a ( i   rj  r ) GN i m  i 1 j r 3   (1 (1 j  r   r ( r r i j  r ) j i ) i e )

N-Body simulation

S 1 G   1 16  R G  2   4 on MOdified Gravity S   g   1 d x,   B   4 B 1   2    S 2   S   1 1 V       g      4 G   G  gd x  2   G  ,  G 2    (MOG=MOdified Gravity? or MOffat Gravity?)    2         V   G G V  G 2      2 V       g 4  d x.

Yamaguchi University Laboratory of Theoretical particle physics and Cosmology

PhD student 2nd grade Takayuki Suzuki鈴木隆之 ○ Basic theory and review of Moffat gravity

Scalar Tensor Vector Gravity(STVG)

ＡＣＴＩＯＮ developed by John Moffat 2005 1 1 S   R  2  gd 4 x, G 16  G

1  1 2   4 S    B B    V   gd x,  4 2 

1 1  G G     V G V   S   g          G  V   gd 4 x. S    2 2    2 2    G 2  G   G  

It have a massive vector field φμ(x)which couples with matter directly, and 3 scalar fields G(x), ω(x) and μ(x). Bμν = ∂μφν – ∂νφμ and

Vφ(φ), VG(G), Vω(ω) and Vμ(μ) denote self-interaction potentials. G(x) is gravitational constant. Weak field approximation • The equation of motion of a test particle is given by  du         m   u u      m B  u .  ds  • For weak fields the MOG acceleration law is d 2r G M   N 1 1  re r , dt2 r 2 1/ 2 1 D  6250M  kpc , M  G  D     1,   , E  25000M 1/ 2 , 2  G    M  E  N  M G  20GN , d 2r G M   eff , G  G 1 1  re r . dt2 r 2 eff N key point : Vector field is couple with matter → Geodesic equation has external force term massive → It has effective range → Yukawa like force In fact, Newton gravity we recognize = (1+α)×Gravitation of the inverse square law － Yukawa like Repulsive force Why can MOG explain galaxy flat rotate curve without dark matter？

11 if galaxy mass ～10 Ｍ☉ →linear plot 70 G  G 1   1 rer  G eff N 60 eff On disk scale 50 Ｇ∝ 15 r 40

10 30

20 5 10

200 400 600 800 1000 0.01 0.1 1 10 100 1000 104 kpc ←disk size→ ←satellite galaxy→

Equation of Kepler motion 2 GM GM v v   r r 2 r if G∝r, v is constant! How are MOG different from MOND?

As it is said from old days, MOND can explain galactic flat rotate curve without dark matter. MOND=MOdified Newton Dynamics：Mordehai Milgrom 1983 BUT, MOG is・・・ • It is not simple-minded phenomenalism.  It is relativistic gravity theory.  It can be derived from an action principle. • It can explain without dark matter from small scale(galaxy) to large scale(cosmology). • It can explain dark energy too. (arXiv:0710.0364) Moffat says －

• A fitting routine has been applied to fit a large number of galaxy rotation curves (101 galaxies), using photometric data (58 galaxies) and a core model (43 galaxies) (J. R. Brownstein and JWM, 2005; J. R. Brownstein, 2009). •The fits to the data are remarkably good for STVG. For the photometric data, only one parameter, the mass-to-light ratio M/L, is used.

But, above verification is the viewpoint from "statics". It is necessary to examine it about the real galactic "kinetic" evolution more. ○ the summary and motivation of my study

My study is verification of Moffat gravity from the viewpoint of N-body simulation. (r  r )m j i i ( rj ri ) Method is very simple. ai  G N  3 1 (1 (1  rj  ri )e ) j rj  ri performing normal N-body simulation after having changed equation of motion. cold collapse ・・・ The gravitational collapse of the isodensity ball. Very simple N-body simulation. This seems to be "an exercise simulation" for beginners But,It's a bare process of the elliptical galaxy formation. If you want to know the details please watch my poster(P31). ○ Acknowledgments Numerical computations were carried out on the general-purpose PC farm at Center for Computational Astrophysics, CfCA, of National Astronomical Observatory of Japan. 数値計算には国立天文台天文シミュレーションプロジェクトの汎用計算機を使わせて頂きました。 Thanks to