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The use of mathematical concepts and formal reasoning is one of the main hurdles for students entering introductory physics courses at university. The ability to apply mathematical tools in the context of physics also relies on the use of multiple representations, i.e., the different forms in which a concept can be expressed, such as words, graphs, numbers and formal language. Based on these considerations, we have developed a multiple-choice test consisting in 34 items aimed at investigating students’ understanding of derivatives, integrals and vectors and their application in the context of introductory classical mechanics. The items were constructed using multiple representational formats and isomorphic items in mathematics, and in physics, in order to explore students’ representational fluency and their ability to transfer knowledge and skills from mathematics to physics. The test has been administered to 1252 students enrolled in introductory courses at the University of Padova in Spring 2018. The results indicate that the test is a valid and reliable instrument and it provides interesting insight into students’ difficulties in the use of mathematical concepts and methods in physics.

We use high-resolution N-body simulations to follow the formation and evolution of tidal streams associated to dwarf spheroidal galaxies (dSphs). The dSph models are embedded in dark matter (DM) haloes with either a centrally-divergent 'cusp', or an homogeneous-density 'core'. In agreement with previous studies, we find that as tides strip the galaxy the evolution of the half-light radius and the averaged velocity dispersion follows well-defined tracks that are mainly controlled by the amount of mass lost. Crucially, the evolutionary tracks behave differently depending on the shape of the DM profile: at a fixed remnant mass, dSphs embedded in cored haloes have larger sizes and higher velocity dispersions than their cuspy counterparts. The divergent evolution is particularly pronounced in galaxies whose stellar component is strongly segregated within their DM halo and becomes more disparate as the remnant mass decreases. Our analysis indicates that the DM profile plays an important role in defining the internal dynamics of tidal streams. We find that stellar streams associated to cored DM models have velocity dispersions that lie systematically above their cuspy counterparts. Our results suggest that the dynamics of streams with known dSph progenitors may provide strong constraints on the distribution of DM on the smallest galactic scales., 5 pages, 4 figures

In the standard model of cosmic structure formation, dark matter haloes form by gravitational instability. The process is hierarchical: smaller systems collapse earlier, and later merge to form larger haloes. The galaxy clusters, hosted by the largest dark matter haloes, are at the top of this hierarchy representing the largest as well as the last structures formed in the universe, while the smaller and first haloes are those Earth-sized dark subhaloes which have been both predicted by theoretical considerations and found in numerical simulations, though it does not exist any observational hints of their existence. The probability that a halo of mass $m$ at redshift $z$ will be part of a larger halo of mass $M$ at the present time can be described in the frame of the extended Press & Schecter theory making use of the progenitor (conditional) mass function. Using the progenitor mass function we calculate analytically, at redshift zero, the distribution of subhaloes in mass, formation epoch and rarity of the peak of the density field at the formation epoch. That is done for a Milky Way-size system, assuming both a spherical and an ellipsoidal collapse model. Our calculation assumes that small progenitors do not lose mass due to dynamical processes after entering the parent halo, and that they do not interact with other subhaloes. For a $\mathrm{\Lambda}$CDM power spectrum we obtain a subhalo mass function $\mathrm{d}n/\mathrm{d}m$ proportional to $m^{- \alpha}$ with a model-independent $\alpha \sim 2$. Assuming the dark matter is a weakly interacting massive particle, the inferred distributions is used to test the feasibility of an indirect detection in the $\gamma$-rays energy band of such a population of subhaloes with a GLAST-like satellite., Comment: 10 pages, 7 figures - submitted to MNRAS

The abundance of galaxy clusters can constrain both the geometry and growth of structure in our Universe. However, this probe could be significantly complicated by recent claims of nonuniversality -- non-trivial dependences with respect to the cosmological model and redshift. In this work we analyse the dependance of the mass function on the way haloes are identified and establish if this can cause departures from universality. In order to explore this dependance, we use a set of different N-body cosmological simulations (Le SBARBINE simulations), with the latest cosmological parameters from the Planck collaboration; this first suite of simulations is followed by a lower resolution set, carried out with different cosmological parameters. We identify dark matter haloes using a Spherical Overdensity algorithm with varying overdensity thresholds (virial, 2000rho_c, 1000rho_c, 500rho_c, 200rho_c and 200rho_b) at all redshifts. We notice that, when expressed in term of the rescaled variable nu, the mass functionfor virial haloes is a nearly universal as a function of redshift and cosmology, while this is clearly not the case for the other overdensities we considered. We provide fitting functions for the halo mass function parameters as a function of overdensity, that allow to predict, to within a few percent accuracy, the halo mass function for a wide range of halo definitions, redshifts and cosmological models. We then show how the departures from universality associated with other halo definitions can be derived by combining the universality of the virial definition with the expected shape of the density profile of halos., 22 pages, 21 figures, accepted for publication in MNRAS - some changes with respect to the previous version and a new appendix

The next generation of large sky photometric surveys will finally be able to use arc statistics as a cosmological probe. Here, we present the first of a series of papers on this topic. In particular, we study how arc counts are sensitive to the variation of two cosmological parameters: the (total) matter density parameter, Ωm, and the normalization of the primordial power spectrum, expressed in terms of σ8. Both these parameters influence the abundances of collapsed structures and their internal structure. We compute the expected number of gravitational arcs with various length-to-width ratios in mock light cones, by varying these cosmological parameters in the ranges 0.1 ≤ Ωm ≤ 0.5 and 0.6 ≤ σ8 ≤ 1. We find that the arc counts dependence on Ωm and σ8 is similar, but not identical, to that of the halo counts. We investigate how the precision of the constraints on the cosmological parameters based on arc counts depend on the survey area. We find that the constraining power of arc statistics degrades critically only for surveys covering an area smaller than 10 per cent of the whole sky. Finally, we consider the case in which the search for arcs is done only in frames where galaxy clusters have been previously identified. Adopting the selection function for galaxy clusters expected to be detected from photometric data in future wide surveys, we find that less than 10 per cent of the arcs will be missed, with only a small degradation of the corresponding cosmological constraints.

