02+6

2010/12/03 String Seminar @ NTU

f(R) Modified Gravity

with Chameleon Mechanism

Cosmological ^& Solar-System Tests

Collaborators : Wei-Ting Lin 林韋廷 @ Phys, NTU

Dark Energy Working Group @ LeCosPA & NCTS-FGCPA

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

arXiv:1009.3488

2010/12/03 String Seminar @ NTU

f(R) Modified Gravity

with Chameleon Mechanism

Cosmological ^& Solar-System Tests

Collaborators : Wei-Ting Lin 林韋廷 @ Phys, NTU

Dark Energy Working Group @ LeCosPA & NCTS-FGCPA

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

arXiv:1009.3488

Outline

 Introduction

 Cosmological & Solar-System Tests of f(R) Modified Gravity

 Modified Gravity: f(R) and Designer f(R)

 f(R) Modified Gravity with Chameleon mechanism : Solar-System Tests

(appendix)

(appendix)

(highlight of our work)

 Coffee Time announcement

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 G

_μν

＝ 8πG

_N

T

_μν

• 

(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 G

_μν

＝ 8πG

_N

T

_μν

• 

(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 G

_μν

＝ 8πG

_N

T

_μν

• 

(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 G

_μν

＝ 8πG

_N

T

_μν

• 

(from vacuum energy)

• Quintessence / Phantom Dark Energy

↑

1. Einstein GR 2. 3+1 space-time 3. RW metric

FLRW

based on

Candidates: Dark Gravity vs. Dark Energy Einstein Equations

Geometry Matter/Energy G

_μν

＝ 8πG

_N

T

_μν

• 

(from vacuum energy)

• Quintessence / Phantom

minimal coupling to gravity

Dark Energy

↑



^ ^

 

^



 d x g V

S^(min) 



⁴   

Candidates: Dark Gravity vs. Dark Energy Einstein Equations

Geometry Matter/Energy G

_μν

＝ 8πG

_N

T

_μν

• 

(from vacuum energy)

• Quintessence / Phantom

minimal coupling to gravity

non-minimal coupling to gravity (inevitably?) Einstein GR + Non-Min. Scalar Field

Dark Energy

↑



^ ^

 

^



 d x g V

S^(min) 



⁴   

 

_^

   





^{    }

 d x g R V

S^{(nm) 4 2}

2 1

Candidates: Dark Gravity vs. Dark Energy Einstein Equations

Geometry Matter/Energy G

_μν

＝ 8πG

_N

T

_μν

• 

(from vacuum energy)

• Quintessence / Phantom

minimal coupling to gravity

non-minimal coupling to gravity (inevitably?) Einstein GR + Non-Min. Scalar Field

Brans-Dicke gravity Scalar-Tensor gravity special case: f(R)

Dark Energy

↑

 

^R

f g x d R

g x d

S_G ^



^{4 }



⁴

: action gravity

Candidates: Dark Gravity vs. Dark Energy Einstein Equations

Geometry Matter/Energy G

_μν

＝ 8πG

_N

T

_μν

• 

(from vacuum energy)

• Quintessence / Phantom

minimal coupling to gravity

non-minimal coupling to gravity (inevitably?) Einstein GR + Non-Min. Scalar Field

Brans-Dicke gravity Scalar-Tensor gravity

1. Modified Gravity (MG)

special case: f(R)

Dark Energy

↑

MG ~

Candidates: Dark Gravity vs. Dark Energy Einstein Equations

Geometry Matter/Energy G

_μν

＝ 8πG

_N

T

_μν

• 

(from vacuum energy)

• Quintessence / Phantom 1. Modified Gravity (MG)

Dark Energy

↑

MG ~

1. Einstein GR 2. 3+1 space-time 3. RW metric

FLRW

Candidates: Dark Gravity vs. Dark Energy Einstein Equations

Geometry Matter/Energy G

_μν

＝ 8πG

_N

T

_μν

• 

(from vacuum energy)

• Quintessence / Phantom 1. Modified Gravity (MG)

