11

THE CUSP/CORE PROBLEM THE CUSP/CORE PROBLEM AND THE SECONDARY INFALL AND THE SECONDARY INFALL MODELMODEL

Antonino Del PopoloAntonino Del Popolo Dep. of Physics and Astronomy, Catania Dep. of Physics and Astronomy, Catania

UniversityUniversityFebruary 1th, Vitoria

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22

OutlineOutline• The small scale problems in The small scale problems in ΛΛCDM modelCDM model

• Proposed solutionsProposed solutions

• Secondary Infall Model (SIM) and the “cusp” problemSecondary Infall Model (SIM) and the “cusp” problem

• Concluding RemarksConcluding Remarks

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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 themost massive subhaloes of the Milky Way are too dense to host any of its brightsatellites

4. Cusp/Core Problem: Dark matter cusps in simulations of galaxy centers, absent in dwarf Irr, dominated by dark matter

Missing Satellite Problem• 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

Mateo 1998 Binggeli et al. 1985

Now and 4 billion years agoBasically 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

• Strigari et al. 2007

Solution: only some satellites are visiblea) 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:

Problem less severe after the discovery of Ultra-Faint-Dwarfs

• 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

Cumulative n. Via Lactea subhalos within r200

MW satellites within 420 kpc(Mateo 1998; Simon & Geha 2007; Munoz et al. 2006; Martin 2007

---- abundance of luminous satellites after correcting for the sky coverage of the SDSS

-----“fossil of reionization”.-.-.- 65 largest Vmax,p subhalos before accretion

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

____ 51 Most Massive Via LActea subhalos at z=0------- 51 subhalos with largest masses at the time they were accreted by the main halo.

The angular momentum catastrophe • The Standard Model of Disk Formation

• Detailed Conservation of Angular Momentum (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)

Problems with the Standard ModelThe 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)

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 lostduring repeated collisions throughdynamical friction or other mechanisms(van den Bosch et al. 2002; Navarro &Steinmetz 2000).

Solution (s): stellar feedback processes (Weilet al. 1998), but part of the angular momentum problem

seemsdue to numerical effects, most likelyrelated to the shock capturing, artificialViscosity used in smoothed particlehydrodynamics (SPH) simulations(Sommer-Larsen &

Dolgov 2001).

Navarro and Steinmetz 2000

Angular momentum catastrophe

Excess of low and high angular momentum material compared to an exponential disk.

log-normal distribution

Μ shape factor

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 willlose 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.

“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 σ denser than every bright MW dwarf spheroidal curves are plotted with solid curves, while the remaining subhalos are plotted as dotted curves.

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

1313

cusp

core

Flores & Primack 1994Flores & Primack 1994Moore 1994Moore 1994Swaters et al. (2003)Swaters et al. (2003)Diemand et al. (2004)Diemand et al. (2004)Simon et al. (2005) Simon et al. (2005) De Blok et al. (2008)De Blok et al. (2008)de Naray and Kaufmann (2011)de Naray and Kaufmann (2011)

CUSP/CORE PROBLEM

• Cusp/Core Problem: Dark matter cusps absent in galaxy centers, LSBs and dwarf Irr (CDM dominated)

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Density profiles of haloes: a short Density profiles of haloes: a short storystory

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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

>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).

Power law profiles:

Non power law profiles:

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

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Navarro, Frenk & White Navarro, Frenk & White (1995, 1996, 1997)(1995, 1996, 1997)

log(

dens

itylo

g(de

nsity

))

log(radius)log(radius)

)]1/()1[ln(3)(

)()()(

)/1)(/()(

30

2

cccc

M

MrMrMc

rrrrr

virc

s

virvir

ss

critc

• Asymptotic outer slope -3; Asymptotic outer slope -3;

inner -1inner -1 • Universal ProfileUniversal Profile

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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)

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Newer Models Newer Models • Navarro et al (200Navarro et al (20044, 2010), 2010)

proposed a new analytic form: proposed a new analytic form:

Where rWhere r-2-2 is defined as the radius is defined as the radius at which: at which:

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•The profile slope is now varying with radius to 0.5% of r200

•Profile becomes shallower, No asymptote

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Observational controversy: LSB rotation curves

• Flores & Primack 1994; Moore 1994: Flat rotation curves of the low : Flat rotation curves of the low surface brightness (LSB) galaxies -> halos are not going to be singularsurface brightness (LSB) galaxies -> halos are not going to be singular

