We employ very high resolution simulations of isolated Milky Way-like galaxies to study the effect of triaxial dark matter haloes on exponential stellar discs. Non-adiabatic halo shape changes can trigger two-armed grand-design spiral structures which extend all the way to the edge of the disc. Their pattern speed coincides with the inner Lindblad resonance indicating that they are kinematic density waves which can persist up to several Gyr. In dynamically cold discs, grand-design spirals are swing amplified and after a few Gyr can lead to the formation of (multi-armed) transient recurrent spirals. Stellar discs misaligned to the principal planes of the host triaxial halo develop characteristic integral shaped warps, but otherwise exhibit very similar spiral structures as aligned discs. For the grand-design spirals in our simulations, their strength dependence with radius is determined by the torque on the disc, suggesting that by studying grand-design spirals without bars it may be possible to set constraints on the tidal field and host dark matter halo shape.
We use N-body simulations to study the tidal evolution of globular clusters (GCs) in dwarf spheroidal (dSph) galaxies. Our models adopt a cosmologically motivated scenario in which the dSph is approximated by a static NFW halo with a triaxial shape. For a large set of orbits and projection angles we examine the spatial and velocity distribution of stellar debris deposited during the complete disruption of stellar clusters. Our simulations show that such debris appears as shells, isolated clumps and elongated over-densities at low surface brightness (>26 mag/arcsec^2), reminiscent of substructu…
The baryonic discs of galaxies are believed to alter the shapes of the dark matter haloes in which they reside. We perform a set of hydrodynamical N-body simulations of disc galaxies with triaxial dark matter haloes, using elliptical discs with a gaseous component as initial conditions. We explore models of different halo triaxiality and also of different initial gas fractions, which allows us to evaluate how each affects the formations of the bar. Due to star formation, models of all halo shapes and of all initial gas fractions reach approximately the same gas content at the end of the simula…
Cosmological N-body simulations indicate that the dark matter haloes of galaxies should be generally triaxial. Yet, the presence of a baryonic disc is believed to alter the shape of the haloes. Here we aim to study how bar formation is affected by halo triaxiality and how, in turn, the presence of the bar influences the shape of the halo. We perform a set of collisionless N-body simulations of disc galaxies with triaxial dark matter haloes, using elliptical discs as initial conditions. We study models of different halo triaxialities and, to investigate the behaviour of the halo shape in the ab…
We construct simple triaxial generalisations of Navarro-Frenk-White haloes. The models have elementary gravitational potentials, together with a density that is cusped like 1/r at small radii and falls off like 1/r^3 at large radii. The ellipticity varies with radius in a manner that can be tailored to the user's specification. The closed periodic orbits in the planes perpendicular to the short and long axes of the model are well-described by epicyclic theory, and can be used as building blocks for long-lived discs. As an application, we carry out the simulations of thin discs of satellites in…
We use N-body simulations to study the tidal evolution of globular clusters (GCs) in dwarf spheroidal (dSph) galaxies. Our models adopt a cosmologically motivated scenario in which the dSph is approximated by a static NFW halo with a triaxial shape. We apply our models to five GCs spanning three orders of magnitude in stellar density and two in mass, chosen to represent the properties exhibited by the five GCs of the Fornax dSph. We show that only the object representing Fornax's least dense GC (F1) can be fully disrupted by Fornax's internal tidal field--the four denser clusters survive even …
We use data from the IAC Stripe82 Legacy Project to study the surface photometry of 22 nearby, face-on to moderately inclined spiral galaxies. The reprocessed and combined Stripe 82 $g'$, $r'$ and $i'$ images allow us to probe the galaxy down to 29-30 $r'$-magnitudes/arcsec$^2$ and thus reach into the very faint outskirts of the galaxies. Truncations are found in three galaxies. An additional 15 galaxies are found to have an apparent extended stellar halo. Simulations show that the scattering of light from the inner galaxy by the Point Spread Function (PSF) can produce faint structures resembl…
The absence of stellar disc truncations in low-inclined spiral galaxies has been a matter of debate in the last decade. Disc truncations are often observed in highly inclined galaxies but no obvious detection of this feature has so far been made in face-on spirals. Here we show, using a simple exponential disc plus stellar halo model based on current observational constraints, that truncations in face-on projections occur at surface brightness levels comparable to the brightness of stellar haloes at the same radial distance. In this sense, stellar haloes outshine the galaxy disc at the expecte…