We use high-resolution hydrodynamic re-simulations to investigate the properties of the thermal Sunyaev-Zel'dovich (SZ) effect from galaxy clusters. We compare results obtained using different physical models for the intracluster medium (ICM), and show how they modify the SZ emission in terms of cluster profiles and scaling relations. We also produce realistic mock observations to verify whether the results from hydrodynamic simulations can be confirmed. We find that SZ profiles depend marginally on the modelled physical processes, while they exhibit a strong dependence on cluster mass. The central and total SZ emission strongly correlate with the cluster X-ray luminosity and temperature. The logarithmic slopes of these scaling relations differ from the self-similar predictions by less than 0.2; the normalization of the relations is lower for simulations including radiative cooling. The observational test suggests that SZ cluster profiles are unlikely to be able to probe the ICM physics. The total SZ decrement appears to be an observable much more robust than the central intensity, and we suggest using the former to investigate scaling relations., 13 pages, 12 figures, accepted by MNRAS

We examine the systematics affecting the X-ray mass estimators applied to a set of five simulated galaxy clusters. They have been processed through the X-ray Map Simulator, X-MAS, to provide Chandra-like long exposures that are analyzed to reconstruct the gas temperature, density, and mass profiles used as input. We find that at R_2500 the mass profile obtained via a direct application of the hydrostatic equilibrium equation is consistent within 1 sigma with the actual mass; although we notice this estimator shows high statistical errors due to high level of Chandra background. Instead, the poorness of the beta-model in describing the gas density profile makes the evaluated masses to be underestimated by \sim 40 per cent with respect to the true mass, both with an isothermal and a polytropic temperature profile. We also test ways to recover the mass by adopting an analytic mass model, such as those proposed by Navarro et al. (1997) and Rasia et al. (2004), and fitting the temperature profile expected from the hydrostatic equilibrium equation to the observed one. These methods and the one of the hydrostatic equilibrium equation provide a more robust mass estimation than the ones based on the beta-model. In the present work the main limitation for a precise mass reconstruction is to ascribe to the relatively high level of the background chosen to reproduce the Chandra one. After artificially reducing it by a factor of 100, we find that the estimated mass significantly underestimates the true mass profiles. This is manly due (i) to the neglected contribution of the gas bulk motions to the total energy budget and (ii) to the bias towards lower values of the X-ray temperature measurements because of the complex thermal structure of the emitting plasma.

Smoothed particle hydrodynamics (SPH) employs an artificial viscosity to properly capture hydrodynamic shock waves. In its original formulation, the resulting numerical viscosity is large enough to suppress structure in the velocity field on scales well above the nominal resolution limit, and to damp the generation of turbulence by fluid instabilities. This could artificially suppress random gas motions in the intracluster medium (ICM), which are driven by infalling structures during the hierarchical structure formation process. We show that this is indeed the case by analysing results obtained with an SPH formulation where an individual, time-variable viscosity is used for each particle, following a suggestion by Morris & Monaghan. Using test calculations involving strong shocks, we demonstrate that this scheme captures shocks as well as the original formulation of SPH, but, in regions away from shocks, the numerical viscosity is much smaller. In a set of nine high-resolution simulations of cosmological galaxy cluster formation, we find that this low-viscosity formulation of SPH produces substantially higher levels of turbulent gas motions in the ICM, reaching a kinetic energy content in random gas motions (measured within a 1-Mpc cube) of up to 5-30 per cent of the thermal energy content, depending on cluster mass. This also has significant effects on radial gas profiles and bulk cluster properties. We find a central flattening of the entropy profile and a reduction of the central gas density in the low-viscosity scheme. As a consequence, the bolometric X-ray luminosity is decreased by about a factor of 2. However, the cluster temperature profile remains essentially unchanged. Interestingly, this tends to reduce the differences seen in SPH and adaptive mesh refinement simulations of cluster formation. Finally, invoking a model for particle acceleration by magnetohydrodynamics waves driven by turbulence, we find that efficient electron acceleration and thus diffuse radio emission can be powered in the clusters simulated with the low-viscosity scheme provided that more than 5-10 per cent of the turbulent energy density is associated with fast magneto-sonic modes. © 2005 RAS.

We analyze the dynamical and thermal evolution of dark matter and ICM in hydrodynamical Tree-SPH simulations of galaxy clusters. Starting from a sample of 17 high-resolution objects, with virial mass ranging from 3 10^{14} to 1.7 10^{15} M_sun, we follow the build-up of the systems in dark matter and hot gas through the repeated merging of satellites along their merging history trees. We measure the self-bound mass fraction of subhaloes as a function of time after the merging, estimate the satellite mean orbital properties as a function of the mass ratio with the main cluster at merging time, and study the evolution of their internal velocity dispersion, gas temperature and entropy as the substructure is disrupted by various dynamical processes, reaching eventually thermo-dynamic equilibrium in the gravitational potential of the main cluster. We model some relevant properties of subhalo orbits, as the time of the first pericentric and apocentric passages, and the typical distances and velocities at the corresponding times. This survival study can be used to interpret the dynamics of observed merging clusters; as an example we apply our results to the system 1E0657-56. We show that, in the light of our results, the most likely interpretation of the data for this cluster points to the merger of a small group with mass M\approx 1 10^{13} M_sun with a massive cluster with M\approx 1.3 10^{15} M_sun., 14 pages, version accepted for publication on MNRAS

We present results on the X-ray properties of clusters and groups of galaxies, extracted from a large hydrodynamical simulation. We used the GADGET code to simulate a LambdaCDM model within a box of 192 Mpc/h on a side, with 480^3 dark matter particles and as many gas particles. The simulation includes radiative cooling, star formation and supernova feedback. The simulated M-T relation is consistent with observations once we mimic the procedure for mass estimates applied to real clusters. Also, with the adopted choices of Omega_m=0.3 and sigma_8=0.8 the resulting XTF agrees with observational determinations. The L-T relation also agrees with observations for clusters with T>2 keV, with no change of slope at the scale of groups. The entropy in central cluster regions is higher than predicted by gravitational heating alone, the excess being almost the same for clusters and groups. The simulated clusters appear to have suffered some overcooling, with f*~0.2, thus about twice as large as the value observed. Interestingly, temperature profiles are found to steadily increase toward cluster centers. They decrease in the outer regions, much like observational data do at r>0.2r_vir, while not showing an isothermal regime followed by a smooth temperature decline in the innermost regions., 20 pages, 14 figures, to appear in MNRAS

