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THE CUSP/CORE PROBLEM AND THE SECONDARY INFALL MODEL

THE CUSP/CORE PROBLEM AND THE SECONDARY INFALL MODEL. Antonino Del Popolo Dep. of Physics and Astronomy, Catania University February 1th, Vitoria. *. Outline. The small scale problems in Λ CDM model Proposed solutions Secondary Infall Model (SIM) and the “ cusp ” problem

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THE CUSP/CORE PROBLEM AND THE SECONDARY INFALL MODEL

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  1. THE CUSP/CORE PROBLEM AND THE SECONDARY INFALL MODEL Antonino Del Popolo Dep. of Physics and Astronomy, Catania University February 1th, Vitoria *

  2. Outline The small scale problems in ΛCDM model Proposed solutions Secondary Infall Model (SIM) and the “cusp” problem Concluding Remarks *

  3. Small Scale Challenges for CDM • Despite successes of ΛCDM on large and intermediate scales, serious issues remain on smaller, galactic and sub-galactic, scales. In particular: • 1. Missing satellite problem:High predicted • number of small haloes • 2. Angular momentum catastrophe: • Low angular momentum of baryons, and • consequent small radius of disks • 3. Too Big To Fail: Dissipationless ΛCDM simulations predict that the majority of the • most massive subhaloes of the Milky Way are too dense to host any of its bright • satellites 4. Cusp/Core Problem: Dark matter cusps in simulations of galaxy centers, absent in dwarf Irr, dominated by dark matter

  4. Missing Satellite Problem • Basically all cosmological simulations predict that there are at least • one order of magnitude more small subhalos (dwarf galaxies) • around Milky Way like galaxies than what is observed (e.g. Via • Lactea simulation Diemand et al. 2007, ApJ 657)->234 million • particles in (90 Mpc/h)³ multimass simulation, mp=20900 Msun Now and 4 billion years ago • Moore et al 1999: Substructure within galactic and cluster mass halos. In MW 500 satellites with circular velocities larger than Draco and Ursa-Minor i.e. bound masses >108M⊙ and tidally limited sizes > kpc Binggeli et al. 1985 Mateo 1998 Problem less severe after the discovery of Ultra-Faint-Dwarfs • Solution: only some satellites are visible • The ones having larger masses before accretion by • MW (LBA) and which could resist tides (Diemand et al. • 2007) • b) The ones which acquired gas before re-ionization and • formed stars: Earliest Forming (EF) (Bullock et al. 2000; • Moore et al. 2006) • Reionization suppression (acquired gas • before recombination and formed stars): Earliest Forming • (EF) (Bullock et al. 2000; Moore et al. 2006) • Models: Strigari et al. 2007

  5. -----“fossil of reionization” .-.-.- 65 largest Vmax,p subhalos before accretion Cumulative n. Via Lactea subhalos within r200 ---- abundance of luminous satellites after correcting for the sky coverage of the SDSS MW satellites within 420 kpc (Mateo 1998; Simon & Geha 2007; Munoz et al. 2006; Martin 2007 • Madau et al. 2008: • Simon & Geha (2007) Ultra-faint dwarfs with M/L~1000 discovered (from SDSS data) help to solve this discrepancy, but not fully (factor of 4 difference). If reionization occurred around redshift 9 − 14 , and dwarf galaxy formation was strongly suppressed thereafter, they have not been able to attract enough  baryonic matter to create a visible dwarf Galaxy the circular velocity function of Milky Way satellite galaxies approximately matches that of CDM subhalos in Via Lactea simulation. (Keck observations of Simon and Geha 2007, astro-ph. 0706.0516 of 8 newly discovered ultra-faint Milky Way dwarf satellites showed that six were around 99.9% dark matter ____ 51 Most Massive Via LActea subhalos at z=0 ------- 51 subhalos with largest masses at the time they were accreted by the main halo. Bullock et al. 2000; Kravtsov et al. 2004; Moore et al. 2006, had also discussed reionization suppression. Other solutions: supernova feedback (e.g. Dekel & Silk 1986; Mac Low & Ferrara 1999; Mori et al. 2002), and gas stripping by ram pressure (e.g. Mayer et al. 2006) as favourite suppression mechanisms

