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