Chap.4 Galactic Dark Matter• Evidence for dark matter in the Milky Way• Properties of a dark matter halo–

Total mass, global shape, density profile,
substructures• Recent progress on small-scale issues––

Missing satellites problem
Core/cusp problem• Future prospects
11. Evidence of dark matter in the Milky Way
In 1932, Jan Oort suggested the presence ofdark matter near the Sun (“missing mass”)
from the dynamical analysis of stellar motions

pressuregravity

8 kpc

Pressure force due to therandom motions of stars
are in balance withgravity exerted from both
visible and invisible matter⇒visible mass is foundto be insufficient⇒missing mass, dark matterDark matter density near the Sun
SDSS

LAMOST

Measured from the dynamical analysis of
the large number of nearby star sample≃0.5 GeV/cm3

Gaia

Sivertsson et al. 2022

Evidence for dark matter from rotation curves
Vrot（Ｒ）(km/s)

Rotation curve of the Milky Way(Sofue et al. 2009)flat curve: Vrot ~ const.
If spherically Vrot2
GM ( r )symmetric,rr2

Vrot  const .  M ( r )  r

Presence of a dark matter halo

Dark matter in an external spiral galaxy
NGC3198

van Albada et al. 1985

Stellar disk

HI gasIf (r) is spherically symmetric
Vrot(r) = (GM(<r)/r)1/2
where M(<r) = r 4r2dr5

Dark matter candidates• Faint compact objects– Brown dwarfs, white dwarfs, neutron stars, stellar BHs– Primordial BHs– MACHOs （Massive Compact Halo Objects ）• Elementary particles （non-baryonic matter）– Neutrino, neutralino, axion…
– Cold Dark Matter: CDM• Massive particles (10~1000 Gev) with small streaming motions
WIMPs （Weakly Interacting Massive Particles）
e.g. neutralino• Axions

CDM-based structure formation
Distribution of CDM particles

time

Cold Dark Matter (CDM): WIMP, Axion
Small-scale halos form first, then larger-scale structures
form subsequently through merging and accretion⇒ successful for reproducing observed structures

Density fluctuations in various scalesCDM
Credit: Tegmark

2. Properties of a dark matter halo
2.1 Total mass
Beyond the rotation curve method

Dark halo
Bulge
たRotating Galactic disk

Dark halo is much larger than the size of the Galactic disk,
where the rotation curve method is applicableHalo objects as tracers of dark-halo mass
Dark halo

Globular clusters

Field halo stars
Satellite galaxies

たSpatial motions (dominated by random motions)reflect a gravitational potential of a dark halo ⇒ mass

Velocity distribution of disk/halo stars near the Sun(VR2+Vz2)1/2

VΦ
Escape velocity near the Sun: Vesc=500~550km/s⇒ Limits on a gravitational potential  at R=Rsun : Vesc=(2(Rsun))1/2

Limits on Φ(r) at other radii based on
rest-frame velocities of distant sample: VRF ≤ Vesc(r)(Sakamoto, Chiba, Beers 2003)

GM  r 2  a2  a (r) 
log


a
r


Case 1：a = 195 kpc
Satellites
GCs
BHBs

/

𝜌 ∝ 𝑟 𝑎𝑡 𝑟 ≫ 𝑎
𝑎: size of a halo
→ total mass M

Case 2: a = 20 kpc(rejected)

filled/open:
w/o PMs

12

Maximum likelihood method to maximize
the probability for getting the observed (ri, vi) i=1,N
assumption: stellar distribution function f(E,L)

Velocity anisotropy parameter 

1 d  vr2
 vr2d
GM ( r )
2


 dr
r
dr
r2

2
 2  GM ( r ) 2   const.   vr  r 
r dr
2
r
r

Total mass = 2.5 × 1012 Msun over ~ 200 kpc
Visible mass =1011 Msun over ~ 15 kpc⇒ We see only 10 % of the total mass

13

Recent results using Gaia PMs
Space motions of Globular Clusters

Gaia + HSTPROMO

Vasiliev (Gaia)

Eadie & Juric 2019 M200 = 0.7+0.11-0.08 x 1012 Msun (r<200kpc)
Other recent results
Sohn et al. 2018
Mvir = 2.05+0.97-0.79 x 1012 Msun
Watkins et al 2019 Mvir = 1.41+0.99-0.52 x 1012 Msun
Posti & Helmi 2019 Mvir = 1.3 ± 0.3 x 1012 Msun

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2.2 Global shape
Sagittarius dwarf galaxy(Ibata, Gilmore, & Irwin 1994)

Sgr dwarfrdd ~ 24 kpc
r ~ 16 kpc

15

Sgr stream:
tracer of the MW dark halo

Sgr stream

Stream is confined onto an orbital plane⇒ round dark halo at 15 < r < 60 kpc

Majewski et al.

