Dark Energy and Modified Gravity

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Dark Energy and Modified Gravity

• Shinji Tsujikawa
• (Gunma National College of Technology)

Collaborations with L. Amendola, S. Capozziello,
R. Gannouji, S. Mizuno, D. Polarski, R. Tavakol,
K. Uddin, J. Yokoyama
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Dark Energy
Komatsu et al, 0803.0547 (astro-ph)
SNe Ia
CMB
LSS
The current universe is accelerating!
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Simplest model of dark energy
Cosmological constant
(Equation of state )
This corresponds to the energy scale
If this originates from vacuum energy in particle
physics,
Huge difference compared to the present value!
Cosmological constant problem
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Are there some other models of dark energy?
There are two approaches to dark energy.
(Einstein equations)

(i) Changing gravity
(ii) Changing matter
f(R) gravity models, Scalar-tensor
models, Braneworld models, Inhomogeneities, ..
Quintessence, K-essence, Tachyon, Chaplygin
gas, ..
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Quintessence, K-essence, Tachyon, phantom field,

Changing matter models
To get the present acceleration most of these
models are based upon scalar fields with a very
light mass
The coupling of the field to ordinary matter
should lead to observable long-range forces.
(Carroll, 1998)
In super-symmetric theories the severe
fine-tuning of the field potential is required.
Flat
(Kolda and Lyth, 1999)
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f(R) gravity, scalar-tensor gravity, braneworld
models,..
Changing gravity models
Dark energy may originate from some
geometric modification from Einstein gravity.
The simplest model
f(R) gravity
R Ricci scalar
model
Starobinskys inflation model
Used for early universe inflation
f(R) modified gravity models can be used for dark
energy ?
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f(R) dark energy Example
Capozziello, Carloni and Troisi (2003) Carroll,
Duvvuri, Trodden and Turner (2003)
(ngt0)
It is possible to have a late-time acceleration
as the second term becomes important as R
decreases.
In the small R region we have

Late-time acceleration is realized.
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Is the model (ngt0) are
cosmologically viable?

No!
This model does not have a standard matter
era prior to the late-time acceleration.

• The f(R) action is transformed to

(Einstein frame)
where
Matter fluid satisfies

Coupled quintessence
Dark matter is coupled to the field (curvature).
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The model
(ngt0)
.
The potential in Einstein frame is
For large field region,
Coupled quintessence with an exponential potential
The standard matter era is replaced by phi
matter dominated era

Jordan frame
Incompatible with observations
L. Amendola, D. Polarski, S.T., PRL (2007)
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What are general conditions for the cosmological
viability of f(R) dark energy models?
L. Amendola, D. Polarski, R. Gannouji and S.T.,
PRD75, 083504 (2007)
For the FRW background with a scale factor a, we
have

Pressure-less Matter
Radiation
We carried out general analysis without
specifying the form of f(R).
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See the review article E. Copeland, M. Sami and
S.T., IJMPD (2006)
Autonomous equations
We introduce the following variables
Then we obtain
and
,
where
and
The above equations are closed.
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The parameter
characterises a deviation from the
model.
model
(a)
(b)
The constant m model corresponds to
(c) The model of Capozziello et al and Carroll et
al
This negative m case is excluded as we will see
below.
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The cosmological dynamics is well understood by
the geometrical approach in the (r, m) plane.
(i) Matter point P
M
From the stability analysis around the fixed
point, the existence of the saddle matter epoch
requires
at
(ii) De-sitter point P
A
For the stability of the de-Sitter point, we
require
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Viable trajectories
Amendola and S.T. (2007)
(another accelerated point)
Constant m model
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More than 200 papers were written about f(R) dark
energy!
Lists of cosmologically non-viable models
(ngt0)
. many !
Lists of cosmologically viable models
(0ltnlt1)
Li and Barrow (2007)
Amendola and S.T. (2007)
Hu and Sawicki (2007)
Starobinsky (2007)
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Local gravity constraints (LGC)
The f(R) models need to satisfy constraints
coming from solar system and equivalence
principle tests.
The f(R) action in the Einstein frame is
where the coupling between dark energy and dark
matter is
(of the order of 1)
Even in this case, LGC can be satisfied provided
that the mass of the field is sufficiently heavy
in high-density regions
( is required)
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Chameleon mechanism
Khoury and Weltman (2003)
Spherically symmetric body
where
Inside and outside the body, the effective
potential has minima at
The body has a thin-shell inside it if the field
is heavy.
The gravitational potential at the surface of
the body
Thin-shell parameter
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f(R) chameleons
In f(R) gravity the effective gravitational
constant and the post-Newtonian parameter are
Faulker, Tegmark et al, Capozziello and S.T.
The tightest solar-system bound is
For the Sun ( ),
High-density (massive)
.
This can be satisfied for the model where
.
.
Low-density
(massless)
is large in the region
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Models that satisfy local gravity constraints
Hu
Starobinsky
Hu and Sawicki
Starobinsky
Cosmological constant disappears in a flat space.
The solar-system constraints are satisfied for
The equivalence principle constraints are
satisfied for
(Capozziello and S.T.)
In these models the deviation from the Lambda CDM
model becomes significant around present on
cosmological scales.
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Observational signatures of viable f(R) models
To confront with SN Ia observations, we write the
equations in the form
present value
where
This satisfies
The equation of state of dark energy is
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The equation of state diverges at
Viable f(R) models satisfy
F increases toward the past.
The divergence of w occurs.
DE
Amendola and S.T. (2007)
The redshift at the divergence can be as close as
z2.
But this behaviour can be still allowed in
current SN Ia observations.
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Are there other observational constraints on f(R)
models?
Matter power spectrum
Under the sub-horizon approximation (kgtgtaH), the
matter density perturbation satisfies
where
(i)
Standard evolution
(early time)
(ii)
Non-standard evolution
(late time)
This enhances the growth rate of matter
perturbations.
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The models of Hu Sawicki and Starobinsky behave
as
for
Modes relevant to matter power spectrum
The crossing occurs during the
matter era.
The time of crossing has a dependence
This leads to the difference of spectral indices
between matter power spectrum and CMB spectrum
Starobinsky S.T.
for
Likelihood analysis using the LSS and CMB data is
necessary.
V. Acquaviva, S. Matarrese, S.T., M. Viel, in
preparation.
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Generalization to scalar-tensor models
One can generalize the analysis to Brans-Dicke
theory with a potential
Setting , this action is
equivalent to
where
and
The f(R) gravity corresponds to
Chiba (2003)
One can search for viable models for general
coupling Q.
S.T., Uddin, Mizuno, Tavakol, Yokoyama, arXiv
0803.1106 astro-ph
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The local gravity constraints can be satisfied if
the field is sufficiently heavy in the
large-curvature region.
We find that
A representative potential

• The divergence of w is generic.
• Q and p are constrained by LGC
• and matter perturbations.

DE
(0ltplt1)
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Conclusions

• We derived conditions for the cosmological
  viability of f(R)
• modified models. This is useful to exclude
  many f(R) models
• e.g.,
  (ngt0).

• The viable f(R) models show peculiar features for
  the
• equation of state of dark energy.
• It can diverge at the redshift around z2.

• We discussed a number of observational and
• experimental signatures of modified gravity
  models
• SN Ia, LGC, Matter power spectra, CMB, .

• We also studied the case of general coupling Q
  and found
• that the results obtained in f(R) gravity
  are generic.