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Transcript of Structure of Dark-Matter Halos - UC Santa Cruz - Physics...
Structure ofDark-Matter Halos
Universal Halo profile
The cusp/core problem
Dynamical friction
Tidal effects
Origin of the cusp in hierarchical clustering
Lecture
N-body simulation of Halo Formation
N-body simulation of Halo Formation
Dark Halo (Moore)
CDM halos (simulations)
• Density profiles are universal shape independent of mass and cosmology.
• Density profiles are cuspy density increases inward down to the innermost resolved
radius. Asymptotic power-law near the center?
• Halos are clumpy~10% of the mass is in self-bound clumps ---the surviving cores of accreted satellites.
The dark-halo cusp/core problem00 3
1)(αα
ρρ+−−
+
=
sss r
rrrr
log r
log ρ
rs
observed core
simulated cuspα≥1
α=0
α=3
Universal Profile
Dark Halos
flat rotation curvev
R
30,0000 ly
dark halo
RRMRRGMV
∝→
=
)(
)(2
v
R
3,000light years RGMV =2
Isothermal Sphere
rr
drdP
rrrGM )()()( 2
2
ρασρ=−=Hydrostatic equilibrium:
)3(00 )3(
4)()( αα ραπ
ρρ −−
−=→= rrMrr
rr
drdPr
mkTnkTP
22 )()( σαρ
σρρ −=→===
isothermal
222
2)(2)( −=→=→ r
Grr
GrM
πσ
ρσ
22 2)()( σ==rrGMrV
Universal Mass Profile of CDM Halos
Mass profile general shapesare independent of halo mass &cosmological parameters
Mass profile general shapesare independent of halo mass &cosmological parameters
RadiusRadius
Den
sity
Den
sity
Density profiles differ frompower law
The profile is shallower thanisothermal near the center
But no obvious flat-densitycore near the center
Density profiles differ frompower law
The profile is shallower thanisothermal near the center
But no obvious flat-densitycore near the center
A cusp; some controversyabout inner slope
A cusp; some controversyabout inner slope
New results for ΛCDM halos
Simulations span ~6 decadesin Mvir, from dwarf galaxies(Vc~ 50 km/s) to galaxyclusters (Vc~1000 km/s)
~million particles within Rvir
Controled numerical effectsvia convergence studies
Simulations span ~6 decadesin Mvir, from dwarf galaxies(Vc~ 50 km/s) to galaxyclusters (Vc~1000 km/s)
~million particles within Rvir
Controled numerical effectsvia convergence studies
RadiusRadius Navarro, Frenk, White, Hayashi,Jenkins, Power, Springel, Quinn, Stadel
Navarro, Frenk, White, Hayashi,Jenkins, Power, Springel, Quinn, Stadel
Den
sity
Den
sity
Recent results for ΛCDM halos
Properly scaled, all haloslook alike: CDM halostructure appears to be“universal”
Properly scaled, all haloslook alike: CDM halostructure appears to be“universal”
Scaled RadiusScaled Radius
Scal
ed D
ensi
tySc
aled
Den
sity
Navarro, Frenk, White, Hayashi,Jenkins, Power, Springel, Quinn, Stadel
Navarro, Frenk, White, Hayashi,Jenkins, Power, Springel, Quinn, Stadel
Universal Profile: NFW
log r/Rvir
log ρ
_=2 at rs Navarro, Frenk & White 95, 96, 97Cole & Lacey 96Moore et al. 98Ghinga et al. 00Klypin et al. 01Power et al. 02Navarro, Hayashi et al. 03, 04Stoehr et al. 04, 05
virvirs
s RrRrrx
xxr <<≡
+= 01.0for
)1()( 2
ρρ
00 3)1()( αα
ρρ −+
=xx
r sgeneralizedcusp:
3≈α
rddrlnln)( ρ
α −=
1≈α
two parameters:
10~s
virvir r
RCM ≡
rs
-2 -1 0
2=α
slope:
Ellipsoidal shape: a3/a1~0.5
Halo Concentration vs Mass and HistorySelf-similar Toy model (Bullock et al. 2001):
)1()(* MfaM c ≡Define ac as the time when typically aconstant fraction f of M is collapsing:
3)3/4(~
ss r
Mπ
ρ ≡
3
333 )()()()(~
cucus a
aaakaak ρρρ Δ=Δ=
