CUSP or CORE CUSP or CORE Antonino Del Popolo Antonino Del Popolo - - PowerPoint PPT Presentation
* CUSP or CORE CUSP or CORE Antonino Del Popolo Antonino Del Popolo Vulcano Workshop 2010 Workshop 2010 Vulcano May 23-29, Vulcano, Italy * Outline Outline The cusp/core problem in CDM haloes The cusp/core problem in CDM haloes
Vulcano Vulcano Workshop 2010 Workshop 2010
May 23-29, Vulcano, Italy
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*
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1.02 0.02 0.73 0.04 0.27 0.04
tot m
± = ± = ±
71 4 / / 13.7 0.2 H km s Mpc t Gyr = ± = ±
0.044 0.04
B
= ±
Structure formation
Despite successes of ΛCDM on large and intermediate scales, serious issues remain
Flores & Flores & Primack Primack 1994 1994: at small radii halos are not going to be singular (analysis of : at small radii halos are not going to be singular (analysis of the flat rotation curves of the low surface brightness (LSB) galaxies). the flat rotation curves of the low surface brightness (LSB) galaxies). Other studies Other studies (Moore 1994; (Moore 1994; Burkert Burkert 1995; 1995; Kravtsov Kravtsov et al. 1998; et al. 1998; Borriello Borriello & & Salucci Salucci 2001; de 2001; de Blok Blok et al. 2001; de et al. 2001; de Blok Blok & & Bosma Bosma 2003, etc.) indicates that the shape of 2003, etc.) indicates that the shape of the density profile is shallower than what is found in numerical simulations the density profile is shallower than what is found in numerical simulations ( = 0.2 ± 0.2 (de ( = 0.2 ± 0.2 (de Blok Blok, , Bosma Bosma, & , & McGaugh McGaugh 2003)) 2003))
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Navarro, Navarro, Frenk Frenk & & White (1997) White (1997)
)] 1 /( ) 1 [ln( 3 ) ( ) ( ) ( ) ( ) / 1 )( / ( ) (
3 2
c c c c M M r M r M c r r r r r
vir c s vir vir s s crit c
+
=
inner slope in higher-resolution simulations is inner slope in higher-resolution simulations is steeper (~ steeper (~ – –1.5) than the NFW value ( 1.5) than the NFW value (– –1.0) 1.0)
Moore et al. (1998) mass resolution
Asymptotic outer slope -3; inner -1 inner -1
Navarro et al. 2004
ln( /2) = (2/)[(r /r
2) 1]
= dlog /dlogr = 2
r r-2
radius at which which
Fitting Formula, Stadel-Moore *
Gentile et al. 2004 (and similarly Gentile et al. 2007): models with a constant density core are preferred.
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Tyson, Kochanski & Dell’Antonio (1998)
with Numerical simulations (Dahle et al 2003; Gavazzi et al. 2003) or finding much shallower Slopes (-0.5) (Sand et al. 2002; Sand et al. 2004)
from: -0.6 (Ettori et al. 2002) to -1.2 (Lewis et al. 2003) till -1.9 (Arabadjis et al. 2002) InnerSlope= 0.57 0.02
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Elliptical potentials can be unphysical (Schramm 1994), so the mass distribution is parameterized as a cluster of mass concentrations (“mascons”). Each mascon is based
model (Schneider, Ehlers, & Falco 1993) for the mass density versus projected radius
Observational problems – – Beam smearing; non-circular motion etc. Beam smearing; non-circular motion etc.