  6. The angular momentum catastrophe • The Standard Model of Disk Formation • DetailedConservationofAngularMomentum (Mestel 1963) • Baryons initially trace dark matter (Fall and Estafiou 1980) • Adiabatic Contraction (Barnes and White 1984, Bluementhal et al 1986) • Realistic Halo Profile (Dalconton et al 1996, Mo et al 1997) • Bulge formation from disk instabilities (Dalconton et al 1996, Mo et al 1997, van der Bosch 1998) • Supernova feedback (van der Bosch et al 2000, 2002)

  7. Problems with the Standard Model The angular momentum catastrophe Hydrodynamical simulations show that the angular momentum of the baryons is not conserved during collapse (Navarro and Benz 1991, Steinmetz and Navarro 1998, 2000; Sommer-Larson et al 2000) The j-profile mismatch The distribution of specific angular momentum in N-body simulations does not agree with observations (Bullock et al 1999, van der Bosch et al 2000) Other problems: The spread in disk sizes seems to be narrower then the spread in λ values. (Lacey and de Jong 2000) Major mergers should lead to spheroids, but they also have the highest λ values. (Gardner 2000, Wechsler 2000)

  8. Angular momentum catastrophe In hydrodynamical simulations baryons have ~10% of the angular momentum of observed disks. This has been associated with the problem of “over-cooling” also seen in hydrodynamical simulations Angular momentum possibly lost during repeated collisions through dynamical friction or other mechanisms (van den Bosch et al. 2002; Navarro & Steinmetz 2000). Solution (s): stellar feedback processes (Weil et al. 1998), but part of the angular momentum problem seems due to numerical effects, most likely related to the shock capturing, artificial Viscosity used in smoothed particle hydrodynamics (SPH) simulations(Sommer-Larsen & Dolgov 2001). Navarro and Steinmetz 2000 Μ shape factor Excess of low and high angular momentum material compared to an exponential disk. log-normal distribution

  9. Maller & Dekel 2002 model Over-cooling in merging satellites -> angular momentum Catastrophe The gas contraction within the incoming satellite makes the gas immune against tidal stripping: it spirals all the way into the halo center while losing all of its orbital angular momentum to the dark halo due to dynamical friction. The dark matter, which dominates the outer regions of the satellite, is gradually stripped in the outer parts of the halo, thus retaining part of its orbital angular momentum. If the baryons cool rapidly and sink to the centers of dark halos, then they will lose their angular momentum. The obvious solution to this problem is some form of heating that will prevent the baryons from contracting to the center of the dark halos. By incorporating a simple recipe of supernova feedback, it is possiible to solve the problems of angular momentum in disk formation.

  10. “Too Big To Fail” The "too big to fail" (TBTF) problem arose from analyses of the Aquarius and Via Lactea (high resolution of MW-like halos). Each simulated halo had ~10 sub-halos that were so massive and dense that they would appear to be too big to fail to form lots of stars. The TBTF problem is that none of the observed satellites of the Milky Way or Andromeda have stars moving as fast as would be expected in these densest sub-halos. Subhalos that are at least 2 σdenser than every bright MW dwarf spheroidal curves are plotted with solid curves, while the remaining subhalos are plotted as dotted curves.

  11. A Baryonic Solution to “Too Big To Fail” • Inclusion of baryonic physics can create shallower slopes of the dark matter densities in the centers of low-mass galaxies reducing or solving the discrepancy between cuspy profile predicted in N-body simulations and flat ones seen in observation. • Zolotov et al. (2012) have suggested a correction to be applied to the central masses of dark matter-only satellites in order to mimic the effect of (1) the flattening of the dark matter cusp due to supernova feedback in luminous satellites, and (2) enhanced tidal stripping due to the presence of a baryonic disk: ∆(v1kpc) = 0.2vinfall − 0.26 km/s for 20 km/s< vinfall < 50 km/s