We compare the structure and kinematics of the 11 known satellites of the Milky Way with high resolution simulations of the formation of its dark halo in a LambdaCDM universe. In contrast to earlier work, we find excellent agreement. The observed kinematics are exactly those predicted for stellar populations with the observed spatial structure orbiting within the most massive ``satellite'' substructures in our simulations. Less massive substructures have weaker potential wells than those hosting the observed satellites. If there is a halo substructure ``problem'', it consists in understanding why halo substructures have been so inefficient in making stars. Suggested modifications of dark matter properties (for example, self-interacting or warm dark matter) may well spoil the good agreement found for standard Cold Dark Matter.

The excursion set approach allows one to estimate the abundance and spatial distribution of virialized dark matter haloes efficiently and accurately. The predictions of this approach depend on how the nonlinear processes of collapse and virialization are modelled. We present simple analytic approximations which allow us to compare the excursion set predictions associated with spherical and ellipsoidal collapse. In particular, we present formulae for the universal unconditional mass function of bound objects and the conditional mass function which describes the mass function of the progenitors of haloes in a given mass range today. We show that the ellipsoidal collapse based moving barrier model provides a better description of what we measure in the numerical simulations than the spherical collapse based constant barrier model, although the agreement between model and simulations is better at large lookback times. Our results for the conditional mass function can be used to compute accurate approximations to the local-density mass function which quantifies the tendency for massive haloes to populate denser regions than less massive haloes. This happens because low density regions can be thought of as being collapsed haloes viewed at large lookback times, whereas high density regions are collapsed haloes viewed at small lookback times. Although we have only applied our analytic formulae to two simple barrier shapes, we show that they are, in fact, accurate for a wide variety of moving barriers. We suggest how they can be used to study the case in which the initial dark matter distribution is not completely cold., 18 pages, submitted to MNRAS

We present a new algorithm for identifying the substructure within simulated dark matter haloes. The method is an extension of that proposed by Tormen et al. (2004) and Giocoli et al. (2008a), which identifies a subhalo as a group of self-bound particles that prior to being accreted by the main progenitor of the host halo belonged to one and the same progenitor halo (hereafter satellite). However, this definition does not account for the fact that these satellite haloes themselves may also have substructure, which thus gives rise to sub-subhaloes, etc. Our new algorithm identifies substructures at all levels of this hierarchy, and we use it to determine the mass function of all substructure (counting sub-haloes, sub-subhaloes, etc.). On average, haloes which formed more recently tend to have a larger mass fraction in substructure and to be less concentrated than average haloes of the same mass. We provide quantitative fits to these correlations. Even though our algorithm is very different from that of Gao et al. (2004), we too find that the subhalo mass function per unit mass at redshift z = 0 is universal. This universality extends to any redshift only if one accounts for the fact that host haloes of a given mass are less concentrated at higher redshifts, and concentration and substructure abundance are anti-correlated. This universality allows a simple parametrization of the subhalo mass function integrated over all host halo masses, at any given time. We provide analytic fits to this function which should be useful in halo model analyses which equate galaxies with halo substructure when interpreting clustering in large sky surveys. Finally, we discuss systematic differences in the subhalo mass function that arise from different definitions of (host) halo mass., 18 pages, 24 figures, accepted for publication on MNRAS

We present a detailed analysis of dark matter halo shapes, studying how the distributions of ellipticity, prolateness and axial ratios evolve as a function of time and mass. With this purpose in mind, we analysed the results of three cosmological simulations, running an ellipsoidal halo finder to measure triaxial halo shapes. The simulations have different scales, mass limits and cosmological parameters, which allows us to ensure a good resolution and statistics in a wide mass range, and to investigate the dependence of halo properties on the cosmological model. We confirm the tendency of haloes to be prolate at all times, even if they become more triaxial going to higher redshifts. Regarding the dependence on mass, more massive haloes are also less spherical at all redshifts, since they are the most recent forming systems and so still retain memory of their original shape at the moment of collapse. We then propose a rescaling of the shape-mass relations, using the variable $\nu = \delta_{c}/\sigma$ to represent the mass, which absorbs the dependence on both cosmology and time, allowing to find universal relations between halo masses and shape parameters (ellipticity, prolateness and the axial ratios) which hold at any redshift. This may be very useful to determine prior distributions of halo shapes for observational studies., Comment: 11 pages, 9 figures, accepted for publication on MNRAS - Fig.1 updated, minor changes

We simulate the assembly of a massive rich cluster and the formation of its constituent galaxies in a flat, low-density universe. Our most accurate model follows the collapse, the star-formation history and the orbital motion of all galaxies more luminous than the Fornax dwarf spheroidal, while dark halo structure is tracked consistently throughout the cluster for all galaxies more luminous than the SMC. Within its virial radius this model contains about 2.0e7 dark matter particles and almost 5000 distinct dynamically resolved galaxies. Simulations of this same cluster at a variety of resolutions allow us to check explicitly for numerical convergence both of the dark matter structures produced by our new parallel N-body and substructure identification codes, and of the galaxy populations produced by the phenomenological models we use to follow cooling, star formation, feedback and stellar aging. This baryonic modelling is tuned so that our simulations reproduce the observed properties of isolated spirals outside clusters. Without further parameter adjustment our simulations then produce a luminosity function, a mass-to-light ratio, luminosity, number and velocity dispersion profiles, and a morphology-radius relation which are similar to those observed in real clusters. In particular, since our simulations follow galaxy merging explicitly, we can demonstrate that it accounts quantitatively for the observed cluster population of bulges and elliptical galaxies.

