We discuss the evolved mass profile near the center of aninitial spherical density perturbation, δ ∝ M–ε, of collision-less particles with non-radialmotions. W consider a scheme inwhich a particle moves on a radial orbit until it reaches itsturnaround radius, r*. At turnaround the particle acquires anangular momentum $L={\cal L} \sqrt{GM_* r_*}$ per unit mass, whereM* is the mass interior to r*. In this scheme, the mass profileis M ∝ r3/(1+3ε) for all ε > 0, in the region r/rt ≪ ${\cal L}$, where rt is the current turnaround radius.If ${\cal L}$ ≪ 1 then the profile in the region ${\cal L}$ ≪ r/rt ≪ is M ∝ r for ε <2/3. We also present a model for the growth of dark matter halos and use it to study their evolved density profiles.In this model, halos are spherical and form byquiescent accretion of matter in clumps, called satellites. The halo mass as a function of redshift is given by the mass of the most massive progenitor, and is determined from Monte-Carlo realizations of the merger-history tree. Inside the halo, satellites move under the action of the gravitational force of the halo and a dynamical friction drag force. The associated equation of motion is solved numerically. The energy lost to dynamical friction is transferred to the halo in the form of kinetic energy. As they sink into the halo, satellites continually lose matter as a result of tidal stripping. The stripped matter moves inside the for mass scales where the effective spectral index of the initial density field is less than –1, the model predicts a profile which can only approximately be matched by the NFW one parameter family of curves. For scale-free power-spectra with initial slope n, the density profile within about 1% of the virial radius is ρ ∝ r–β, with 3(3+n)/(5+n) ≤ β ≤ 3(3+n)/(4+n).

N-body simulations find a universal structure for the halos whichresult from the nonlinear growth of Gaussian-random-noise densityfluctuations in the CDM universe. This talk summarized our attempts toderive and explain this universal structure by analyticalapproximation and simplified models. As an example, we show here thata 1D spherical infall model involving a fluid approximation derivedfrom the Boltzmann equation can explain not only the halo density profilebut its phase-space density profile, as well.


We investigated the structure of the dark matter halo formed in thecold dark matter scenarios by N-body simulations with paralleltreecode on GRAPE cluster systems (Fukushige et al. 2004, ApJ,606, 625). We simulated 8 halos with the mass of 4.4 × 1014 M๏ to 1.6 × 1015 M๏ in the SCDM and LCDMmodel using up to 30 million particles. With the resolution of oursimulations, the density profile is reliable down to 0.2 percent ofthe virial radius. Our results show that the slope of inner cuspwithin 1 percent virial radius is shallower than -1.5, and theradius where the shallowing starts exhibits run-to-run variation.



The formation and structure of dark matter halos is studied by constrained simulations. A series of experiments of theformation of a 1012h-1 M๏ halo is designed to study the dependenceof the density profile on its merging history.We find that the halo growth consist of several quiescent phases intermitted by violent events, with the density well approximated by the NFWprofile during the former phases. We find that (1) the NFW scale radiusRs  stays constant during the quiescent phase and grows abruptly during theviolent one. In contrast, the virial radiusgrows linearly during the quiescent and abruptly during the violentphases. (2) The central density stays unchanged during the quiescentphase while dropping abruptly during the violent phase, and it does not reflectthe formation time of the halo.(3) The clear separation of the evolution of an individual halo into quiescent and violent phases implies that its entire evolution cannot be fitted by simple scaling relations.


Caustics are formally singular structures that frequently form in collisionless media. The non-negligible velocity dispersion of dark matterparticles renders their density finite. We evaluated the maximum density ofthe caustics within the framework of secondary infall model of formation ofdark matter haloes. The result isthen used to demonstrate that caustics can be probedby properly stacking the weak-lensing signal of about 600 haloes. CFHTLS accompanied by X-ray observations and the space-based experiments like SNAP or DUNE can provide us with the required statistics.The extension of our results to more realistic models including the effects of mergers is outlined.