Dark Energy

↑

1. Einstein GR 2. 3+1 space-time 3. RW metric

FLRW

2. Extra Dimensions

Candidates: Dark Gravity vs. Dark Energy Einstein Equations

Geometry Matter/Energy G

_μν

＝ 8πG

_N

T

_μν

• 

(from vacuum energy)

• Quintessence / Phantom 1. Modified Gravity (MG)

Dark Energy

↑

1. Einstein GR 2. 3+1 space-time 3. RW metric

FLRW 2. Extra Dimensions

? Is FLRW a good approximation ??

isotropic homogeneous

Candidates: Dark Gravity vs. Dark Energy Einstein Equations

Geometry Matter/Energy G

_μν

＝ 8πG

_N

T

_μν

• 

(from vacuum energy)

• Quintessence / Phantom 1. Modified Gravity (MG)

Dark Energy

↑

1. Einstein GR 2. 3+1 space-time 3. RW metric

FLRW 2. Extra Dimensions

3. Averaging Einstein Equations for an inhomogeneous universe

? Is FLRW a good approximation ??

Candidates: Dark Gravity vs. Dark Energy Einstein Equations

Geometry Matter/Energy

Dark Geometry

↑

G

_μν

＝ 8πG

_N

T

_μν

3. Averaging Einstein Equations 2. Extra Dimensions

Non-FLRW

for an inhomogeneous universe

• 

(from vacuum energy)

(Gravity)

• Quintessence 1. Modified Gravity (MG)

Dark Energy

↑

1. Einstein GR 2. 3+1 space-time 3. RW metric

FLRW

Candidates: Dark Gravity vs. Dark Energy Einstein Equations

Geometry Matter/Energy

Dark Geometry

↑

G

_μν

＝ 8πG

_N

T

_μν

• Extra Dimensions

• 

(from vacuum energy)

• Quintessence

• Averaging Einstein Equations for an inhomogeneous universe

Back 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:0712.2441 Gu, arXiv:0801.4737

Gu, 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 WG LeCosPA

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 Lee

Chia-Chun Chang (DE & Structures)

Wetty Chao ^(DE)

Vincent Chu (Inhomogeneous Cosmo.)

[ NCTU ]

Tzuu-Kang Chyi

Friends & "historically" active members:

Feng-Yin Chang (LeCosPA, NTU)

Pisin Chen (LeCosPA Director, NTU)

Fei-Hung Ho ^(NCU) Pei-Ming 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)

Dark Energy (Modified Gravity) Working Group

2008.09 - present

2008.

03-08

[ NDHU ]

Tao-Mao Chuang ^(MG)

• Meeting time in 2010 Fall Semester: Wednesday, 4 pm — (indefinite)

• Webpage: http://lecospa.ntu.edu.tw/wg_list.php?wgid=2

DE / MG Working Group meeting

 Meeting time in 2010 Fall Semester: Wednesday, 4 pm — (indefinite)

 Webpage: http://lecospa.ntu.edu.tw/wg_list.php?wgid=2

DE / MG Working Group meeting

 Meeting time in 2010 Fall Semester: Wednesday, 4 pm — (indefinite)

 Webpage: http://lecospa.ntu.edu.tw/wg_list.php?wgid=2

DE / MG Working Group meeting

DE / MG Working Group meeting

Something you may never enjoy unless you attend this meeting !!

 Meeting time in 2010 Spring Semester: Wednesday, 4:30 pm — (indefinite)

 Webpage: http://lecospa.ntu.edu.tw/wg_list.php?wgid=2

Cosmological Models vs.

Dark Energy Modified Gravity

Dark Matter Inflation

Observations SNe Ia

LSS Lensing

CMB Inter-medium

(physical quantities)

d

_L

(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)

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 Gravity with 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].