+
−+=CCCs 1
)1ln(3ρDefine a characteristic halo density: forNFW
Assume additional contraction ofinner halo by a constant factor k:
)2(),(caakaC =µ
s
vir
rRC ≡
)(/)( * aMaM≡µ
ααα µσ )(0
1/1* f
aaaMM c =→∝→∝ − αµµ −= )(),( fkaCEdS
Pk∝kn
13.0401.0~ ≈≈ αkfDetermine parameters from simulations:
Excellent fit!caaaC 4)01.0(4),( 13.0 ≈≈ −µµ
Concentration vs Mass
Bullock et al. 2001
caaaC 4)01.0(4),( 13.0 ≈≈ −µµ
Concentration vs time, given mass
caaaC 4)01.0(4),( 13.0 ≈≈ −µµ
Bullock et al. 2001
Distribution of C: log-normal
NFW Rotation Curve
xxA
CACVxV vir
)()(
)( 22 =
)(216.016.2 2
2
CAC
VVrrvir
maxsmax ≈≈
CCCCACArM ss +
−+≡=1
)1ln()()(4 3πρ
low C
high C
Mass Assembly HistoryWechsler et al. 2002
zaceaM 2)( −∝
caadMd defines2
loglog
=
Mass Assembly History
zaceaM 2)( −∝
caadMd defines2
loglog
=
a0.1 0.3 1
Wechsler et al. 2002
M>3x1013
M<4x1012
a a
M>3x1013
M<4x1012
Mass dependence of History and ConcentrationWechsler et al. 2002
caaaC 4)01.0(4),( 13.0 ≈≈ −µµ
Concentration vs History
recent majormerger z<1
smoothaccretion z<1
Wechsler et al. 2002
caaaC 4)01.0(4),( 13.0 ≈≈ −µµ
History vs MassWechsler et al. 2002
caaaC 4)01.0(4),( 13.0 ≈≈ −µµ
Concentration of LSB galaxies and ΛCDM halos
Meandensitycontrastwithinr(Vmax/2)
Meandensitycontrastwithinr(Vmax/2)
Maximum Rotation SpeedMaximum Rotation Speed
The averageintermediate-scaleconcentration andscatter of ΛCDMhalos is roughlyconsistent withobservations of LSBand dwarf galaxies
The averageintermediate-scaleconcentration andscatter of ΛCDMhalos is roughlyconsistent withobservations of LSBand dwarf galaxies
Alam et al 2001 Hayashi et al 2003Alam et al 2001
Hayashi et al 2003
Simulated Cusp
Recent results for ΛCDM halos
No obvious convergenceto a power law:profiles get shallowerall the way in.
Innermost slopes areshallower than –1.5
No obvious convergenceto a power law:profiles get shallowerall the way in.
Innermost slopes areshallower than –1.5
Radius Radius
Loga
rithm
ic S
lope
Loga
rithm
ic S
lope
Navarro, Frenk, White, Hayashi,Jenkins, Power, Springel, Quinn, Stadel
Navarro, Frenk, White, Hayashi,Jenkins, Power, Springel, Quinn, Stadel
rddrlnln)( ρ
α =−
β
β
ρα
=−≡
srr
rddr 2lnln)(
−
−=
12ln
β
β
βρ
ρ
ss rr
2.01.0~ −β
Improved profile:
Improved Cusp Profiles
Stoehr et al. 20042
maxmax
loglog
−=
rra
VV
-slope _
-slope _
Improved Cusp Profiles:extrapolated to the inner cusp
-slope _core
Moore
NFW
Maximum Asymptotic Inner Slope
M(r) is robustly measured inthe simulations.
With the local density, itprovides an upper limit tothe inner asymptotic logslope
°There is not enough massin cusp to sustain a power-law as steep as ρ~r-1.5
M(r) is robustly measured inthe simulations.
With the local density, itprovides an upper limit tothe inner asymptotic logslope
°There is not enough massin cusp to sustain a power-law as steep as ρ~r-1.5
Radius Radius Navarro, Hayashi, Frenk, Jenkins,White, Power, Springel, Quinn, Stadel
Navarro, Hayashi, Frenk, Jenkins,White, Power, Springel, Quinn, Stadel
p
r
p
rrrr
rrdrrr
rrrr
<−=→
−==→<= −− ∫
inslopeforlimitupper)](/)(1[333)'(''4
)3/4(1)(
0
23
ρρα
αρπ
πρρ αα
How good or bad are simple fits?
Over the well resolvedregions, both NFW andMoore functionsexhibit comparablesystematic deviationswhen fitted tosimulated CDM halos.
Over the well resolvedregions, both NFW andMoore functionsexhibit comparablesystematic deviationswhen fitted tosimulated CDM halos.