Using high resolution DM simulations we study the shape of dark matter halos. Halos become more spherical with decreasing mass. This trend is even more pronounced for the inner part of the halo. Angular momentum and shape are correlated. The angular momenta of neighboring halos are correlated.


The hierarchical nature of halo formation in the Cold Dark Matter (CDM) model suggests that there should be mass dependent trends in the radial velocity andvelocity anisotropy profiles of dark matter haloes, with more massive systems exhibiting the effects of more pronounced infall. We present results from a sample of high resolution cosmological N-body simulations of individual dwarf, galaxy and cluster mass haloes, and demonstrate that such trends with mass areindeed observed.


One can solve the Jeans equation analytically forequilibrated dark matter structures, once given two pieces of inputfrom numerical simulations. These inputs are 1) a connection betweenphase-space density and radius, and 2) a connection between velocityanisotropy and density slope, the α – β relation. The first(phase-space density vs. radius) has been analysed through severaldifferent simulations, however the second (α – β relation) has notbeen quantified yet. We perform a large set of numerical experimentsin order to quantify the slope and zero-point of the α – βrelation. This allows us to conclude that equilibrated dark matterstructures indeed have zero central velocity anisotropy,central densityslope of α0 ≈ –0.8, and outer anisotropy ofapproximately β∞ ≈ 0.5.

A new description of the halo's dynamics is introduced. The response of haloes interacting with the accretion and the tidal field is described analytically in the non-linear and secular regime via a perturbative approach. It relates the dynamical properties of a population of such objects to the statistics of their environment. The latter are deduced from cosmological simulations and first results on the anisotropy of accretion, its kinematic properties and the two-point correlation of the tidal field and the density flux of mass are described. This approach should allow to predict the statistical properties of the dark matter tracers within the galaxy (e.g. Lensing, X Emission) or conversely, use these observational probes to constrain the characteristics of haloes' environments.

I review recent results on the angular distribution of substructure in CDM halos. I show that the subhalo distribution isanisotropic and generally prolate with a long axis that is closely aligned with the long axis of the total mass distribution of the host halo. I show that the correlation between the subhalo distribution and the long axis of the host halo is strong both in dissipationless and dissipational gasdynamical simulations. More massive subhalos tend to be more strongly clustered along the major axis of the host halo reflecting filamentary accretion. The anisotropy of subhalos has potential implications for the interpretation of several observations. For example, I show that while the mean projected mass fraction in substructure in the central regions of CDM halos is fsub ≈ 0.4%, fsub is a strong function of projection angle and is ~6 times higher for projections nearly collinear with the major axis of the host.

I consider a very simple model for the evolution of the real space and phase-space density profiles of a dark matter halo as it acquires its mass. I show that this model, which assumes slow growth through gradual accretion, predicts structures much more concentrated than those found in self-consistent simulations of halo formation. By implication, major mergers must act to reduce the concentration of dark matter haloes, adding the orbital energy of the infalling satellite to the internal energy of the pre-existing system and causing it to expand at intermediate radii. This effect should be easy to detect in numerical simulations.

N-body simulations predict that CDM halo-assembly occurs in twophases: a rapid accretion phase dominated by major mergers and a slow accretion phase characterised by a gentle addition of mass to the outer halo. We demonstrate that this two-phase accretion leads to CDM halos of the NFW form. During the fast accretionphase fluctuations in the gravitational potential effectivelyisotropise the velocities of CDM particles and we show that this leadsto an inner profile ρ(r) ∝ r-1. Slow accretion onto anestablished potential well leads to an outer profile with ρ(r) ∝ r-3. Using a simple analytic model thatcaptures much of the important physics we show thatthe inner r-1 profile of CDM halos is a natural result of hierarchical mass assembly with a initial phase of rapid accretion.

N-body simulations show that dark matter haloes develop from theinside out in phases of smooth accretion between major mergers. Inthese conditions it can be shown that the density profile ofellipsoidal anisotropic rotating haloes is completely determined bythe current values of their extensive properties and theircorresponding accretion rates. This allows us to build a simplephysical model for the evolution of the NFW scale radius. Moreover,since we can arbitrarily assume that a halo has grown by pureaccretion, the inside-out growth condition automatically leads to thehalo density profile. We find that the typical accretion ratepredicted by the extended Press-Schechter formalism in any givencosmology leads to a typical density profile of the NFW form for thecorrect mass-concentration relation.

