observed in numerical simulations, and are well parametrized by a total of 4 or 6 model parameters. We do not consider dynamical evolution of halos; instead, taking advantage of the fact that the algebraic Jeans equation for equilibrium halos puts relations (i)-(iii) on the same footing, we study the (approximate) solutions of this equation in the 4 and 6 dimensional spaces. We argue that the distribution of best solutions in these parameter spaces is inconsistent with $\rho/\sigma^3\propto r^{-\alpha}$ being an fundamental property of gravitational evolution, and conclude that the scale-free nature of this quantity is likely to be a fluke.

Halo density profiles and baryon physics

The radial dependence of the pseudo phase-space density, ρ(r)/σ 3(r) is studied. We find that the pseudo phase-space density for halos consisting both of dark matter and baryons is approximately a power-law only down to 0.1% of the virial radius while it has a non-power law behavior below the quoted scale, with inner profiles changing with mass. Halos consisting just of dark matter, as the one in dark matter only simulations, are characterized by an approximately power-law behavior. The results argue against universality of the pseudo phase-space density, when the baryons effect are included, and as a consequence argue against universality of density profiles constituted by dark matter and baryons as also discussed in [1].

On the apparent power law in CDM halo pseudo-phase space density profiles

Abstract We investigate the apparent power-law scaling of the pseudo-phase space density (PPSD) in cold dark matter (CDM) haloes. We study fluid collapse, using the close analogy between the gas entropy and the PPSD in the fluid approximation. Our hydrodynamic calculations allow for a precise evaluation of logarithmic derivatives. For scale-free initial conditions, entropy is a power law in Lagrangian (mass) coordinates, but not in Eulerian (radial) coordinates. The deviation from a radial power law arises from incomplete hydrostatic equilibrium (HSE), linked to bulk inflow and mass accretion, and the convergence to the asymptotic central power-law slope is very slow. For more realistic collapse, entropy is not a power law with either radius or mass due to deviations from HSE and scale-dependent initial conditions. Instead, it is a slowly rolling power law that appears approximately linear on a log–log plot. Our fluid calculations recover PPSD power-law slopes and residual amplitudes similar to N-body simulations, indicating that deviations from a power law are not numerical artefacts. In addition, we find that realistic collapse is not self-similar; scalelengths such as the shock radius and the turnaround radius are not power-law functions of time. We therefore argue that the apparent power-law PPSD cannot be used to make detailed dynamical inferences or extrapolate halo profiles inwards, and that it does not indicate any hidden integrals of motion. We also suggest that the apparent agreement between the PPSD and the asymptotic Bertschinger slope is purely coincidental.

Deep spectroscopy in nearby galaxy clusters – III. Orbital structure of galaxies in Abell 85

Galaxies in clusters are strongly affected by their environment. They evolve according to several physical mechanisms that are active in clusters. Their efficiency can strongly depend on the orbital configuration of the galaxies. Our aim is to analyse the orbits of the galaxies in the cluster Abell 85, based on the study of the galaxy velocity anisotropy parameter. We have solved the Jeans equation under the assumption that the galaxies in A85 are collisionless objects, within the spherically symmetric gravitational potential of the virialized cluster. The mass of the cluster was estimated with X-ray and caustic analyses. We find that the anisotropy profile of the full galaxy population in A85 is an increasing monotonic function of the distance from the cluster centre: on average, galaxies in the central region (r/r200 < 0.3) are on isotropic orbits, while galaxies in the outer regions are on radial orbits. We also find that the orbital properties of the galaxies strongly depend on their stellar colour. In particular, blue galaxies are on less radial orbits than red galaxies. The different families of cluster galaxies considered here have the pseudo phase-space density profiles Q(r) and Qr(r) consistent with the profiles expected in virialized dark matter halos in $N$-body simulations. This result suggests that the galaxies in A85 have reached dynamical equilibrium within the cluster potential. Our results indicate that the origin of the blue and red colour of the different galaxy populations is the different orbital shape rather than the accretion time.