We generate stellar distribution functions (DFs) in triaxial haloes in order to examine the reliability of slopes $Γ\equiv Δ{\rm log} M / Δ{\rm log} r$ inferred by applying mass estimators of the form $M\propto R_eσ^2$ (i.e. assuming spherical symmetry, where $R_e$ and $σ$ are luminous effective radius and global velocity dispersion, respectively) to two stellar sub-populations independently tracing the same gravitational potential. The DFs take the form $f(E)$, are dynamically stable, and are generated within triaxial potentials corresponding directly to subhaloes formed in cosmological dark-…
Using N-body simulations we study the phenomenon of radial orbit instability occurring in dark matter haloes of the size of a dwarf galaxy. We carried out simulations of seven spherical models, with the same standard NFW density profile but different anisotropy profiles of particle orbits. Four of them underwent instability: two with a constant positive anisotropy, one with an anisotropic core and an isotropic envelope and one with a very small isotropic core and an anisotropic envelope. Haloes affected by the instability become approximately axisymmetric and prolate, with the profile of the s…
Using $N$-body simulations ($N\sim 10^6 - 10^7$), we examine how a non-axisymmetric dark halo affects the dynamical evolution of the structure in collisionless (stellar) discs. We demonstrate how the model parameters such as mass of the halo, initial conditions in the disc and the halo axes ratio affect morphology and kinematics of the stellar discs. We show that a non-axisymmetric halo can generate a large-scale spiral density pattern in the embedded stellar disc. The pattern is observed in the disc for many periods of its revolution, even if the disc is gravitationally over-stable. The growt…
We utilize Young's algorithm to model the adiabatic compression of the dark matter haloes of galaxies in the THINGS survey to determine the relationship between the halo fit to the rotation curve and the corresponding primordial halo prior to compression. Young's algorithm conserves radial action and angular momentum, resulting in less halo compression than more widely utilized approximations. We find that estimates of the parameters of the NFW haloes fit to the current dark matter distribution systematically overestimate the concentration and underestimate the virial velocity of the correspon…
We follow the formation and evolution of bars in N-body simulations of disc galaxies with gas and/or a triaxial halo. We find that both the relative gas fraction and the halo shape play a major role in the formation and evolution of the bar. In gas-rich simulations, the disc stays near-axisymmetric much longer than in gas-poor ones, and, when the bar starts growing, it does so at a much slower rate. Due to these two effects combined, large-scale bars form much later in gas-rich than in gas-poor discs. This can explain the observation that bars are in place earlier in massive red disc galaxies …
An analytic galaxy model with $ρ\sim r^{-1}$ at small radii and $ρ\sim r^{-3.5}$ at large radii is presented. The asymptotic density fall-off is slower than the Hernquist model, but faster than the Navarro-Frenk-White (NFW) profile for dark matter haloes, and so in accord with recent evidence from cosmological simulations. The model provides the zeroth-order term in a biorthornomal basis function expansion, meaning that axisymmetric, triaxial and lopsided distortions can easily be added (much like the Hernquist model itself which is the zeroth-order term of the Hernquist-Ostriker expansion). T…
Numerical models of disc galaxy formation predict the existence of extended, hot ( T~10^6 K) gas haloes around present day spirals. The X-ray luminosity of these haloes is predicted to increase strongly with galaxy mass. However, searches for their X-ray emission have not been successful so far. We calculate the all sky O VII column density distributions for the haloes of three Milky Way like disc galaxies, resulting from cosmological high-resolution, N-body/gasdynamical simulations. We perform calculations both including the disc gas and without it, so the disc contribution to the column dens…
Thanks to deep UV observations with GALEX and Swift, diffuse UV haloes have recently been discovered around galaxies. Based on UV-optical colours, it has been advocated that the UV haloes around spiral galaxies are due to UV radiation emitted from the disc and scattered off dust grains at high latitudes. Detailed UV radiative transfer models that take into account scattering and absorption can explain the morphology of the UV haloes, and they require the presence of an additional thick dust disc next the to traditional thin disc for half of the galaxies in their sample. We test whether such an…