17Formation of stellar streams (by tidal force)

Obs.

Helmi 04

14<r<58kpc
=0.9Gyr, t=5q=10.90 < q < 0.95 is most likely(0.83<q<0.92)q=0.85

Johnston+ 05
19

However, CDM halos are generally triaxial / prolate.
(Jing & Suto 2000, 2002)

Hayashi+07: (c/a) = 0.72, (b/a) = 0.78 in central parts

20

Gas cooling makes CDM halos rounder(Kazantzidis et al. 2004)

0.1

r / rvir

121

2.3 Density profile
van Albada et al. 1985

If (r) is spherically symmetric
Vrot(r) = (GM(<r)/r)1/2
where M(<r) = r 4r2dr
Vrot(r) = const.
(r) ∝ 1 / r2
(Singular) isothermal sphere22

Prediction of CDM models
Virialized dark halos and their density profiles(Navarro, Frenk, & White 1997)

NFW profiles r 1  r 
 r 
rs 
 s 

2ρ∝ r-1 at r << rsρ∝ r-3 at r >> rs

23

NFW or Moore et al. profile?
(Navarro et al. 2004)-ρ∝r -
at inner parts
= 1: NFW
= 1.5: Moore
et al.
No universal 
1 <  < 1.5
Einasto profile:ln  (r ) /   2 (2 /  ) r / r 2   124

Core/cusp problem
Density distribution of a dark halo

Rotation curves of (external)
gas-rich dwarf galaxies

Inner profile: ρ(r) ∝ rα
Log ρ

NFW (Cuspy):
CDM prediction
α = -1
Cored
α=0

NFW

Cored

Log r

Core/cusp problem
Density distribution of a dark halo

Oh et al. 2015

Inner profile: ρ(r) ∝ rα
Log ρCored?
CDM crisis?

NFW (Cuspy):
CDM prediction
α = -1
Cored
α=0

α

Log r

Density profiles of Galactic dwarf spheroidal (dSph) satellites(Walker et al. 2009)
Dark halo model:  3




r
r 
   0   1    
 r0    r0  
NFW: ==1
Plummer for stars:
22 5 / 2
 (r )  1  r / rh

Jeans eq. (spherical sym.)
l-o-s velocity dispersion profile

Some dSphs havecore-like profile27

Dark matter in the MW satellites
dwarf spheroidal (dSph) galaxies
Sculptor

CarinaFornaxLeo II

Leo I

Total stellar mass = 104~106Msun
Random motion ～10 km/s
Self-gravity of the stellar system
alone cannot bind member stars
Dark matter

Massive dark matter neededDark matter in the MW satellites
Mass ratio between DM and stars(Mass enclosed within stellar extent ~ 4 x 107M)The MW satellites are largely
dark matter dominated!

(old data)29

“Too big to fail” problem
Rotation curvesof most massive
subhalos in theMW-like halos

Most massive subhalos in ΛCDM simulation
are denser than those in most luminous satellites.

Filled circles:
Galactic satellites
rotation velocities
at half-light radius

30

Bolyan-Kolchin et al. 2012

2.4 Substructures
Missing satellites problem
CDM-based simulation result
for the MW-mass halo
(Bullock & Boylan-Kolchin 2017)

MW satellites

106~109 Msun

Hundreds of subhalos
Only ~ 50 satellites are known
250kpcPower spectrum for DMs
２ (k) ∝

BB-K2017
CDM

P(k) T2(k)WDM

Dark halos on scalesof dwarf galaxies aremost important keys

CDM

WDM

Alternative dark matter models
CDM・Self-Interacting DM (SIDM)
Interaction among DM particles
cross section: σ/mCored profile is reproduced
・Warm Dark Matter (WDM)
m ~ O(keV) e.g. sterile neutrino
Number of subhalos is reduced

SIDM

WDM

Density distribution

Number of subhalos

SIDM
WDM

Host halo

Various dark matter candidates
Ferreira 2020

Unsolved big issue!

