2010/11/06 DM-DE-BA Workshop @ NTHU

f(R) Modified GravityCosmological & Solar-System Tests

Je-An Gu 顧哲安臺灣大學梁次震宇宙學與粒子天文物理學研究中心

Leung Center for Cosmology and Particle Astrophysics (LeCosPA), NTU

Collaborators : Wei-Ting Lin 林韋廷 @ Phys, NTU Dark Energy Working Group @ LeCosPA & NCTS-FGCPA

arXiv:1009.3488

2010/09/27 COSMO/CosPA @ Tokyo Univ.

f(R) Modified GravityCosmological & Solar-System Tests

Je-An Gu 顧哲安臺灣大學梁次震宇宙學與粒子天文物理學研究中心

Leung Center for Cosmology and Particle Astrophysics (LeCosPA), NTU

Collaborators : Wei-Ting Lin 林韋廷 @ Phys, NTU Dark Energy Working Group @ LeCosPA & NCTS-FGCPA

arXiv:1009.3488

Friends & "historically" active members:

Feng-Yin Chang (LeCosPA, NTU)Pisin Chen (LeCosPA Director, NTU)Fei-Hung Ho (NCU)Pei-Min Ho (NTU)Qing-Guo Huang (KIAS, Korea)Kwang-Chang Lai (NCTU)Seokcheon (Sky) Lee (IoPAS)Guo-Chin Liu (TKU)Debaprasad Maity (LeCosPA, NTU)Kin-Wang Ng (IoPAS) (DE WG Leader)Hau-Yu Liu (NTU & ASIAA)Yen-Wei Liu (NTU)Tao-Tao Qiu (CYCU)Yong Tian (NCU)Keiichi Umetsu (ASIAA)I-Chin Wang (NTNU)

Current active members: [ NTU ]Je-An Gu (LeCosPA) (DE WG server)Chien-Wen Chen (LeCosPA)(DE)A. E. Romano (LeCosPA)(Inhomog., Inflation)Huitzu Tu (LeCosPA)(DM)Florian Borchers (Germany)Wei-Ting Lin (MG key worker)Pao-Yu Wang (DE)Tse-Chun Wang (MG)Yen-Tin Wu (MG)[ NTNU ]Wolung LeeChia-Chun Chang (DE & Structures)Wetty Chao (DE)Vincent Chu (Inhomogeneous Cosmo.)[ NCTU ]Tzuu-Kang Chyi

Dark Energy (Modified Gravity) Working Group

2008.09 - present

2008.03-08

[ NDHU ]Tao-Mao Chuang (MG)

Introduction

Concordance: = 0.73 , M = 0.27

Accelerating Expansion

(homogeneous & isotropic)Based on FLRW Cosmology

Observations (which are driving Modern Cosmology)

Candidates: Dark Gravity vs. Dark Energy

Einstein Equations

Geometry Matter/Energy

Gμν ＝ 8πGNTμν

• (from vacuum energy)Many success factors: fit data connections: - to famous person: Einstein- to well known theory: QFT- to nature of creator: simple

Candidates: Dark Gravity vs. Dark Energy

Einstein Equations

Geometry Matter/Energy

Gμν ＝ 8πGNTμν

• (from vacuum energy)Many success factors: fit data connections: - to famous person: Einstein- to well known theory: QFT- to nature of creator: simple

Issues: (why small) problem (why now) coincidence problem

To avoid the issues,necessary condition: Energy density changes with time.

Candidates: Dark Gravity vs. Dark Energy

Einstein Equations

Geometry Matter/Energy

Gμν ＝ 8πGNTμν

• (from vacuum energy)

Issues: (why small) problem (why now) coincidence problem

• Quintessence / Phantom(a simple realization)

To avoid the issues,necessary condition: Energy density changes with time.

Candidates: Dark Gravity vs. Dark Energy

Einstein Equations

Geometry Matter/Energy

Gμν ＝ 8πGNTμν

• (from vacuum energy)

• Quintessence / Phantom

Dark Energy↑

1. Einstein GR2. 3+1 space-time3. RW metric

FLRWbased on

Candidates: Dark Gravity vs. Dark Energy

Einstein Equations

Geometry Matter/Energy

Gμν ＝ 8πGNTμν

• (from vacuum energy)

• Quintessence / Phantomminimal coupling to gravity

Dark Energy↑

VgxdS 4(min)

Candidates: Dark Gravity vs. Dark Energy

Einstein Equations

Geometry Matter/Energy

Gμν ＝ 8πGNTμν

• (from vacuum energy)

• Quintessence / Phantomminimal coupling to gravity

non-minimal coupling to gravity

(inevitably?)

