Phase-space shapes of clusters and rich groups of galaxies

Clusters and groups of galaxies are highly aspherical, with shapes approximated by nearly prolate ellipsoids of revolution. An equally fundamental property is the shape of these objects in velocity space which is the anisotropy of the global velocity dispersion tensor. Here we make use of kinematical data comprising around 600 nearby clusters and rich groups of galaxies from the SDSS to place constraints on the phase-space shapes of these objects, i.e. their shapes in both position and velocity space. We show that the line of sight velocity dispersion normalised by a mass dependent velocity scale correlates with the apparent elongation, with circular (elongated) clusters exhibiting an excessive (decremental) normalised velocity dispersion. This correlation holds for dynamically young or old clusters and, therefore, it originates from projecting their intrinsic phase-space shapes rather than from dynamical evolution. It signifies that clusters are preferentially prolate not only in position space, but also in velocity space. The distribution of the axial ratios in position space is found to be well approximated by a Gaussian with a mean 0.66+/-0.01 and a dispersion 0.07+/-0.008. The velocity ellipsoids representing the shapes in velocity space are more spherical, with a mean axial ratio of 0.78+/-0.03. This finding has important implications for mass measurements based on the line of sight velocity dispersion profiles in individual clusters. For typical axial ratios of the velocity ellipsoids in the analysed cluster sample, systematic errors on the mass estimates inferred from the line of sight velocity dispersions become comparable to statistical uncertainties for galaxy clusters with as few as 40 spectroscopic redshifts.Comment: 9 pages, 7 figures; published in A&A; typo in eq. 5 correcte

Effect of asphericity in caustic mass estimates of galaxy clusters

The caustic technique for measuring mass profiles of galaxy clusters relies on the assumption of spherical symmetry. When applied to aspherical galaxy clusters, the method yields mass estimates affected by the cluster orientation. Here we employ mock redshift catalogues generated from cosmological simulations to study the effect of clusters intrinsic shape and surrounding filamentary structures on the caustic mass estimates. To this end, we develop a new method for removing perturbations from large-scale structures, modelled as the two-halo term, in a caustic analysis of stacked cluster data. We find that the cluster masses inferred from kinematical data of ~10^14 Msun clusters observed along the major axis are larger than masses from those observed along the minor axis by a factor of 1.7 within the virial radius, increasing to 1.8 within three virial radii. This discrepancy increases by 20% for the most massive clusters. In addition a smaller but still significant mass discrepancy arises when filamentary structures are present near a galaxy cluster. We find that the mean cluster mass from random sightlines is unbiased at all radii and their scatter ranges from 0.14 to 0.17 within one and three virial radii, with a 40% increase for the most massive clusters. We provide tables which estimate the caustic mass bias given observational constraints on the cluster orientation.Comment: 19 pages, 9 figures, 6 tables, accepted for publication in MNRA

Statistical mechanics of collisionless orbits. III. Comparison with N-body simulations

We compare the DARKexp differential energy distribution, N(E) \propto \exp(\phi_0-E)-1, obtained from statistical mechanical considerations, to the results of N-body simulations of dark matter halos. We first demonstrate that if DARKexp halos had anisotropic velocity distributions similar to those of N-body simulated halos, their density and energy distributions could not be distinguished from those of isotropic DARKexp halos. We next carry out the comparison in two ways, using (1) the actual energy distribution extracted from simulations, and (2) N-body fitting formula for the density distribution as well as N(E) computed from the density using the isotropic Eddington formula. Both the methods independently agree that DARKexp N(E) with \phi_0\approx 4-5 is an excellent match to N-body N(E). Our results suggest (but do not prove) that statistical mechanical principles of maximum entropy can be used to explain the equilibrated final product of N-body simulations.Comment: 17 pages, 7 figures; ApJ, in pres