  12. * CUSP/CORE PROBLEM • Cusp/Core Problem: Dark matter cusps absent in galaxy centers, • LSBs and dwarf Irr (CDM dominated) Flores & Primack 1994 Moore 1994 Swaters et al. (2003) Diemand et al. (2004) Simon et al. (2005) De Blok et al. (2008) de Naray and Kaufmann (2011) cusp core 13

  13. Density profiles of haloes: a short story * Power law profiles: 1970:Peebles - N-body simulation (N=300). 1972:Gunn & Gott - dissipationless collapse of a spherical homogeneous perturbation in a expanding Friedmann universe. 1977:Gott - secondary infall model ρ ∝ r -9/4. Spherical inhomogeneous perturbation + shell crossing 1984, 1985:Fillmore & Goldreich; Bertschinger - self-similar solutions collapse scale-free spherical perturbation 1985:Hoffman & Shaham - Predicted that density profile around density peaks is ρ ∝ r –3(n+3)/(n+4) 1986:Quinn, Salmon & Zurek - N-body simulations (N～10000), confirmed ρ ∝ r –3(n+3)/(n+4) 1988:Frenk, White, Davis & Efstathiou - N-body simulations (N=323) - CDM flat rotation curve out to 100kpc 1990:Hernquist - Analytic model with a central cusp for elliptical galaxies - ρ ∝ r –1(r + rs) –3 Non power law profiles: >1990:Dubinski & Carlberg (1991), Lemson (1995), Cole & Lacey (1996), Navarro et al. (1995; 1996, 1997)(NFW)... + Moore et al. (1998), Jing & Suto (2000), Klypin et al. (2001), Bullock et al. (2001), Power et al. (2003); Navarro et al. (2004, 2010), Stadel et al (2008).

  14. Asymptotic outer slope -3; inner -1 Universal Profile * Navarro, Frenk & White (1995, 1996, 1997) log(density) HIGHER RESOLUTION: Slope -1.5 (Moore et al. 1998, Fukushige & Makino 2001) NON-UNIVERSALITY: slope -1.5, -1.3, -1.1 (Jing & Suto 2000, Ricotti 2003, Ricotti & Wilkinson 2004; Ricotti et al. 2007) log(radius)

  15. Newer Models Navarro et al (2004, 2010) proposed a new analytic form: Where r-2 is defined as the radius at which: *- • The profile slope is now varying with • radius to 0.5% of r200 • Profile becomes shallower, • No asymptote

  16. Observationalcontroversy: LSB rotation curves Flores & Primack 1994; Moore 1994: Flat rotation curves of the low surface brightness (LSB) galaxies -> halos are not going to be singular * Predictions from CDM simulations (two different normalizations) Observed profile Moore et al. 1999 Other studies: Burkert 1995; de Blok & Bosma 2003, Gentile et al. 2008, Spano et al. 2008, de Blok et al. 2009, Oh et al. 2010; de Naray & Kaufmann 2011 Shape of the density profile is shallower than that found in numerical simulations (e.g., 0.2 ± 0.2 (de Blok, Bosma, & McGaugh 2003))

  17. * • Gentile et al. 2004, 2006 • Rotational curves of spiral galaxies : stellar, + gaseous + dark matter • Fitting the density with various models • Constant density core models preferred. • Burkert: with a DM core • = rs/(1+r/rs)(1+(r/rs)2) • NFW • r = rs/(r/rs)(1+r/rs)2 • Moore • r = rs/(r/rs)1.5(1+(r/rs)1.5) • HI-scaling const. factor, • MOND without DM Mass models for the galaxy Eso 116-G12. Solid line: best fit, long-dashed line: DM halo; dotted: stellar; dashed: gaseous disc. 1kpc = 13.4 arcsec. Below: residuals: (Vobs-Vmodel)

  18. * Oh et al. (2010) However always from THINGS: (high mass spirals) galaxies having MB < −19 -> NFW profile or an PI profile; (low mass spirals) MB > −19 -> PI model

