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 atgalactic and extra-galactic scales

・ EoM is changed if acceleration is lower than criticalvalue g0

Fmg = 280 scm101 −−×≅< ggif

Fggm =02

ｇ0 : constant

Radius where MOND becomes important

02 grM

= 280 scm101 −−×≅g0gMr =

AUr 8000≈ solar system sunM

kpc10≈ galaxy sunM1110

kpc400≈ galaxy cluster sunM1410

MOND-gravity regime outside the above radius

Galactic rotation curves

（Sanders & McGaugh 2003）

MONDstar （Newton）gas （Newton）

Free parameter is only mass-to-light ratio

Rotation curveeq. of motion

M

vr m

Fggm =02

centrifugal force gravity

rvg2

= 2rMmGF =

circular motion

const.=v 4vLM ∝∝（flat rotation curve）

Tully-Fisher relation2/1−∝ rv

（Kepler motion）

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

MOND can not explain temperature profiles of clusters

hydrostatic equilibrium

( )ρµ pmkTP = [ ] rGMgg 2/10=gdrdP ρ−= with 0for gg <

[ ] 2/10ln

lnlnln GMg

kTm

rdTd

rdd pµρ

−=+ : mean molecular weightµ)(rM : mass inside a radius r

If and are power law, left hand side is constant)(rρ )(rT

2/1MT ∝ Isothermal suggested by observations

Virgo Cluster

Including dark mass component at center（Sanders 2003）

const.=M2/1MT ∝ outside core

const.=T

dashed line : only gas component（ isothermal β model ）

solid & dotted lines : including dark halo

Original MOND was modification of 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 MONDby Bekenstein （2004）

merging cluster 1E 0657-558 x-ray

200kpc

green contour : surface density detected by weak lensing

baryonic mass peak total mass peak

8σ spatial offset

black lines : convergence （= surface density） map in MONDred dotted lines : observationgreen region : neutrino halo

including neutrino dark halo eV2=νm

DM to baryon mass ratio is 2.4

Weak Lensing in Cluster

shear of background galaxies

gravitational potentialof cluster

cluster A1689

Weak Lensing in MOND

MOND radius Einstein radius28

0 scm101 −−×≅g02 g

rM

=

0gMrM = kpc400≈

sunM1410

comparable

≈ 0HMrE ≈

cluster00 Hg ≈since

MOND-gravity regimeoutside this radius

Weak lensing provides important method toprove which MOND is valid or not

deflection anglel

o

b r

light path

lens

∫=rrgdlb )(2α

Ngg =0ggfor N > M（<r）: lens mass

inside a radius r

Ngg0= 2

)()(r

rMrgN<

=0ggfor N <

（Bekenstein 2004; Zhao et al. 2006）

shear γ＆ convergence κ

GR

MOND

0gg > 0gg <

lens mass γ0≥pprrM ∝< )( with

2−∝∝ pθκγ12/ −∝ pθ

0ggfor >

0ggfor <θ

θ：angle from cluster center

slope in MOND is shallower than that in GR

Because gravitational force in MOND decreases slowly

Cluster A1689mass profile mass-to-light ratio

galaxies

101 102 103

1010

1012

1014

r (kpc)

A1689

gas

dark halo

galaxies

M(<

r) (

M )

(a)SUNLM )(8

（Zekser et al. 2006）

gas X-ray obs.（Andersson & Madejski 2004）

need to fit shear data dark halo（Hernquist profile）

We calculate shear using above mass profile, and compare observational data

)1( κγ −reduced shear （Broadhurst et al. 2005）

100 101

10−2

10−1

100

γ / (

1−κ)

θ (arcmin)

gas+galaxies

with dark halo

g0=4 10−7cm/s2

(c) solid line : MOND prediction（gravitational sources are

galaxies & gas）

It is clearly smaller than data.

dotted line ： 40×g0

too shallow to fit data

dashed line：including dark halo

Dark halo is needed in MOND

Other Clusters

CL0024+1654 CL1358+6245

100 10110−3

10−2

10−1

γ / (

1−κ)

θ (arcmin)

gas+galaxies

with dark halo

g0=5 10−8cm/s2

(c)

10010−2

10−1

γ / (

1−κ)

θ (arcmin)

gas+galaxies

with dark halo

g0=3 10−8cm/s2

(c)

Conclusion・ MOND cannot explain weak lensing data

irrespective of g0.

・ Dark matter halo is needed in MOND.

・ Above results are consistent with previous studies. （Aguirre, Schaye & Quanaert 2001; Sanders 2003）

05.017.078.0 =Ω=Ω=ΩΛ Bν

05.095.0 =Ω=ΩΛ B

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

ΛCDM

MOND はWMAP の結果を再現できる

（Skordis et al. 2006）

WMAP data

MOND

MOND は power spectrum も再現

大規模構造（Skordis et al. 2006）

05.017.078.0 =Ω=Ω=ΩΛ Bν

05.095.0 =Ω=ΩΛ B

ΛCDM

SDSS data