The Press--Schechter, excursion set approach allows one to make predictions about the shape and evolution of the mass function of bound objects. It combines the assumption that objects collapse spherically with the assumption that the initial density fluctuations were Gaussian and small. While the predicted mass function is reasonably accurate at the high mass end, it has more low mass objects than are seen in simulations of hierarchical clustering. This discrepancy can be reduced substantially if bound structures are assumed to form from an ellipsoidal, rather than a spherical collapse. In the spherical model, a region collapses if the initial density within it exceeds a threshold value, delta_sc. This value is independent of the initial size of the region, and since the mass of the collapsed object is related to its initial size, delta_sc is independent of final mass. In the ellipsoidal model, the collapse of a region depends on the surrounding shear field, as well as on its initial overdensity. Therefore, there is a relation between the density threshold value required for collapse, and the mass of the final object. We provide a fitting function to this delta_ec(m) relation for initially Gaussian fields which simplifies the inclusion of ellipsoidal dynamics in the excursion set approach. We discuss the relation between the excursion set predictions and the halo distribution in high resolution N-body simulations, and show that our simple parametrization of the ellipsoidal collapse model represents a significant improvement on the spherical model on an object-by-object basis. Finally, we show that the associated statistical predictions, the mass function and the large scale halo-to-mass bias relation, are also more accurate than the standard predictions., Comment: 12 pages, 6 figures, submitted to MNRAS

The hierarchical growth of dark matter haloes, in which galaxies are hosted, has been studied and modeled using various approaches. In this paper we use a modified version the Sheth & Lemson algorithm for a $\mathrm{\Lambda}$ cold dark matter power spectrum, and model the growth of a Milky-Way sized halo with microsolar mass resolution, corresponding to the typical Jeans mass for a dark matter Weakly Interacting Massive Particle with mass of 100 GeV. We then compute the \emph{unevolved} subhalo mass function and build-up a Milky-Way halo placing and evolving its satellites. This subhalo population is used to study the $\gamma$-ray emission from dark matter annihilation. In this case, the subhaloes which populate the host halo have been computed considering only progenitor haloes accreted by the main branch of the tree, so as to correctly treat the embedding of sub-subhaloes inside subhaloes. Each subhalo will indeed host atthe present-time sub-subhaloes accreted when it was an isolated system. In order to compute the sub-subhalo population of a Milky-Way dwarf galaxy, like Draco, and to study its $\gamma$-ray emission, we first estimate the Draco virial mass at merging redshift $z_m$ and then we run the merger tree from $z_m$ following the halo down to the dark matter Jeans mass. We then study the effect on the Fermi-LAT (GLAST) detectability for both subhaloes in the Milky-Way and in Draco, and we show how subhaloes cannot be responsible for the boost factor needed for detection., Comment: Accepted for publication by MNRAS - 12 pages, 7 figures. Text improved and comments added. Name of a author fixed

Monthly Notices of the Royal Astronomical Society, 359 (3), ISSN:0035-8711, ISSN:1365-2966, ISSN:1365-8711

We describe an algorithm for identifying ellipsoidal haloes in numerical simulations, and quantify how the resulting estimates of halo mass and shape differ with respect to spherical halo finders. Haloes become more prolate when fit with ellipsoids, the difference being most pronounced for the more aspherical objects. Although the ellipsoidal mass is systematically larger, this is less than 10% for most of the haloes. However, even this small difference in mass corresponds to a significant difference in shape. We quantify these effects also on the initial mass and deformation tensors, on which most models of triaxial collapse are based. By studying the properties of protohaloes in the initial conditions, we find that models in which protohaloes are identified in Lagrangian space by three positive eigenvalues of the deformation tensor are tenable only at the masses well-above $M_*$. The overdensity $\delta$ within almost any protohalo is larger than the critical value associated with spherical collapse (increasing as mass decreases); this is in good qualitative agreement with models which identify haloes requiring that collapse have occured along all three principal axes, each axis having turned around from the universal expansion at a different time. The distributions of initial values are in agreement with the simplest predictions associated with ellipsoidal collapse, assuming initially spherical protohaloes, collapsed around random positions which were sufficiently overdense. However, most protohaloes are not spherical and departures from sphericity increase as protohalo mass decreases. [Abridged], Comment: 18 pages, 17 figures, accepted for publication in MNRAS

We combine the physics of the ellipsoidal collapse model with the excursion set theory to study the shapes of dark matter halos. In particular, we develop an analytic approximation to the nonlinear evolution that is more accurate than the Zeldovich approximation; we introduce a planar representation of halo axis ratios, which allows a concise and intuitive description of the dynamics of collapsing regions and allows one to relate the final shape of a halo to its initial shape; we provide simple physical explanations for some empirical fitting formulae obtained from numerical studies. Comparison with simulations is challenging, as there is no agreement about how to define a non-spherical gravitationally bound object. Nevertheless, we find that our model matches the conditional minor-to-intermediate axis ratio distribution rather well, although it disagrees with the numerical results in reproducing the minor-to-major axis ratio distribution. In particular, the mass dependence of the minor-to-major axis distribution appears to be the opposite to what is found in many previous numerical studies, where low-mass halos are preferentially more spherical than high-mass halos. In our model, the high-mass halos are predicted to be more spherical, consistent with results based on a more recent and elaborate halo finding algorithm, and with observations of the mass dependence of the shapes of early-type galaxies. We suggest that some of the disagreement with some previous numerical studies may be alleviated if we consider only isolated halos.

