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.

???

Beyond GR

(ii) Solar system scales (small scales)

The models need to be close to GRfrom solar system experiments.

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

Inflation is realized by the R term.2

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. To avoid ghosts

2.

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.

For the presence of the matter era and for consistency with LGC.

4. The presence of a stable late-time de Sitter point

(R : present cosmological Ricci scalar)

0

To avoid tachyonic instability

Viable f(R) dark energy models

1. Hu and Sawicki, 2007

2. Starobinsky, 2007

3. S.T., 2007

Cosmological constant disappearsin 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 interestingobservational signatures:

• Phantom equation of state of DE

• Modified matter power spectrum

• Modified weak lensing spectrum

Dark energy equation of state in f(R) modelsFriedmann equations in the FLRW background

where

€

F =∂f

∂R

€

3H 2 = κ 2(ρ DE + ρ m )

€

2 ˙ H = −κ 2(ρ DE + pDE + ρ m )

where

€

κ 2ρ DE = (1/2)(FR − f ) − 3H ˙ F + 3H 2(1− F)

€

κ 2 pDE = ˙ ̇ F + 2H ˙ F − (1/2)(FR − f ) − (2 ˙ H + 3H 2)(1− F)

€

˙ ρ DE + 3H(ρ DE + pDE ) = 0This satisfies

€

wDE =pDE

ρ DE

=weff

1− FΩm

€

weff = −1− (2 ˙ H /3H 2)

€

Ωm = ρ m /3FH 2

€

(ρ rad = 0)€

κ 2 = 8πG

Phantom equation of state in f(R) modelsPhantom equation of state can be realized without the appearanceof ghosts and instabilities.

This property is useful to discriminate f(R) models from future SN Ia observations.g

(Starobinsky’s model)

Matter perturbations in viable f(R) models

where (S.T., 2007)

is the deviation parameter from the LCDM.

(i)

€

k 2

a2Rm <<1 Standard evolution:

€

δm ∝ t 2 / 3

(early time: ‘GR regime’)

(ii)

€

k 2

a2Rm >>1 Non-standard evolution:

€

δm ∝ t( 33−1)/ 6

(late time: ‘Scalar tensor regime’)

This enhances the growth rate of matter perturbations.

( )

Large-scalestructure

The transition point from the ‘GR regime’ to the ‘scalar tensor regime’ is characterized by

€

k 2

a2Rm =1

For the k modes relevant to matter power spectrum,this occurs during the matter era at

€

tk ∝ k−3 /(6n +4 ) For the Starobinsky’ model:

This leads to the difference of spectral indices between the matter power spectrum and the CMB spectrum:

€

Δns =33 − 5

6n + 4

Starobinsky (2007)Numerically confirmed by S.T. (2007)

Matter power spectra

LCDM

Starobinsky’sf(R) model withn=2

[h/Mpc]

Small-scale spectraare modified.

It will be of interestto 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

Massive(local region)

where

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

Dvali, Gabadadze, Porrati (DGP) model

3-brane is embedded in the5-dimensional bulk

Bulk

3-brane

(for the flat case)

(self acceleration)

5-th dimension

On the 3-brane the Friedmann equation is

where

There is a de Sitter attractor with

• DGP model is disfavored from observations ．

BAOSN Ia

Even in the presence of cosmic curvature K, the DGP model isin high tension with observations.

• Moreover the DGP model contains a ghost mode.

The DGP model is disfavoredfrom both theoretical and observational point of view.

Theoretical curve

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

Exponential growth Steady state

Papers including ‘dark energy’ in title: 2620

Papers including, ‘cosmological constant’ in title:1853

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