  19. Five galaxies: a NGC 2976 0.01 NGC 6689 0.80 NGC 5949 0.88 NGC 4605 0.88 NGC 5963 1.28 Is There a Universal Density Profile? Simon et al. 2005 • No evidence for a universal density profile • large scatter compared to simulations • mean slope shallower than simulations • Also different from previous observations, though • e.g., a = 0.2 ± 0.2 (de Blok, Bosma, & McGaugh 2003)

  20. WHAT ABOUT CLUSTERS?

  21. * This is one of large number of clusters for which measurements like this have been made. Clusters like Abell 1689 and Abell 2218 are particularly good, because they had gravitational arcs near the center. So the results can be calibrated by strong gravitational lensing (the green points in the figure). The dark matter often has structure, sometimes with lumps that are quite massive but have no optical galaxies (for example Abell 1942; Erben et al. 2000, A&A, 355, 23) ACS=dvanced Camera for Surveys

  22. * Donnarumma et al. (2011) A611: Chandra X-ray data+ lensing. Central mass reconstructed through parametric analysis of the SL system with the software Lenstool

  23. Newman et al. (2012) Newman et al. 2012 Gravitational lensing yield conflicting estimates sometime in agreement withNumerical simulations (Dahle et al 2003; Gavazzi et al. 2003; Donnaruma et al. 2011) or finding much shallower slopes (-0.5) (Sand et al. 2002; Sand et al. 2004; Newman et al. 2009, 2011, 2012) X-ray analyses have led to wide ranging of value of the slope from: -0.6 (Ettori et al. 2002) to -1.2 (Lewis et al. 2003) till -1.9 (Arabadjis et al. 2002), or in agreement with the NFW profile (Schmidt & Allen 2007; 34 Chandra X-ray observatory Clusters)

  24. * PROPOSED SOLUTIONS • Observational problems: • Beam smearing; non-circular motion, off-centring, systematic effects in observations • High resolution observations can distinguish cored and cuspy haloes by deriving their asymptotic inner slopes from rotation curve data (de Naray & Kaufmann 2011) • Failure of the CDM model or problems with simulations?(del Blok et al 2001, 2003; Borriello & Salucci 2001) (resolution; relaxation; overmerging; BARYONS NOT TAKEN INTO ACCOUNT !) Convergence Tests (Dieman et al. 2004) New Physics ? • WDM (disperson velocity (now) 100 m/s; reduce the small scale power, fewer low mass halos, all halos have less steep inner profile ) • Self-interacting DM (SIDM) (QCD interaction but no EM. scattering strips the halos from small clumps of dark matter orbiting larger structures, making them vulnerable to tidal stripping and reducing their number. Q-BALLS) • Repulsive Dark matter (RDM) • Fuzzy DM (FDM) (ultra-light scalar particles, similar to axion dark matter models ) • Decaying DM (DDM) • Self-Annihilating DM (SADM) • Modified gravity (F(R); F(T); …MOND) • Possible Solutions in ΛCDM (“heating” of dark matter) • Rotating bar • Passive evolution of cold lumps (El Zant et al., 2001) • AGN, Spernovas (Governato et al.. 2010) • “Maximal stellar feedback”/“blowout” A 1 pc core requires a 0.1 keV thermal candidate. Candidates satisfying large scale structure constrains (mνlarger than 1-2 keV) the expected size of the core is of the order of 10 (20) pc