We reassess the estimate of the cross-correlation of the spatial distribution of the NRAO VLA Sky Survey (NVSS) radio sources with that of Cosmic Microwave Background (CMB) anisotropies from the Wilkinson Microwave Anisotropy Probe (WMAP). This re-analysis is motivated by the fact that most previous studies adopted a redshift distribution of NVSS sources inconsistent with recent data. We find that the constraints on the bias-weighted redshift distribution, b(z)xN(z), of NVSS sources, set by the observed angular correlation function, w(theta), strongly mitigate the effect of the choice of N(z). If such constraints are met, even highly discrepant redshift distributions yield NVSS-WMAP cross-correlation functions consistent with each other within statistical errors. The models favoured by recent data imply a bias factor, b(z), decreasing with increasing z, rather than constant, as assumed by most previous analyses. As a consequence, the function b(z)xN(z) has more weight at z, 6 pages, 2 figures, submitted to MNRAS

By means of high-resolution N-body simulations in a LambdaCDM cosmology, we verify that scaling relations similar to those observed for nearby galaxy clusters are also defined by their dark matter hosts; the slopes, however, are not the same. We then show that the scaling relations of galaxy clusters can be explained as the result of the cosmological collapse of density fluctuations at the appropriate scales, plus a systematic trend of the M/L ratio with cluster mass. The empirical fact that the exponent of the Faber-Jackson relation of elliptical galaxies is significantly different (higher) than that of clusters, force us to conclude that the galaxy scaling laws might derive from the cosmological collapse of density fluctuations at the epoch when galactic scales became non-linear, plus modifications afterward due to early-time dissipative merging., Comment: 2 pages, 1 fig., in press on the Proceedings of the ESO-USM-MPE Workshop "Multiwavelength mapping of galaxy formation and evolution", ESO Astrophysics Symposia, R. Bender and A. Renzini eds

A dark matter halo is said to have formed when at least half its mass hass been assembled into a single progenitor. With this definition, it is possible to derive a simple but useful analytic estimate of the distribution of halo formation times. The standard estimate of this distribution depends on the shape of the conditional mass function--the distribution of progenitor masses of a halo as a function of time. If the spherical collapse model is used to estimate the progenitor mass function, then the formation times one infers systematically underestimate those seen in numerical simulations of hierarchical gravitational clustering. We provide estimates of halo formation which may be related to an ellipsoidal collapse model. These estimates provide a substantially better description of the simulations. We also provide an alternative derivation of the formation time distribution which is based on the assumption that haloes increase their mass through binary mergers only. Our results are useful for models which relate halo structure to halo formation., Comment: 8 pages, 5 figures. Replaced to match version accepted for publication in MNRAS

We examine the systematics affecting the X-ray mass estimators applied to Chandra -like long exposures images of five simulated clusters.

We present a study of the turbulent velocity fields in the intracluster medium (ICM) of a sample of 21 galaxy clusters simulated by the smoothed particle hydrodynamics code GADGET2, using a new numerical scheme where the artificial viscosity is suppressed outside shocks. The turbulent motions in the ICM of our simulated clusters are detected with a novel method devised to better disentangle laminar bulk motions from chaotic ones. We focus on the scaling law between the turbulent energy content of the gas particles and the total mass, and find that the energy in the form of turbulence scales approximately with the thermal energy of clusters. We follow the evolution with time of the scaling laws and discuss the physical origin of the observed trends. The simulated data are in agreement with independent semi-analytical calculations, and the combination between the two methods allows one to constrain the scaling law over more than two decades in cluster mass. © 2006 The Authors. Journal compilation © 2006 RAS.

The detection of excess of soft X-ray or Extreme Ultraviolet (EUV) radiation, above the thermal contribution from the hot intracluster medium (ICM), has been a controversial subject ever since the initial discovery of this phenomenon. We use a large--scale hydrodynamical simulation of a concordance $��$CDM model, to investigate the possible thermal origin for such an excess in a set of 20 simulated clusters having temperatures in the range 1--7 keV. Simulated clusters are analysed by mimicking the observational procedure applied to ROSAT--PSPC data, which for the first time showed evidences for the soft X-ray excess. For cluster--centric distances $0.4< R/R_{\rm vir}< 0.7$ we detect a significant excess in most of the simulated clusters, whose relative amount changes from cluster to cluster and, for the same cluster, by changing the projection direction. In about 30 per cent of the cases, the soft X-ray flux is measured to be at least 50 per cent larger than predicted by the one--temperature plasma model. We find that this excess is generated in most cases within the cluster virialized regions. It is mainly contributed by low--entropy and high--density gas associated with merging sub--halos, rather than to diffuse warm gas. Only in a few cases the excess arises from fore/background groups observed in projection, while no evidence is found for a significant contribution from gas lying within large--scale filaments. We compute the distribution of the relative soft excess, as a function of the cluster--centric distance, and compare it with the observational result by Bonamente et al. (2003) for the Coma cluster. Similar to observations, we find that the relative excess increases with the distance from the cluster center, with no significant excess detected for $R, 10 pages, to appear in A&A

We study the efficiency for producing strong gravitational lensing events of galaxy clusters numerically simulated in different dark-energy cosmologies with constant and time-variable equation of state, and we compare the results with those obtained in standard ACDM and OCDM models. Our main results are (1) that the expected abundance of gravitational arcs with large length-to-width ratio depends on the equation of state of dark energy at the epoch of formation of the halo; and (2) that the high sensitivity of strong-lensing cross sections of galaxy clusters to dynamical processes like mergers, which was found in earlier studies, varies substantially between different cosmologies, being stronger for models in which halos are less concentrated. As expected, the largest differences in the lensing optical depth occur at intermediate and high redshift.

The thermal properties of hydrodynamical simulations of galaxy clusters are usually compared to observations by relying on the emission-weighted temperature T_ew, instead of on the spectroscopic X-ray temperature T_spec, which is obtained by actual observational data. In a recent paper Mazzotta et al. show that, if the cluster is thermally complex, T_ew fails at reproducing T_spec, and propose a new formula, the spectroscopic-like temperature, T_sl, which approximates T_spec better than a few per cent. By analyzing a set of hydrodynamical simulations of galaxy clusters, we find that T_sl is lower than T_ew by 20-30 per cent. As a consequence, the normalization of the M-T_sl relation from the simulations is larger than the observed one by about 50 per cent. If masses in simulated clusters are estimated by following the same assumptions of hydrostatic equilibrium and \beta--model gas density profile, as often done for observed clusters, then the M-T relation decreases by about 40 per cent, and significantly reduces its scatter. Based on this result, we conclude that using the observed M-T relation to infer the amplitude of the power spectrum from the X-ray temperature function could bias low \sigma_8 by 10-20 per cent. This may alleviate the tension between the value of \sigma_8 inferred from the cluster number density and those from cosmic microwave background and large scale structure., Comment: Submitted for publication in APJL; 4 pages, 3 color figures