Failure of the CDM model or problems with simulations (del Blok et al (del Blok et al 2001, 2003; 2001, 2003; Borriello Borriello & & Salucci Salucci 2001) (resolution; relaxation; 2001) (resolution; relaxation; overmerging
)
New physics – – WDM (Colin et al. 2000; WDM (Colin et al. 2000; Sommer Sommer-Larsen &
Dolgov 2001) 2001) – – Self-interacting DM ( Self-interacting DM (Spergel Spergel & Steinhardt 2000; Yoshida et al. 2000; & Steinhardt 2000; Yoshida et al. 2000; Dave et al. 2001) Dave et al. 2001) – – R Repulsive epulsive DM (Goodman 2000) DM (Goodman 2000) – – Fluid DM (Peebles 2000), Fluid DM (Peebles 2000), – – Fuzzy DM ( Fuzzy DM (Hu Hu et al. 2000), et al. 2000), – – Decaying DM ( Decaying DM (Cen Cen 2001), 2001), – – Self- Self-Annihilating DM ( Annihilating DM (Kaplinghat Kaplinghat et al. 2000), et al. 2000), – – Modified gravity Modified gravity
Solutions within standard Λ ΛCDM CDM (requires (requires “ “heating heating” ” of dark matter)
– – Rotating bar Rotating bar – – Passive evolution of cold lumps (e.g., El Passive evolution of cold lumps (e.g., El Zant Zant et al., 2001) et al., 2001) – – AGN AGN
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Gunn & & Gott Gott’ ’s SIM s SIM ( (Ryden Ryden & & Gunn Gunn 1987; 1987; Avila-Reese Avila-Reese 1998; 1998; DP2000; DP2000; Lokas Lokas 2000; 2000; Nusser Nusser 2001; 2001; Hiotelis Hiotelis 2002; Le 2002; Le Delliou Delliou Henriksen Henriksen 2003; 2003; Ascasibar Ascasibar et et al. 2003; Williams
et al. 2004).
DP2000, Lokas Lokas 2000 2000 reproduced reproduced the NFW the NFW profile profile considering considering radial radial collapse collapse. .
The other
authors authors in the in the above above list list studied studied the the effect effect of
angular momentum momentum, L, and , L, and non-radial non-radial motions motions in SIM in SIM showing showing a a flattening flattening
inner profile profile with with increasing increasing L. L.
El-Zant et et al. (2001)
proposed proposed a a semianalytial semianalytial model: model: dynamical dynamical friction friction dissipate dissipate orbital
energy energy of gas
distributed in in clumps clumps depositing depositing it it in dark in dark matter matter with with the the result result of
erasing the the cusp cusp. .
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– – computationally efficient (it takes about 10 s to compute the computationally efficient (it takes about 10 s to compute the density profile of a given object at a given epoch on a density profile of a given object at a given epoch on a desktop PC) desktop PC) – – flexible flexible (one can (one can study study the the effects effects of
physical processes processes one
at a time) at a time) – – can incorporate many physical effects in at least a schematic can incorporate many physical effects in at least a schematic manner manner
– – treatment of physical processes is only approximate treatment of physical processes is only approximate (but (but SIM provides a viable dynamical model for predicting the SIM provides a viable dynamical model for predicting the structure and evolution of the density profile of dark matter structure and evolution of the density profile of dark matter haloes ( haloes (Toth Toth & & Ostriker Ostriker 1992; 1992; Huss Huss et al. 1999; et al. 1999; Ascasibar Ascasibar et et
– – “ “exact exact” ”, detailed treatment of physical processes (but this, , detailed treatment of physical processes (but this, somehow, turns to be a disadvantage*) somehow, turns to be a disadvantage*)
– – computationally expensive, so unfeasible to explore large computationally expensive, so unfeasible to explore large parameter space or to simulate large volumes at high resolution parameter space or to simulate large volumes at high resolution – – No baryons physics No baryons physics – – difficulty to incorporate additional physical processes (e.g. BH difficulty to incorporate additional physical processes (e.g. BH growth, AGN feedback) growth, AGN feedback) – – currently, even currently, even ‘ ‘best best’ ’ simulations do poorly at matching simulations do poorly at matching fundamental galaxy observations (e.g. overcooling problem, fundamental galaxy observations (e.g. overcooling problem, angular momentum problem) angular momentum problem) – – * So rich in dynamical processes that it is hard to disentangle * So rich in dynamical processes that it is hard to disentangle and interpret in terms of underlying physics. and interpret in terms of underlying physics.