  25. Alternatives to (C)CDM WDM: (Warm) disperson velocity (now) 100 m/s; reduce the small scale power, fewer low mass halos, all halos have less steep inner profile SIDM: (Self-Interacting) QCD interaction but no EM; Interaction strength: comparable to neutron-neutron; significant self-scatering cross section; central density declines in the desidered fashion; scattering strips the halos from small clumps of dark matter orbiting larger structures, making them vulnerable to tidal stripping and reducing their number. Difficulty: make spherical clusters: against lensing RDM: (Repulsive): DM consist of a condensate of massive bosons with a short range repulsive potential. The inner parts of dark matter halos would behave like a superfluid and be less cuspy. FDM: (Fuzzy): ultra-light scalar particles whose Compton wavelength (effective size) is the size of galaxy core. DM cannot be concentrated on smaller scales, resulting is softer cores and reduce small-scale structure. SADM: (Self-Annihilating): DM in dense regions may collide and annihilate, liberating radiation. This reduces the density in the central regions of clusters for two reasons: direct removal of particles from the center and re-expansion of the remainder as the cluster adjusts to the reduced central gravity. DDM: (Decaying): if early dense halos decay into relativistic particles and lower mass remnants, then core densities, which form early, are significantly reduced without altering large scale structure

  26. Theory Vs. observations: what could have gone wrong? Beam smearing tend to systematically lower slopes. Error bars large enough so that the cores are favoured but cusps can usually not be ruled out (Hayashi et al. 2004). Also: Non-circular motions, off-centring, spatial snoothing. Power et al. , Hayashi et al. and Navarro et al. 2003 suggest ways to reconcile with the observations: With better simulations, the halos become progressively shallow from the virial radius inwards and show no sign of turning into a power-law. Simulated haloes Vc are consistent with 70% of LSB rotation curves. Observational disagreement is with the fitting formulae, rather than with simulated haloes. CDM haloes are non-spherical and 3D. Comparing rotation speeds of gaseous disks to spherically-averaged circular velocity of DM haloes, differences should be expected. * Hayashi et al. 2004

  27. NUMERICAL ISSUES:Overmerging, two body relaxation, softening length; multi-mass simulations In cosmological simulations of the dark matter each particle represents a coarse grained sampling of phase space which sets a mass and spatial resolution. Unfortunately these super-massive particles will undergo two body encounters that lead to energy transfer as the system tends towards equipartition. In the real Universe the dark matter particles are essentially collisionless and pass unperturbed past each other. The artificial smoothing of the density distribution in these regions (disruption of dark matter halos within dense environments) is referred to as `overmerging‘ (for a review of the problem see Moore 2000). A modification of the 1/r^2 law through a softening length diminishes two body scattering and relaxation and allows larger time steps. Simulation results are least reliable in the densest parts of the halo, where the dynamical timescale is shortest and artificial heating has the greatest effect. In early simulations of the formation of galaxy clusters, overmerging erased substructure completely (e.g. White 1976). When simulations reached sufficient resolution to resolve roughly as many subsystems as there are galaxies in a cluster, the problem was considered `solved' (e.g. Ghigna et al. 2000), although the scale invariance of halo properties quickly lead to an excess dwarf satellite problem in galaxy haloes. The assumption that the overmerging problem is now solved has not been fully tested (Taylor et al. 2003). The effects of particle discreteness in N-body simulations of L CDM are still an intensively debated issue (Romeo et al. 2008). The processes of relaxation is difficult to quantify, but in the large N limit the discreteness effects inherent to the N-body technique vanish, so one tries to use as large a number of particles as computationally possible. Increasing N helps, but slowly ( N -0.25) *

  28. low mass/force resolutions ⇒ shallower potential than real ⇒ artificial disruption/overmerging (especially serious for small systems) central 500kpc region of a simulated halo in SCDM e = 1kpc e = 7.5kpc Moore (2001)

  29. Convergence tests in CDM clusters Numerical flattening due to two body relaxation: slow convergence,1 million particles to resolve 1% of R virial,1000 to resolve 10% !(Moore et al. 1998; Diemand et al. 2004) 14 million6 million1.7 million0.2 million

  30. ALTERNATIVE APPROACH TO N-BODY SIMULATIONS * Controversy regarding central slope and universality of the density profile has stimulated several analytical works : Gunn & Gott’s SIM (Ryden & Gunn 1987; Avila-Reese 1998; DP2000; Lokas 2000; Nusser 2001; Hiotelis 2002; Le Delliou Henriksen 2003; Ascasibar et al. 2003; Williams et al. 2004; Del Popolo 2009). Del Popolo 2000, Lokas 2000 reproduced the NFW profile considering radial collapse. SIM is improved by calculating the initial overdensity from the perturbation spectrum and eliminating limits of previous SIM’s works. Other authors studied the effect of angular momentum, L, and non-radial motions in SIM showing a flattening of the inner profile with increasing L. El-Zant et al. (2001) proposed a semianalytial model: dynamical friction dissipate orbital energy of gas distributed in clumps depositing it in dark matter with the result of erasing the cusp.