We study the efficiency of numerically simulated galaxy clusters for producing strong gravitational lensing events in different dark-energy cosmologies with constant and time-variable equation of state and we compare it with ``standard'' $\Lambda$CDM and OCDM models. Our main results are that: (1) the expected abundance of gravitational arcs with large length-to-width ratio depends on the equation of state of dark energy at the epoch of formation of the halo; (2) as found in previous studies, the strong lensing cross section of galaxy clusters is very sensitive to dynamical processes like mergers, whose impact however is different for different cosmologies, being stronger for models in which halos are less concentrated. As expected, the largest differences in the lensing optical depth occur at moderate and high redshift., Comment: 10 pages, 6 figures, submitted to A&A

High-resolution, numerical simulations of 17 cluster-sized dark-matter haloes in eight different cosmologies with and without dynamical dark energy confirm the picture that core halo densities are imprinted early during their formation by the mean cosmological density. Quite independent of cosmology, halo concentrations have a log-normal distribution with a scatter of ~0.2 about the mean. We propose a simple scaling relation for halo concentrations in dark-energy cosmologies., proceedings contribution to "Dark Matter/Dark Energy 2004"

Using high-resolution non radiative SPH simulations we study the scaling laws of galaxy clusters. In particular, we analyze the relations existing between total mass, gas temperature, X-ray luminosity and dark matter velocity dispersion. Our aim is to understand how these scaling laws depend on numerical, statistical and physical parameters, in order to model and thus reduce potential systematical uncertainties. The simulations were performed by the GADGET code in a $\Lambda CDM$ cosmology, applying resimulation techniques; their mass resolution ranges between $2 \times 10^9$ and $6.8 \times 10^9 h^{-1}M_{\odot}$ for dark matter particles and between $3 \times 10^8$ and $7 \times 10^8 h^{-1}M_{\odot}$ for gas particles. To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

The most commonly used definition of halo formation is the time when a halo's most massive progenitor first contains at least half the final mass of its parent. Reasonably accurate formulae for the distribution of formation times of haloes of fixed mass have been available for some time. We use numerical simulations of hierarchical gravitational clustering to test the accuracy of formulae for the mass at formation. We also derive and test a formula for the joint distribution of formation masses and times. The structure of a halo is expected to be related to its accretion history. Our tests show that our formulae for formation masses and times are reasonably accurate, so we expect that they will aid future analytic studies of halo structure., Comment: 5 pages, 5 figures, MNRAS, in press

We study the concentration parameters, their mass dependence and redshift evolution, of dark-matter halos in different dark-energy cosmologies with constant and time-variable equation of state, and compare them with "standard'' Lambda-CDM and OCDM models. We find that previously proposed algorithms for predicting halo concentrations can be well adapted to dark-energy models. When centred on the analytically expected values, halo concentrations show a log-normal distribution with a uniform standard deviation of ~0.2. The dependence of averaged halo concentrations on mass and redshift permits a simple fit of the form (1+z) c=c0 (M/M0)^a, with a~-0.1 throughout. We find that the cluster concentration depends on the dark energy equation of state at the cluster formation redshift z_{coll} through the linear growth factor D_+(z_{coll}). As a simple correction accounting for dark-energy cosmologies, we propose scaling c0 from Lambda-CDM with the ratio of linear growth factors, c0 -> c0 D_+(z_{coll})/D_{+,Lambda-CDM}(z_{coll})., 11 pages, submitted to Astronomy & Astrophysics

We study the properties of the diffuse light in galaxy clusters forming in a large hydrodynamical cosmological simulation of the Lambda-CDM cosmology. The simulation includes a model for radiative cooling, star formation in dense cold gas, and feedback by SN-II explosions. We select clusters having mass M>10^(14) h^(-1) Msun and study the spatial distribution of their star particles. While most stellar light is concentrated in gravitationally bound galaxies orbiting in the cluster potential, we find evidence for a substantial diffuse component, which may account for the extended halos of light observed around central cD galaxies. We find that more massive simulated clusters have a larger fraction of stars in the diffuse light than the less massive ones. The intracluster light is more centrally concentrated than the galaxy light, and the stars in the diffuse component are on average older than the stars in cluster galaxies, supporting the view that the diffuse light is not a random sampling of the stellar population in the cluster galaxies. We thus expect that at least ~10% of the stars in a cluster may be distributed as intracluster light, largely hidden thus far due to its very low surface brightness., 4 pages, 3 figures

Theoretical studies of the physical processes in clusters of galaxies are mainly based on the results of numerical simulations, which in turn are often directly compared to X-ray observations. Although trivial in principle, these comparisons are not always simple. We show that the projected spectroscopic temperature of clusters obtained from X-ray observations is always lower than the emission-weighed temperature. This bias is related to the fact that the emission-weighted temperature does not reflect the actual spectral properties of the observed source. This has implications for the study of thermal structures in clusters, especially when strong temperature gradients, like shock fronts, are present. In real observations shock fronts appear much weaker than what is predicted by emission-weighted temperature maps. We propose a new formula, the spectroscopic-like temperature function that better approximates the spectroscopic temperature, making simulations more directly comparable to observations

A generic prediction of hierarchical gravitational clustering models is that the distribution of halo formation times should depend relatively strongly on halo mass, massive haloes forming more recently, and depend only weakly, if at all, on the large scale environment of the haloes. We present a novel test of this assumption which uses a statistic that proves to be particularly well-suited to detecting and quantifying weak correlations with environment. We find that close pairs of haloes form at slightly higher redshifts than do more widely separated halo pairs, suggesting that haloes in dense regions form at slightly earlier times than do haloes of the same mass in less dense regions. The environmental trends we find are useful for models which relate the properties of galaxies to the formation histories of the haloes which surround them., Comment: 7 pages, 10 figures, submitted to MNRAS

