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Giuseppe Murante

Publications and source records attributed to Giuseppe Murante.

At least 73 records · Page 4Linked to original sources

Temperature Structure of the Intra-Cluster Medium from SPH and AMR simulations

Analyses of cosmological hydrodynamic simulations of galaxy clusters suggest that X-ray masses can be underestimated by 10% to 30%. The largest bias originates by both violation of hydrostatic equilibrium and an additional temperature bias caused by inhomogeneities in the X-ray emitting intra-cluster medium (ICM). To elucidate on this large dispersion among theoretical predictions, we evaluate the degree of temperature structures in cluster sets simulated either with smoothed-particle-hydrodynamics (SPH) and adaptive-mesh-refinement (AMR) codes. We find that the SPH simulations produce larger temperature variations connected to the persistence of both substructures and their stripped cold gas. This difference is more evident in no-radiative simulations, while it is reduced in the presence of radiative cooling. We also find that the temperature variation in radiative cluster simulations is generally in agreement with the observed one in the central regions of clusters. Around R_500 the temperature inhomogeneities of the SPH simulations can generate twice the typical hydrostatic-equilibrium mass bias of the AMR sample. We emphasize that a detailed understanding of the physical processes responsible for the complex thermal structure in ICM requires improved resolution and high sensitivity observations in order to extend the analysis to higher temperature systems and larger cluster-centric radii.

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Simulating the evolution of disc galaxies in a group environment. II. The influence of close-encounters between galaxies

We study the evolution of disc galaxies in group environments under the effect of both the global tidal field and close-encounters between galaxies, using controlled N-body simulations of isolated mergers. We find that close-range encounters between galaxies are less frequent and less damaging to disc galaxies than originally expected, since they mostly occur when group members have lost a significant fraction of their initial mass to tidal stripping. We also find that group members mostly affect disc galaxies indirectly by modifying their common global tidal field. Different initial orbital parameters of group members introduce a significant "scatter" in the evolution of general properties of disc galaxies around a "median" evolution that is similar to when only the effect of the global tidal field is included. Close-encounters introduce a high variability in the properties of disc galaxies, even slowing their evolution in some cases, and could wash out correlations between galaxy properties and the group total mass. The combined effect of the global tidal field and close-encounters appears to be inefficient at forming/enhancing central stellar bulges. This implies that bulges of S0 galaxies should be mostly composed by young stars, which is consistent with recent observations.

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Kinetic or thermal AGN feedback in simulations of isolated and merging disk galaxies calibrated by the M-sigma relation

(Abridged) We investigate two modes of coupling the feedback energy from a central AGN to the neighboring gas in galaxy simulations: kinetic - velocity boost, and thermal - heating. We formulate kinetic feedback models for energy-driven wind (EDW) and momentum-driven wind (MDW), using two free parameters: feedback efficiency epsilon_f, and AGN wind velocity v_w. A novel numerical algorithm is implemented in the SPH code GADGET-3, to prevent the expansion of a hole in the gas distribution around the BH. We perform simulations of isolated evolution and merger of disk galaxies, of Milky-Way mass as well as lower and higher masses. We find that in the isolated galaxy BH kinetic feedback generates intermittent bipolar jet-like gas outflows. We infer that current prescriptions for BH subgrid physics in galaxy simulations can grow the BH to observed values even in an isolated disk galaxy. The BH growth is enhanced in a galaxy merger. Comparing the [M_BH - sigma_star] relation obtained in our simulations with observational data, we conclude that it is possible to find parameter sets for a fit in all the models, except for the case with MDW feedback in a galaxy merger, in which the BH is always too massive. The BH thermal feedback implementation of Springel, Di Matteo & Hernquist (2005) within the multiphase star-formation model is found to have negligible impact on gas properties; and the effect claimed in all previous studies is attributed to gas depletion around the BH by the creation of an artificial hole. The BH mass accretion rate in our simulations exhibit heavy fluctuations. The star formation rate is quenched with feedback by removal of gas. The CGM gas at galactocentric distances (20 - 100)/h kpc are found to give the best metallicity observational diagnostic to distinguish between BH models.

