*
THE CUSP/CORE PROBLEM
AND THE SECONDARY INFALL MODEL
Antonino Del Popolo
Dep. of Physics and Astronomy, Catania University
February 1th, Vitoria
1
*
Outline
• The small scale problems in ΛCDM model
• Proposed solutions
• Secondary Infall Model (SIM) and the “cusp” problem
• Concluding Remarks
2
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
•
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
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
Mateo 1998
Binggeli et al. 1985
• Problem less severe after the discovery of
Ultra-Faint-Dwarfs
Solution: only some satellites are visible
a) 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)
Strigari
et al. 2007
• Madau et al. 2008:
-----“fossil of reionization”
.-.-.- 65 largest Vmax,p
subhalos before accretion
• 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
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
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
____ 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
•
Supernova feedback (van der Bosch et al 2000, 2002)
1997, van der Bosch 1998)
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
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)
simulations
In hydrodynamical simulations baryons
have ~10% of the angular momentum
of observed disks.
Angular momentum catastrophe
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(SommerLarsen & Dolgov 2001).
Navarro and Steinmetz 2000
Μ shape factor
log-normal distribution
Excess of low and high angular
momentum material compared
to an exponential disk.
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.
“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
*
•
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
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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
log(density)
*
Navarro, Frenk & White
(1995, 1996, 1997)
• Asymptotic outer slope -3;
inner -1
 c  crit
 (r ) 
( r / rs )(1  r / rs ) 2
r (M )
• Universal Profile
cvir ( M )  vir
rs ( M )
vir 0 c 3
 c( M ) 
3[ln(1  c )  c /(1  c )]
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:
ln(ra / r-2 ) = (-2/a )[(r /r-2 )a -1]
Where r-2 is defined as the radius at
which:
b = -dlog r /dlogr = 2
•The profile slope is now varying with
radius to 0.5% of r200
•Profile becomes shallower,
No asymptote
16
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Observational controversy: 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
= 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
Mass models for the galaxy Eso 116-G12. Solid line: best fit, longdashed 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
Is There a Universal Density
Profile?
Five galaxies:
a
NGC 2976
0.01
NGC 6689
0.80
NGC 5949
0.88
NGC 4605
0.88
NGC 5963
1.28
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
*
Donnarumma et al. (2011) A611: Chandra X-ray data+ lensing. Central mass
reconstructed through parametric analysis of the SL system with the software Lenstool
Newman et al. (2012)
Newman et al. 2012
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)
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. QBALLS)
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) A 1 pc core requires a 0.1 keV thermal candidate.
AGN, Spernovas (Governato et al.. 2010)
“Maximal stellar feedback”/“blowout”
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
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
*
•
•
•
•
1.
2.
3.
4.
Theory Vs. observations: what could have gone
Hayashi et al. 2004
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, offcentring, 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 powerlaw.
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 sphericallyaveraged circular velocity of DM
haloes, differences should be
expected.
*
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)
• low mass/force resolutions
⇒ shallower potential than real
⇒ artificial disruption/overmerging
(especially serious for small
systems)
e = 1kpc
e = 7.5kpc
central
500kpc
region of a
simulated
halo in
SCDM
Moore (2001)
29
Convergence tests in CDM clusters

Numerical flattening due to two body
relaxation:
r  N 1/ 3 slow convergence,
1 million particles to resolve 1% of R virial,
1000 to resolve 10% !
(Moore et al. 1998; Diemand et al. 2004)
14 million
6 million
1.7 million
0.2 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
*
• 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)
Hiotelis 2002
• Improvement of Gunn 1977; ZH93
• Angular momentum introduced
at turn-around as:
where
L is obtained by the spin parameter
9.8 1012 M ;
(Case A:
12
Case B: 4.3 10 M )
* 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)
• 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.
*-
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 subuniverse (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
38
*
The Model: SIM + L+DF+DC
•
(Del Popolo 2009, 2011, 2012)
r
r
A bound mass shell having initial comoving radius
will expand to a maximum radius
i
m
(apapsis) of a shell:
rm  g ( ri )  ri
where
i 
1 i
 i ( i 1 1)
Eq. 1
ri
3
2

(
y
)
y
dy
3 
ri 0
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):
2
 ri  dri
 ta ( rm )   ( ri )  
 rm  drm
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)
39
*
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:
dvr h(r , )2
dr L


G
(
r
)


 r
*
3
dt
r
dt 3
Kandrup 1980
*
Eq. 7
h (specific angular momentum)
(specific coefficient of dynamical friction)
DF
L from
TTT
Random angular momentum
or ja
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->
40
*
Mi(ri) = initial mass distribution; Mdm= final distribution of dissipationless halo particles
If there is no crossing ->
Eq. 9
Eqs 8 and 9 can be solved to calculate the final radial distribution of halo particles:
ta ( xm )  d ln f ( xi ) 
 ( x) 
1

3 
f ( xi )  d ln g ( xi ) 
•
The collapse factor, f, of a shell with initial radius
1977; FG84; ZH93):
r  f ( ri )rm 
1
Eq. 10
ri and apapsis rm is given by (Gunn
m p ( rm )
mP ( rm )  mM ( rm )
The problem of determining the density profile is then solved
fixing the initial conditions (  i ), the angular momentum h(r),
and the coefficient of dynamical friction.
41
*
SOME BASIC RESULTS
42
GALAXY DENSITY PROFILES
1010 M
108 M
1011 M
1012 M
43
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 = 0,12 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 profile44
reproducing the NFW halo, as descried in Section 4.
*
109 M
Evolution with z
45
*
1014 M
Evolution with z
46
*
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
*
---…..
___
108 M
Effect of Environment
---
9
10 M
1010 M
L
L/2
2L
L->0
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. 48
*
Effect of Environment
L, fd/3
L, 3fd
L/2, fd/3
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
Rotation curve of NGC 2976 (Simon et al. 2003). Solid line is a powerlaw fit corresponding to a density profile ∝ r0.01, obtained by S03.
Dashed line plots the result of my model (Del Popolo 2011)
50
*
NGC5949
_ _ _ SIM
_____ NFW
………. Power law
NGC 5963
_ _ _ SIM
_____ NFW
- - - - PI
………. Power law
- - - - PI
51
*
PROFILES OF
CLUSTERS
52
*
A611
MACS J1423
A383
RXJ 1133
53
*
COMPARISON WITH SPHSIMULATIONS
Governato et al. (2010)
Model : 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: LSBs with DM core
54
55
*
Summary & Conclusions
– ΛCDM model problems at small scales (satellites, cusp/core, L, TBTF)
– Solvable introducing baryon physics
– Dissipationless numerical simulations -> cusps
– Galactic rotation curves -> usually cores
– Clusters of galaxies-> mainly cuspy profiles
– Cusp/Core problem GENUINE (the disagreement between observations and N-body
simulations is not due to numerical artifacts or problems with simulations)
– BUT APPARENT (disagreement related to the fact that the simulations are not taking
account of baryons physics)
– Comparing pears and apple: dissipationless (i.e., DM) and the other dissipational (i.e.,
inner part of structures)
– Perspectives: in the future it necessary to run SPH simulations that repeat the mass
modeling including a self-consistent treatment of the baryons and DM component in a
larger extent than it was done.
56