Lecture 3:Modified matter 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:

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Λ Originally introduced by Einstein to realize the static Universe ．

1917 (38 old) 1945 (66 old)

‘Biggest Blunder in my life’

1998 (119 old:heaven)

In 1929Hubble found the expansion of the Universe.

Static Universe

Big Bang Cosmology

Big Bang cosmology+cosmic acceleration

Cosmological constant problem

The energy scale of dark energy today is

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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

String Landscape

We may live in a vacuum with a small energy density (related with anthropic selection).

10 upliftedvacua!

500

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,(including mass varying neutrinos)…..

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’

Quintessence: French wine!

_____________________________

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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.

Dynamical system approach to quintessence

Dynamical equations

The fixed point responsible for the cosmic acceleration is

Phase space

Attractor(cosmic acceleration)

Saddle(matter point)

General potentials

where

(tracking condition)

Tracking always occurs.

Numerical simulations for

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 condition 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)

Chaplygin gas model

Chaplygin gas Generalized Chaplygin gas

This corresponds to unified dark energy models in which darkmatter and dark energy are explained as a single component.

(pressureless matter)

(dark energy)

Continuity equation:

Past:

Future:

Chaplygin gas satisfies observational constrants ? No!

Matter power spectrum

_____________________

The sound speed term prevents the growth oflarge-scale structure.

Observational constraints

This cannot be distinguishedfrom the LCDM.