Ultralight DM：Fuzzy Dark Matter (FDM)
Hu, Barkana, Gruzinov (2000)

m = 1 x 10-22 eV

core

NFW

core mass

Chan, Ferreira, May, Hayashi, Chiba (2022)

Jowett Chan

Diversity inouter-halo profiles
Relation adopted
so farQuantum pressure vs. gravity

halo mass
A few kpc scales: dwarf galaxy sales

Diversity in core-halo mass
relationship is discovered.

Probing dark matter substructures• Dynamical effects on galactic structure– Star clusters and stellar streams– Stellar disks• Effects on gravitational lensing– Anomalous flux ratios between lensed images– Effects on extended lensed images

36

Probing evidence for CDM subhalos
from their gravitational effects on a stellar stream(Carlberg 2011)
dynamical effects
on stellar stream(Mstar=106Msun

CDM halo in a galaxy

No subhalos

Showing gaps

star cluster
Moore

1000 subhalos, M-1.9

Perturbation in the MW stream
Bonaca et al. 2019 GD-1 stream selected with Gaia PMs

Perturbation by a subhalo?

38

Limits on the abundance of DM subhalos from GD-1 and Pal 5 streams
Banik+2019
arXiv:1911.02663
Banik+2021
MN, 502, 2364

39

Lens mapping of CDM subhalos
Moore“Anomalous Flux Ratios”
for multiply lensed QSOs
(Metcalf & Madau 2001, Chiba 2002,
Dalal & Kochanek 2002)

These are hardly explainedby smooth lens models.

40

PG1115+080(radio quiet)
zs=1.72, zL=0.31
C(0.24)

Smooth lens model(Singular Isothermal Ellipsoid+ External Shear)(Chiba 2002)

sourceA2(0.59)

A1(1.00)

critical curve
caustic

B(0.16): obs
 : model

Iwamuro et al. 2000
Model: A2/A1  1 (fold caustic)
Observed A2/A1 (near-IR):  0.59 – 0.67 (anomalous)

41

B1422+231(radio loud)zs=3.62, zL=0.34
A(0.90)B(1.00)D(0.02)C(0.53)Smooth lens model(Singular Isothermal Ellipsoid+ External Shear)

critical curve
caustic

source: obs
 : model
CASTLES

Model: (A+C)/B  1 (cusp caustic)
Observed (A+C)/B (radio):  1.42 – 1.50 (anomalous)42

Elliptical Lens
lens planesource planelens planesource plane

critical lines

caustics

critical lines

caustics

Fold singularity

Cusp singularity
43

Anomalous Flux Ratios
 Implausible by luminous GCs and satellites,

CDM subhalos are most likely (Chiba 2002)
 Mass fraction of CDM subhalos ~ a few %(Dalal & Kochanek 2002)
 Flux anomaly depends on image parities,
being consistent with substructure lensing(Kochanek & Dalal 2004)⇒ Evidence for many CDM subhalos!?
44

Limits on the abundance of WDM subhalos from lensing
Schutz 2021 (arXiv: 2001.05503)

45

Summary• The Milky Way is dominated by a dark halo
– Halo tracers suggest Mtot (MW) = 1 ~ 2 x 1012 Msun
– Sgr stream suggests a nearly spherical shape at 15 < r < 60 kpc,
not clear beyond– Flat rotation curve suggests ρtot(r) ∝ r-2 in the inner part (where adisk dominates), not clear beyond• Satellite galaxies and small-scale issues– Largely dark-matter dominated: (M/L) = 10 ~ 1000
– Contradiction to CDM predictions:• Cored in some galaxies (Core/cusp problem)• Mean density is small (Too big to fail problem)• Total number is small (Missing satellites problem)46

Supplementary slides

47

Unsolved issues
Other causes for anomalous flux ratios
 Differential dust extinction?
 Stellar microlensing?

Limits on the mass of lens substructure Mass of a subhalo?
 How many subhalos?