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The dynamical evolution of matter at the distance xi
i from the peak is determined by
from the peak is determined by the mean cumulative density perturbation within x the mean cumulative density perturbation within xi
i and the maximum radius of
and the maximum radius of expansion can be obtained knowing x expansion can be obtained knowing xi
i and the mean cumulative density of the
and the mean cumulative density of the perturbation. perturbation.
After reaching maximum radius, a shell collapses and will start oscillating and it will contribute to the inner shells with the result that energy will not be an integral of contribute to the inner shells with the result that energy will not be an integral of motion any longer. The dynamics of the motion any longer. The dynamics of the infalling infalling shells is obtained by assuming that shells is obtained by assuming that the potential well near the center varies adiabatically (Gunn 1977; FG84; the potential well near the center varies adiabatically (Gunn 1977; FG84; Ryden Ryden & & Gunn 1987). Gunn 1987).
Initial density peak are smooth, but contain many smaller scale positive and negative perturbations that originate in the same Gaussian random perturbations that originate in the same Gaussian random fi field eld producing the main producing the main
particles from their otherwise purely radial orbits. particles from their otherwise purely radial orbits.
Ordered angular momentum was calculated by means of the standard theory of acquisition of angular momentum through tidal torques, while the random part of acquisition of angular momentum through tidal torques, while the random part of angular momentum was assigned to angular momentum was assigned to protostructures protostructures according to Avila-Reese et al. according to Avila-Reese et al. (1998) scheme. (1998) scheme.
Dynamical friction was calculated dividing the gravitational fi field eld into an average and into an average and a random component generated by the clumps constituting hierarchical universes. a random component generated by the clumps constituting hierarchical universes.
The baryonic dissipative collapse (adiabatic contraction) was taken into account by means of means of Gnedin Gnedin et al. (2004) model and et al. (2004) model and Klypin Klypin et al. (2002) model taking also et al. (2002) model taking also account of exchange of angular momentum between baryons and DM. account of exchange of angular momentum between baryons and DM.
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Dark matter haloes generated with the model described. In panels (a)-(d) the solid line represents the NFW model while the dotted line the density profile obtained with the model
The dashed line in panel (b) represents the density profile obtained reducing the magnitude of h, j and mu.
11
10 M
12
10 M
a b a *
c d * (panel c), (panel d), . The dashed line in panel (c) represents the Burkert fit to the halo.
8
10 M
10
10 M
8
10 M
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,
distribution for the density profile reproducing the NFW halo.
12
10 M
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Density profile evolution of a halo. The solid line represents the profile at z=10. The profile at z=5, z=3, z=2, z=1, z=0 is represented by the uppermost dashed line, long-dashed line, short-dashed line, dot-dashed line, dotted line, respectively.
9
10 M
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Comparison of the rotation curves obtained our model (solid lines) with rotation curves Of four LSB galaxies studied by Gentile et al. (2004). The dotted line represents the fit With a NFW model. *
Comparison of the rotation curves obtained with the model of the present paper (solid lines) with the rotation curves of four LSB galaxies studied by de Blok & Bosma (2002). The dashed line represents the fit with NFW model.
The density profile evolution of a halo. The (uppermost) dot-dashed line represents the total density profile of a halo at z=0. The profile at z=3, z=1.5, z=1 and z=0 is represented by the solid line, dotted-line, short-dashed-line, long-dashed-line, respectively.