Mass, velocity anisotropy, and pseudo phase-space density profiles of Abell 2142(Corrigendum)

The pseudo phase-space density profile of Abell 2142, defined with the galaxy number density profile ν instead of the cluster mass density profile ρ, was shown in Fig. 12 of Munari et al. (2014). This graph was erroneous (because we had incorrectly considered the projected number density profile instead of the 3D profile). The correct figure is shown below. The values in Table 4 are changed, and the correct ones are reported in the table below. While Fig. 12 of Munari et al. (2014) indicated that the PPSDs computed with the number density profile are significantly shallower than the theoretical relation of Dehnen & McLaughlin (2005), the corrected version of the figure shown here indicates that the PPSDs computed with the number density profile are now either consistent with the relation of Dehnen & McLaughlin (Q(r) for BLUE sample) or only slightly shallower, but not less consistent with that relation than found for the PPSDs computed with the mass density profile. Therefore, the statement in Munari et al. (2014) that the mass density profile represents the PPSD and β − γ relations better is no longer correct. Indeed, Figs. 11 and 13 of Munari et al. show that the β − γ relations obtained using the mass density or tracer number density are indistinguishable for the RED and ALL samples. And therefore, the entire discussion of the greater relevance of the mass density profile relative to the galaxy number density profile must be dismissed. Fig. 12. Same as Fig. 10 of Munari et al. (2014), but now using the radial profiles of galaxy number density instead of total mass density to estimate the PPSD.

Relaxation of N-body systems with additive r−α interparticle forces

In Newtonian gravity the final states of cold dissipationless collapses are characterized by several structural and dynamical properties remarkably similar to those of observed elliptical galaxies. Are these properties a peculiarity of the Newtonian force or a more general feature of long-range forces? We study this problem by means of $N-$body simulations of dissipationless collapse of systems of particles interacting via additive $r^{-\alpha}$ forces. We find that most of the results holding in Newtonian gravity are also valid for $\alpha\neq2$. In particular the end products are triaxial and never flatter than an E7 system, their surface density profiles are well described by the S\'ersic law, the global density slope-anisotropy inequality is obeyed, the differential energy distribution is an exponential over a large range of energies (for $\alpha\geq1$), and the pseudo phase-space density is a power law of radius. In addition, we show that the process of virialization takes longer (in units of the system's dynamical time) for decreasing values of $\alpha$, and becomes infinite for $\alpha=-1$ (the harmonic oscillator). This is in agreement with the results of deep-MOND collapses (qualitatively corresponding to $\alpha=1$) and it is due to the fact the force becomes more and more similar to the $\alpha=-1$ case, where as well known no relaxation can happen and the system oscillates forever.

The phase-space density of fermionic dark matter haloes

We have performed a series of numerical experiments to investigate how the primordial thermal velocities of fermionic dark matter particles affect the physical and phase space density profiles of the dark matter haloes into which they collect. The initial particle velocities induce central cores in both profiles, which can be understood in the framework of phase space density theory. We find that the maximum coarse-grained phase space density of the simulated haloes (computed in 6 dimensional phase space using the EnBid code) is very close to the theoretical fine-grained upper bound, while the pseudo phase space density, Q ~ {\rho}/{\sigma}^3, overestimates the maximum phase space density by up to an order of magnitude. The density in the inner regions of the simulated haloes is well described by a 'pseudo-isothermal' profile with a core. We have developed a simple model based on this profile which, given the observed surface brightness profile of a galaxy and its central velocity dispersion, accurately predicts its central phase space density. Applying this model to the dwarf spheroidal satellites of the Milky Way yields values close to 0.5 keV for the mass of a hypothetical thermal warm dark matter particle, assuming the satellite haloes have cores produced by warm dark matter free streaming. Such a small value is in conflict with the lower limit of 1.2 keV set by observations of the Lyman-{\alpha} forest. Thus, if the Milky Way dwarf spheroidal satellites have cores, these are likely due to baryonic processes associated with the forming galaxy, perhaps of the kind proposed by Navarro, Eke and Frenk and seen in recent simulations of galaxy formation in the cold dark matter model.