We present a 2D kinematic analysis out to ~2-5 effective radii (Re) of 33 massive elliptical galaxies with stellar velocity dispersions larger than 150 km/s. Our observations were taken using the Mitchell Spectrograph (formerly VIRUS-P), a spectrograph with a large 107 x 107 square arcsecond field-of-view that allows us to construct robust, spatially resolved kinematic maps of the mean velocity and velocity dispersion for each galaxy extending to at least 2 Re. Using these maps we study the radial dependence of the stellar angular momentum and other kinematic properties. We see the familiar di…
Dark matter haloes are formed through hierarchical mergers of smaller haloes in large-scale cosmic environments, and thus anisotropic subhalo accretion through cosmic filaments have some impacts on halo structures. Recent studies using cosmological simulations have shown that the orientations of haloes correlate with the direction of cosmic filaments, and these correlations significantly depend on the halo mass. Using high-resolution cosmological $N$-body simulations, we quantified the strength of filamentary subhalo accretion for galaxy- and group-sized host haloes ($M_{\rm host}=5\times10^{1…
The dark matter halos in $Λ$CDM cosmological simulations are triaxial and highly flattened. In many cases, these triaxial equilibria are also tumbling slowly, typically about their short axes, with periods of order a Hubble time. Halos may therefore exert a slowly changing external torque on spiral galaxies that can affect their dynamical evolution in interesting ways. We examine the effect of the external torques exerted by a tumbling quadrupolar tidal field on the evolution of spiral galaxies using N-body simulations with realistic, disk galaxy models. We measure the amplitude of the externa…
We investigate the alignment of galaxies and haloes relative to cosmic web filaments using the EAGLE hydrodynamical simulation. We identify filaments by applying the NEXUS+ method to the mass distribution and the Bisous formalism to the galaxy distribution. Both web finders return similar filamentary structures that are well aligned and that contain comparable galaxy populations. EAGLE haloes have an identical spin alignment with filaments as their counterparts in dark matter only simulations: a complex mass dependent trend with low mass haloes spinning preferentially parallel to and high mass…
The dynamics of a galactic disk in a non-axisymmetric (triaxial) dark halo is studied in detail using high-resolution, numerical, hydrodynamical models. A long-lived, two-armed spiral pattern is generated for a wide range of parameters. The spiral structure is global, and the number of turns can be two or three, depending on the model parameters. The morphology and kinematics of the spiral pattern are studied as functions of the halo and disk parameters. The spiral structure rotates slowly, and its angular velocity varies quasi-periodically. Models with differing relative halo masses, halo sem…
We use zoom hydrodynamical simulations to investigate the properties of satellites within galaxy-sized dark-matter haloes with different assembly histories. We consider two classes of haloes at redshift $z=0$: `stalled' haloes that assembled at $z>1$ and `accreting' ones that are still forming nowadays. Previously, we showed that the stalled haloes are embedded within thick filaments of the cosmic web while the accreting ones lie where multiple thin filaments converge. We find that satellites in the two classes have both similar and different properties. Their mass spectra, radial count profil…
Beginning with a geometric motivation for dark matter going back to the axioms of general relativity, we show how scalar field dark matter, which naturally forms dark matter density waves due to its wave nature, may cause the observed barred spiral pattern density waves in many disk galaxies and triaxial shapes with plausible brightness profiles in many elliptical galaxies. If correct, this would provide a unified explanation for spirals and bars in spiral galaxies and for the brightness profiles of elliptical galaxies. We compare the results of preliminary computer simulations with photos of …
It is known observationally that the major axes of galaxy clusters and their brightest cluster galaxies are roughly aligned with each other. To understand the origin of the alignment, we identify 40 cluster-sized dark matter (DM) haloes with masses higher than $5\times10^{13}~M_{\odot}$ and their central galaxies (CGs) at $z\approx 0$ in the Horizon-AGN cosmological hydrodynamical simulation. We trace the progenitors at 50 different epochs between $0<z<5$. We then fit their shapes and orientations with a triaxial ellipsoid model. While the orientations of both DM haloes and CGs change signific…