Dark Energy

Modified Gravity

f(R) Modified Gravity

Cosmological ^& Solar-System Tests

(our work)

f(R) Modified Gravity (MG):

^{Sg } ₁₆¹_G



^{d4x  g}



^{R  f}

^{ }

^R



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):

^{Sg } ₁₆¹_G



^{d4x  g}



^{R  f}

^{ }

^R



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 APPROACH

Consider the expansion H(t) parametrized via

the Chevallier-Polarski-Linder w_eff(z):

  ^{z w w z}  ^z 

w

_CPL



₀



_a

1 

with

current observational constraints (WMAP7+BAO+SN):

72 . 0

71 . 0

0 0.93 0.13, 0.41

053 . 0 980 . 0



 

















a eff

w w

w

(2)

constant (1)

 

 ^{R w w}

_a

^f

_ini

^q

_j



f ;

₀

, , ,

 construct

q_j : other cosmological parameters f_ini : initial condition of f(R)

f(R) Modified Gravity (MG):

^{Sg } ₁₆¹_G



^{d4x  g}



^{R  f}

^{ }

^R



Cosmic Expansion Cosmic Structure Solar-System Test Cosmological Test Local Test

q_j : other cosmological parameters f_ini : initial condition of f(R)

Example w_eff = 1

For a given expansion history H(t), one can reconstruct f(R)

which generates the required H(t).

FACT

“designer f(R)”

OUR APPROACH

Consider the expansion H(t) parametrized via

the Chevallier-Polarski-Linder w_eff(z):

  ^{z w w z}  ^z 

w

_CPL



₀



_a

1 

 

 ^{R w w}

_a

^f

_ini

^q

_j



f ;

₀

, , ,

 construct

f/H 02 +6 DE

2

H0

R

f(R) Modified Gravity (MG):

^{Sg } ₁₆¹_G



^{d4x  g}



^{R  f}

^{ }

^R



Cosmic Expansion Cosmic Structure Solar-System Test Cosmological Test Local Test

q_j : other cosmological parameters f_ini : initial condition of f(R)

Then, proceed to the other two tests of

 

 R w w

a

f

ini

q

j



f ;

₀

, , ,

“designer f(R)”

OUR APPROACH

Consider the expansion H(t) parametrized via

the Chevallier-Polarski-Linder w_eff(z):

  ^{z w w z}  ^z 

w

_CPL



₀



_a

1 

with

observational constraints (WMAP7+BAO+SN):

72 . 0

71 . 0

0 0.93 0.13, 0.41

053 . 0 980 . 0



 

















a eff

w w

w

(2)

constant (1)

 

 ^{R w w}

_a

^f

_ini

^q

_j



f ;

₀

, , ,

 construct

f(R) Modified Gravity (MG):

^{Sg } ₁₆¹_G



^{d4x  g}



^{R  f}

^{ }

^R



Cosmic Expansion Cosmic Structure Solar-System Test Cosmological Test Local Test

 Key quantities distinguishing GR & MG





G G

_eff

m m m

m 



   ^

 Perturbed metric:

  ^{dt a}  

_ij

^dx

ⁱ

^dx

^j

ds

²

  1  2 

²



²

1  2  

 Evolution eqn. of matter density perturbation:

defined in :

0 4

2  



_{m eff m m}

m

H   G  

 ^{ }

^late-time,

sub-horizon

f(R) Modified Gravity (MG):

^{Sg } ₁₆¹_G



^{d4x  g}



^{R  f}

^{ }

^R



Cosmic Expansion Cosmic Structure Solar-System Test Cosmological Test Local Test

GR

1





1 G G_eff

 

R RR

R RR

R eff

f f a k

f f a k f

G a k G

 

 

 

3 1 1

4 1 1 1

1 ,

2 2 2 2

 

R RR

R RR

f f a k

f f a k a

k

 

 

 



2 1 1

4 1 1 ,

2 2 2 2

f(R) MG

^initial

;

, ₂

2

R Ri RR

R f f

R f f

R

f f 



 



 

late-time, sub-horizon

f(R) Modified Gravity (MG):

^{Sg } ₁₆¹_G



^{d4x  g}



^{R  f}

^{ }

^R



Cosmic Expansion Cosmic Structure Solar-System Test Cosmological Test Local Test

GR

1





1 G G_eff

 

R RR

R RR

R eff

f f a k

f f a k f

G a k G

 

 

 

3 1 1

4 1 1 1

1 ,

2 2 2 2

 

R RR

R RR

f f a k

f f a k a

k

 

 

 