Magnification of a source with radius Rs
compared with Einstein radius RE(∝ M1/2)

48

Panoramic views of a QSO center
1. Mid-IR imaging of a dust torus(Near-IR at rest) Extinction free Microlensing free Radio quiet QSOs
are available
 Source size is available
 Hot dust torus
at sublimation T of ~1800K
 Size (inner radius) Rs (~1pc) ∝ L1/2
from dust reverberation mapping
 Einstein radius RE (∝ M1/2) vs Rs⇒ limits on M

RS
RE
49

Panoramic views of a QSO center
2. Spectroscopy of NLR and BLR

NLR: microlensing free
BLR: affected by microlensing
NLR

BLR

Hβ

[OIII]

Moustakas & Metcalf 2003

Selective magnification
depending on RE vs Rs⇒ limits on M

50

Subaru observations ofquadruple lenses
 Mid-IR imaging with COMICS(Chiba et al. 2005; Minezaki et al. 2009) FOV=38”×30”, 0.”129/pix
 N band, λ=11.7μm,
continuum emission from dust torus

 IFS observation with Kyoto 3DII
(Sugai et al. 2007)
 FOV=3”×3”,0.”096 lenslet-1,37×37lenslets
 0.730 <λ< 0.915μm,
line emission from NLR and BLR

51

Subaru image @ 11.7μm
PG1115+080

Chiba et al. 2005

B1422+231

C
A
A2
A1

B

Total flux = 17.5 mJy
A2/A1 (Mid-IR) = 0.930.06(model)  0.92 fold caustic(near-IR) = 0.59 ~ 0.67

BC

Total flux = 19.2 mJy(A+C)/B (Mid-IR) = 1.510.06(model)  1.25 cusp caustic(radio) = 1.42 ~ 1.50

52

Subaru image @ 11.7μm

Minezaki et al. 2007

MG0414+0534

Q2237+030

Total flux = 39.2 mJy
A2/A1 (Mid-IR) = 0.900.04(model)  1.1 fold caustic(near-IR) = 0.4 ~ 0.8

Total flux = 22.2 mJy
B/A (Mid-IR) = 0.840.05,
C/A=0.460.02, D/A=0.870.05
B/A (model) = 0.87,
C/A=0.46,
D/A=0.86

53

IFS data of RXJ1131-1231
Sugai et al. 2007
B

A
C

Selective
Magnification !

Model:
B~C~ 0.5AA
B
C
Wavelength54

Hβline flux

[O III] line flux

A/B=1.74
C/B=0.46

A/B=1.63
C/B=1.19

Smooth model
A/B  1.7
C/B  1.0
• NLR([OIII]) is OK• BLR (Hβ) of Image C is
microlensed[O III] line flux
(point source subtracted)

Rs (NLR)  90pc

55

Limits on substructure lensing

microlens

PG1115+080(A1, A2)
Rs ~ 1 pc
Mid-IR flux ratio
ME < 16 Msun

subhaloMG0414+0534(A1, A2)
Rs ~ 2 pc
Mid-IR flux ratio
ME > 200 Msun

microlenssubhalo

B1422+231(A, B, C)
Rs ~ 3 pc
Mid-IR flux ratio
ME > 200 Msun

microlens

Q2237+030(A, B, C, D)
Rs ~ 2 pc
Mid-IR flux ratio
ME < 10 Msun

RXJ1131-1231
Rs (BLR) ~ 0.01 pc, Rs (NLR) ~ 100pc
ME < 105 Msun for NLR

56

ALMA observation ofgravitationally-lensed, extended images
 Direct imaging of subhalo-lensed images withhigh resolution observation (10mas)
 Determination of subhalo masses
 Spatial distribution of subhalos

 Source image： sub-millimeter
continuum radiation from dust
 T=30~60 K, L=102~103 pc
 S at 850μm=several tens mJy

Test for CDM models

57

ALMA: lensing galaxy SDP.81
Hezaveh et al. 2016

Subhalo with 109 Msun?

Testing ΛCDM with gravitationallens is on going

Inoue, K. T., Minezaki, Matsushita, Chiba 2016:
showing the effect of under-dense large-scale structures on lensed image
This issue is yet unsettled.

58