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Brightest cluster galaxies in cosmological simulations: achievements and limitations of AGN feedback models

We analyze the basic properties of Brightest Cluster Galaxies (BCGs) produced by state of the art cosmological zoom-in hydrodynamical simulations. These simulations have been run with different sub-grid physics included. Here we focus on the results obtained with and without the inclusion of the prescriptions for supermassive black hole (SMBH) growth and of the ensuing Active Galactic Nuclei (AGN) feedback. The latter process goes in the right direction of decreasing significantly the overall formation of stars. However, BCGs end up still containing too much stellar mass, a problem that increases with halo mass, and having an unsatisfactory structure. This is in the sense that their effective radii are too large, and that their density profiles feature a flattening on scales much larger than observed. We also find that our model of thermal AGN feedback has very little effect on the stellar velocity dispersions, which turn out to be very large. Taken together, these problems, which to some extent can be recognized also in other numerical studies typically dealing with smaller halo masses, indicate that on one hand present day sub-resolution models of AGN feedback are not effective enough in diminishing the global formation of stars in the most massive galaxies, but on the other hand they are relatively too effective in their centers. It is likely that a form of feedback generating large scale gas outflows from BCGs precursors, and a more widespread effect over the galaxy volume, can alleviate these difficulties.

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An improved prescription for merger time-scales from controlled simulations

We compare three analytical prescriptions for merger times available from the literature to simulations of isolated mergers. We probe three different redshifts, and several halo concentrations, mass ratios, orbital circularities and orbital energies of the satellite. We find that prescriptions available in the literature significantly under-predict long timescales for mergers at high redshift. We argue that these results have not been highlighted previously either because the evolution of halo concentration of satellite galaxies has been neglected (in previous isolated merger simulations), or because long merger times and mergers with high initial orbital circularities are under-represented (for prescriptions based on cosmological simulations). Motivated by the evolution of halo concentration at fixed mass, an explicit dependence on redshift added as t_merger,modified(z) = (1+z)^0.44 t_merger to the prescription based on isolated mergers gives a significant improvement in the predicted merger times up to ~20 t_dyn in the redshift range 0<z<2. When this modified prescription is used to compute galaxy stellar mass functions, we find that it leads up to a 25 per cent increase in the number of low mass galaxies surviving at z=0, and a 10 per cent increase for more massive galaxies. This worsen the known over-prediction in the number of low mass galaxies by hierarchical models of galaxy formation.

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The habitable zone of Earth-like planets with different levels of atmospheric pressure

As a contribution to the study of the habitability of extrasolar planets, we implemented a 1-D Energy Balance Model (EBM), the simplest seasonal model of planetary climate, with new prescriptions for most physical quantities. Here we apply our EBM to investigate the surface habitability of planets with an Earth-like atmospheric composition but different levels of surface pressure. The habitability, defined as the mean fraction of the planet's surface on which liquid water could exist, is estimated from the pressure-dependent liquid water temperature range, taking into account seasonal and latitudinal variations of surface temperature. By running several thousands of EBM simulations we generated a map of the habitable zone (HZ) in the plane of the orbital semi-major axis, a, and surface pressure, p, for planets in circular orbits around a Sun-like star. As pressure increases, the HZ becomes broader, with an increase of 0.25 AU in its radial extent from p=1/3 bar to p=3 bar. At low pressure, the habitability is low and varies with a; at high pressure, the habitability is high and relatively constant inside the HZ. We interpret these results in terms of the pressure dependence of the greenhouse effect, the effciency of horizontal heat transport, and the extent of the liquid water temperature range. Within the limits discussed in the paper, the results can be extended to planets in eccentric orbits around non-solar type stars. The main characteristics of the pressure-dependent HZ are modestly affected by variations of planetary properties, particularly at high pressure.

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The relation between velocity dispersion and mass in simulated clusters of galaxies: dependence on the tracer and the baryonic physics

