銀河団の弱い重力レンズを用いた重力理論（MOND）への制限

高橋龍一（国立天文台）、千葉剛（日大文理）

● MOND （MOdified Newtonian Dynamics : Milgrom 1983）
・ It changes equation of motion （EoM） to explain thegalactic dynamics without dark matter
・ Gravitational sources are only baryonic components（stars and gas）
・ It effectively makes gravitational force stronger at
galactic and extra-galactic scales
・ EoM is changed if acceleration is lower than criticalvalue g0

mg = Fm g 2 g0 = F−8if g < g 0 ≅ 1× 10 cm s−2ｇ 0 : constantRadius where MOND becomes importantM= g02rr ≈ 8000 AU≈ 10 kpc≈ 400 kpc

r=g 0 ≅ 1× 10 −8 cms −2

M g0

solar system M sun
galaxy1011 M sun

galaxy cluster 1014 M sun

MOND-gravity regime outside the above radius

Galactic rotation curves

MOND
star （Newton）
gas （Newton）
Free parameter is only mass-to-light ratio（Sanders & McGaugh 2003）

● Rotation curve
eq. of motion

m g 2 g0 = Fv
Mrmcentrifugal force

circular motionv = const.
（flat rotation curve）

v ∝ r −1/ 2（Kepler motion）

v2
g=rgravitymMF =G 2rM ∝L∝v

4

Tully-Fisher relation

● Galaxy Clusters（Aguirre et al. 2001; Sanders 2003）

MOND can not explain temperature profiles of clustershydrostatic equilibriumdP dr = − ρ gwith P = ( kT µ m p ) ρµ mpd ln ρ d ln T
[g 0GM ] 1/ 2+=−d ln r d ln r
kTg = [g 0GM ]1/ 2

rfor g < g 0µ : mean molecular weight
M (r ) : mass inside a radius r

If ρ (r ) and T (r ) are power law, left hand side is constantT ∝M1/ 2Isothermalsuggested by observations

Virgo Cluster

Including dark mass component at center（Sanders 2003）

T ∝M

1/ 2

M = const. outside coreT = const.

dashed line : only gas component（ isothermal β model ）solid & dotted lines :including dark halo

Original MOND was modificationof Newtonian dynamics

Bekenstein （2004） proposed relativistic MOND.
After his work, we can discuss relativistic
phenomena such as CMB, large scale
structure, gravitational lens.

Citation history for Milgrom （1983）

relativistic MOND
by Bekenstein （2004）

merging cluster 1E 0657-558

x-ray

200kpcgreen contour : surface density detected by weak lensing

baryonic mass peaktotal mass peak

8σ spatial offset

including neutrino dark halomν = 2 eV

black lines : convergence （= surface density） map in MOND
red dotted lines : observation
green region : neutrino halo

DM to baryon mass ratio is 2.4

Weak Lensing in Cluster

shear of backgroundgalaxies
gravitational potentialof cluster

cluster A1689

Weak Lensing in MOND
MOND radius
M= g02rrM =Einstein radiusg 0 ≅ 1× 10 −8 cms −2

M g 0 ≈ 400 kpc
cluster 1014 M sun≈comparablerE ≈ M H 0

since g 0 ≈ H 0

MOND-gravity regime
outside this radius

Weak lensing provides important method toprove which MOND is valid or notdeflection angle
llight pathg (r )α = 2b ∫ dlr
g = gNbrolensfor g N > g 0

M（<r）: lens mass
inside a radius r= g0 g N

for g N < g 0g N (r ) =M (< r )r2（Bekenstein 2004; Zhao et al. 2006）

shear γ ＆ convergence κg > g0lens mass

M (< r ) ∝ r pwithγ ∝ κ ∝ θ p−2∝ θ p / 2−1p≥0

g < g0γ
MOND
GRfor g > g 0
for g < g 0θθ：angle from cluster center

slope in MOND is shallower than that in GR
Because gravitational force in MOND decreases slowly

Cluster A1689
mass profile
galaxies(a)M(<r) (M )10dark halo

14

mass-to-light ratio
8 ( M L) SUN（Zekser et al. 2006）

galaxiesgas
10

X-ray obs.

（Andersson & Madejski 2004）

12

gas

10

dark halo

10A1689
10110

210

need to fit shear data（Hernquist profile）

3r (kpc)
We calculate shear using above mass profile,and compare observational datareduced shear γ (1 − κ )10（Broadhurst et al. 2005）

0with dark halosolid line : MOND prediction(c)γ / (1−κ)（gravitational sources aregalaxies & gas）
10−72

g0=4 10 cm/s

−1gas+galaxies

It is clearly smaller than data.
dotted line ： 40×g0
too shallow to fit data

10−2

dashed line：including dark halo
100101θ (arcmin)

Dark halo is needed in MOND

Other Clusters
CL0024+1654CL1358+6245

with dark halo
−1−82

g0=5 10 cm/s

gas+galaxies
10−2γ / (1−κ)γ / (1−κ)10with dark halo−110gas+galaxies(c)10−8(c)−3−2100θ (arcmin)

101102g0=3 10 cm/s010

θ (arcmin)

● Conclusion
・ MOND cannot explain weak lensing datairrespective of g0.
・ Dark matter halo is needed in MOND.
・ Above results are consistent with previous studies.
（Aguirre, Schaye & Quanaert 2001; Sanders 2003）

● CMB
バリオンのみだと Silk damping で小スケールの揺らぎが抑えられる（Skordis et al. 2006）Ω Λ = 0.78 Ων = 0.17 Ω B = 0.05
MOND

Ω Λ = 0.95 Ω B = 0.05

ΛCDM

WMAP data

MOND は WMAP の結果を再現できる

● 大規模構造（Skordis et al. 2006）Ω Λ = 0.78 Ων = 0.17 Ω B = 0.05

Ω Λ = 0.95 Ω B = 0.05

ΛCDM

SDSS data

MOND は power spectrum も再現