Theoretical dark matter halo kinematics and triaxial shape

In a recent paper, Salvador-Sol\'e et al. (2012) have derived the typical inner structure of dark matter haloes from that of peaks in the initial random Gaussian density field, determined by the power-spectrum of density perturbations characterising the hierarchical cosmology under consideration. In the present paper, we extend this formalism to the typical kinematics and triaxial shape of haloes. Specifically, we establish the link between such halo properties and the power-spectrum of density perturbations through the typical shape of peaks. The trends of the predicted typical halo shape, pseudo phase-space density and anisotropy profiles are in good agreement with the results of numerical simulations. Our model sheds light on the origin of the power-law-like pseudo phase-space density profile for virialised haloes.

SELF-SIMILAR DYNAMICAL RELAXATION OF DARK MATTER HALOS IN AN EXPANDING UNIVERSE

We investigate the structure of cold dark matter halos using advanced models of spherical collapse and accretion in an expanding Universe. These base on solving time-dependent equations for the moments of the phase-space distribution function in the fluid approximation; our approach includes non-radial random motions, and most importantly, an advanced treatment of both dynamical relaxation effects that takes place in the infalling matter: phase-mixing associated to shell crossing, and collective collisions related to physical clumpiness. We find self-similar solutions for the spherically-averaged profiles of mass density rho(r), pseudo phase-space density Q(r) and anisotropy parameter beta(r). These profiles agree with the outcomes of state-of-the-art N-body simulations in the radial range currently probed by the latter; at smaller radii, we provide specific predictions. In the perspective provided by our self-similar solutions we link the halo structure to its two-stage growth history, and propose the following picture. During the early fast collapse of the inner region dominated by a few merging clumps, efficient dynamical relaxation plays a key role in producing a closely universal mass density and pseudo phase-space density profiles; in particular, these are found to depend only weakly on the detailed shape of the initial perturbation and the related collapse times. The subsequent inside-out growth of the outer regions feeds on the slow accretion of many small clumps and diffuse matter; thus the outskirts are only mildly affected by dynamical relaxation but are more sensitive to asymmetries and cosmological variance.

Non-power law behavior of the radial profile of phase-space density of halos

We study the pseudo phase-space density, ρ(r)/σ3(r), of ΛCDM dark matter halos with and without baryons (baryons+DM, and pure DM), by using the model introduced in Del Popolo (2009), which takes into account the effect of dynamical friction, ordered and random angular momentum, baryons adiabatic contraction and dark matter baryons interplay. We examine the radial dependence of ρ(r)/σ3(r) over 9 orders of magnitude in radius for structures on galactic and cluster of galaxies scales. We find that ρ(r)/σ3(r) is approximately a power-law only in the range of halo radius resolved by current simulations (down to 0.1% of the virial radius) while it has a non-power law behavior below the quoted scale, with inner profiles changing with mass. The non-power-law behavior is more evident for halos constituted both of dark matter and baryons while halos constituted just of dark matter and with angular momentum chosen to reproduce a Navarro-Frenk-White (NFW) density profile, are characterized by an approximately power-law behavior. The results of the present paper lead to conclude that density profiles of the NFW type are compatible with a power-law behavior of ρ(r)/σ3(r), while those flattening to the halo center, like those found in Del Popolo (2009) or the Einasto profile, or the Burkert profile, cannot produce radial profile of the pseudo-phase-space density that are power-laws at all radii. The results argue against universality of the pseudo phase-space density and as a consequence argue against universality of density profiles constituted by dark matter and baryons as also discussed in Del Popolo (2009).

Non-spherical similarity solutions for dark halo formation

We carry out fully 3-dimensional simulations of evolution from self-similar, spherically symmetric linear perturbations of a Cold Dark Matter dominated Einstein-de Sitter universe. As a result of the radial orbit instability, the haloes which grow from such initial conditions are triaxial with major-to-minor axis ratios of order 3:1. They nevertheless grow approximately self-similarly in time. In all cases they have power-law density profiles and near-constant velocity anisotropy in their inner regions. Both the power-law index and the value of the velocity anisotropy depend on the similarity index of the initial conditions, the former as expected from simple scaling arguments. Halo structure is thus not "universal" but remembers the initial conditions. On larger scales the density and anisotropy profiles show two characteristic scales, corresponding to particles at first pericentre and at first apocentre after infall. They are well approximated by the NFW model only for one value of the similarity index. In contrast, at all radii within the outer caustic the pseudo phase-space density can be fit by a single power law with an index which depends only very weakly on the similarity index of the initial conditions. This behaviour is very similar to that found for haloes formed from LCDM initial conditions and so can be considered approximately universal.