RadiusRadius
resi
dual
sre
sidu
als
Navarro, Frenk, White,Hayashi, Jenkins, Power,Springel, Quinn, Stadel
Navarro, Frenk, White,Hayashi, Jenkins, Power,Springel, Quinn, Stadel
Den
sity
Den
sity
RadiusRadius
resi
dual
sre
sidu
als
Navarro, Frenk, White,Hayashi, Jenkins, Power,Springel, Quinn, Stadel
Navarro, Frenk, White,Hayashi, Jenkins, Power,Springel, Quinn, Stadel
Circ
ular
Vel
ocity
Circ
ular
Vel
ocity
How good or bad are simple fits?
Over the well resolvedregions, both NFW andMoore functionsexhibit comparablesystematic deviationswhen fitted tosimulated CDM halos.
Over the well resolvedregions, both NFW andMoore functionsexhibit comparablesystematic deviationswhen fitted tosimulated CDM halos.
Origin of the Halo inner Cusp?Dynamical Friction and Tidal Effects
Dekel, Arad, Devor, et al. 2003
peri 1
apo 1 apo 2
apo 0
apo 3 final
Dekel, Devor & Hetzroni 2003
Halo Bulidup by Mergers
tidal stripping &dynamical friction
Dynamical Friction and Tidal stripping
Moore et al.
Dynamical Friction
Dynamical Friction
Mv
m
m<<M
Dynamical Friction
wake
Dynamical Friction
satX
sat MmeXXerfvvMvG
dtvd
<<
−<Λ−= − 2
2/132 2)()(ln4
πρπ
rr
sat
halo
sat
max
MM
GMvb
≈=Λ20
Chandrasekhar formula:
Coulomb logarithm:
σ2vX ≡
drag proportional to _ but independent of m
acceleration propto M (because wake density propto M)
peri 1
apo 1 apo 2
apo 0
apo 3 final
Dekel, Devor & Hetzroni 2003
Halo Bulidup by Mergers
tidal stripping &dynamical friction
Tidal Effects
12-hour period
Tidal interaction & Merger
The Antenna
Tidal stripping of a satellite?
Ibata et al. 2001
M31M32
stream
The tidal disruption of an NFW Satellite halo
Harrasment of a satellite
Moore et al.
Sagitarius Dwarf
Tidal Force by a Point Mass
l
rM 2)( lr
GMF−
−=2r
GMFcm −= 2)( lrGMF+
−=
Inertial frame
3222
)( rlGM
rGM
lrGMFt ≈−−
=
Satellite frame
3222
)( rlGM
rGM
lrGMFt ≈++
−=
rl <<tidal limit
Tidal Radius of a Satellite
32)(2)(
rlrGM
llGm t
t
t =self-gravityforce
tidal force
)(~)(~)(~)( 33 rrrM
llml halot
ttsat ρρ
resonance ω=Ω
rl
)(~)( rtlt halotsat
2/12/13
2/1)/(−∝
∝∝∝ ρMR
RGMR
VRt
Density Profiles of stripped NFW halos
2
maxmax
loglog
−=
rra
VV
Profiles of sub-halosStoehr et al 2004:
7.045.0 ≈↔≈ βa
Origin of a cusp:tidal effects in mergers
Dekel, Devor, Arad et al.
a. If satellites settle in halo core → steepening to a cusp α≥1
b. Mass-transfer recipe → convergence to a universal slope α>1
c. Flat-density core? Only if satellites are puffed up, e.g. by gas blowout
Tidal force on a satellite
core α=0
rdrdr
ln)(ln)( ρ
α −≡α=0 1 23
ρ)(rr αρ −∝
r
( )3213tidal ]1)([)(lr
lr
lrr
−−−∝ rrrGMF α
rˆ1 =l
2l
isothermal α=2cusp α=1 point mass α=3
+halocenter
→ no mass transfer where α<1
rr lr
( )3213tidal ]1)([)(lr
lr
lrr
−−−∝ rrrGMF α
rˆ1 =l
2l
Impulsive stripping and deposit
pericenterstripping
apocenterdeposit
Dekel, Devor & Hetzroni 2003
Impulsive stripping and deposit
pericenterstripping
apocenterdeposit
Dekel, Devor & Hetzroni 2003
Adiabatic evolution of satellite profile
stripping
no stripping tidalcompression inhalo core
Merger of a compact satellite
no masstransfer
satellitedecays intactto halo center
Dekel, Devor &Hetzroni 03
N-bodysimulation
Tandem mergers with compact satellites
The cusp is stable!