Radial velocity moments of dark matter haloes

Using cosmological N-body simulations we study the radial velocity distribution in dark matter haloes focusing on the lowest-order even moments, dispersion and kurtosis. We determine the properties of ten massive haloes in the simulation box approximating their density distribution by the NFW formula characterized by the virial mass and concentration. We also calculate the velocity anisotropy parameter of the haloes and find it mildly radial and increasing with distance from the halo centre. The radial velocity dispersion of the haloes shows a characteristic profile with a maximum, while the radial kurtosis profile decreases with distance starting from a value close to Gaussian near the centre. We therefore confirm that dark matter haloes possess intrinsically non-Gaussian, flat-topped velocity distributions. We find that the radial velocity moments of the simulated haloes are quite well reproduced by the solutions of the Jeans equations obtained for the halo parameters with the anisotropy measured in the simulations. We also study the radial velocity moments for a composite cluster made of ten haloes out to ten virial radii. In this region the velocity dispersion decreases systematically to reach the value of the background, while kurtosis increases from below to above the Gaussian value of 3 signifying a transition from a flat-topped to a strongly peaked velocity distribution with respect to the Gaussian, which can be interpreted as the dominance of ordered flow with a small dispersion. We illustrate the transition by showing explicitly the velocity distribution of the composite cluster in a few radial bins.Comment: 5 pages, 4 figures, minor changes, accepted for publication in MNRAS Letter

Velocity moments of dark matter haloes

Using cosmological N-body simulations we study the line-of-sight velocity distribution of dark matter haloes focusing on the lowest-order even moments, dispersion and kurtosis, and their application to estimate the mass profiles of cosmological structures. For each of the ten massive haloes selected from the simulation box we determine the virial mass, concentration and the anisotropy parameter. In order to emulate observations from each halo we choose randomly 300 particles and project their velocities and positions along the line of sight and on the surface of the sky, respectively. After removing interlopers we calculate the profiles of the line-of-sight velocity moments and fit them with the solutions of the Jeans equations. The estimates of virial mass, concentration parameter and velocity anisotropy obtained in this way are in good agreement with the values found from the full 3D analysis.Comment: 2 pages, 1 figure, poster contribution to the proceedings of the XXIst IAP Colloquium "Mass Profiles and Shapes of Cosmological Structures", Paris 4-9 July 2005, Editors: G. Mamon, F. Combes, C. Deffayet, B. Fort, EDP Sciences, in pres

Mass distribution in nearby Abell clusters

We study the mass distribution in six nearby (z<0.06) relaxed Abell clusters of galaxies A0262, A0496, A1060, A2199, A3158 and A3558. Given the dominance of dark matter in galaxy clusters we approximate their total density distribution by the NFW formula characterized by virial mass and concentration. We also assume that the anisotropy of galactic orbits is reasonably well described by a constant and that galaxy distribution traces that of the total density. Using the velocity and position data for 120-420 galaxies per cluster we calculate, after removal of interlopers, the profiles of the lowest-order even velocity moments, dispersion and kurtosis. We then reproduce the velocity moments by jointly fitting the moments to the solutions of the Jeans equations. Including the kurtosis in the analysis allows us to break the degeneracy between the mass distribution and anisotropy and constrain the anisotropy as well as the virial mass and concentration. The method is tested in detail on mock data extracted from N-body simulations of dark matter haloes. We find that the best-fitting galactic orbits are remarkably close to isotropic in most clusters. Using the fitted pairs of mass and concentration parameters for the six clusters we conclude that the trend of decreasing concentration for higher masses found in cosmological N-body simulations is consistent with the data. By scaling the individual cluster data by mass we combine them to create a composite cluster with 1465 galaxies and perform a similar analysis on such sample. The estimated concentration parameter then lies in the range 1.5 < c < 14 and the anisotropy parameter in the range -1.1 < \beta < 0.5 at the 95 percent confidence level.Comment: 11 pages, 9 figures, final version accepted for publication in MNRA

A new method to measure the mass of galaxy clusters

The mass measurement of galaxy clusters is an important tool for the determination of cosmological parameters describing the matter and energy content of the Universe. However, the standard methods rely on various assumptions about the shape or the level of equilibrium of the cluster. We present a novel method of measuring cluster masses. It is complementary to most of the other methods, since it only uses kinematical information from outside the virialized cluster. Our method identifies objects, as galaxy sheets or filaments, in the cluster outer region, and infers the cluster mass by modeling how the massive cluster perturbs the motion of the structures from the Hubble flow. At the same time, this technique allows to constrain the three-dimensional orientation of the detected structures with a good accuracy. We use a cosmological numerical simulation to test the method. We then apply the method to the Coma cluster, where we find two galaxy sheets, and measure the mass of Coma to be Mvir=(9.2\pm2.4)10^{14} Msol, in good agreement with previous measurements obtained with the standard methods.Comment: 10 pages, 12 figures, submitted to MNRA