[Abridged] We present an analysis of the relation between the masses of cluster- and group-sized halos, extracted from $Λ$CDM cosmological N-body and hydrodynamic simulations, and their velocity dispersions, at different redshifts from $z=2$ to $z=0$. The main aim of this analysis is to understand how the implementation of baryonic physics in simulations affects such relation, i.e. to what extent the use of the velocity dispersion as a proxy for cluster mass determination is hampered by the imperfect knowledge of the baryonic physics. In our analysis we use several sets of simulations with different physics implemented. Velocity dispersions are determined using three different tracers, DM particles, subhalos, and galaxies. We confirm that DM particles trace a relation that is fully consistent with the theoretical expectations based on the virial theorem and with previous results presented in the literature. On the other hand, subhalos and galaxies trace steeper relations, and with larger values of the normalization. Such relations imply that galaxies and subhalos have a $\sim10$ per cent velocity bias relative to the DM particles, which can be either positive or negative, depending on halo mass, redshift and physics implemented in the simulation. We explain these differences as due to dynamical processes, namely dynamical friction and tidal disruption, acting on substructures and galaxies, but not on DM particles. These processes appear to be more or less effective, depending on the halo masses and the importance of baryon cooling, and may create a non-trivial dependence of the velocity bias and the $\soneD$--$\Mtwo$ relation on the tracer, the halo mass and its redshift. These results are relevant in view of the application of velocity dispersion as a proxy for cluster masses in ongoing and future large redshift surveys.

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Baryon Census in Hydrodynamical Simulations of Galaxy Clusters

We carry out an analysis of a set of cosmological SPH hydrodynamical simulations of galaxy clusters and groups aimed at studying the total baryon budget in clusters, and how this budget is shared between the hot diffuse component and the stellar component. Using the TreePM+SPH GADGET-3 code, we carried out one set of non-radiative simulations, and two sets of simulations including radiative cooling, star formation and feedback from supernovae (SN), one of which also accounting for the effect of feedback from active galactic nuclei (AGN). The analysis is carried out with the twofold aim of studying the implication of stellar and hot gas content on the relative role played by SN and AGN feedback, and to calibrate the cluster baryon fraction and its evolution as a cosmological tool. We find that both radiative simulation sets predict a trend of stellar mass fraction with cluster mass that tends to be weaker than the observed one. However this tension depends on the particular set of observational data considered. Including the effect of AGN feedback alleviates this tension on the stellar mass and predicts values of the hot gas mass fraction and total baryon fraction to be in closer agreement with observational results. We further compute the ratio between the cluster baryon content and the cosmic baryon fraction, Y_b, as a function of cluster-centric radius and redshift. At R_500 we find for massive clusters with M_500>2\times10^{14} h^{-1} M_sun that Y_b is nearly independent of the physical processes included and characterized by a negligible redshift evolution: Y_{b,500}=0.85+/-0.03 with the error accounting for the intrinsic r.m.s. scatter within the set of simulated clusters. At smaller radii, R_2500, the typical value of Y_b slightly decreases, by an amount that depends on the physics included in the simulations, while its scatter increases by about a factor of two.

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The thick disk rotation-metallicity correlation as a fossil of an "inverse chemical gradient" in the early Galaxy

The thick disk rotation--metallicity correlation, \partial V_ϕ/\partial[Fe/H] =40÷50 km s^{-1}dex^{-1} represents an important signature of the formation processes of the galactic disk. We use nondissipative numerical simulations to follow the evolution of a Milky Way (MW)-like disk to verify if secular dynamical processes can account for this correlation in the old thick disk stellar population. We followed the evolution of an ancient disk population represented by 10 million particles whose chemical abundances were assigned by assuming a cosmologically plausible radial metallicity gradient with lower metallicity in the inner regions, as expected for the 10-Gyr-old MW. Essentially, inner disk stars move towards the outer regions and populate layers located at higher |z|. A rotation--metallicity correlation appears, which well resembles the behaviour observed in our Galaxy at a galactocentric distance between 8 kpc and 10 kpc. In particular,we measure a correlation of \partial V_ϕ/\partial[Fe/H]\simeq 60 km s^{-1}dex^{-1} for particles at 1.5 kpc < |z| < 2.0 kpc that persists up to 6 Gyr. Our pure N-body models can account for the V_ϕvs. [Fe/H] correlation observed in the thick disk of our Galaxy, suggesting that processes internal to the disk such as heating and radial migration play a role in the formation of this old stellar component. In this scenario, the positive rotation-metallicity correlation of the old thick disk population would represent the relic signature of an ancient "inverse" chemical (radial) gradient in the inner Galaxy, which resulted from accretion of primordial gas.

