Dark energy II :Models of dark energy

Shinji Tsujikawa(Tokyo University of Science)

What is the origin of dark energy?

The simplest candidate: Cosmological constant However this suffers from a fine-tuning problem

if it originates from a vacuum energy.

Dynamical dark energy models

Quintessence, k-essence, chaplygin gas, tachyon, f (R) gravity, scalar-tensor theories, Braneworld, Galileon, …

Cosmological constant problem

The energy scale of dark energy today is

€

or, Cosmo-illogical constant problem (by Rocky Kolb)

If we take the Planck scale as a cut-off scale, the energy scale of the vacuum energy is

Problem even before 1998

See my review in 1989. by Steven Weinberg

The cosmological constant is (i) sufficiently small to explain the energy scale of dark energy?(ii) or, completely zero?

Case (i): Both the cosmological constant and the dark energy problems are solved at the same time.

Economical

Case (ii): The cosmological constant problem is solved, but the 　　　　 dark energy problem has to be addressed.

This possibility remains.

`Modified matter’ (such as a scalar field) is introduced, or gravity is modified from Einstein gravity (Dynamical dark energy) ．

Example of case (i): de-Sitter vacua in string theory

Kachru-Kallosh-Linde-Trivedi (KKLT) scenario

Type II string theory compactified on a Calabi Yau manifold with a flux.

The KKLT scenario consists of three steps.

Potential: where

We add uplifting potential generated by anti-D3 braneat the tip of warped throat:

uplifting

It is possible to explain dark energy if

The total potential is

AdS

dS

Example of case (ii) [vanishing cosmological constant]

_________________ ______K: Kahler potentialW: Superpotential

In supersymmetric theories the vacuum energy is zero if supersymmetry is unbroken, but in real word supersymmetry is broken.

Cancellation is required

　　

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We can classify the models into two classes ．

(i) Modified gravity (ii) Modified matter

f(R) gravity,Scalar-tensor theory,Braneworlds,Gauss-Bonnet gravity,Galileon gravity,…..

Quintessence,K-essence,Chaplygin gas,Coupled dark energy,…..

Dynamical dark energy models

(Einstein equation)

Modified matter models based on scalar fields

• Quintessence (‘fifth element’):

Chiba, Sugiyama, Nakamura (1997) ‘X matter’

Caldwell, Dave, Steinhardt (1998) ‘Quintessence’

• K-essence:

Accelerated expansion based on the potential energy

where

Chiba, Okabe, Yamaguchi (1999) ‘Kinetically driven quintessence’

Accelerated expansion based on the kinetic energy

Armendariz-Picon, Mukhanov, Steinhardt (2000) ‘k-essence’

Piorneering papers were written by Fujii (1982), Wetterich (1988),Ratra and Peebles (1988).

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Potentials of Quintessence

As long as the potential is sufficiently flat, cosmic acceleration can be realized.

Energy density:

Pressure:

Equation of state for Quintessence

Quintessencephantom

Quintessence can be distinguishedfrom the LCDM.

Particle physics models of quintessence

(i) Fermion condensate in globally supersymmetric QCD theories (Binetruy)

The inverse power-law potential was derived.

where

(ii) Supergravity models (Brax and Martin, Copeland et al)

The field potential in SUGRA theories is

(iii) Pseudo-Nambu Goldston Boson (PNGB) models (Friemann et al)

The filed starts to evolve only recently.

Classification of Quintessence potentials (Caldwell and Linder, 2003)

(A) Freezing models:

Since the potential tends to be flatter, the evolutionof the field slows down.

(B) Thawing models:

The field has been nearly frozen in the past, but it starts to evolve around today

.

.Example

Example

Quintessence in the (w,w’) plane

.

LCDM

The current observations are not still enough tofind the evidence for the variation of the equation of state.

K-essenceK-essence is described by the action

where

The models that belong to k-essence is

Conformal transformation

or

Equation of state for k-essence

Stability conditions for k-essence

Some people tried to solve the coincidence problem of dark energy by considering a specific Lagrangian

However it is difficult to construct such models theoretically. Moreover they typically have the superluminal propagation speed.

k-essence density parameter

Armendariz-Picon, Mukhanov, Steinhardt (2000)

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 modelsMore than 700 papers, see the Living Review of De Felice and S.T. (2010).

Capozziello Turner

The first dark energy model is

Capozziello, Carloni and Troisi (2003)Carroll, Duvvuri, Trodden and Turner (2003)

_____________

This term leads to the late-time acceleration.

However this model is not valid because of the following reasons.

(I) Incompatible with local gravity tests　　　 Chiba, Dolgov and Kawasaki, …(II) Instability of cosmological perturbations Hu, Tegmark, Trodden,…(III) Absence of the matter era　　　 Amendola, Polarski and S.T,…

(n > 0)

The main reason why the model does not work is

Conditions for the cosmological viability of f(R) dark energy 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)

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

The most general single-field scalar-tensor theories having second-order equations of motion:

Horndeski (1974) Deffayet et al (2011)

This action covers most of the dark energy models proposed in literature. Quintessence and K-essence

Non-minimal coupling models

Scalar-tensor theories (including f(R) gravity, Brans-Dicke theory)

Field-derivative coupling models

Galileon

(Kobayashi, Yamaguchi, Yokoyama, arXiv: 1105.5723)

Full background and linear perturbation equations were recently derived in the Horndeski’s most general scalar-tensor theories.

A. De Felice, T. Kobayashi, S.T., arXiv:1108.4242

On sub-horizon scales the matter perturbations satsisfies

Using our general formula, we can estimate the growth rate of perturbations in each theory.

in f(R) gravity

at late times

Summary of dark energy models

(1) Cosmological constant

Observationally favored, but theoretically further progress is required.

(2) Modified matter models

Quintessence, k-essence: these are not distinguished from the LCDM observationally. Chaplygin gas :× Excluded from the observations

of large-scale structure.(3) Modified gravity models

f(R) gravity, scalar-tensor theories: the models need to be carefully constructed to satisfy all the required constraints. DGP braneworld:× Galileon model:

Ruled out from the observations and the ghost problem.

Strongly constrained from the LSS and CMB observations.