2 1 1

4 1 1 ,

2 2 2 2

f(R) MG

^initial

;

, ₂

2

R Ri RR

R f f

R f f

R

f f 



 



 

late-time, sub-horizon

 

 R w w

a

f

ini

q

j



f ;

₀

, , ,

“designer f(R)”



^,



^;



^;



0^{, , ;}

 





 





j Ri

a f q

w w R f a

 function of k

f(R) Modified Gravity (MG):

^{Sg } ₁₆¹_G



^{d4x  g}



^{R  f}

^{ }

^R



Cosmic Expansion Cosmic Structure Solar-System Test Cosmological Test Local Test

E.g. w

_eff

= 1

For the present time and k = 0.01h / Mpc.

Mpc 1

01 .

0 ^

 h

k

10

³¹

f

Ri

 / (now)

403 . 1

1G_eff G  1   1.996

Observational constraint (Giannantonio et al, 2009):

initial

2 2

R Ri

RR R

f f

R f f

R f f





 



  73

. 0

27 .

0

 

eff m

f(R) Modified Gravity (MG):

^{Sg } ₁₆¹_G



^{d4x  g}



^{R  f}

^{ }

^R



Cosmic Expansion Cosmic Structure Solar-System Test Cosmological Test Local Test

E.g. w

_eff

= 1

For the present time and k = 0.01h / Mpc.

Mpc 1

01 .

0 ^

 h

k

10

³¹

f

Ri

 / (now)

403 . 1

1G_eff G  1   1.996

Observational constraint (Giannantonio et al, 2009):

initial

2 2

R Ri

RR R

f f

R f f

R f f





 



 

GR

most f (R)

Similar behavior for other w_eff(z).

73 . 0

27 .

0

 

eff m

f(R) Modified Gravity (MG):

^{Sg } ₁₆¹_G



^{d4x  g}



^{R  f}

^{ }

^R



Cosmic Expansion Cosmic Structure Solar-System Test Cosmological Test Local Test

initial

2 2

R Ri

RR R

f f

R f f

R f f





 



 

E.g. w_eff = constant

 / (now)

GR

most f (R)

Similar behavior for other w_eff(z).

f(R) Modified Gravity (MG):

^{Sg } ₁₆¹_G



^{d4x  g}



^{R  f}

^{ }

^R



Cosmic Expansion Cosmic Structure Solar-System Test Cosmological Test Local Test

initial

2 2

R Ri

RR R

f f

R f f

R f f





 



 

E.g. w_eff = CPL best fit

 / (now)

GR

most f (R)

Similar behavior for other w_eff(z).

 ^{z w w z}  ^z

w_CPL  ₀  _a 1

f(R) Modified Gravity (MG):

^{Sg } ₁₆¹_G



^{d4x  g}



^{R  f}

^{ }

^R



Cosmic Expansion Cosmic Structure Solar-System Test Cosmological Test Local Test

f

Ri

viable

w

eff

constant

initial

2 2

R Ri

RR R

f f

R f f

R f f





 



 

f(R) Modified Gravity (MG):

^{Sg } ₁₆¹_G



^{d4x  g}



^{R  f}

^{ }

^R



Cosmic Structure

Cosmic Expansion Solar-System Test

Cosmological Test Local Test

 Constraint on

5 2

0

5 2

0 15

10

, 5 2 0

10

, 0 10













 ^

H R Rf

H R f

RR R

f(R) MG with

Chameleon Mechanism

f(R) Modified Gravity (MG):

^{Sg } ₁₆¹_G



^{d4x  g}



^{R  f}

^{ }

^R



Cosmic Structure

Cosmic Expansion Solar-System Test

Cosmological Test Local Test

parameter space

survey around GR point

f = constant



^1,0



 Viable ^f  ^{R ;}  ^w

_eff

^{, f}

_ini

 

 Constraint on

5 2

0

5 2

0 15

10

, 5 2 0

10

, 0 10













 ^

H R Rf

H R f

RR R

f(R) MG with

Chameleon Mechanism

(constant w_eff)

f(R) Modified Gravity (MG):