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Tomographic weak lensing shear spectra from large N-body and hydrodynamical simulations

Forthcoming experiments will enable us to determine tomographic shear spectra at a high precision level. Most predictions about them have until now been biased on algorithms yielding the expected linear and non-linear spectrum of density fluctuations. Even when simulations have been used, so-called Halofit (Smith et al 2003) predictions on fairly large scales have been needed. We wish to go beyond this limitation. We perform N-body and hydrodynamical simulations within a sufficiently large cosmological volume to allow a direct connection between simulations and linear spectra. While covering large length-scales, the simulation resolution is good enough to allow us to explore the high-l harmonics of the cosmic shear (up to l ~ 50000), well into the domain where baryon physics becomes important. We then compare shear spectra in the absence and in presence of various kinds of baryon physics, such as radiative cooling, star formation, and supernova feedback in the form of galactic winds. We distinguish several typical properties of matter fluctuation spectra in the different simulations and test their impact on shear spectra. We compare our outputs with those obtainable using approximate expressions for non--linear spectra, and identify substantial discrepancies even between our results and those of purely N-body results. Our simulations and the treatment of their outputs however enable us, for the first time, to obtain shear results taht are fully independent of any approximate expression, also in the high-l range, where we need to incorporate a non-linear power spectrum of density perturbations, and the effects of baryon physics. This will allow us to fully exploit the cosmological information contained in future high--sensitivity cosmic shear surveys, exploring the physics of cosmic shears via weak lensing measurements.

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Working out density fluctuation spectra from shear spectra

Forthcoming experiments will enable us to determine high precision tomographic shear spectra. Matter density fluctuation spectra, at various $z$, should then be worked out of them, in order to constrain the model and determine the DE state equation. Available analytical expressions, however, do the opposite, enabling us to derive shear spectra from fluctuation spectra. Here we find the inverse expression, yielding density fluctuation spectra from observational tomographic shear spectra. The procedure involves SVD techniques for matrix inversion. We show in detail how the approach works and provide a few examples.

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The effects of baryons on the halo mass function

We present an analysis of the effects of baryon physics on the halo mass function. The analysis is based on simulations of a cosmological volume. Besides a Dark Matter (DM) only simulation, we also carry out two other hydrodynamical simulations. We identified halos using a spherical overdensity algorithm and their masses are computed at three different overdensities (with respect to the critical one), $Δ_c=200$, 500 and 1500. We find the fractional difference between halo masses in the hydrodynamical and in the DM simulations to be almost constant, at least for halos more massive than $\log (M_{Δ_c} / \hMsun)\geq 13.5$. In this range, mass increase in the hydrodynamical simulations is of about 4-5 per cent at $Δ_c=500$ and $\sim 1$ - 2 per cent at $Δ_c=200$. Quite interestingly, these differences are nearly the same for both radiative and non-radiative simulations. Such variations of halo masses induce corresponding variations of the halo mass function (HMF). At $z=0$, the HMFs for GH and CSF simulations are close to the DM one, with differences of $\mincir 3$ per cent at $Δ_c = 200$, and $\simeq 7$ per cent at $Δ_c=500$, with $\sim 10$ - 20 per cent differences reached at $Δ_c = 1500$. At this higher overdensity, the increase of the HMF for the radiative case is larger by about a factor 2 with respect to the non--radiative case. Assuming a constant mass shift to rescale the HMF from the hydrodynamic to the DM simulations, brings the HMF difference with respect to the DM case to be consistent with zero. Our results have interesting implications to bracket uncertainties in the mass function calibration associated to the uncertain baryon physics, in view of cosmological applications of future large surveys of galaxy clusters. (Abridged)

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A warm mode of gas accretion on forming galaxies

We present results from high--resolution cosmological hydrodynamical simulations of a Milky--Way-sized halo, aimed at studying the effect of feedback on the nature of gas accretion. Simulations include a model of inter-stellar medium and star formation, in which SN explosions provide effective thermal feedback. We distinguish between gas accretion onto the halo, which occurs when gas particles cross the halo virial radius, and gas accretion onto the central galaxy, which takes place when gas particles cross the inner one-tenth of the virial radius. Gas particles can be accreted through three different channels, depending on the maximum temperature value, $T_{\rm max}$, reached during the particles' past evolution: a cold channel for $T_{\rm max}<2.5 \times 10^5$ K, a hot one for $T>10^6$K, and a warm one for intermediate values of $T_{\rm max}$. We find that the warm channel is at least as important as the cold one for gas accretion onto the central galaxy. This result is at variance with previous findings that the cold mode dominates gas accretion at high redshift. We ascribe this difference to the different supernova feedback scheme implemented in our simulations. While results presented so far in the literature are based on uneffective SN thermal feedback schemes and/or the presence of a kinetic feedback, our simulations include only effective thermal feedback. We argue that observational detections of a warm accretion mode in the high--redshift circum-galactic medium would provide useful constraints on the nature of the feedback that regulates star formation in galaxies.