We investigate the correlation between nine different dark matter halo properties using a rank correlation analysis and a Principal Component Analysis for a sample of haloes spanning five orders of magnitude in mass. We consider mass and dimensionless measures of concentration, age, relaxedness, sphericity, triaxiality, substructure, spin, and environment, where the latter is defined in a way that makes it insensitive to mass. We find that concentration is the most fundamental property. Except for environment, all parameters are strongly correlated with concentration. Concentration, age, subst…
We present an update to the multiphase SPH galaxy formation code by Scannapieco et al. We include a more elaborate treatment of the production of metals, cooling rates based on individual element abundances, and a scheme for the turbulent diffusion of metals. Our SN feedback model now transfers energy to the ISM in kinetic and thermal form, and we include a prescription for the effects of radiation pressure from massive young stars on the ISM. We calibrate our new code on the well studied Aquarius haloes and then use it to simulate a sample of 16 galaxies with halo masses between 1x10^11 and 3…
We recover spiral and barred spiral patterns in disk galaxy simulations with a Wave Dark Matter (WDM) background (also known as Scalar Field Dark Matter (SFDM), Ultra-Light Axion (ULA) dark matter, and Bose-Einstein Condensate (BEC) dark matter). Here we show how the interaction between a baryonic disk and its Dark Matter Halo triggers the formation of spiral structures when the halo is allowed to have a triaxial shape and angular momentum. This is a more realistic picture within the WDM model since a non-spherical rotating halo seems to be more natural. By performing hydrodynamic simulations,…
We search for detectable signatures of f(R) gravity and its chameleon screening mechanism in the baryonic and dark matter (DM) properties of simulated void galaxies. The enhancement of the gravitational acceleration can have a meaningful impact on the scaling relations as well as on the halo morphology. The galaxy rotational velocity field (calculated with the velocity of the gas disc and the acceleration fields) deviates from the typical values of the Tully-Fisher Relation (TFR) in GR. For a given stellar mass, f(R) gravity tends to produce greater maximum velocities. On the other hand, the m…
We provide a family of action-based distribution functions (DFs) for the double-power law family of densities often used to model galaxies. The DF itself is a double-power law in combinations of the actions, and reduces to the known limits in the case of a pure power-law at small and large radii. Our method enables the velocity anisotropy of the model to be tuned, and so the anisotropy in the inner and outer parts can be specified for the application in hand. We provide self-consistent DFs for the Hernquist and Jaffe models - both with everywhere isotropic velocity dispersions, and with kinema…
(Abridged) We use N-body simulations to study the effects that a divergent (i.e. "cuspy") dark matter (DM) profile introduces on the tidal evolution of dwarf spheroidal galaxies (dSphs). Our models assume cosmologically-motivated initial conditions where dSphs are DM-dominated systems on eccentric orbits about a host galaxy composed of a dark halo and a baryonic disc. We find that the resilience of dSphs to tidal stripping is extremely sensitive to the halo cuspiness; whereas dwarfs with a cored profile can be easily destroyed by the host disc, those with cusps always retain a bound remnant. F…
Based on a combined analysis of SDSS imaging and CALIFA integral field spectroscopy data, we report on the detection of faint (24 < μ$_r$ mag/arcsec$^2$ < 26) star-forming spiral-arm-like features in the periphery of three nearby early-type galaxies (ETGs). These features are of considerable interest because they document the still ongoing inside-out growth of some local ETGs and may add valuable observational insight into the origin and evolution of spiral structure in triaxial stellar systems. A characteristic property of the nebular component in the studied ETGs, classified i+, is a two-rad…
The results of numerical simulations of the dynamics of galactic disks, which are submerged into nonaxisymmetric dark massive halo are discussed. Galaxy disks dynamics in the nonaxisymmetric (triaxial) dark halo were investigated in detail by the high resolution numerical hydrodynamical methods (TVD \& SPH) and N-body models. The long-lived two arms spiral structure generates for a wide range of parameters. The spiral structure is global and number of turns can be 2-3 in depends of model parameters. Morphology and kinematics of spiral structures were investigated in depends of the halo and the…
Within the standard hierarchical structure formation scenario, Milky Way-mass dark matter haloes have hundreds of dark matter subhaloes with mass $\gtrsim 10^8 \, {\rm M_{\odot}}$. Over the lifetime of a galactic disc a fraction of these may pass close to the central region and interact with the disc. We extract the properties of subhaloes, such as their mass and trajectories, from a realistic cosmological simulation to study their potential effect on stellar discs. We find that massive subhalo impacts can generate disc heating, rings, bars, warps, lopsidedness as wells as spiral structures in…