^{Sg } ₁₆¹_G



^{d4x  g}



^{R  f}

^{ }

^R



Cosmic Structure

Cosmic Expansion Solar-System Test

Cosmological Test Local Test

36 9

10 10 1











Ri eff

f w

parameter space

survey around GR point

f = constant



^1,0



 Viable ^f  ^{R ;}  ^w

_eff

^{, f}

_ini

 

6 6

10 1

10 1











 



G G_eff

very small viable region

 Constraint on

5 2

0

5 2

0 15

10

, 5 2 0

10

, 0 10













 ^

H R Rf

H R f

RR R

f(R) MG with

Chameleon Mechanism

(constant w_eff)

f(R) Modified Gravity (MG):

^{Sg } ₁₆¹_G



^{d4x  g}



^{R  f}

^{ }

^R



Cosmic Structure

Cosmic Expansion Solar-System Test

Cosmological Test Local Test

36 9

10 10 1











Ri eff

f

w closely

mimicking GR + 

parameter space

survey around GR point

f = constant



^1,0



 Viable ^f  ^{R ;}  ^w

_eff

^{, f}

_ini

 

6 6

10 1

10 1











 



G G_eff

indistinguishable from GR !!

very small viable region

 Constraint on

5 2

0

5 2

0 15

10

, 5 2 0

10

, 0 10













 ^

H R Rf

H R f

RR R

f(R) MG with

Chameleon Mechanism

(constant w_eff)

f(R) Modified Gravity (MG):

^{Sg } ₁₆¹_G



^{d4x  g}



^{R  f}

^{ }

^R



Cosmic Structure

Cosmic Expansion Solar-System Test

Cosmological Test Local Test

The viable f(R) models in the parameter space (w_eff, f_Ri) around the GR point (



1,0) for constant w_eff.

w

eff

 1

f

Ri GR

 Constraint on

5 2

0

5 2

0 15

10

, 5 2 0

10

, 0 10













 ^

H R Rf

H R f

RR R

f(R) MG with

Chameleon Mechanism

^initial

2 2

R Ri

RR R

f f

R f f

R f f





 



 

Conclusion

Cosmic Expansion

Solar-System Test Cosmic Structure

 The existence of the designer models which pass the cosmic-structure test

would require fine-tuning of initial condition f

_ini

.

 

 ^{R w w}

_a

^f

_ini



f ;

₀

, ,

 Designer w.r.t. the constraint on {w

₀

,w

_a

} (by design) can pass the cosmic-expansion test.

 

 ^{R w w}

_a

^f

_ini



f ;

₀

, ,

(observational)

 Among the designer models,

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

 

 ^{R w w}

_a

^f

_ini



f ;

₀

, ,



As a result,

the solar-system test rules out

the frequently studied designer models with that are distinct from  CDM in .

  ^R

f w

_eff

  1

G G

_eff

 ,



i.e., same as CDM in cosmic expansion

Appendix

Modified Gravity :

f(R) ^and designer f(R)

(Appendix)

Candidates: Modified Gravity vs. Dark Energy Einstein Equations

Modified Gravity

G

_μν

＝ 8πG

_N

T

_μν

Dark Energy

Different effects (predictions) on:

 Cosmic expansion (background evolution): a(t)

 Evolution of cosmic perturbations: 

_m

, ,  (k,a)

matter density perturb. metric perturb.

CL WL ISW SN BAO CMB

∞ tests !!

one tes t

Another motivation/goal of MG investigations:

Cosmological test of alternative gravity theories



Modified Gravity : L

G

= R + f (R) with flat RW

 Cosmic expansion (background evolution): a(t) one test

  _ ^

    



 G

_m

Rf

_R

f H f

_R

H f

_R

H

^{2 2}



6 1 3

8  

 

 _{m DE DE}

N

DE m

N

G p

G a

H a

3 3 4 a

a

3 8

: GR

2 2











 



 







 



 





 

     





 G

_m

f H f

_R

H f

_R

f

_R

a

a   

2 1 2

1 6

1 3

4  

₂

(modified gravity action)

2 2

,

R

f f R

f R

f

_R

f

_RR



 

 

 







²



2 6 2

6 H H H

a

R a   



 

 

  