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:

Originally introduced by Einstein to realize the static Universe．
1917 (38 old)

1945 (66 old)
1998 (119 old:heaven)
In 1929
Hubble found
the expansion
of the Universe.
‘Biggest Blunder
in my life’
Static
Universe
Big Bang
Cosmology
Big Bang cosmology+
cosmic acceleration
Cosmological constant problem
or, Cosmo-illogical constant problem (by Rocky Kolb)
The energy scale of dark energy today is
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 brane
at the tip of warped throat:
The total potential is
It is possible to explain
dark energy if
dS
uplifting
AdS
String Landscape
We may live in a vacuum with a small energy density
(related with anthropic selection).
500
10 uplifted
vacua!
Example of case (ii) [vanishing cosmological constant]
In supersymmetric theories the vacuum energy is zero if supersymmetry is
unbroken, but in real word supersymmetry is broken.
_________________
______
K: Kahler potential
W: Superpotential
Cancellation is required
Dynamical dark energy models
We can classify the models into two classes．
(Einstein equation)
(i) Modified gravity
f(R) gravity,
Scalar-tensor theory,
Braneworlds,
Gauss-Bonnet gravity,
Galileon gravity,
…..
(ii) Modified matter
Quintessence,
K-essence,
Chaplygin gas,
Coupled dark
energy,
(including mass varying neutrinos)
…..
Modified matter models based on scalar fields
• Quintessence (‘fifth element’):
Accelerated expansion based on the potential energy
Chiba, Sugiyama, Nakamura (1997)
‘X matter’
Caldwell, Dave, Steinhardt (1998)
‘Quintessence’
• K-essence:
where
Accelerated expansion based on the kinetic energy
Chiba, Okabe, Yamaguchi (1999) ‘Kinetically driven quintessence’
Armendariz-Picon, Mukhanov, Steinhardt (2000)
‘k-essence’
Quintessence: French wine!
_____________________________
Potentials of Quintessence
ÉXÉJÉâÅ[èÍÇÃÉ|ÉeÉìÉVÉÉÉã
Energy density:
Pressure:
Equation of state for Quintessence
phantom Quintessence
ÉtÉ@ÉìÉgÉÄ
ÉNÉCÉìÉeÉbÉZÉìÉX

As long as the potential is
sufficiently flat, cosmic
acceleration can be realized.
Quintessence can be distinguished
from 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:
Example
Since the potential tends to be flatter, the evolution
of the field slows down.
.
(B) Thawing models:
Example
The field has been nearly frozen in the past,
but it starts to evolve around today
.
Quintessence in the (w,w’) plane
The current observations
are not still enough to
find the evidence for
the variation of the
equation of state.
.
LCDM
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-essence
K-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
Armendariz-Picon, Mukhanov,
Steinhardt (2000)
 However it is difficult to construct such models theoretically.
 Moreover they typically have the superluminal propagation speed.
k-essence density parameter
Chaplygin gas model
Chaplygin gas
Generalized Chaplygin gas
This corresponds to unified dark energy models in which dark
matter and dark energy are explained as a single component.
(pressureless matter)
(dark energy)
Continuity equation:
Past:
Future:
Chaplygin gas satisfies observational constrants ?
_________________
No!
____
The sound speed term
prevents the growth of
large-scale structure.
Observational constraints
This cannot be distinguished
from the LCDM.
Matter power spectrum