14
10 M
14
10 M
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L_in L_in+L_out L_in+L_out+ DF+BDC
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– – CDM struggles to answer questions of galaxy formation, including missing satellites, CDM struggles to answer questions of galaxy formation, including missing satellites, cusps vs. cores, and structure in voids. cusps vs. cores, and structure in voids. – – Numerical simulations for Numerical simulations for collisionless collisionless dark matter consistently suggest the formation dark matter consistently suggest the formation
flat core rather than a cusp. flat core rather than a cusp. – – SIM has proven to predict correctly density profiles. It agrees with simulations over all SIM has proven to predict correctly density profiles. It agrees with simulations over all radial ranges if the collapse is purely spherical. radial ranges if the collapse is purely spherical. – – SIM with L SIM with Lin
in , L
, Lout
, DF, Baryons AC agrees with simulations except in the inner part of the density profile where predicts core-like profiles (different slopes for galaxies and the density profile where predicts core-like profiles (different slopes for galaxies and clusters). clusters). – – On galactic scales, where DM dynamics and baryons dynamics are entangled, the On galactic scales, where DM dynamics and baryons dynamics are entangled, the cusp/core problem seems to be a cusp/core problem seems to be a “ “genuine genuine” ” one, in the sense that the disagreement
between observations and N-body simulations is not due to numerical artifacts or between observations and N-body simulations is not due to numerical artifacts or problems with simulations. problems with simulations. – – At the same time it is an apparent problem, since the disagreement between At the same time it is an apparent problem, since the disagreement between
dissipationless simulations is related to the fact that the latter are simulations is related to the fact that the latter are not taking account of baryons physics. This means that we are comparing two not taking account of baryons physics. This means that we are comparing two different systems, one different systems, one dissipationless dissipationless (i.e., DM) and the other (i.e., DM) and the other dissipational dissipational (i.e., inner (i.e., inner part of structures), and we cannot expect them to have the same behavior. part of structures), and we cannot expect them to have the same behavior.
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Del Del Popolo Popolo A., 2009, A., 2009, ApJ ApJ 698:2093-2113 698:2093-2113 Del Del Popolo Popolo A., A., Kroupa Kroupa P., 2009, A&A 502, 733-747 P., 2009, A&A 502, 733-747
– – CDM struggles to answer questions of galaxy formation, including missing satellites, cusps vs. CDM struggles to answer questions of galaxy formation, including missing satellites, cusps vs. cores, and structure in voids. cores, and structure in voids. – – Numerical simulations for Numerical simulations for collisionless collisionless dark matter consistently suggest the formation of a central dark matter consistently suggest the formation of a central cusp ( cusp (∝ ∝ r r-1.5
) rather than a core. – – The Universality of the profile is no longer so sure . The Universality of the profile is no longer so sure . – – Galactic rotation curves indicate a relatively flat core rather than a cusp, but gravitational Galactic rotation curves indicate a relatively flat core rather than a cusp, but gravitational lensing lensing, , in some cases, indicates the contrary. in some cases, indicates the contrary. – – Some kind of DM can erase the central cusp but new complications appear (e.g., Some kind of DM can erase the central cusp but new complications appear (e.g., collisional collisional dark dark matter with an appropriate cross section can erase the central cusp but result in too spherical matter with an appropriate cross section can erase the central cusp but result in too spherical halos.) halos.) – – Baryons-DM Baryons-DM interaction interaction has has the power the power to to flatten flatten the the inner inner density density profile profile but but it it is is not not clear clear how how they they can can affect affect the the profile profile of
LSBs. . – – SIM has proven to predict correctly density profiles. It agrees with simulations over all radial SIM has proven to predict correctly density profiles. It agrees with simulations over all radial ranges if the collapse is purely spherical. ranges if the collapse is purely spherical. – – SIM with non-radial collapse agrees with simulations except in the inner part of the density profile SIM with non-radial collapse agrees with simulations except in the inner part of the density profile where predicts a core. where predicts a core. – – A A simple simple solution solution for for a a complicated complicated problem problem? ? – – In any case several others problems affect LCDM, difficult to solve all together by only one model In any case several others problems affect LCDM, difficult to solve all together by only one model
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(0)
Initial density profiles of dark matter haloes for Different peak for 2, 3, 4 peaks (2, 3, 4 sigma) ( )
Specific angular momentum for the previously quoted density peaks: 2 sigma (dotted); 3 sigma (solid); 4 sigma dashed)
Left: Time evolution of the scale parameter for pure sphericall collapse (solid line), and taking account of angular momentum (dashed), and angular momentum and dynamical friction (dotted). Peak: 2 sigma Right: Accreted mass
The 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.