DISTRIBUTION FUNCTION IN THE CENTER OF THE DARK MATTER HALO

N-body simulations of dark matter halos show that the density profiles of the halos behave as ρ(r) ∝ r-α(r), where the density logarithmic slope α ≃ 1–1.5 in the center and α ≃ 3–4 in the outer parts of the halos. However, some observations are not in agreement with simulations in the very central region of the halos. The simulations also show that the velocity dispersion anisotropy parameter β ≈ 0 in the inner part of the halo and the so-called pseudo–phase-space density ρ/σ3 behaves as a power law in radius r. With these results in mind, we study the distribution function and the pseudo–phase-space density ρ/σ3 of the center of dark matter halos and find that they are closely related.

THE DYNAMICAL STRUCTURE OF DARK MATTER HALOS WITH UNIVERSAL PROPERTIES

$N$-body simulations have unveiled several apparently universal properties of dark matter halos, including a cusped density profile, a power-law pseudo phase-space density $\rho/\sigma_r^3$, and a linear $\beta-\gamma$ relation between the density slope and the velocity anisotropy. We present a family of self-consistent phase-space distribution functions $F(E,L)$, based on the Dehnen-McLaughlin Jeans models, that incorporate these universal properties very accurately. These distribution functions, derived using a quadratic programming technique, are analytical, positive and smooth over the entire phase space and are able to generate four-parameter velocity anisotropy profiles $\beta(r)$ with arbitrary asymptotic values $\beta_0$ and $\beta_\infty$. We discuss the orbital structure of six radially anisotropic systems in detail and argue that, apart from its use for generating initial conditions for $N$-body studies, our dynamical modeling provides a valuable complementary approach to understand the processes involved in the formation of dark matter halos.

Alas, the Dark Matter Structures Were Not That Trivial

The radial density profile of dark matter structures has been observed to have an almost universal behaviour in numerical simulations, however, the physical reason for this behaviour remains unclear. It has previously been shown that if the pseudo phase-space density, rho/sigma_d^epsilon, is a beautifully simple power-law in radius, with the values epsilon=3 and d=r (i.e., the phase-space density is only dependent on the radial component of the velocity dispersion), then one can analytically derive the radial variation of the mass profile, dispersion profile etc. That would imply, if correct, that we just have to explain why rho/sigma^3_r ~r^{-alpha}, and then we would understand everything about equilibrated DM structures. Here we use a set of simulated galaxies and clusters of galaxies to demonstrate that there are no such golden values, but that each structure instead has its own set of values. Considering the same structure at different redshifts shows no evolution of the phase-space parameters towards fixed points. There is also no clear connection between the halo virialized mass and these parameters. This implies that we still do not understand the origin of the profiles of dark matter structures.

Power Laws and Non–Power Laws in Dark Matter Halos

Simulated dark matter profiles are often modelled as a `NFW' density profile rather than a single power law. Recently, attention has turned to the rather rigorous power-law behaviour exhibited by the `pseudo phase-space density' of the dark matter halo, which is defined dimensionally in terms of the local density and velocity dispersion of the dark matter particles. The non-power-law behaviour of the density profile is generally taken to exclude simple scale-free, in-fall models; however the power-law behaviour of the `pseudo-density' is a counter indication. We argue in this paper that both behaviours may be at least qualitatively understood in terms of a dynamically evolving self-similarity, rather than the form for self-similar infall that is fixed by cosmological initial conditions. The evolution is likely due to collective relaxation such as that provided by the radial-orbit instability on large scales. We deduce, from a distribution function given by first order coarse-graining, both the NFW-type density profile and the power-law pseudo-density profile. The results are not greatly sensitive to variation about 3 in the power of the velocity dispersion used in the definition of the phase space pseudo-density. We suggest that the power 2 may create the more physical quantity, whose deviations from a power-law are a diagnostic of incomplete relaxation.