The anisotropic distribution of satellites around the central galaxy of their host halo is well-documented. However the relative impact of baryons and dark matter in shaping this distribution is still debated. Using the simulation Horizon-AGN, the angular distribution of satellite galaxies with respect to their central counterpart and halo is quantified. Below one Rvir, satellites cluster more strongly in the plane of the central, rather than merely tracing the shape of their host halo. This is due to the increased isotropy of inner haloes acquired through their inside-out assembly in vorticit…
Efficient expansions of the gravitational field of (dark) haloes have two main uses in the modelling of galaxies: first, they provide a compact representation of numerically-constructed (or real) cosmological haloes, incorporating the effects of triaxiality, lopsidedness or other distortion. Secondly, they provide the basis functions for self-consistent field expansion algorithms used in the evolution of $N$-body systems. We present a new family of biorthogonal potential-density pairs constructed using the Hankel transform of the Laguerre polynomials. The lowest-order density basis functions a…
We present radio continuum maps of 12 nearby ($D\leq 27~\rm Mpc$), edge-on ($i\geq 76^{\circ}$), late-type spiral galaxies mostly at $1.4$ and 5 GHz, observed with the Australia Telescope Compact Array, Very Large Array, Westerbork Synthesis Radio Telescope, Effelsberg 100-m and Parkes 64-m telescopes. All galaxies show clear evidence of radio haloes, including the first detection in the Magellanic-type galaxy NGC 55. In 11 galaxies, we find a thin and a thick disc that can be better fitted by exponential rather than Gaussian functions. We fit our SPINNAKER (SPectral INdex Numerical Analysis o…
Spectroscopic observations at the Russian 6-m telescope are used to study the two polar ring galaxies (PRGs) from the catalogue by Moiseev et al.: SPRC-7 and SPRC-260. We have analyzed the kinematics of the stellar component of the central galaxies as well as the ionized gas kinematics in the external ring structures. The disc-halo decomposition of rotation curves in two perpendicular directions are considered. The observed 2D velocity fields are compared with the model predictions for different dark halo shapes. Based on these data, we constrain that for potential of DM halo semiaxis ratios i…
New photometric and long-slit spectroscopic observations are presented for NGC 7113, PGC 1852, and PGC 67207 which are three bright galaxies residing in low-density environments. The surface-brightness distribution is analysed from the K_S-band images taken with adaptive optics at the Gemini North Telescope and the ugriz-band images from the Sloan Digital Sky Survey while the line-of-sight stellar velocity distribution and line-strength Lick indices inside the effective radius are measured along several position angles. The age, metallicity, and alpha-element abundance of the galaxies are esti…
Triaxiality of the Milky Way dark matter halo and the microlensing optical depth towards the Large Magellanic Cloud
Institute for Astronomy, University of Edinburgh（爱丁堡大学天体物理研究所）；Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK（英国爱丁堡黑福德山皇家天文台）；Instituto de Física Teórica UAM/CSIC, Universidad Autónoma de Madrid（马德里自治大学理论物理研究所）
We present measurements of the morphology of star-forming gas in galaxies from the EAGLE simulations, and its alignment relative to stars and dark matter (DM). Imaging of such gas in the radio continuum enables weak lensing experiments that complement traditional optical approaches. Star-forming gas is typically more flattened than its associated stars and DM, particularly for present-day subhaloes of total mass $\sim$$10^{ 12-12.5} \mathrm{M_{ \odot}}$, which preferentially host star-forming galaxies with rotationally-supported stellar discs. Such systems have oblate, spheroidal star-forming …
The dynamical friction timescale of massive globular clusters (GCs) in the inner regions of cuspy dark haloes in dwarf spheroidal (dSph) galaxies can be much shorter than the Hubble time. This implies that a small fraction of the GCs is expected to be caught close to the centre of these galaxies. We compare the radial distribution of GCs predicted in simple Monte Carlo models with that of a sample of $38$ spectroscopically confirmed GCs plus 17 GC candidates, associated mainly to low-luminosity dSph galaxies. If dark matter haloes follow an NFW profile, the observed number of off-center GCs at…
In the first part of this article we expand three fundamental aspects of the methodology connected to the determination of a relation among the spatial density and the gravitational potential that can be specialised to distinct mass density agglomerations. As a consequence, we obtain general relations for the diagonal entries of a square symmetric matrix without zeros, we provide an expression of the gravitational potential, suitable, to represent several different mass density configurations, and we determine relations for the semi-axes of a triaxial spheroidal mass distribution, as a functio…