Einstein GR + Non-Min. Scalar Field

Dark Energy↑

VgxdS 4(min)

VRgxdS 24)nm( 2

1

Candidates: Dark Gravity vs. Dark Energy

Einstein Equations

Geometry Matter/Energy

Gμν ＝ 8πGNTμν

• (from vacuum energy)

• Quintessence / Phantomminimal coupling to gravity

non-minimal coupling to gravity

(inevitably?)

Einstein GR + Non-Min. Scalar Field

Brans-Dicke gravityScalar-Tensor gravity

special case: f(R)

Dark Energy↑

RfgxdRgxdSG 44 :actiongravity

Candidates: Dark Gravity vs. Dark Energy

Einstein Equations

Geometry Matter/Energy

Gμν ＝ 8πGNTμν

• (from vacuum energy)

• Quintessence / Phantomminimal coupling to gravity

non-minimal coupling to gravity

(inevitably?)

Einstein GR + Non-Min. Scalar Field

Brans-Dicke gravityScalar-Tensor gravity

1. Modified Gravity (MG)

special case: f(R)

Dark Energy↑

MG ~

Candidates: Dark Gravity vs. Dark Energy

Einstein Equations

Geometry Matter/Energy

Gμν ＝ 8πGNTμν

• (from vacuum energy)

• Quintessence / Phantom

1. Modified Gravity (MG)

Dark Energy↑

MG ~

1. Einstein GR2. 3+1 space-time3. RW metric

FLRW

Candidates: Dark Gravity vs. Dark Energy

Einstein Equations

Geometry Matter/Energy

Gμν ＝ 8πGNTμν

• (from vacuum energy)

• Quintessence / Phantom

1. Modified Gravity (MG)

Dark Energy↑

1. Einstein GR2. 3+1 space-time3. RW metric

FLRW

2. Extra Dimensions

Candidates: Dark Gravity vs. Dark Energy

Einstein Equations

Geometry Matter/Energy

Gμν ＝ 8πGNTμν

• (from vacuum energy)

• Quintessence / Phantom

1. Modified Gravity (MG)

Dark Energy↑

1. Einstein GR2. 3+1 space-time3. RW metric

FLRW

2. Extra Dimensions

? Is FLRW a good approximation ??

isotropichomogeneous

Candidates: Dark Gravity vs. Dark Energy

Einstein Equations

Geometry Matter/Energy

Gμν ＝ 8πGNTμν

• (from vacuum energy)

• Quintessence / Phantom

1. Modified Gravity (MG)

Dark Energy↑

1. Einstein GR2. 3+1 space-time3. RW metric

FLRW

2. Extra Dimensions

3. Averaging Einstein Equationsfor an inhomogeneous universe

? Is FLRW a good approximation ??

Candidates: Dark Gravity vs. Dark Energy

Einstein Equations

Geometry Matter/Energy

Dark Geometry↑

Gμν ＝ 8πGNTμν

3. Averaging Einstein Equations

2. Extra Dimensions

Non-FLRWfor an inhomogeneous universe

• (from vacuum energy)

(Gravity)

• Quintessence

1. Modified Gravity (MG)

Dark Energy↑

1. Einstein GR2. 3+1 space-time3. RW metric

FLRW

Candidates: Dark Gravity vs. Dark Energy

Einstein Equations

Geometry Matter/Energy

Dark Geometry↑

Gμν ＝ 8πGNTμν

• Extra Dimensions

• (from vacuum energy)

• Quintessence• Averaging Einstein Equationsfor an inhomogeneous universeBack reaction of inhomogeneities

(Gravity)

Gu and Hwang, Phy.Rev.D (2002); Gu, Hwang and Tsai, Nucl.Phys.B (2004)

Chuang, Gu and Hwang, Class.Quant.Grav (2008)

Gu and Hwang, Phys.Rev.D (2006)Chen and Gu, arXiv:0712.2441Gu, arXiv:0801.4737Gu, Nucl.Phys.A (2010)Shao, Chen and Gu, Phys.Rev.D (2009)

Gu and Hwang, Phys.Lett.B (2001)Gu, Mod.Phys.Lett.A (2008)Gu, arXiv:0711.36

Dark Energy↑

• Modified Gravity (MG)DE/MG WGLeCosPA

vs.Cosmological Models

Dark EnergyModified Gravity

Dark MatterInflation

Observations

SNe IaLSS

LensingCMB

Inter-medium(physical quantities)

dL(z)

matter power spectrum

CMB A&P spectrum

Theoretical prediction Observational info

Technique / Know-how

(Dark Energy & Modified Gravity) Phenomenology

(not limited to DE models)

(Dark Energy & Modified Gravity) Phenomenology Je-An Gu, Chien-Wen Chen, and Pisin Chen,

“A new approach to testing dark energy models by observations,”New Journal of Physics 11 (2009) 073029 [arXiv:0803.4504].

