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)

COSMO11, 22­26 Aug 2011 @ Porto, Portugal

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

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 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.

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 a fixed mass and the 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)

Varying      with a fixed mass and the object size

(correction to Poisson)

Inside the object, indicates quite large non-linear effects. Neverthelessindicates only 4% of . We are trying to address this intuitive inconsistency...

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 non-linear effects of the brane-bending mode in two body system.

We found - Strong non-linear effects emerge at the toroidal region around Object B.- For large , and small , the profile of tend to converge, obtaining a robust prediction even in the realistic case ( , )- tends to compensate - (5th-forth) is, however, only 4% of (contrary to intuition, 5th-force is not significant)

Issues to be addressed are- how to explain why is small, although is large.

- giving observational implications, e.g. modification of Lue & Starkmann's prediction of precession, and proposals for new type observations using, e.g., Lunar Laser Ranging Experiment and/or lunar orbiters.

Consequently, even the Moon cannot be treated asa test particle orbiting the Earth.

?

16/15On the Vainshtein mechanismfor two body system in DGP model

Takashi Hiramatsu

Appendix

17/15On the Vainshtein mechanismfor two body system in DGP model

Takashi HiramatsuDependence on box size

(inhomogeneous)

Basically, the results do not depend on the box size.Notice that the inhomogeneous coordinate seems not suitable for P1,particularly inner the object.

18/15On the Vainshtein mechanismfor two body system in DGP model

Takashi HiramatsuCoordinate transform

DENCE

SPARSE

SPARSE

To reduce the contamination from the boundaries, we have to set a large box.

19/15On the Vainshtein mechanismfor two body system in DGP model

Takashi HiramatsuVainshtain mechanism

Vainshtein radius

GR5DST

< Here !

20/15On the Vainshtein mechanismfor two body system in DGP model

Takashi HiramatsuVainshtain mechanism

Vainshtein radius

GR ST 5D

< Here !

21/15On the Vainshtein mechanismfor two body system in DGP model

Takashi Hiramatsu'Static' solution for two body system in DGP

Top-hat profile

axially-symmetric

Strong interference

A

B

A

B

22/15On the Vainshtein mechanismfor two body system in DGP model

Takashi Hiramatsu“Initial”/Boundary conditions

As a initial ansatz for the iterations,

23/15On the Vainshtein mechanismfor two body system in DGP model

Takashi HiramatsuResults : Strength of interference (density)

In interior, non-linear effect contributes ~4% of A In interior, the contribution of A iscompletely cancelled by non-linear effect.The correction to the Poisson equationfrom A vanished inner the object B.

Varying the object size (    ) with a fixed mass

Slices at

(5th-forth) (correction to Poisson)