The ratio of the ordered, h, to 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.
Variation of the dynamical friction coefficient mu with nuc for a system having Rsys = 5 Mpc and for Rf = 0,12 Mpc (solid line), Rf = 0,3 Mpc (dot- dashed line), Rf = 0,5 Mpc (short-dashed line), Rf = 0,7 Mpc (long-dashed line). Units have been expressed in terms of Msolar, Mpc, Tc0.
i
m
1
i m i i i i
3
i
x i i
m
i m
*
2
i i ta m i m m
2
m
x x =
2 2
i i i
However, after , after reaching reaching maximum maximum radius radius, a , a shell shell collapse collapse and and it it will will contribute contribute to to the the inner inner shells shells: Energy : Energy not not conserved conserved and f and f not not constant constant and so and so even even energy energy is is not not an an integral integral of
motion anymore anymore and f and f is is no no longer longer constant constant. .
The effect effect of the
infalling
shells shells on the
dynamics of a
given shell shell can can be be described described assuming assuming that that the the potential potential near near the center the center varies varies adiabatically adiabatically ( (Gunn Gunn 1977; FG84; ZH93) 1977; FG84; ZH93)
Using this this assumption assumption, the total mas inside , the total mas inside apapsis apapsis is is
summing summing the mass the mass contained contained in in shells shells with with apapsis apapsis smaller smaller than than ( (permanent permanent component component, ) and the , ) and the second second ( (additional additional mass, ) mass, ) is is the the contribution contribution of the
shells shells passing passing momentarily momentarily through through the the shell shell : :
m
x ( ) ( ) ( )
T m p m add m
m x m x m x = +
m
p
add
m
m
R add m rm x
Where the additional component is given by: *
m(x) given by Eq. 4
(probability probability to to find find the the shell shell with with apapsis apapsis x inside x inside radius radius ) ) is is calculated calculated as as the ratio of the time the the ratio of the time the outer
shell shell ( (with with apapsis apapsis x) x) spends spends inside inside radius radius to to its its period period: :
xm
m p p
x x x xm x x x
m
Where xp is the pericenter of the shell with apapsis x and
x
is the radial Velocity of the shell with apapsis x as it passes from the radius The radial velocity is obtained by integrating the equation of motion of the shell: (specific coefficient of dynamical friction) h (specific angular momentum) G (acceleration)
2 * 3
r
*
*
The collapse collapse factor factor, f, of a , f, of a shell shell with with initial initial radius radius and and apapsis apapsis x xm
m
is is given given by by ( (Gunn Gunn 1977; FG84; ZH93): 1977; FG84; ZH93):
and the final density profile profile is is: :
The problem problem of
determining the density the density profile profile is is then then solved solved fixing the fixing the initial initial conditions conditions ( ), the ( ), the angular angular momentum momentum h(r), and the h(r), and the coefficient coefficient of
dynamical friction friction. .
i
p m i p m add m
1 3
ta m i i i
the velocity and then the probability that the particle (shell) inside radius r, then one calculates the mass added, , and the collapse factor f. Eq. 12 will give the final density profile.
m
x
add
*
Overmerging Overmerging, , two two body body relaxation relaxation, , softening softening length length; ; multi-mass multi-mass simulations simulations
In cosmological cosmological simulations simulations of the dark
matter each each particle particle represents represents a a coarse coarse grained grained sampling sampling of
phase space space which which sets sets a mass and a mass and spatial spatial resolution resolution. .
Unfortunately these these super-massive super-massive particles particles will will undergo undergo two two body body encounters encounters that that lead lead to to energy energy transfer transfer as as the system the system tends tends towards towards equipartition
. In the real real Universe Universe the dark the dark matter matter particles particles are are essentially essentially collisionless collisionless and pass and pass unperturbed unperturbed past past each each
.
The artificial artificial smoothing smoothing of the density
distribution in in these these regions regions ( (disruption disruption of dark
matter halos halos within within dense dense environments environments) ) is is referred referred to to as as `overmerging `overmerging‘ ‘ ( (for for a a review review of the
problem problem see see Moore Moore 2000). 2000).
