f(R) Modified Gravity
Cosmological & Solar-System Tests
arXiv:1009.3488
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
2010/09/27 COSMO/CosPA @ Tokyo Univ.
f(R) Modified Gravity (MG):
Sg 
1
d 4 x  g R  f R 

16G
Purposes
Explain
cosmic acceleration
Model (parameterize)
deviation from GR
as an essence of cosmology,
need to pass
as a theory of modified gravity,
need to pass
Cosmological Test
Cosmic Expansion
Cosmic Structure
Local Test
Solar-System Test
f(R) Modified Gravity (MG):
Sg 
1
d 4 x  g R  f R 

16G
Cosmological Test
Cosmic Expansion
Local Test
Cosmic Structure
Solar-System Test
FACT
For a given expansion history H(t),
one can reconstruct f(R)
“designer f(R)”
which generates the required H(t).
OUR APPROACH
with
Consider the expansion H(t)
parametrized via
the Chevallier-Polarski-Linder weff(z):
current observational constraints
wCPL z   w 0  w a z 1  z 
(WMAP7+BAO+SN):
(1) w eff  constant  0.980  0.053
72
(2) w 0  0.93  0.13, w a  0.4100..71
 construct f R;w 0 ,w a , fini , q j 
fini : initial condition of f(R)
qj : other cosmological parameters
f(R) Modified Gravity (MG):
Sg 
1
d 4 x  g R  f R 

16G
Cosmological Test
Cosmic Expansion
Local Test
Cosmic Structure
Solar-System Test
FACT
OUR APPROACH
Consider the expansion H(t)
parametrized via
the Chevallier-Polarski-Linder weff(z):
wCPL z   w 0  w a z 1  z 
f / H02 + 6DE
For a given expansion history H(t),
one can reconstruct f(R)
“designer f(R)” w = 1 Example
eff
which generates the required H(t).
R H02
 construct f R;w 0 ,w a , fini , q j 
fini : initial condition of f(R)
qj : other cosmological parameters
f(R) Modified Gravity (MG):
Sg 
1
d 4 x  g R  f R 

16G
Cosmological Test
Cosmic Expansion
Local Test
Cosmic Structure
Solar-System Test
Then, proceed to the other two tests of
“designer f(R)” f R; w 0 ,w a , fini , q j
 
OUR APPROACH
Consider the expansion H(t)
parametrized via
the Chevallier-Polarski-Linder weff(z):
wCPL z   w 0  w a z 1  z 

with
observational constraints
(WMAP7+BAO+SN):
(1) w eff  constant  0.980  0.053
72
(2) w 0  0.93  0.13, w a  0.4100..71
 construct f R;w 0 ,w a , fini , q j 
fini : initial condition of f(R)
qj : other cosmological parameters
f(R) Modified Gravity (MG):
Sg 
1
d 4 x  g R  f R 

16G
Cosmological Test
Cosmic Expansion
Local Test
Cosmic Structure
Solar-System Test
 Key quantities distinguishing GR & MG


Geff
G
defined in :
 Perturbed metric:
ds 2  1 2 dt 2  a2 1 2 ij dx i dx j
 Evolution eqn. of matter density perturbation:
late-time,
  2H  4G    0
m
m
eff
m m
sub-horizon
f(R) Modified Gravity (MG):
Sg 
1
d 4 x  g R  f R 

16G
Cosmological Test
Cosmic Expansion
Cosmic Structure
GR
“designer f(R)”
f R; w 0 ,w a , fini , q j
 

Local Test

1

Geff
1
G
Solar-System Test
f(R) MG
k 2 fRR
1 4 2

a 1  fR
k, a  
k 2 fRR

1 2 2
a 1  fR
late-time,
sub-horizon
k 2 fRR
1 4 2
Geff k, a 
1
a 1  fR

k 2 fRR
G
1  fR
1 3 2
a 1  fR




 function of k , a  ; f R; w 0 ,w a , fRi ;q j  
f
 2f
fR 
, fRR 
; fRi  fR initial
R
R 2
f(R) Modified Gravity (MG):
Sg 
1
d 4 x  g R  f R 

16G
Cosmological Test
Cosmic Expansion
Local Test
Cosmic Structure
Solar-System Test
E.g. weff = 1
 /  (now)
most
f (R)
 m  0.27
 eff  0.73
For the present time
and k = 0.01h / Mpc.
Similar
behavior
for other
weff(z).
GR
k  0.01 h Mpc 1
1032 fRi
Observational constraint (Giannantonio et al, 2009):
1  Geff G  1.403
1     1.996
f
R
 2f
fRR 
R 2
fRi  fR initial
fR 
f(R) Modified Gravity (MG):
Sg 
1
d 4 x  g R  f R 

16G
Cosmological Test
Cosmic Expansion
Local Test
Cosmic Structure
Solar-System Test
viable fRi
f
R
 2f
fRR 
R 2
fRi  fR initial
fR 
constant w eff
f(R) Modified Gravity (MG):
Sg 
1
d 4 x  g R  f R 

16G
Cosmological Test
Cosmic Expansion
Local Test
Cosmic Structure
Solar-System Test
 Constraint on
survey
parameter
space
around GR point
 1, 0
f = constant
f(R) MG with
Chameleon Mechanism
 10 15  fR  0
0  RfRR  2 5
 Viable f R;w eff , fini 
10
1  w eff  10
fRi  10 37
very small viable region

 1  10  6

Geff
 1  10  6
G
closely
GR + 
mimicking
indistinguishable from GR !!
f(R) Modified Gravity (MG):
Sg 
1
d 4 x  g R  f R 

16G
Cosmological Test
Cosmic Expansion
Local Test
Cosmic Structure
Solar-System Test
 Constraint on
fRi
f(R) MG with
Chameleon Mechanism
GR
1  w eff
The viable f(R) models in the parameter space (weff, fRi)
around the GR point (1,0) for constant weff.
 10 15  fR  0
0  RfRR  2 5
Conclusion
Cosmic Expansion
(observational)
 Designer f R;w 0 ,w a , fini  w.r.t. the constraint on {w0,wa}
(by design) can pass the cosmic-expansion test.
Cosmic Structure
 The existence of the designer f R;w 0 ,w a , fini  models
which pass the cosmic-structure test
would require fine-tuning of initial condition fini.
Solar-System Test
 Among the designer f R;w 0 ,w a , fini  models,
only those closely mimicking GR +  (in all the 3 tests)
can pass the solar-system test.
 As a result, the solar-system test rules out
the frequently studied w eff  1 f R ; fini  models
that are distinct from CDM in   , Geff G .

2015-2016NEWモデル試乗機リスト(タナベスポーツin戸隠) H27.3/28;pdf