Result:No mass transfer in core →
rapid steepening to a cusp of α≥1
Tidal mass-transfer recipe at α>1
)()()( f rmrm ll →=
final initial satellite profile
)(f rm
)(lm
Deposit radius
Dekel & Devor 2003
Tidal mass-transfer recipe at α>1
)]([)]([)( rrr
αψσρ
=l
initial satellite
halo)()()( f rmrm ll →=
final initial satellite profile
=1? resonance
ω=Ω
Tidal mass-transfer recipe at α>1
ααψ
σρ
21)]([
)]([)(
≈= rrr
l
satellite
halo
)()()( rmrmf ll →=final initial sat. profile
merger simulations
→ stripping efficiency grows with α
ψ
smaller
denserψ
steepen
flatten
Steepening / flattening
αψ
αψσρ
21
)]([)]([)(
≈
= rrr
l
6/)5(2/)3(:scaling
satellite and halo homologousn
sn
s mrm ++− ∝∝ρ
Adding satellite to halo profile
3
3
oldnew )()()(r
rr llσρρ +=
−∝Δ 3
3
)()()(rrdr
dr ll
ρσ
α
linear perturbation analysis ⇒ α → αasymptotic
⇒
srr <<at
Convergence to an asymptotic slope
Dekel, Arad, Devor, Birnboim 03
Linear perturbation analysis
__1.3
Summary: Cusp
Dark-matter halos in CDMnaturally form cusps due tomerging compact satellites
Observed Core
‰‡Ò˯ÂÂÌ ‰˙¯ÎÈ (‰ÒÈÍ ‰˜ËÔ)
Low Surface Brightness Galaxies
Compare simulated Vc(r) withrotation curves of dark-matterdominated LSB galaxies
Observations:de Blok et al (2001) (B01),de Blok & Bosma (2002) (B02),and Swaters et al (2003) (S03)
Peak velocities range from 25km/s to 270 km/s
These measurements are hard!These measurements are hard!
DDO154 (a dwarf LSB)DDO154 (a dwarf LSB)
Observed cores vs. simulated cusps
core α=0
cusp α≥1 V
r
2/1
2 )(
α−∝→
=
rVrrGMV
Marchesini, D’Onghia, et al.
LSB rotation curves and CDM halos
Two problems:
The shape of LSB galaxy rotationcurves is inconsistent with thecircular velocity curves of CDMhalos.
The concentration of dark matterhalos is inconsistent with rotationcurve data: there is too much darkmatter in the inner regions of LSBgalaxies. McGaugh & de Block 1998
see also Moore 1994Flores & Primack 1994
McGaugh & de Block 1998see also Moore 1994
Flores & Primack 1994
LSB rotation curves (McGaugh et al sample)R
otat
ion
Spee
dR
otat
ion
Spee
d
RadiusRadius
The shape of V(r) variesfrom galaxy to galaxy
A fitting function:Vγ(r)=V0 (1+(r/rt)-γ)-1/γ
The parameter γ is a goodindicator of the shape ofthe rotation curve, therate of change fromrising to flat.
The shape of V(r) variesfrom galaxy to galaxy
A fitting function:Vγ(r)=V0 (1+(r/rt)-γ)-1/γ
The parameter γ is a goodindicator of the shape ofthe rotation curve, therate of change fromrising to flat.
Hayashi et al 2003Hayashi et al 2003
Scaled LSB rotation curves: a representative sampleR
otat
ion
Spee
dR
otat
ion
Spee
d
RadiusRadiusHayashi et al 2003Hayashi et al 2003
75% of LSB have 0.5<γ<2(~CDM halos)
25% have γ>>2(in conflict with CDM halos)
75% of LSB have 0.5<γ<2(~CDM halos)
25% have γ>>2(in conflict with CDM halos)
Scaled LSB rotation curvesR
otat
ion
Spee
dR
otat
ion
Spee
d
RadiusRadiusHayashi et al 2003Hayashi et al 2003
75% of LSB have 0.5<γ<2(~CDM halos)
25% have γ>>2(in conflict with CDM halos)
75% of LSB have 0.5<γ<2(~CDM halos)
25% have γ>>2(in conflict with CDM halos)
Rotation Curves Inconsistent with CDM Halos
Three categories of rotationcurves:
A) Well fit by Vg with LCDMcompatible parameters(70%)
•
B) Poorly fit by Vg with LCDM-compatible parameters(10%)
C) Poorly fit by Vg with anyparameters (20%)
Only 10% of LSB rotationcurves are robustlyinconsistent with LCDMhalo structure
The dark-halo cusp/core problem00 3
1)(αα
ρρ+−−
+
=
sss r
rrrr
log r
log ρ
rs
observed core
simulated cuspα≥1
α=0
α=3
How to make and maintaina core?
must suppress satellitemergers with the halo core!