A modification modification of the 1/r^2
law through a through a softening softening length length diminishes diminishes two two body scattering body scattering and and relaxation relaxation and and allows allows larger larger time time steps steps. .
Simulation results results are are least least reliable reliable in the in the densest densest parts parts of the
halo, , where where the the dynamical dynamical timescale timescale is is shortest shortest and and artificial artificial heating heating has has the the greatest greatest effect
. In early early simulations simulations of the
formation formation of
galaxy clusters clusters, , overmerging
erased erased substructure substructure completely completely (e.g. White 1976). (e.g. White 1976). When When simulations simulations reached reached sufficient sufficient resolution resolution to to resolve resolve roughly roughly as as many many subsystems subsystems as as there there are are galaxies galaxies in a cluster, the in a cluster, the problem problem was was considered considered `solved' (e.g. Ghigna `solved' (e.g. Ghigna et et al. al. 2000), 2000), although although the scale the scale invariance invariance of
halo properties properties quickly quickly lead lead to to an an excess excess dwarf dwarf satellite satellite problem problem in in galaxy galaxy haloes haloes. .
The assumption assumption that that the the overmerging
problem problem is is now now solved solved has has not not been been fully fully tested tested (Taylor (Taylor et et al. 2003).
The The effects effects of
particle discreteness discreteness in in N-body N-body simulations simulations of
Λ CDM are CDM are still still an an intensively intensively debated debated issue issue (Romeo (Romeo et et al.
2008).
The processes processes of
relaxation is is difficult difficult to to quantify quantify, , but but in the in the large large N N limit limit the the discreteness discreteness effects effects inherent inherent to to the the N-body N-body technique technique vanish vanish, so one , so one tries tries to to use use as as large large a a number number of
particles as as computationally computationally possible possible. . Increasing N helps, but slowly ( N Increasing N helps, but slowly ( N -0.25
)
*
*
15 yr
65 yr
N =10 000 Equilibrium Hernquist Model with massless tracers (green) in a plane
cluster with 650'000 particles log density log "relaxation"
cluster with 650'000 particles
N=4'000 Hernquist halo, after 1 and 3 mean relaxation times:
merging... final profile also shallower
Numerical flattening due to two body relaxation: relaxation: slow convergence, slow convergence, 1 million to resolve 1% of 1 million to resolve 1% of Rvirial Rvirial, , 1000 to resolve 10% ! 1000 to resolve 10% ! (Moore et al. 1998; (Moore et al. 1998; Diemand Diemand et al. 2004) et al. 2004)
To have such a high number of particles per halo the use of a special technique is per halo the use of a special technique is
simulating with high mass resolution only simulating with high mass resolution only the particles that will end up in the halo of the particles that will end up in the halo of interest at interest at z z = 0, while having less mass = 0, while having less mass resolution for the particles that end up far resolution for the particles that end up far away from the halo of interest. This has away from the halo of interest. This has the drawback that each simulation can the drawback that each simulation can resolve only one halo at a time, therefore resolve only one halo at a time, therefore selection effect biases and a poor selection effect biases and a poor statistical sample could affect the reliability statistical sample could affect the reliability
very accurate and has high resolution very accurate and has high resolution
14 million 6 million 1.7 million 0.2 million
3 / 1
r *
Refinement: Resimulating Resimulating halos with better halos with better mass resolution mass resolution
300 Mpc 3 Mpc
Highest resolution numerical simulations of the structure of dark matter halos 105 steps 108 particles High mass and force resolution
Lokas 2000; scale free spectrum
NFW1: values of of c_vir calculated from a model based
formalism provided by NFW that describes better their N-body Simulations. NFW2: c is obtained from NFW fitting formula *
Ascasibar et et al. 2003 (
radial density density profile profile from from a 3 sigma a 3 sigma fluctuation fluctuation on 1
h^(-1) (-1) Mpc Mpc scale). scale).