We study the impact of merger events on the strong lensing properties of galaxy clusters. Previous lensing simulations were not able to resolve dynamical time scales of cluster lenses, which arise on time scales which are of order a Gyr. In this case study, we first describe qualitatively with an analytic model how some of the lensing properties of clusters are expected to change during merging events. We then analyse a numerically simulated lens model for the variation in its efficiency for producing both tangential and radial arcs while a massive substructure falls onto the main cluster body. We find that: (1) during the merger, the shape of the critical lines and caustics changes substantially; (2) the lensing cross sections for long and thin arcs can grow by one order of magnitude and reach their maxima when the extent of the critical curves is largest; (3) the cross section for radial arcs also grows, but the cluster can efficiently produce this kind of arcs only while the merging substructure crosses the main cluster centre; (4) while the arc cross sections pass through their maxima as the merger proceeds, the cluster's X-ray emission increases by a factor of $\sim5$. Thus, we conclude that accounting for these dynamical processes is very important for arc statistics studies. In particular, they may provide a possible explanation for the arc statistics problem., 16 pages, submitted to MNRAS, revised version after referee' Comments. Gzipped file including full resolution images can be downloaded at http://dipastro.pd.astro.it/~cosmo/massimo/high-res-images.tar.gz

Using high-resolution non-radiative hydrodynamic simulations of galaxy clusters we obtain simple analytic formulae for DM and gas distribution in the spherical approximation. We derive fits for the DM density, velocity dispersion and velocity anisotropy. We use these models to test the dynamical equilibrium hypothesis through the Jeans equation: we find that this is satisfied to good accuracy. This result show that our fits constitute a self-consistent dynamical model. We then extend our analysis to the gas, studying its density, temperature and velocity structure, with no further hypothesis on the dynamical status or state equation. Gas and DM show self-similar density profiles down to 0.06 R_vir, while at smaller radii the gas produces a central core. Gas temperatures are almost isothermal out to 0.2 R_vir, then steeply decrease reaching at R_vir a value a factor of 2 lower. We find that the gas is not at rest inside R_vir: velocity dispersions are increasing functions of the radius, motions are isotropic to slightly tangential, and contribute to the total pressure support. We test our model using a generalization of the hydrostatic equilibrium equation, where the gas motion is properly taken into account. We find that our fits provide an accurate description of the system: the gas is in equilibrium and is a good tracer the overall cluster potential if all terms are taken into account, while simpler assumptions, as the beta-model, cause systematic mass underestimates. We also find that, if gas velocities are neglected, then a simple isothermal model fares better at large radii than a non-isothermal one. (abridged), 16 pages, MNRAS in press. added discussion on the dependence of profiles on the dynamical and environmental status

We analyse the properties of substructures within dark matter halos (subhalos) using a set of high-resolution numerical simulations of the formation of structure in a Lambda-CDM Universe. Our simulation set includes 11 high-resolution simulations of massive clusters as well as a region of mean density, allowing us to study the spatial and mass distribution of substructures down to a mass resolution limit of 10^9 h^(-1)Mo. We also investigate how the properties of substructures vary as a function of the mass of the `parent' halo in which they are located. We find that the substructure mass function depends at most weakly on the mass of the parent halo and is well described by a power-law. The radial number density profiles of substructures are steeper in low mass halos than in high mass halos. More massive substructures tend to avoid the centres of halos and are preferentially located in the external regions of their parent halos. We also study the mass accretion and merging histories of substructures, which we find to be largely independent of environment. We find that a significant fraction of the substructures residing in clusters at the present day were accreted at redshifts z < 1. This implies that a significant fraction of present-day `passive' cluster galaxies should have been still outside the cluster progenitor and more active at z~1., 13 pages, 15 figure. Accepted to MNRAS

If the dark matter in the Universe is made of weakly self-interacting particles, they may self-annihilate and emit gamma-rays. We use high resolution numerical simulations to estimate directly the annihilation flux from the central regions of the Milky Way and from dark matter substructures in its halo. Although such estimates remain uncertain because of their strong dependence on the structure of the densest regions, our numerical experiments suggest that less direct calculations have overestimated the emission both from the centre and from halo substructure. We estimate a maximal enhancement of at most a factor of a few with respect to a smooth spherical halo of standard Navarro-Frenk-White (NFW) structure. We discuss detection strategies for the next generation of gamma-ray detectors and find that the annihilation flux may be detectable, regardless of uncertainties about the densest regions, for the annihilation cross-sections predicted by currently popular elementary particle models for the dark matter., Comment: 10 pages, 8 figures, replaced with version accepted by MNRAS (very minor changes)

Numerical hydro-N-body simulations are very important tools for making theoretical predictions for the formation of galaxy clusters. They show that the atmospheres of clusters of galaxies have quite complex angular and thermal structures. The full understanding of the physical processes behind these features can be only achieved by direct comparison of observations to hydro-N-body simulations. Although simple in principle, these comparisons are not always trivial. In fact, real data are convolved with the instrument response which may substantially influence the apparent properties of the studied features. To overcome this problem we build the software package X-MAS devoted to simulate X-ray observations of galaxy clusters obtained from hydro-N-body simulations. In this paper we present how this software package works and discuss its application to the simulation of Chandra ACIS-S3 observations. We compare some of the main physical properties of the input data to the ones derived from simulated observations after performing a standard imaging and spectral analysis. We show that, if the thermal structure of the cluster along a particular line of sight is quite complex, the projected spectroscopic temperature obtained from the observation is significantly lower than the emission-weighed value inferred directly from hydrodynamical simulation. This implies that much attention must be paid in the theoretical interpretation of observational temperatures., Comment: Submitted for publication in MNRAS; 9 pages, 8 color figures and 2 BW figures,mn2e.cls