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Simulating the evolution of disc galaxies in a group environment. I. The influence of the global tidal field

We present the results of a series of numerical simulations aimed to study the evolution of a disc galaxy within the global tidal field of a group environment. Both the disc galaxy and the group are modelled as multi-component, collision-less, N-body systems, composed by both dark matter and stars. In our simulations, the evolution of disc galaxies is followed as their orbits sink towards the group centre, under the effect of dynamical friction. We explore a broad parameter space, covering several aspects of the galaxy-group interaction that are potentially relevant to galaxy evolution. Namely, prograde and retrograde orbits, orbital eccentricities, disc inclination, role of a central bulge in discs, internal disc kinematics, and galaxy-to-group mass ratios. We find that significant disc transformations occur only after the mean density of the group, measured within the orbit of the galaxy, exceeds ~0.3-1 times the central mean density of the galaxy. The morphological evolution of discs is found to be strongly dependent on the initial inclination of the disc with respect to its orbital plane. That is, discs on face-on and retrograde orbits are shown to retain longer their disc structures and kinematics, in comparison to prograde discs. This suggests that after interacting with the global tidal field alone, a significant fraction of disc galaxies should be found in the central regions of groups. Prominent central bulges are not produced, and pre-existing bulges are not enhanced in discs after the interaction with the group. Assuming that most S0 are formed in group environments, this implies that prominent bulges should be formed mostly by young stars, created only after a galaxy has been accreted by a group. Finally, contrary to some current implementations of tidal stripping in semi-analytical models of galaxy evolution, we find that more massive galaxies suffer more tidal stripping.

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Schmidt-Kennicutt relations in SPH simulations of disc galaxies with effective thermal feedback from supernovae

We study several versions of the Schmidt-Kennicutt (SK) relation obtained for isolated spiral galaxies in TreeSPH simulations run with the GADGET3 code including the novel MUlti-Phase Particle Integrator (MUPPI) algorithm for star formation and stellar feedback. [...] The standard SK relation between surface densities of cold (neutral+molecular) gas and star formation rate of simulated galaxies shows a steepening at low gas surface densities, starting from a knee whose position depends on disc gas fraction: for more gas-rich discs the steepening takes place at higher surface densities. Because gas fraction and metallicity are typically related, this environmental dependence mimics the predictions of models where the formation of H2 is modulated by metallicity. The cold gas surface density at which HI and molecular gas surface densities equate can range from ~10 up to 34 Msun/pc^2. As expected, the SK relation obtained using molecular gas shows much smaller variations among simulations. We find that disc pressure is not well represented by the classical external pressure of a disc in vertical hydrostatic equilibrium. Instead is well fit by the expression P_fit = Sigma_cold sigma_cold kappa / 6, where the three quantities on the right-hand side are cold gas surface density, vertical velocity dispersion and epicyclic frequency. When the "dynamical" SK relation, i.e. the relation that uses gas surface density divided by orbital time, is considered, we find that all of our simulations stay on the same relation. We interpret this as a manifestation of the equilibrium between energy injection and dissipation in stationary galaxy discs, when energetic feedback is effective and pressure is represented by the expression given above. These findings further support the idea that a realistic model of the structure of galaxy discs should take into account energy injection by SNe. [Abridged]

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Hydrodynamic simulations with the Godunov SPH