  31. * • Lokas 2000; • scale free • spectrum • Improvement • of HS model. • Radial collapse • NFW1: values of of • c_vir calculated • from a model based • on merging • formalism provided • by NFW that describes • better their N-body • Simulations. • NFW2: c is obtained • from NFW fitting • formula

  32. * Ascasibar et al. 2003 (radial density profile from a 3 sigma fluctuation on 1 h^(-1) Mpc scale). Angular momentum introduced as the eccentricity parameter Changing the orbit eccentricity e (proportional to L) produces a flattening of the inner profile. Radial orbits gives rise to a steep profile similar to that proposed by Moore et al. (1999) Numerical experiments, in the same paper shows that central slopes in relaxed haloes could be less steep than the NFW fit in agreement with analytical models based on the velocity dispersion profile (Taylor & Navarro 2003;Hoeft et al. 2003)

  33. (Case A: ; Case B: ) • Hiotelis 2002 • Improvement of Gunn 1977; ZH93 • Angular momentum introduced • at turn-around as: where • L is obtained by the spin parameter

  34. * Williams et al. (2004) • Follows Ryden & Gunn (1987): only random angular momentum is taken into account • More massive galaxy halos • tend to be more centrally • concentrated, and have • flatter rotation curves • Specific angular momentum • (only random L) in Williams • haloes is less centrally • concentrated and • larger than in N-body • simulations (e.g. van den • Bosch et al. 2002). • In order to reproduce a NFW profile one has to reduce random velocities by a factor of two. As suggested by Williams haloes in N-body simulations lose angular momentum between 0.1 and 1 R_vir. It is well known that numerical haloes have too little angular momentum vs. real disk galaxies (L catastrophe)

  35. El-Zant et al. 2003. • Transfer of energy from baryons to DM by DF. • Initial and final DM rotation curve and profile. Dashed line is a fit using Burkert profile. • Bottom left: DM profile in absence of energy feedback.

  36. How does SCM-SIM work? *- In the evolution of a density perturbation in the NL phase one may use semianalytical models (spherical or ellipsoidal collapse) • A slightly overdense sphere, embedded in the Universe, is a useful non-linear model, as it behaves exactly as a closed sub-universe (if density is > criticical density) • The overdensity expands with Hubble flow till a maximum radius (turn-around). (r=rmax, dr/dt=0) occurs at dlin ~1.06 • Then it collapses to a singularity. (r=0): dlin ~ 1.69 • Collapse to a point will never occurr in practice; dissipative physics and the process of violent relaxationwill eventually intervene and convert the kinetic energy of collapse into random motions. This is named: virialization (occurs at 2tmax, and rvir = rmax/2) • Once a non-linear object has formed, it will ontinue to attract matter in its neighbourhood ant its mass will grow by accretion of new material (secondary infall). • Through dissipative processes, baryons lose energy and fall deeper in the potential well of DM. • If the cooling time of the baryon gas is smaller than the collapse time, fragmentation will take place and smaller units can collapse

  37. A bound mass shell having initial comoving radius will expand to a maximum radius (apapsis) of a shell: * The Model: SIM + L+DF+DC (Del Popolo 2009, 2011, 2012) Eq. 1 where Eq. 2 If mass is conserved and each shell is kept at its turn-around radius,then the shape of the density profile is given by (Peebles 1980; HS; White & Zaritsky 1992): Eq. 3 After turn-around, a shell collapse, reexpands, recollapse (oscillation). This shell will cross other shells collapsing and oscillating like itself. Energy is not conserved and it is not an integral of motion anymore dynamics studied assuming that the potential near the center varies adiabatically (Gunn 1977; Filmore & Goldreich 1984; Zaritski & Hoffman 1993)