The cooling of baryons in the centers of dark matter halos leads to a more concentrated dark matter distribution. This effect has traditionally been calculated using the model of adiabatic contraction, which assumes spherical symmetry, while in hierarchical formation scenarios halos grow via multiple violent mergers. We test the adiabatic contraction model in high-resolution cosmological simulations and find that the dissipation of gas indeed increases the density of dark matter and steepens its radial profile compared to the case without cooling. Although the standard model systematically overpredicts the increase of dark matter density, a simple modification of the assumed invariant from M(r)r to $M(\bar{r})r$, where $\bar{r}$ is the orbit-averaged particle position, reproduces the simulated profiles within 10%.


Using hydrodynamical cosmological simulations we investigate the effect of baryonic dissipation on halo shapes. We show that dissipational simulations produce significantly rounder halos than those formed in equivalent dissipationless simulations. Gas cooling causes an average increase in halo principal axis ratios of ~0.2−0.4 in the inner regions and a systematic shift that persists out to the virial radius, alleviating any tension between theory and observations. Although the magnitude of the effect may be overestimated due to overcooling, cluster formation simulations designed to reproduce the observed fraction of cold baryons still produce substantially rounder halos. Subhalos also exhibit a trend of increased axis ratios in dissipational simulations. Moreover, we demonstrate that subhalos are generally rounder than corresponding field halos even in dissipationless simulations. All of these results highlight the vital role of baryonic processes in comparing theory with observations and warn against over-interpreting discrepancies with collisionless simulations on small scales.

The rotation curves of spiral galaxies obey strong scaling relations.These include the Tully-Fisher and baryonic Tully-Fisher relations, and themass discrepancy–acceleration relation.These relations can be used to place constraints on the mass-to-lightratios of stars. Once the stellar mass is constrained, the distribution ofdark matter follows. The shape of the dark matter distribution is consistentwith the expectations of NFW halos exterior to 1 kpc, but the amplitudeis wrong. This is presumably related to the long-standing problemof the normalization of the Tully-Fisher relation and may imply a downturn in the amplitude of the power spectrum at small scales. More fundamentally, the persistent success of MOND remains a troubling fact.


In the past years a wealth of observationsallowed to unravel the structural properties of the Dark Matter Halos around spirals. First, their rotation curves followan Universal profile (URC) that can be described in terms of an exponential thin stellar disk and adark halo with aconstant density core, whose relative importance increases with galaxy luminosity. Careful studies of individual objects, from dwarfs to giants, reveal that dark halos have a core, whose size r0 correlates with the central density ρ0.These properties are in serious discrepancy with the cuspy density distribution predicted by N-body simulations in collisionless ΛCDM Cosmology.

We discuss the accuracy for the recovery of mass profiles in simulated dwarf galaxies. We also present cosmologically motivated, high resolution N-body models for two nearby dwarf galaxies: NGC 3109 and NGC 6822. The rotation curves in our models are measured using standard observational techniques, projecting velocities along the line of sight of an artificial observer and performing a tilted ring analysis. Our models reproduce the rotation curves of both galaxies, the disk surface brightness profiles as well as the profile of isophotal ellipticity and position angle. We show that non-circular motions combined with gas pressure support and projection effects systematically underestimate by up to 50% the rotation velocity of cold gas in the central ~1 kpc region, creating the illusion of a constant density core.

In its first part, this paper summarizes recent work on the mass andshape of the Galactic dark halo. The second part presents a review ofthe large-scale structure of the Milky Way, and of the evidence thatthe inner Galaxy is dominated by baryonic matter. This is brieflycompared with the predictions of ΛCDM and MOND. Finally, asummary is given of bulge formation from clumpy, gas-rich disks, aprocess which may give rise to old, disk-like, α-rich bulgessimilar to the Galactic bulge.