Modified Gravity

Takeshi ChibaNihon University

Why?

Why?1. A theory predicts the modification!: Scalar-Tensor Gravity2. The Nature of Dark Matter is unkown: MOdified Newtonian Dynamics(MOND)3. The Nature of Dark Energy is completely unkow

n: F(R) type gravityWe simply do not know the correct gravity theory in the large (and small) scale

Gravity is Probed at …

10-3cm 1AU 1kpc 1Mpc 1000Mpc

large extra dimensions?

MOND?Modified Gravity?

Modified Gravity I

Theory Motivated: String theory → Scalar-tensor Gravity If dilaton is (almost) massless, then cosmolo

gy and gravity can be different (time varying G)

Brans-Dicke parameter: ω0 > 20000 (Cassin

i satellite,2004)

Scalar-Tensor Cosmology Scalar-Tensor Gravity Consequence: Varying G Constraints: z=1010 (BBN) -0.15<(GBBN-G0)/G0<0.21 (Copi-Davis-Krauss,2004) z=0 (LRR) dG/dt/G<4x10-13 yr-1

What else?

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(Bh2 ) Diffusion damping(D H-1lmfp ) (damping factor:exp(-2/D

2) ) Decay of gravitational potential (0, ISW)

Nagata-TC-Sugiyama(2002)

Grecom-G0/G0<0.05 (Nagata-TC=Sugiyama,2004)

Scalar-Tensor Cosmology

Scalar-Tensor Gravity Consequence: Varying G Constraints: z=1010 (BBN)

-0.15<(GBBN-G0)/G0<0.21

(Copi-Davis-Krauss,2004)

z=1100 (CMB) (Grecom-G0)/G0<0.05

z=0 (LRR) dG/dt/G<4x10-13 yr-1

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 ( (v2/r)2/a0=GM/r2 )

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

MOND

Bekenstein’s Tensor-Scalar-Vector theory for MOND

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)

MOND CMB peaks: sensitive to baryon and dark matter

B h2 (shift of zero point of oscillation)

→ first peak height second peak height Mh2 (increases the depth of potential well decreases radiation relative to matter(ISW))→ first peak height second peak → third peak height

MOND trouble with higher (second and third)

peaks of CMB(Slosar-Melchiorri-Silk,2005)

(← 　 Silk damping for baryons)WMAP

WMAP/Boomerang

Modified Gravity III Recent acceleration of the Universe (SNIa)1.Dark energy: modify RHS of Einstein

equation2.Modify LHS instead ⇒ Modified gravity

modification should be significant only recently

→ 　 1/R gravity (Carroll et al., 2003)

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) masslessThis is the case for 1/R gravity :m ~ H0

1/R gravity modifies gravity not only at large scales but also at local scale

Einstein

1/R

ずれ

scale

Rise and Fall of 1/R Gravity

So much ado… F(R,P,Q) gravity?(Carroll et al.,2004) P=RabRab,Q=RabcdRabcd

→ higher derivative (4th order) theory→ Ghosts (Stelle 1977,Nunes,Chiba)propagator:

cross coupling:

The situation is much worse!!

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?)　　　　　　　　　　

Gravity is Probed at …

10-3cm 1AU 1kpc 1Mpc 1000Mpc

large extra dimensions?

MOND?Modified Gravity?

ガモフの飛躍！（ 100 億年→ 3分間）

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 specifyi

ng 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)

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

　　

~H

~2 ~ 2/2

nonlinear

linear

Newton

post Newton

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

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)

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

Searching for alternatives

is important to reinforce the evidence for DM and DE

to check the internal consistency of cosmological data (→ understand systematics)