Angular momentum momentum introduced introduced as as the the eccentricity eccentricity parameter parameter
Changing the the orbit
eccentricity eccentricity e e ( (proportional proportional to to L) L) produces produces a a flattening flattening of the
inner profile profile. . Radial Radial
gives gives rise rise to to a a steep steep profile profile similar similar to to that that proposed proposed by by Moore Moore et et al. (1999)
Numerical experiments experiments, in the , in the same same paper paper shows shows that that central central slopes slopes in in relaxed relaxed haloes haloes could could be be less less steep steep than than the NFW the NFW fit fit in in agreement agreement with with analytical analytical models models based based on the
velocity dispersion dispersion profile profile (Taylor & (Taylor & Navarro 2003; Navarro 2003;
Hoeft
Hoeft et et al. 2003)
*
(Case A: ; (Case A: ; Case B: ) Case B: )
12
9.8 10 M
4.3 10 M
at turn-around as: where L is obtained by the spin parameter
Williams et al. (2004)
random angular momentum is taken into account
tend to be more centrally concentrated, and have flatter rotation curves
(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).
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)
*
*
baryons to DM by DF.
curve and profile. Dashed line is a fit using Burkert profile.
absence of energy feedback.
In order
to to describe describe the the evolution evolution of a density
perturbation in the in the nonlinear nonlinear phase phase we we may may use use some some analytical analytical models models ( (e.g e.g., ., spherical spherical or
ellipsoidal, , collapse collapse) )
A slightly slightly
sphere sphere, , embedded embedded in the in the Universe Universe, , is is a a useful useful non-linear non-linear model, model, as as it it behaves behaves exactly exactly as as a a closed closed sub-universe sub-universe. .
The overdensity
expands expands with with Hubble Hubble flow flow till till a a maximum maximum radius radius ( (turn-around turn-around). ). (r= (r=rmax rmax, , dr/dt dr/dt=0) occurs at =0) occurs at δ δlin lin ~1.06 ~1.06
Then it it collapses collapses to to a a singularity singularity. . (r=0): (r=0): δ δlin lin ~ 1.69 ~ 1.69
Collapse to to a a point point will will never never
in practice practice; ; dissipative dissipative physics physics and and the process of violent relaxation will will eventually eventually intervene and intervene and convert convert the the kinetic kinetic energy energy of
collapse collapse into into random random motions motions. . This This is is named named: : virialization virialization ( (occurs at 2tmax, and
rvir = rmax/2) = rmax/2)
Once a non-linear non-linear
has has formed formed, , it it will will
to to attract attract matter matter in in its its neighbourhood neighbourhood ant ant its its mass mass will will grow grow by by accretion accretion of new material (
secondary infall infall). ).
well of DM.
will take place and smaller units can collapse
A good good deal of deal of insight insight into into this this process process can can be be gained gained by by considering considering the the spherical spherical symmetric symmetric case ( case (Gunn Gunn & & Gott Gott) ) with with further further important important extensions extensions ( (Bertschinger Bertschinger 1985). 1985).
*-
uniform, spherical perturbation
δ δi
i =
= ρ ρ(t (ti
i)/
)/ρ ρb
b(t
(ti
i)-1
)-1 M = M = ρ ρb
b(4
(4π πr ri
i 3 3/3)(1+
/3)(1+ δ δi
i)
)
For simplicity, we are neglecting radiation and Lambda, since structure formation (probably) kicks in mainly after epoch of radiation and matter equality and before Lambda comes into play
*
i
2 2
i i
) (4 ) n n
+
(3 ) n i
i
m
1
i m i i i i
3
i
x i i
m
i m
*
2
i i ta m i m m
2
m
x x =
2 2
i i i
*
However, after , after reaching reaching maximum maximum radius radius, a , a shell shell collapse collapse and and it it will will contribute contribute to to the the inner inner shells shells: Energy : Energy not not conserved conserved and f and f not not constant constant and so and so even even energy energy is is not not an an integral integral of
motion anymore anymore and f and f is is no no longer longer constant constant. .