Like early-type galaxies, also nearby galaxy clusters define a Fundamental Plane, a luminosity-radius, and a luminosity-velocity dispersion relations, whose physical origin is still unclear. By means of high resolution N--body simulations of massive dark matter halos in a Lambda-CDM cosmology, we find that scaling relations similar to those observed for galaxy clusters are already defined by their dark matter hosts. The slopes however are not the same, and among the various possibilities in principle able to bring the simulated and the observed scaling relations in mutual agreement, we show that the preferred solution is a luminosity dependent mass-to-light ratio (M/L ~ L^0.3), that well corresponds to what inferred observationally. We then show that at galactic scales there is a conflict between the cosmological predictions of structure formation, the observed trend of the mass-to-light ratio in ellipticals, and the slope of their luminosity-velocity dispersion relation (that significantly differs from the analogous one followed by clusters). The conclusion is that the scaling laws of elliptical galaxies might be the combined result of the cosmological collapse of density fluctuations at the epoch when galactic scales became non-linear, plus important modifications afterward due to early-time dissipative merging. Finally, we briefly discuss the possible evolution of the cluster scaling relations with redshift., Comment: 18 pages, 3 figures, ApJ, accepted version. A new Table added, some extended discussion, conclusions unchanged

We carry out ray tracing through five high resolution simulations of a galaxy cluster to study how its ability to produce giant gravitationally lensed arcs is influenced by the collision cross-section of its dark matter. In three cases typical dark matter particles in the cluster core undergo between 1 and 100 collisions per Hubble time; two more explore the long (``collisionless'') and short (``fluid'') mean free path limits. We study the size and shape distributions of arcs and compute the cross-section for producing ``extreme'' arcs of various sizes. Even a few collisions per particle modify the core structure enough to destroy the cluster's ability to produce long, thin arcs. For larger collision frequencies the cluster must be scaled up to unrealistically large masses before it regains the ability to produce giant arcs. None of our models with self-interacting dark matter (except the ``fluid'' limit) is able to produce radial arcs; even the case with the smallest scattering cross-section must be scaled to the upper limit of observed cluster masses before it produces radial arcs. Apparently the elastic collision cross-section of dark matter in clusters must be very small, below 0.1 cm^2 g^-1, to be compatible with the observed ability of clusters to produce both radial arcs and giant arcs., Comment: 8 pages, Latex using MN style, accepted version, to appear in MNRAS. Other figures not included in the paper can be found at http://www.mpa-garching.mpg.de/~massimo/SIDMlens.html

We study how the internal structure of dark halos is affected if Cold Dark Matter particles are assumed to have a large cross-section for elastic collisions. We identify a cluster halo in a large cosmological N-body simulation and resimulate its formation with progressively increasing resolution. We compare the structure found in the two cases where dark matter is treated as collisionless or as a fluid. For the collisionless case our results agree with those of other workers. Collisional dark matter results in a cluster which is more nearly spherical and has a more singular central density profile. Substructure within the cluster is only weakly suppressed relative to the collisionless case. The observed structure of dwarf galaxies argues against self-interacting dark matter if, as seems likely, intermediate cross-sections produce structure lying between the extremes we have simulated., Revised, Figure3 replaced. ApJLetters in press

We study the formation of dark halos in a $\Lambda$CDM universe under the assumption that Cold Dark Matter particles have a finite cross-section for elastic collisions. We compare evolution when CDM mean free paths are comparable to halo sizes with the collisionless and fluid limits. We show that a few collisions per particle per Hubble time at halo centre can substantially affect the central density profile. Cross-sections an order of magnitude larger produce sufficient relaxation for rich clusters to develop core radii in the range 100-200 $h^{-1}$kpc. The structural evolution of halos is a competition between collisional relaxation caused by individual particle interactions and violent relaxation resulting from the infall and merging processes by which clusters grow. Although our simulations concentrate on systems of cluster size, we can scale our results to address the halo structure expected for dwarf galaxies. We find that collision cross-sections sufficiently large to significantly modify the cores of such galaxies produce cluster cores which are too large and/or too round to be consistent with observation. Thus the simplest model for self-interacting dark matter is unable to improve fits to published dwarf galaxy rotation curves without violating other observational constraints., Comment: Revised, accepted for publication in ApJ Letters. Figure1 replaced

Dark matter haloes are biased tracers of the underlying dark matter distribution. We use a simple model to provide a relation between the abundance of dark matter haloes and their spatial distribution on large scales. Our model shows that knowledge of the unconditional mass function alone is sufficient to provide an accurate estimate of the large scale bias factor. Then we use the mass function measured in numerical simulations of SCDM, OCDM and LCDM to compute this bias. Comparison with these simulations shows that this simple way of estimating the bias relation and its evolution is accurate for less massive haloes as well as massive ones. In particular, we show that haloes which are less/more massive than typical M* haloes at the time they form are more/less strongly clustered than formulae based on the standard Press-Schechter mass function predict., 8 pages, 6 figures, submitted to MNRAS corrected y-label for fig.4 (newlabel = 1 + oldlabel)

We present the results of a set of numerical simulations evaluating the effect of cluster galaxies on arc statistics. We perform a first set of gravitational lensing simulations using three independent projections for each of nine different galaxy clusters obtained from N-body simulations. The simulated clusters consist of dark matter only. We add a population of galaxies to each cluster, mimicking the observed luminosity function and the spatial galaxy distribution, and repeat the lensing simulations including the effects of cluster galaxies, which themselves act as individual lenses. Each galaxy is represented by a spherical Navarro, Frenk & White (1997) density profile. We consider the statistical distributions of the properties of the gravitational arcs produced by our clusters with and without galaxies. We find that the cluster galaxies do not introduce perturbations strong enough to significantly change the number of arcs and the distributions of lengths, widths, curvature radii and length-to-width ratios of long arcs. We find some changes to the distribution of short-arc properties in presence of cluster galaxies. The differences appear in the distribution of curvature radii for arc lengths smaller than 12'', while the distributions of lengths, widths and length-to-width ratios are significantly changed only for arcs shorter than 4''., 11 pages, Latex using MN style, 6 figures enclosed. Version accepted for publication in MNRAS. Gzipped file including a full resolution version of Figures 2a, 2b, 3a, 3b, 3c can be downloaded by anonymous ftp at http://gt.pd.astro.it/~bepi/PAPERS/lensing.ps.gz