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
NT
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
NT
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
NT
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
NT
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
NT
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
NT
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
NT
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
↑
Rf g x d R
g x d
SG
4
4: action gravity
Candidates: Dark Gravity vs. Dark Energy Einstein Equations
Geometry Matter/Energy G
μν= 8πG
NT
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
NT
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
NT
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
NT
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
NT
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
NT
μν3. 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
NT
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 161G
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 161G
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 weff(z):
z w w z z
w
CPL
0
a1
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
af
iniq
j
f ;
0, , ,
construct
qj : other cosmological parameters fini : initial condition of f(R)
f(R) Modified Gravity (MG):
Sg 161G
d4x g
R f
R
Cosmic Expansion Cosmic Structure Solar-System Test Cosmological Test Local Test
qj : other cosmological parameters fini : initial condition of f(R)
Example weff = 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 weff(z):
z w w z z
w
CPL
0
a1
R w w
af
iniq
j
f ;
0, , ,
construct
f/H 02 +6 DE
2
H0
R
f(R) Modified Gravity (MG):
Sg 161G
d4x g
R f
R
Cosmic Expansion Cosmic Structure Solar-System Test Cosmological Test Local Test
qj : other cosmological parameters fini : initial condition of f(R)
Then, proceed to the other two tests of
R w w
af
iniq
j
f ;
0, , ,
“designer f(R)”
OUR APPROACH
Consider the expansion H(t) parametrized via
the Chevallier-Polarski-Linder weff(z):
z w w z z
w
CPL
0
a1
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
af
iniq
j
f ;
0, , ,
construct
f(R) Modified Gravity (MG):
Sg 161G
d4x g
R f
R
Cosmic Expansion Cosmic Structure Solar-System Test Cosmological Test Local Test
Key quantities distinguishing GR & MG
G G
effm m m
m
Perturbed metric:
dt a
ijdx
idx
jds
2 1 2
2
21 2
Evolution eqn. of matter density perturbation:
defined in :
0 4
2
m eff m mm
H G
late-time,sub-horizon
f(R) Modified Gravity (MG):
Sg 161G
d4x g
R f
R
Cosmic Expansion Cosmic Structure Solar-System Test Cosmological Test Local Test
GR
1
1 G Geff
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 161G
d4x g
R f
R
Cosmic Expansion Cosmic Structure Solar-System Test Cosmological Test Local Test
GR
1
1 G Geff
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
af
iniq
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 161G
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
31f
Ri / (now)
403 . 1
1Geff 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 161G
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
31f
Ri / (now)
403 . 1
1Geff 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 weff(z).
73 . 0
27 .
0
eff m
f(R) Modified Gravity (MG):
Sg 161G
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. weff = constant
/ (now)
GR
most f (R)
Similar behavior for other weff(z).
f(R) Modified Gravity (MG):
Sg 161G
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. weff = CPL best fit
/ (now)
GR
most f (R)
Similar behavior for other weff(z).
z w w z z
wCPL 0 a 1
f(R) Modified Gravity (MG):
Sg 161G
d4x g
R f
R
Cosmic Expansion Cosmic Structure Solar-System Test Cosmological Test Local Test
f
Riviable
w
effconstant
initial
2 2
R Ri
RR R
f f
R f f
R f f
f(R) Modified Gravity (MG):
Sg 161G
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 161G
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 weff)
f(R) Modified Gravity (MG):
Sg 161G
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 Geff
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 weff)
f(R) Modified Gravity (MG):
Sg 161G
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 Geff
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 weff)
f(R) Modified Gravity (MG):
Sg 161G
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 (weff, fRi) around the GR point (
1,0) for constant weff.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
af
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
af
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
af
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
NT
μνDark 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
mRf
Rf H f
RH f
RH
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
mf H f
RH f
Rf
Ra
a
2 1 2
1 6
1 3
4
2(modified gravity action)
2 2
,
R
f f R
f R
f
Rf
RR
2
2 6 2
6 H H H
a
R a