Lecture 4:
Modified gravity
models of dark energy
Shinji Tsujikawa
(Tokyo University
of Science)
Modified gravity models of dark energy
This corresponds to large distance modification of gravity.
(i) Cosmological scales (large scales)
Modification from General Relativity (GR)
can be allowed.
(ii) Solar system scales (small scales)
The models need to be close to GR
from solar system experiments.
Beyond GR
???
GR+small corrections
Concrete modified gravity models
or
f (R) gravity
GR Lagrangian:
(R is a Ricci scalar)
Extensions to arbitrary function f (R)
f(R) gravity
The first inflation model (Starobinsky 1980)
Starobinsky
2
Inflation is realized by the R term.
Favored from CMB observations
Spectral index:
Tensor to scalar ratio:
N: e-foldings
f(R) dark energy models (more than 500 papers)
Capozziello
Turner
Capozziello, Carloni and Troisi (2003)
Carroll, Duvvuri, Trodden and Turner (2003)
Please see the review article:
A. De Felice and S. Tsujikawa, Living Reviews in Relativity, 13, 3 (2010)
Conditions for the cosmological viability of f(R) models
1.
2.
To avoid ghosts
To avoid tachyonic instability
 The mass M of a scalar-field degree of freedom needs to be
positive for consistency with local gravity constraints (LGC).
 This condition is also required for the stability of perturbations.
3.
(R0 : present cosmological
Ricci scalar)
For the presence of the matter era and for consistency with LGC.
4. The presence of a stable late-time de Sitter point
Viable f(R) dark energy models
1.
Hu and Sawicki, 2007
2.
Starobinsky, 2007
3.
S.T., 2007
Cosmological constant disappears
in flat space-time.
The models approach the LCDM for
.
(for the models
1 and 2)
The local gravity constraints can be satisfied for
(Capozziello and S.T., 2008)
Cosmology of viable f(R) models
(i) During radiation and deep matter eras (
the models are close to the LCDM model:
),
‘GR regime’
(ii) Around the end of the matter era to the accelerated epoch,
the deviation from the LCDM model becomes important.
‘Scalar-tensor regime’
The existence of this regime leaves several interesting
observational signatures:
• Phantom equation of state of DE
• Modified matter power spectrum
• Modified weak lensing spectrum
Dark energy equation of state in f(R) models
Friedmann equations in the FLRW background
where
  8 G
f
2
F 
3H   (  DE   m )
2
2

(  rad  0 )

2
2 HÝ    (  DE  p DE   m )




R
where
2
2
  DE  (1 / 2)( FR  f )  3HFÝ  3H (1  F )
2
Ý 2 H FÝ  (1 / 2)( FR  f )  (2 HÝ  3H 2 )(1  F )
 p DE  FÝ
This satisfies
w DE 

p DE
 DE
ÝDE  3H (  DE  p DE )  0


w eff
1  F m
2
w eff  1  (2 HÝ/ 3H )
 m   m / 3FH
2
Phantom equation of state in f(R) models
Phantom equation of state can be realized without the appearance
of ghosts and instabilities.
(Starobinsky’s model)
g
This property is useful to
discriminate f(R) models from
future SN Ia observations.
Matter perturbations in viable f(R) models
Large-scale
structure
where
(S.T., 2007)
is the deviation parameter from the LCDM.
(
(i)
k
2
m  1
2
a R

(ii)
k
2

Standard evolution:   t 2 / 3
m
(early time: ‘GR regime’)
Non-standard evolution:  m  t
(late
time: ‘Scalar tensor regime’)
This enhances the growth rate of
matter perturbations.
(
2
a R
)
m  1
33 1) / 6
The transition point from the ‘GR regime’ to the
‘scalar tensor regime’ is characterized by
k
2
2
m 1
a R
For the k modes relevant to matter power spectrum,
this occurs during the matter era at

t
k
k
3 /( 6 n  4 )
For the Starobinsky’ model:
This leads to the difference of spectral indices between
 the matter power spectrum and the CMB spectrum:
n s 

33  5
6n  4
Starobinsky (2007)
Numerically confirmed by S.T. (2007)
Matter power spectra
Small-scale spectra
are modified.
Starobinsky’s
f(R) model with
n=2
LCDM
[h/Mpc]
It will be of interest
to see whether the
signature of f(R)
gravity can seen in
future observations.
Conformal transformation in f (R) gravity
where
where
where we used
_________________
Chameleon mechanism
where
Massive
(local region)
Massless
(Cosmological
region)
.
.
In the local region with high density, the field does not
propagate freely because of a large effective mass.
where
Because of the presence of a matter coupling with the field,
the field is nearly frozen with a large mass.
High-density
(massive)
Low-density
.
The field is nearly frozen.
The detailed calculation shows
that the solar-system
constraints are satisfied for
Braneworld models of dark energy
5-th dimension
Dvali, Gabadadze, Porrati (DGP) model
3-brane is embedded in the
5-dimensional bulk
Bulk
3-brane
On the 3-brane the Friedmann equation is
(for the flat case)
where
There is a de Sitter attractor with
(self acceleration)
• DGP model is disfavored from observations．
Even in the presence of cosmic
curvature K, the DGP model is
in high tension with observations.
SN Ia
BAO
• Moreover the DGP model
contains a ghost mode.
Theoretical curve
The DGP model is disfavored
from both theoretical and
observational point of view.
Galileon gravity
Galileon cosmology
: five covariant Galileon Lagrangians
(second-order)
Cosmological evolution in Galileon cosmology
De Felice and S.T.,
PRL (2010)
Tracker solution
Gauss-Bonnet gravity
A. De Felice, D. Mota, S.T. (2009)
where
Considering the perturbations of a
perfect fluid with an equation of
state w, the speed of propagation is
Excluded!
Summary of modified gravity models of dark energy
Papers including ‘dark energy’ in title: 2620
Papers including, ‘cosmological constant’ in title:1853
Exponential growth
Steady state
謝謝
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