We present results based on an implementation of the Godunov Smoothed Particle Hydrodynamics (GSPH), originally developed by Inutsuka (2002), in the GADGET-3 hydrodynamic code. We first review the derivation of the GSPH discretization of the equations of moment and energy conservation, starting from the convolution of these equations with the interpolating kernel. The two most important aspects of the numerical implementation of these equations are (a) the appearance of fluid velocity and pressure obtained from the solution of the Riemann problem between each pair of particles, and (b the absence of an artificial viscosity term. We carry out three different controlled hydrodynamical three-dimensional tests, namely the Sod shock tube, the development of Kelvin-Helmholtz instabilities in a shear flow test, and the "blob" test describing the evolution of a cold cloud moving against a hot wind. The results of our tests confirm and extend in a number of aspects those recently obtained by Cha (2010): (i) GSPH provides a much improved description of contact discontinuities, with respect to SPH, thus avoiding the appearance of spurious pressure forces; (ii) GSPH is able to follow the development of gas-dynamical instabilities, such as the Kevin--Helmholtz and the Rayleigh-Taylor ones; (iii) as a result, GSPH describes the development of curl structures in the shear-flow test and the dissolution of the cold cloud in the "blob" test. We also discuss in detail the effect on the performances of GSPH of changing different aspects of its implementation. The results of our tests demonstrate that GSPH is in fact a highly promising hydrodynamic scheme, also to be coupled to an N-body solver, for astrophysical and cosmological applications. [abridged]

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Measuring the escape velocity and mass profiles of galaxy clusters beyond their virial radius

The caustic technique uses galaxy redshifts alone to measure the escape velocity and mass profiles of galaxy clusters to clustrocentric distances well beyond the virial radius, where dynamical equilibrium does not necessarily hold. We provide a detailed description of this technique and analyse its possible systematic errors. We apply the caustic technique to clusters with mass M_200>=10^{14}h^{-1} M_sun extracted from a cosmological hydrodynamic simulation of a LambdaCDM universe. With a few tens of redshifts per squared comoving megaparsec within the cluster, the caustic technique, on average, recovers the profile of the escape velocity from the cluster with better than 10 percent accuracy up to r~4 r_200. The caustic technique also recovers the mass profile with better than 10 percent accuracy in the range (0.6-4) r_200, but it overestimates the mass up to 70 percent at smaller radii. This overestimate is a consequence of neglecting the radial dependence of the filling function F_beta(r). The 1-sigma uncertainty on individual escape velocity profiles increases from ~20 to ~50 percent when the radius increases from r~0.1 r_200 to ~4 r_200. Individual mass profiles have 1-sigma uncertainty between 40 and 80 percent within the radial range (0.6-4) r_200. We show that the amplitude of these uncertainties is completely due to the assumption of spherical symmetry, which is difficult to drop. Alternatively, we can apply the technique to synthetic clusters obtained by stacking individual clusters: in this case, the 1-sigma uncertainty on the escape velocity profile is smaller than 20 percent out to 4 r_200. The caustic technique thus provides reliable average profiles which extend to regions difficult or impossible to probe with other techniques.

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The universal distribution of halo interlopers in projected phase space. Bias in galaxy cluster concentration and velocity anisotropy?

When clusters of galaxies are viewed in projection, one cannot avoid picking up foreground/background interlopers (FBIs), that lie within the virial cone (VC), but outside the virial sphere. Structural & kinematic deprojection equations are not known for an expanding Universe, where the Hubble flow (HF) stretches the line-of-sight (LOS) distribution of velocities. We analyze 93 mock relaxed clusters, built from a cosmological simulation. The stacked mock cluster is well fit by an m=5 Einasto DM density profile (but only out to 1.5 virial radii [r_v]), with velocity anisotropy (VA) close to the Mamon-Lokas model with VA radius equal to that of density slope -2. The surface density of FBIs is nearly flat out to r_v, while their LOS velocity distribution shows a dominant gaussian cluster-outskirts component and a flat field component. This distribution of FBIs in projected phase space is nearly universal in mass. A local k=2.7 sigma velocity cut returns the LOS velocity dispersion profile (LOSVDP) expected from the NFW density and VA profiles measured in 3D. The HF causes a shallower outer LOSVDP that cannot be well matched by the Einasto model for any k. After this velocity cut, FBIs still account for 23% of DM particles within the VC (close to the observed fraction of cluster galaxies lying off the Red Sequence). The best-fit projected NFW/Einasto models underestimate the 3D concentration by 6+/-6% (16+/-7%) after (before) the velocity cut, unless a constant background is included in the fit. Assuming the correct mass profile, the VA profile is well recovered from the measured LOSVDP, with a slight bias towards more radial orbits in the outer regions. These small biases are overshadowed by large cluster-cluster variations caused by cosmic variance. An appendix provides an analytical approximation to the surface density, projected mass and tangential shear profiles of the Einasto model.

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