The halo shape plays a central role in determining important observational properties of the haloes such as mass, concentration and lensing cross-sections. The triaxiality of lensing galaxy clusters has a substantial impact on the distribution of the largest Einstein radii, while weak lensing techniques are sensitive to the intrinsic halo ellipticity. In this work, we provide scaling relations for the shapes of dark matter haloes as a function of mass (peak height) and redshift over more than four orders of magnitude in halo masses, namely from $10^{11.5}$ to $10^{15.8}~h^{-1}~$M$_\odot$. We h…
Massive and diffuse haloes of hot gas (coronae) are important intermediaries between cosmology and galaxy evolution, storing mass and angular momentum acquired from the cosmic web until eventual accretion on to star-forming discs. We introduce a method to reconstruct the rotation of a galactic corona, based on its angular momentum distribution (AMD). This allows us to investigate in what conditions the angular momentum acquired from tidal torques can be transferred to star forming discs and explain observed galaxy-scale processes, such as inside-out growth and the build-up of abundance gradien…
Cosmological simulations predict that galaxies are embedded into triaxial dark matter haloes, which appear approximately elliptical in projection. Weak gravitational lensing allows us to constrain these halo shapes and thereby test the nature of dark matter. Weak lensing has already provided robust detections of the signature of halo flattening at the mass scales of groups and clusters, whereas results for galaxies have been somewhat inconclusive. Here we combine data from five surveys (NGVSLenS, KiDS/KV450, CFHTLenS, CS82, and RCSLenS) in order to tighten observational constraints on galaxy-s…
The cold dark matter theory predicts triaxial dark matter haloes. The radial distribution of halo ellipticity depends on baryonic processes and the nature of dark matter particles (collisionless or collisional). Here we show that we can use lensing flexion ratios to measure the halo ellipticity as a function of radius. We introduce a weight function and study the relationship between the first and second order statistics of flexion ratios, both of which can be used to reduce the bias in the estimate of ellipticity. we perform numerical tests for our method, and demonstrate that it can reduce t…
In the present paper, we extend the study of Del Popolo (2010) to determine the slope of the inner density profile of galaxy haloes with different morphologies. We study how galaxy morphology changes the relation between the inner slope of the galaxy halo density profile, $α$, and the stellar mass, $M_{*}$, or rotation velocity $V_{\rm rot}$. For this, we use the model of Del Popolo (2009) in combination with observed data from the Romanowsky \& Fall (2012) sample of elliptical and spiral galaxies, the Local Group sample compiled by McConnachie (2012), and the simulation results by Cloet-Ossel…
We report the alignment and shape of dark matter, stellar, and hot gas distributions in the EAGLE and cosmo-OWLS simulations. The combination of these state-of-the-art hydro-cosmological simulations enables us to span four orders of magnitude in halo mass ($11 < log_{10}(M_{200}/ [h^{-1}M_\odot]) < 15$), a wide radial range ($-2.3 < log_{10}(r/[h^{-1}Mpc ]) < 1.3$) and redshifts $0 < z < 1$. The shape parameters of the dark matter, stellar and hot gas distributions follow qualitatively similar trends: they become more aspherical (and triaxial) with increasing halo mass, radius and redshift. We…
A family of spherical halo models with flat circular velocity curves is presented. This includes models in which the rotation curve has a finite central value but declines outwards (like the Jaffe model). It includes models in which the rotation curve is rising in the inner parts, but flattens asymptotically (like the Binney model). The family encompasses models with both finite and singular (cuspy) density profiles. The self-consistent distribution function depending on binding energy $E$ and angular momentum $L$ is derived and the kinematical properties of the models discussed. These really …
We review most dynamical constraints on the gravitational field of spiral galaxies in general, and of the Milky Way in particular. Such constraints are of prime importance for determining the characteristics of the putative dark matter haloes of galaxies. For the Milky Way, we review observational constraints in the inner parts (cored or cusped dark matter distribution, maximum disk or not), in the solar neighbourhood (local dark matter density) and in the outer parts (virial mass and triaxial shape of the dark matter halo). We also point out various caveats, systematic effects, and large curr…