The effect effect of the
infalling
shells shells on the
dynamics of a
given shell shell can can be be described described assuming assuming that that the the potential potential near near the center the center varies varies adiabatically adiabatically ( (Gunn Gunn 1977; FG84; ZH93) 1977; FG84; ZH93)
Using this this assumption assumption, the total mas inside , the total mas inside apapsis apapsis is is
summing summing the mass the mass contained contained in in shells shells with with apapsis apapsis smaller smaller than than ( (permanent permanent component component, ) and the , ) and the second second ( (additional additional mass, ) mass, ) is is the the contribution contribution of the
shells shells passing passing momentarily momentarily through through the the shell shell : :
m
x ( ) ( ) ( )
T m p m add m
m x m x m x = +
m
p
add
m
m
R add m rm x
Where the additional component is given by: *
m(x) given by Eq. 4
(probability probability to to find find the the shell shell with with apapsis apapsis x inside x inside radius radius ) ) is is calculated calculated as as the ratio of the time the the ratio of the time the outer
shell shell ( (with with apapsis apapsis x) x) spends spends inside inside radius radius to to its its period period: :
xm
m p p
x x x xm x x x
m
Where xp is the pericenter of the shell with apapsis x and
x
is the radial Velocity of the shell with apapsis x as it passes from the radius The radial velocity is obtained by integrating the equation of motion of the shell: (specific coefficient of dynamical friction) h (specific angular momentum) G (acceleration)
2 * 3
r
*
*
The collapse collapse factor factor, f, of a , f, of a shell shell with with initial initial radius radius and and apapsis apapsis x_m x_m is is given given by by ( (Gunn Gunn 1977; FG84; ZH93): 1977; FG84; ZH93):
and the final density profile profile is is: :
The problem problem of
determining the density the density profile profile is is then then solved solved fixing the fixing the initial initial conditions conditions ( ), the ( ), the angular angular momentum momentum h(r), and the h(r), and the coefficient coefficient of
dynamical friction friction. .
i
p m i p m add m
1 3
ta m i i i
the velocity and then the probability that the particle (shell) inside radius r, then one calculates the mass added, , and the collapse factor f. Eq. 12 will give the final density profile.
m
x
add
*
2 2
1 2
k l CDM
2( ) CDM
Gravitatinal force on the central region isgiven by (Ryden 1988; Eisenstein & Loeb 1995): The tidal moments are While the density profile of each protostructure is approximated by the Superposition of a spherical density profile and a random CDM distribution
The tidal torque is Whose integral gives the acquired angular momentum:
can be represented as a collisionless medium made of a hierarchy of density
groups and so on.
from the smoothed out distribution of mass, and a stochastic component, Fstoch(r), generated from the fluctuations in number of the field particles.
(Holtsmark 1919; Chandrasekhar & von Neumann 1942).
derivatives (Kandrup 1980):
dark matter orbit changes slowly compared to the orbital period.
(Blummenthal et al. 1986; Ryden & Gun 1987)
till they are on circular orbits
compact distribution , and the new potential is
apocenter untill the orbits of DM in have the same value of which they had in the predissipation contribution.
that and are conserved.
radius:
and
0( ) B
M FM r = (1 ) ( )
D
M F M r =
B
M r j
1( )
r
j
1 D
M
r
j j
2( )
r
j j
1 10 100
Fukushige & Makino (1997)
r[kpc]
Moore et al. (1998)
Moore et al. (1998)
force resolution mass resolution
*-
Navarro 2004 4 proposed a proposed a new analytic form: new analytic form: Where r Where r-2
is defined as the radius at which: radius at which:
2) 1]
Shown below: Logarithmic slope of the density
model, fitting (from left to right) dwarf, galaxy- sized, cluster-sized halos. Dot-dashed line:NFW. Dotted lines: M99. *-
(a few 106p) Navarro et al 2004
radius to 0.5% of r200
profile to better than 10% over the reliably resolved radial range
have no cusps….