銀河団の弱い重力レンズを用いた 重力理...
TRANSCRIPT
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