On the Vainshtein mechanismfor two body system in DGP model
Yukawa Institute for Theoretical Physics (YITP)Kyoto University
Takashi Hiramatsu
Collaboration withWayne Hu (Chicago), Fabian Schmidt (Caltech), Kazuya Koyama (Portsmouth)
Midterm TAP workshop, 68 Sep 2011 @ Kyoto
2/15On the Vainshtein mechanismfor two body system in DGP model
Takashi HiramatsuModified gravity
Accelerating expansion of present universe
Possibility to need the modification of gravity theory
Dvali-Gabadadze-Porrati model 5D braneworld model, and our brane has 4D Ricci scalar.
Non-linear interactions of scalar recovering GR at short distances. Vainshtein mechanism.
Unfortunately, original DGP model is no longer acceptable
(observational inconsistency, ghost problem)e.g., Xia, PRD79 (2009) 103527 Koyama, CQG24 (2007) R231
Dvali, Gabadadze, Porrati, PLB485 (2000) 208
cf. massive gravity, Galileon
3/15On the Vainshtein mechanismfor two body system in DGP model
Takashi HiramatsuField equation of brane-bending mode
quasi-static limit= neglecting all time derivatives
Gravity potential
Perturbed metric brane-bending mode
(matter on brane)
correction
Self-acceleration branchNormal branch
Koyama, Silva, PRD75 (2007) 084040
4/15On the Vainshtein mechanismfor two body system in DGP model
Takashi HiramatsuSpherically-symmetric solution
Top-hat profile
Vainshtein radius
GRST 5D
: Schwartzschild radius
: Radius
5/15On the Vainshtein mechanismfor two body system in DGP model
Takashi HiramatsuMotivation
GR is recovered with help of strong non-linear interaction of extra scalar d.o.f of gravity, Vainshtein mechanism.
Vainshtein radius is ~pc even for the Earth.
We live in the many-body system.
How does the scalar field interfere with those sourced by other stars ?
Many studies have focused on the scalar field of a single source.
For simplicity, here we focus on
Static Two-body case~ snapshot of the Earth-Moon system.
e.g. extra periastron precession per orbital period :
cf. massive gravity, Galileon
Lue and Starkmann PRD67 (2003) 064002
6/15On the Vainshtein mechanismfor two body system in DGP model
Takashi Hiramatsu'Static' solution for two body system in DGP
Top-hat profile
Axial symmetry
A
B
A
B
7/15On the Vainshtein mechanismfor two body system in DGP model
Takashi Hiramatsu“Initial”/Boundary conditions
superposition
superpo sition
superposition
Superposed solution
At boundaries,
As a initial ansatz for the iterations,
Iterative scheme
Solve the field equation with Conjugate Gradient Squared + Successive Over-Relaxation.Shibata, PRD55 (1997) 2002
8/15On the Vainshtein mechanismfor two body system in DGP model
Takashi HiramatsuBox size/parameters
Minimum requirement on boxsize :
The box should be as large as possible so thatthe superposed solution at boundaries remainsto be good approximation, but we have to keepthe high resolution near sources.
separation Near the sources, we keep
Fixed the mass ratio .
A trick : inhomogeneous coordinate
9/15On the Vainshtein mechanismfor two body system in DGP model
Takashi HiramatsuExpectation : cancellation of
(near source B)
is a special solution
Non-linear interactions work to cancel .
constants
10/15On the Vainshtein mechanismfor two body system in DGP model
Takashi HiramatsuResults : Significance of cancellation
A
B
A
B
superposition
interfered
11/15On the Vainshtein mechanismfor two body system in DGP model
Takashi HiramatsuResults : Significance of cancellation
A
B
A
B
superposition
Completelycancelled!
Near Object B, the scalar field tends to erase
12/15On the Vainshtein mechanismfor two body system in DGP model
Takashi HiramatsuResults : Significance of cancellation ( )
Slices at
Inside Object B
Inside Object B
: almost insensitive to (discrete change comes from numerical error) : convergence property as is large, small deviation from superposition at
Varying with fixed mass and object size
(5th-forth) (correction to Poisson)
INSIDE
OUTSIDE
13/15On the Vainshtein mechanismfor two body system in DGP model
Takashi HiramatsuResults : Significance of cancellation ( )
Slices at
Inside ObjectB
Inside ObjectB
In exterior, indicates non-linear effects contribute ~10% of .
(5th-forth) (correction to Poisson)
Inside the object, indicates quite large non-linear effects. Neverthelessindicates only 4% of . We are trying to address this intuitive inconsistency...
Varying with fixed mass and object size
14/15On the Vainshtein mechanismfor two body system in DGP model
Takashi HiramatsuResults : Significance of cancellation ( )
Slices at
: for small , tends to converge to a constant value.: no dependence on mass density
: for small , the non-linear effect gets to completely cancel .: no dependence on mass density
Use only homogeneous
coordinate
Varying the object size ( ) with a fixed mass
(5th-forth) (correction to Poisson)
15/15On the Vainshtein mechanismfor two body system in DGP model
Takashi HiramatsuSummary
Study the nonlinear effects of the branebending mode in two body system.
Strong non-linear effects emerge at the toroidal region around B.- For large , and small , the profile of tend to converge, obtaining robust (?) predictions even in the realistic case ( , )- ~ 100% of , and tends to compensate - is, however, only 4% of
- how to explain why is small, although is large.
- giving observational implications (precession and/or new types) using, e.g., Lunar Laser Ranging Experiment and/or lunar orbiters.
Consequently, even the Moon cannot be treated asa test particle orbiting the Earth.
We found
Issues to be considered