1. Modified Gravity Takeshi Chiba Nihon University

2. Why?

3. 1. A theory predicts the modification!: Scalar-Tensor Gravity 2. The Nature of Dark Matter is unkown: MOdified Newtonian Dynamics(MOND) 3. The Nature of Dark Energy is completely unkown: F(R) type gravity We simply do not know the correct gravity theory in the large (and small) scale

4. Gravity is Probed at … 10 -3 cm 1AU 1kpc 1Mpc 1000Mpc large extra dimensions? MOND? Modified Gravity?

5. Modified Gravity I Theory Motivated: String theory → Scalar-tensor Gravity If dilaton is (almost) massless, then cosmology and gravity can be different (time varying G) Brans-Dicke parameter: ω 0 > 20000 (Cassini satellite,2004)

7. Scalar-Tensor Cosmology Scalar-Tensor Gravity Consequence: Varying G Constraints: z=10 10 (BBN) -0.15<(G BBN -G 0 )/G 0 <0.21 (Copi-Davis-Krauss,2004) z=0 (LRR) dG/dt/G<4x10 -13 yr -1 What else?

8. Scalar-Tensor Cosmology z=1100 (CMB) (Nagata-TC-Sugiyama,2004) Effect of G  Projection effect(first acoustic peak, H -1  ) Shift of zero point of oscillation(  B h 2  ) Diffusion damping( D  H -1 l mfp  ) (damping factor:exp(- 2 / D 2 )  ) Decay of gravitational potential (  0, ISW)

9. Nagata-TC-Sugiyama(2002)

10. G recom -G 0 /G 0 <0.05 (Nagata-TC=Sugiyama,2004)

11. Scalar-Tensor Cosmology Scalar-Tensor Gravity Consequence: Varying G Constraints: z=10 10 (BBN) -0.15<(G BBN -G 0 )/G 0 <0.21 (Copi-Davis-Krauss,2004) z=1100 (CMB) (G recom -G 0 )/G 0 <0.05 z=0 (LRR) dG/dt/G<4x10 -13 yr -1

12. Modified Gravity II Observation motivated(Phenomenology?) Flat rotation curve → MOdified Newtonian Dynamics (MOND)(Miligrom,1986): alternative to dark matter v is constant at large scale ( (v 2 /r) 2 /a 0 =GM/r 2 )

13. MOND Problems(so far): no relativistic formulation ← ✖ light propagation(gravitational lensing) ✖ large scale structure ✖ cosmology (only recently) relativistic formulation by Bekenstein(2003): vector-scalar-tensor gravity

14. MOND Bekenstein’s Tensor-Scalar-Vector theory for MOND

15. MOND CMB and LSS by Bekenstein ’ s model (Skordis et al.,2005): consistent with obs.(WMAP,SDSS) if neutrino is massive (  ~0.17) ( ← first peak location)

16. MOND CMB peaks: sensitive to baryon and dark matter  B h 2  (shift of zero point of oscillation) → first peak height  second peak height   M h 2  (increases the depth of potential well decreases radiation relative to matter(ISW)) → first peak height  second peak → third peak height 

17. MOND trouble with higher (second and third) peaks of CMB(Slosar-Melchiorri-Silk,2005) ( ← Silk damping for baryons) WMAP WMAP/Boomerang

18. Modified Gravity III Recent acceleration of the Universe (SNIa) 1.Dark energy: modify RHS of Einstein equation 2.Modify LHS instead ⇒ Modified gravity modification should be significant only recently → 1/R gravity (Carroll et al., 2003)

19. Rise and Fall of 1/R Gravity F(R) gravity is equivalent to Scalar-Tensor Gravity(Higgs,Whitt,Wands,Chiba): Scalar-tensor with vanishing Brans-Dicke parameter: ω=0 can be in conflict with solar system experiments (ω>20000, Cassini satellite) if Brans-Dicke scalar is (almost) massless This is the case for 1/R gravity :m ~ H 0

20. 1/R gravity modifies gravity not only at large scales but also at local scale Einstein 1/R scale Rise and Fall of 1/R Gravity

21. So much ado … F(R,P,Q) gravity?(Carroll et al.,2004) P=R ab R ab,Q=R abcd R abcd → higher derivative (4th order) theory → Ghosts (Stelle 1977,Nunes,Chiba) propagator: cross coupling: The situation is much worse!!

22. But … This does not mean all attempts at modifying gravity in the large scale are in trouble (eg. DGP model) We simply do not know the correct gravity theory in the large (and small) scale （ → cosmological PPN formalism?)

23. Gravity is Probed at … 10 -3 cm 1AU 1kpc 1Mpc 1000Mpc large extra dimensions? MOND? Modified Gravity? ガモフの飛躍！（ 100 億年 → 3 分 間）

24. PPN(Parameterized Post-Newtonian) PPN formalism: expand the metric around the Minkowski up to post-Newtonian order ( (v/c)^4 ) parametrize possible form of the metric without specifying the gravitational theory solve the motions of planets and light using the metric and compare them with the observations |  - 1|<4.4 x 10 -5 (Cassini,2003), |  - 1|< 2.3 x 10 -4 (LLR,2004)

25. Cosmological PPN(or constructing approximate geometry of the universe) Newtonian gauge: Cosmological  ( ,x): Lessons from scalar-tensor gravity: for large  and  is constant if  H -1

26.   ~ H  ~  2  ~  2 /  2 nonlinear linear Newton post Newton

27. Cosmological PPN(or constructing approximate geometry of the universe) Cosmological metric (valid for  H -1 )  Bad: a(  ) is model dependent (H(  )) Good: (once H is specified) we only have to solve the same linear equations What about  ? -> second order perturbation

28. But … This does not mean all attempts at modifying gravity in the large scale are in trouble (eg. DGP model) We simply do not know the correct gravity theory in the large (and small) scale （ → cosmological PPN formalism?) In this respect, various consistency checks among cosmological observations are important (eg. growth rate, H via lensing and SNIa)

29. Ishak,Upadhye,Spergel(2005) see also Knox,Song,Tyson(2005)

30. Searching for alternatives is important to reinforce the evidence for DM and DE to check the internal consistency of cosmological data ( → understand systematics)