  38. Total mass in r is * Eq. 4 Eq. 5 and Eq. 6 The radial velocity is obtained by integrating the equation of motion of the shell: Eq. 7 Kandrup 1980 h (specific angular momentum) (specific coefficient of dynamical friction) L from TTT DF Random angular momentum or Avila-Resse et al (1998) Adiabatic contraction of DM: Gnedin et al. 2004; Del Popolo 2009 Baryons cool and fall into their final mass distribution Mb(r ), initial distribution of L. DM particle at ri moves to r<ri. Adiabatic invariant->

  39. The collapse factor, f, of a shell with initial radius ri and apapsis rm is given by (Gunn 1977; FG84; ZH93): * Mi(ri) = initial mass distribution; Mdm= final distribution of dissipationless halo particles If there is no crossing -> Eqs 8 and 9 can be solved to calculate the final radial distribution of halo particles: Eq. 9 Eq. 10 The problem of determining the density profile is then solved fixing the initial conditions ( ), the angular momentum h(r), and the coefficient of dynamical friction.

  40. * SOME BASIC RESULTS

  41. GALAXY DENSITY PROFILES * 43

  42. L from tidal torques Random L The tidal,h, and random specific angular momentum, j for three values of the parameter nu(nu= 2 solid line, nu= 3 dotted line, nu= 4 dot-dashed line) and for Rf = 0,12Mpc. Theradius r is connected to the mass M as described in the text. Distribution of the total specific angular momentum, JTot. The dotted-dashed and dashed line represents the quoted distribution for the halo n. 170 and n. 081, respectively, of van den Bosch et al. (2002). The dashed histogram is the distribution obtained from our model for the $10^{12} Msunhalo and the solid one the angular momentum distribution for the density profile reproducing the NFW halo, as descried in Section 4. 44

  43. * Evolution with z 45

  44. * Evolution with z 46

  45. * Rotation curves obtained with SIM (solid lines) and rotation curves of 4 LSBs Gentile et al. (2004). The dotted lines represent the fit with the NFW model 47

  46. Effect of Environment ---- * --- ….. ___ 2 L L L/2 L->0 Figure 1.DM halos shapechanges with angular momentum. DM haloes generated with the model of Section 2. In panels (a)(c), the dashed line, the dotted line and the solid line represent the density profile for a halo of 10^8M_sun, 10^9M_sun, and 10^10M_sun, respectively. In case (a), that is our reference case, the specific angularmomentum was obtained using the tidal torque theory as described in Del Popolo (2009). The specific angular momentum, h, for the halo of mass 10^9M_sun is 400 kpc km/s (spin= 0.04) and the baryon fraction fd = 0.04. In panel (b) we increased the value of specific ordered angular momentum, h, to 2h leaving unmodified the baryonic fraction to fd and in panel (c) the specific ordered angular momentum is h/2 and the baryonic fraction equal to the previous cases, namely fd . Panel (d) shows the density profile of a halo of 10^10M_sun with zero ordered angular momentum and no baryons (solid line), while the dashed line is the Einasto profile. 48

  47. Effect of Environment * L, fd/3 L/2, fd/3 L, 3fd L/2, 3fd Figure 2 DM halos shape changes with baryon fraction. Same as previous figure, Fig. 1, but in panel (a) we reduced the value of baryonic fraction of Fig. 1a (h, fd) to fd/3, and in panel (b) we reduced the value of baryonic fraction of Fig. 1c to fd/3; in panel (c) we increased the value of baryonic fraction of Fig. 1a (h/2, fd) to 3fd, and in panel (d) we increased the value of baryonic fraction of Fig. 1c to 3fd. 49

  48. Rotation curve of NGC 2976 (Simon et al. 2003). Solid line is a power-law fit corresponding to a density profile ∝ r0.01, obtained by S03. Dashed line plots the result of my model (Del Popolo 2011) 50

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