Chien-Wen Chen, Je-An Gu, and Pisin Chen,“Consistency test of dark energy models,”Modern Physics Letters A 24 (2009) 1649 [arXiv:0903.2423].

Chien-Wen Chen, Pisin Chen, and Je-An Gu,“Constraints on the phase plane of the dark energy equation of state,”Physics Letters B 682 (2009) 267 [arXiv:0905.2738].

Chien-I Chiang, Je-An Gu, and Pisin Chen,“Constraining the Detailed Balance Condition in Hořava Gravitywith Cosmic Accelerating Expansion,”

to appear in JCAP [arXiv:1007.0543].

Wei-Ting Lin, Je-An Gu, and Pisin Chen,“Cosmological and Solar-System Tests of f (R) Modified Gravity,”submitted for publication [arXiv:1009.3488].

f(R) Modified GravityCosmological & Solar-System Tests

(our work)

f(R) Modified Gravity (MG): RfRgxdGSg4

161

Cosmic Expansion Cosmic Structure Solar-System Test

Cosmological Test Local Test

as an essence of cosmology, need to pass

Purposes

as a theory of modified gravity, need to pass

Explain cosmic acceleration

Model (parameterize)deviation from GR

f(R) Modified Gravity (MG): RfRgxdGSg4

161

Cosmic Expansion Cosmic Structure Solar-System Test

Cosmological Test Local Test

For a given expansion history H(t),one can reconstruct f(R) which generates the required H(t).

FACT

“designer f(R)”

OUR APPROACHConsider the expansion H(t) parametrized viathe Chevallier-Polarski-Linder weff(z):

zzwwzw aCPL 10

withcurrent observational constraints (WMAP7+BAO+SN):

72.071.00 41.0,13.093.0

053.0980.0

a

eff

ww

w

(2)

constant (1)

jinia qfwwRf ,,,; 0 constructqj : other cosmological parametersfini : initial condition of f(R)

f(R) Modified Gravity (MG): RfRgxdGSg4

161

Cosmic Expansion Cosmic Structure Solar-System Test

Cosmological Test Local Test

qj : other cosmological parametersfini : initial condition of f(R)

Exampleweff = 1

For a given expansion history H(t),one can reconstruct f(R) which generates the required H(t).

FACT

“designer f(R)”

OUR APPROACHConsider the expansion H(t) parametrized viathe Chevallier-Polarski-Linder weff(z):

zzwwzw aCPL 10

jinia qfwwRf ,,,; 0 construct

f/H

02+

6D

E20HR

f(R) Modified Gravity (MG): RfRgxdGSg4

161

Cosmic Expansion Cosmic Structure Solar-System Test

Cosmological Test Local Test

qj : other cosmological parametersfini : initial condition of f(R)

Then, proceed to the other two tests of jinia qfwwRf ,,,; 0“designer f(R)”

OUR APPROACHConsider the expansion H(t) parametrized viathe Chevallier-Polarski-Linder weff(z):

zzwwzw aCPL 10

withobservational constraints (WMAP7+BAO+SN):

72.071.00 41.0,13.093.0

053.0980.0

a

eff

ww

w

(2)

constant (1)

jinia qfwwRf ,,,; 0 construct

f(R) Modified Gravity (MG): RfRgxdGSg4

161

Cosmic Expansion Cosmic Structure Solar-System Test

Cosmological Test Local Test

Key quantities distinguishing GR & MG

GGeff

Perturbed metric: jiij dxdxadtds 2121 222

Evolution eqn. of matter density perturbation:

defined in :

042 mmeffmm GH late-time,sub-horizon

f(R) Modified Gravity (MG): RfRgxdGSg4

161

Cosmic Expansion Cosmic Structure Solar-System Test

Cosmological Test Local Test

GR

1

1G

Geff

R

RR

R

RR

R

eff

ff

ak

ff

ak

fGakG

131

141

11,

2

2

2

2

R

RR

R

RR

ff

ak

ff

ak

ak

121

141

,

2

2

2

2f(R) MG

initial ; , 22

RRiRRR ffRff

Rff

late-time,sub-horizon

f(R) Modified Gravity (MG): RfRgxdGSg4

161

Cosmic Expansion Cosmic Structure Solar-System Test

Cosmological Test Local Test

GR

1

1G

Geff

R

RR

R

RR

R

eff

ff

ak

ff

ak

fGakG

131

141

11,

2

2

2

2

R

RR

R

RR

ff

ak

ff

ak

ak

121

141

,

2

2

2

2f(R) MG

initial ; , 22

RRiRRR ffRff

Rff

late-time,sub-horizon

jinia qfwwRf ,,,; 0“designer f(R)”

;,, ; ; , 0

jRia qfwwRfak function of

f(R) Modified Gravity (MG): RfRgxdGSg4

161

Cosmic Expansion Cosmic Structure Solar-System Test

Cosmological Test Local Test

E.g. weff = 1

For the present timeand k = 0.01h / Mpc.