Compact vs. puffy satellite
1/3 densitycompact puffy
strippedoutside
Dekel, Devor & Hetzroni 2003
Adiabatic Contraction
2/12/1
3 )( RMRM
RGMI ∝
≈−
1−∝MR
gas(1+f)
/(1+f)R
M
DM
TvdtvIT
2
0
2 ≈≈ ∫
Periodic motion under a slowly varying potential
Adiabatic invarinat:
2/12/132/1 )(~)/(~
)/(~~ −− ρGRGM
RGMR
VRtdyn
Instant Blowout
22
21MV
RGMEbefore +−=
0)2/(21)2/( 2
2
=+−= VMRMGEafter
Lose M/2 while V2 is unchanged:
unbound!
Feedback
Instant blowout: Puffed up
DM core
Adiabatic contraction:
baryons
DM
DM-halo reaction to blowout
only 1/6 in density (Gnedin & Zhao 02)
not enough in big galaxies? Enough in satellites?
by supernova feedback
Halo core
DM strippinggas contraction
compressionstar formation
blowoutpuffing
stripping
Satellite disruptionby stimulated feedback
Compression in core
compression α<1
Summary: Core
Feedback may lead to a coreby puffing small satellites
Caveats
Cores detected in big galaxies and clusters (?)
Puffing-up of satellite halos is necessary for cores,but perhaps not sufficient
•
•
Cusps (though flatter) form also in simulationswhere satellites are suppressed
Other scenarios for core formation
• Disruption of satellites by a massive black hole (Merritt & Cruz 01)
• Warm dark matter, Interacting dark matter → suppress satellites
• Angular-momentum transfer from a big bar to the halo core (Weinberg & Katz 02)
• Heating of the cusp by merging clouds (El-Zant, Shlosman & Hoffman 02)
• Delicate resonant tidal reaction of halo-core orbits if the system is noise-less (Katz & Weinberg 02)
Origin of Core:Disk in Triaxial Halo
Disk Rotation curve is NOT V2=GM(r)/r
Hayashi, Navarro et al.
Disks in realistic dark matter halos
Hayashi et al 2003Hayashi et al 2003
Massless isothermal gaseous disk in the non-spherical DM halopotential tracks the closed orbits within this potential
Massless isothermal gaseous disk in the non-spherical DM halopotential tracks the closed orbits within this potential
Disks in realistic dark matter halos
Hayashi et al 2003Hayashi et al 2003
Massless isothermal gaseous disk in the DM halo potentialMassless isothermal gaseous disk in the DM halo potential
Dynamics of a Gaseous Disk
Closed orbits intriaxial potentialsare not circular,and not limited toa plane.
High _?
Disks in triaxial dark matter halos
Hayashi et al 2003Hayashi et al 2003
Inferred rotationspeeds may differsignificantly fromactual circularvelocity.
Inferred rotationspeeds may differsignificantly fromactual circularvelocity.
Inclination: 50 degrees 67 degreesInclination: 50 degrees 67 degrees
Circular velocity
Rotationspeed
γ=4.7 γ=3.5
Scaled Rotation Curves: disk in CDM halo vs LSBs
Scal
ed R
otat
ion
Spee
dSc
aled
Rot
atio
n Sp
eed
Scaled radiusScaled radius
All LSB rotation curveshapes may beaccounted for byvarious projections ofa disk in a single CDMhalo
All LSB rotation curveshapes may beaccounted for byvarious projections ofa disk in a single CDMhalo
Hayashi et al 2003Hayashi et al 2003
Scaled LSB rotation curves: a representative sampleR
otat
ion
Spee
dR
otat
ion
Spee
d
RadiusRadiusHayashi et al 2003Hayashi et al 2003
LSB rotation curveshapes may be accountedfor by variousprojections of a disk in asingle CDM halo
Triaxiality in the halopotential may be enoughto explain the “cusp-core” discrepancy.
LSB rotation curveshapes may be accountedfor by variousprojections of a disk in asingle CDM halo
Triaxiality in the halopotential may be enoughto explain the “cusp-core” discrepancy.
Halo Shape