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Shape of the density profile is shallower than that found in numerical simulations (e.g., 0.2 ± 0.2 (de Blok, Bosma, & McGaugh 2003))

Observed profile

Predictions from CDM simulations(two different normalizations)

Moore et al. 1999Other 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

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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= s/(1+r/rs)(1+(r/rs)2)

NFW = s/(r/rs)(1+r/rs)2

Moore = s/(r/rs)1.5(1+(r/rs)1.5)HI-scaling const. factor,MOND without DM

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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)

1919

Oh et al. (2010)

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However always from THINGS:(high mass spirals) galaxies having MB < −19 -> NFW profile or an PI profile;

(low mass spirals) MB > −19 -> PI model

2020

•No evidence for a universal density profile– large scatter compared

to simulations– mean slope shallower

than simulations

Is There a Universal Density Profile?

NGC 2976

NGC 6689

NGC 5949

NGC 4605

NGC 5963

Five galaxies:

• Also different from previous observations, though e.g., = 0.2 ± 0.2 (de Blok, Bosma, & McGaugh 2003)

Simon et al. 2005

2121

WHAT ABOUT CLUSTERS?WHAT ABOUT CLUSTERS?

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)

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ACS=dvanced Camera for Surveys

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

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Gravitational lensing yield conflicting estimates sometime in agreement with Numerical 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)

Newman et al. (2012)

Newman et al. 2012

2525

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)

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New Physics ?New Physics ?

Possible Solutions in Possible Solutions in ΛΛCDM (“heating” of dark matter)

Rotating barPassive evolution of cold lumps (El Zant et al., 2001)AGN, Spernovas (Governato et al.. 2010)“Maximal stellar feedback”/“blowout”

WDM (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) 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)Repulsive Dark matter (RDM)Fuzzy DM (FDM) (ultra-light scalar particles, similar to axion

dark matter models ) Decaying DM (DDM)Decaying DM (DDM) Self-Annihilating DM (SADM) Self-Annihilating DM (SADM) Modified gravity (F(R); F(T); …MOND)Modified gravity (F(R); F(T); …MOND)

A 1 pc core requires a 0.1 keV thermal candidate. Candidates satisfying large scale structure constrains (mν larger than 1-2 keV) the expected size of the core is of the order of 10 (20) pc

Alternatives to (C)CDMWDM: (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 lensingRDM: (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, aresignificantly reduced without altering large scale structure

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:

1. With better simulations, the halos become progressively shallow from the virial radius inwards and show no sign of turning into a power-law.

2. Simulated haloes Vc are consistent with 70% of LSB rotation curves.

3. Observational disagreement is with the fitting formulae, rather than with simulated haloes.

4. 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*

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 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)

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2929

• low mass/force resolutions ⇒ shallower potential than real ⇒ artificial disruption/overmerging (especially serious for small systems)

Moore (2001)

central central 500kpc 500kpc region of a region of a simulated simulated halo in halo in SCDM SCDM

= 1kpc = 7.5kpc

3030

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

3/1 Nr

3131

ALTERNATIVE APPROACH TO N-BODY SIMULATIONSALTERNATIVE APPROACH TO N-BODY SIMULATIONS• Controversy regarding central slope and universality of the density Controversy regarding central slope and universality of the density

profile has stimulated several analytical works : 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.

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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-bodySimulations. NFW2: c is obtained from NFW fitting formula

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• 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)

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(Case A: ; Case B: )

129.8 10 M 124.3 10 M

Hiotelis 2002• Improvement of Gunn 1977; ZH93• Angular momentum introduced at turn-around as: where L is obtained by the spin parameter

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)

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• 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.

3838

How does SCM-SIM work?How does SCM-SIM work?