1Mpc 01.0 hk Rif3110

/

(now

)

403.11 GGeff 996.11

Observational constraint (Giannantonio et al, 2009):

initial2

2

RRi

RR

R

ffR

ff

Rff

73.027.0

eff

m

f(R) Modified Gravity (MG): RfRgxdGSg4

161

Cosmic Expansion Cosmic Structure Solar-System Test

Cosmological Test Local Test

E.g. weff = 1

For the present timeand k = 0.01h / Mpc.

1Mpc 01.0 hk Rif3110

/

(now

)

403.11 GGeff 996.11

Observational constraint (Giannantonio et al, 2009):

initial2

2

RRi

RR

R

ffR

ff

Rff

GR

mostf (R)

Similarbehaviorfor other weff(z).

73.027.0

eff

m

f(R) Modified Gravity (MG): RfRgxdGSg4

161

Cosmic Expansion Cosmic Structure Solar-System Test

Cosmological Test Local Test

initial2

2

RRi

RR

R

ffR

ff

Rff

E.g. weff = constant

/

(now

)

GR

mostf (R)

Similarbehaviorfor other weff(z).

f(R) Modified Gravity (MG): RfRgxdGSg4

161

Cosmic Expansion Cosmic Structure Solar-System Test

Cosmological Test Local Test

initial2

2

RRi

RR

R

ffR

ff

Rff

E.g. weff = CPL best fit

/

(now

)

GR

mostf (R)

Similarbehaviorfor other weff(z).

f(R) Modified Gravity (MG): RfRgxdGSg4

161

Cosmic Expansion Cosmic Structure Solar-System Test

Cosmological Test Local Test

Rif viable

effwconstant initial

2

2

RRi

RR

R

ffR

ff

Rff

f(R) Modified Gravity (MG): RfRgxdGSg4

161

Cosmic StructureCosmic Expansion Solar-System Test

Cosmological Test Local Test

Constraint on

520

520

15

10 ,52 010 ,010

HRRfHRf

RR

R

f(R) MG withChameleon Mechanism

f(R) Modified Gravity (MG): RfRgxdGSg4

161

Cosmic StructureCosmic Expansion Solar-System Test

Cosmological Test Local Test

parameter space

survey around GR point

f = constant 0 , 1

Viable inieff fwRf ,;

Constraint on

520

520

15

10 ,52 010 ,010

HRRfHRf

RR

R

f(R) MG withChameleon Mechanism

(constant weff)

f(R) Modified Gravity (MG): RfRgxdGSg4

161

Cosmic StructureCosmic Expansion Solar-System Test

Cosmological Test Local Test

36

9

10 101

Ri

eff

fw

parameter space

survey around GR point

f = constant 0 , 1

Viable inieff fwRf ,;

6

6

10 1

10 1

GGeff

very small viable region

Constraint on

520

520

15

10 ,52 010 ,010

HRRfHRf

RR

R

f(R) MG withChameleon Mechanism

(constant weff)

f(R) Modified Gravity (MG): RfRgxdGSg4

161

Cosmic StructureCosmic Expansion Solar-System Test

Cosmological Test Local Test

36

9

10 101

Ri

eff

fw closely

mimicking GR +

parameter space

survey around GR point

f = constant 0 , 1

Viable inieff fwRf ,;

6

6

10 1

10 1

GGeff

indistinguishable from GR !!

very small viable region

Constraint on

520

520

15

10 ,52 010 ,010

HRRfHRf

RR

R

f(R) MG withChameleon Mechanism

(constant weff)

f(R) Modified Gravity (MG): RfRgxdGSg4

161

Cosmic StructureCosmic Expansion Solar-System Test

Cosmological Test Local Test

The viable f(R) models in the parameter space (weff, fRi) around the GR point (1,0) for constant weff.

effw1

f Ri

GR

Constraint on

520

520

15

10 ,52 010 ,010

HRRfHRf

RR

R

f(R) MG withChameleon Mechanism

ConclusionCosmic Expansion

Solar-System Test

Cosmic Structure The existence of the designer models

which pass the cosmic-structure testwould require fine-tuning of initial condition fini.

inia fwwRf ,,; 0

Designer w.r.t. the constraint on {w0,wa} (by design) can pass the cosmic-expansion test.

inia fwwRf ,,; 0(observational)

Among the designer models,only those closely mimicking GR + (in all the 3 tests)can pass the solar-system test.

inia fwwRf ,,; 0

As a result, the solar-system test rules outthe frequently studied modelsthat are distinct from CDM in .

inifRf ; 1effwGGeff ,

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