• 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 lin ~1.06

• Then it collapses to a singularity. (r=0): lin ~ 1.69

• Collapse to a point will never occurr in practice; dissipative physics and the process of violent relaxation will 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

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In the evolution of a density perturbation in the NL phase one may use semianalytical models (spherical or ellipsoidal collapse)

3939

• A bound mass shell having initial comoving radius will expand to a maximum radius (apapsis) of a shell:

where

The Model: SIM + L+DF+DC (Del Popolo 2009, 2011, 2012)*

Eq. 1

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

ir mr

11( )( 1)

im i i

i ir g r r

23

0

3 ( )ir

ii

y y dyr

2

( ) ( ) i ita m i

m m

r drr rr dr

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)

4040

Total mass in r is

and and

*

Eq. 4

Eq. 5

The radial velocity is obtained by integrating the equation of motion of the shell:

(specific coefficient of dynamical friction)h (specific angular momentum)

2

*3

( , ) ( )3

rdv h r drG r rdt r dt

*

Eq. 7

Eq. 6

DF

L fromTTT

Random angular momentumjor Avila-Resse et al (1998)

Kandrup 1980

Adiabatic contraction of DM: Gnedin et al. 2004; Del Popolo 2009Baryons cool and fall into their final mass distribution Mb(r ), initial distribution of L. DM particle at ri moves to r<ri. Adiabatic invariant->

4141

• The collapse factor, f, of a shell with initial radius ri and apapsis rm is given by (Gunn 1977; FG84; ZH93):

*

1

3

( ) ln ( )( ) 1( ) ln ( )ta m i

i i

x d f xxf x d g x

( )( )

( ) ( )p m

i mP m M m

m rr f r r

m r m r

Mi(ri) = initial mass distribution; Mdm= final distribution of dissipationless halo particlesIf there is no crossing ->

Eqs 8 and 9 can be solved to calculate the final radial distribution of halo particles:

Eq. 9

Eq. 10

iThe problem of determining the density profile is then solved fixing the initial conditions ( ), the angular momentum h(r), and the coefficient of dynamical friction.

4242

SOME BASIC RESULTS

*

4343

GALAXY DENSITY PROFILES

810 M1010 M

1110 M

1210 M

4444

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 = 0,12 Mpc. The radius 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} Msun halo and the solid one the angular momentum distribution for the density profile reproducing the NFW halo, as descried in Section 4.

L from tidal torques Random L

4545

*910 M Evolution with z

4646

*

1410 M Evolution with z

4747

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

*

4848

Figure 1. DM halos shape changes 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.

* ---810 M----910 M

…..___ 1010 M L

2 L

L/2 L->0

Effect of Environment

4949

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.

*L, fd/3 L/2, fd/3

L, 3fd L/2, 3fd

Effect of Environment

5050

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)

5151

*

NGC 5963_ _ _ SIM_____ NFW

………. Power law

- - - - PI

NGC5949_ _ _ SIM_____ NFW

………. Power law

- - - - PI

5252

* PROFILES OF CLUSTERS

5353

* A611A383

MACS J1423 RXJ 1133

5454

COMPARISON WITH COMPARISON WITH SPHSIMULATIONSSPHSIMULATIONSGovernato et al. (2010)ModelModel : : LCDM

• Gas outflows from supernovas remove the low angular momentum gas, stopping bulge formation and decreasing the central DM

• In the inner 1 kpc the density is less than ½ of that in absence of outflows

• Result: Result: LSBs with DM core

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– ΛΛCDM model problems at small scales (satellites, cusp/core, L, TBTF)CDM model problems at small scales (satellites, cusp/core, L, TBTF)– Solvable introducing baryon physics Solvable introducing baryon physics

– Dissipationless numerical simulations -> cuspsDissipationless numerical simulations -> cusps

– Galactic rotation curves -> usually cores Galactic rotation curves -> usually cores

– Clusters of galaxies-> mainly cuspy profilesClusters of galaxies-> mainly cuspy profiles

– Cusp/Core problem GENUINE (the disagreement between observations and Cusp/Core problem GENUINE (the disagreement between observations and NN--body simulations is not due to numerical artifacts or problems with simulations) body simulations is not due to numerical artifacts or problems with simulations)

– BUT APPARENT (disagreement related to the fact that the simulations are not BUT APPARENT (disagreement related to the fact that the simulations are not taking account of baryons physics)taking account of baryons physics)

– Comparing pears and apple: dissipationless (i.e., DM) and the other Comparing pears and apple: dissipationless (i.e., DM) and the other dissipational (i.e., inner part of structures)dissipational (i.e., inner part of structures)

– Perspectives: in the future Perspectives: in the future it necessary to run SPH simulations that repeat the it necessary to run SPH simulations that repeat the mass modeling including a self-consistent treatment of the baryons and DM mass modeling including a self-consistent treatment of the baryons and DM component in a larger extent than it was done.component in a larger extent than it was done.

Summary & Conclusions*