1. ``Welcome to the dark side of the world.”
A unified approach to scaling solutions and its applications to dark energy
Shinji Tsujikawa （辻川　信二）　
Gunma National college of Technology,Japan(日本）

2. Dark energy (暗黒エネルギー）
Observations suggest that more than 70% of the energy
density of the current universe is dark energy that gives
an accelerated expansion.
Bright side of the world
万里長城

3. Models of dark energy Quintessence (Potential energy of scalar fields)
Cosmo-illogical constant
Quintessence (Potential energy of scalar
fields)
K-essence (Kinematic energy of scalar
Tachyon （Energy density：
(Equation of state：w < -1)
Chaplygin gas sometime known as
Ghost condensate… )
-gas
For other models, see the review: E. Copeland, M. Sami and S.T., hep-th/

4. Cosmological scaling solutions
In constructing viable models of dark energy, it is convenient
if we know cosmological scaling solutions.
(Dark energy density is proportional
to fluid energy density)
Scaling solutions:
For a minimally coupled scalar field, the exponential potential
corresponds to scaling solutions.
In this case
(i)
(scaling solutions)
and
(ii)
It is possible to explain the late-time acceleration if the slope
of the potential changes at late-times: e.g.,

5. Barreiro, Copeland and Nunes, PRD61, 127301 (2000)
Scaling regime
=20,
=0.5
Exit from
scaling regime
Acceleration!

6. Scaling solutions: example

7. A unified approach to scaling solutions
F. Piazza and S.T. (2004)
S.T. and M. Sami (2004)
We wish to know the condition for a scalar-field
Lagrangian for the existence of scaling solutions.
Let us start with a general Lagrangian:
This includes (coupled)-quintessence, phantom,
ghost condensate, tachyon, k-essence, …
We consider a general cosmological background:
General Relativity (GR)
( H: Hubble rate)
Randall-Sundrum (RS) braneworld
3-brane
Gauss-Bonnet (GB) braneworld

8. Existence of scaling solutions
In FRW background we have
Scalar field:
Coupled!
Fluid:
Q: coupling between dark energy and fluid (dark matter)
Scaling solutions:
where
and
and
Q is assumed to be constant.

9. Lagrangian for the existence of scaling solutions
By using
Japonais (日本人)
we find
Beautiful!
where
One can also derive
arbitrary function
for scaling solutions.

10. Application to dark energy models
(A) Quintessence (sometimes known as ‘French wine’)
When p is written in the form:
we obtain ,
In the case of GR (n=1), this corresponds to
an exponential potential with a canonical field.
Defining a new field:
, we get
where
(corresponding to the choice
where
　　　　 )
Canonical field with potential
for RS braneworld
for GB braneworld

12. The Lagrangian for tachyon is
(B) Tachyon
The Lagrangian for tachyon is
Substituting
for
, we get
where
Defining
, we find
Therefore the tachyon has scaling solutions for the potential
for GR
When Q=0, w < 0 for the existence of scaling solutions. m
Scaling solutions exist for w > 0 in the presence of the coupling Q. m
See: B. Gumjudpai, T. Naskar, M. Sami, S.T., JCAP 0506, 007 (2005)

13. (c) Dilatonic ghost condensate
Phantom is plagued by quantum vacuum instability, but this
is overcome by accounting for higher-order kinematic terms:
F. Piazza and S.T. (2004)
This is obtained by substituting
for .
For scaling solutions, we obtain
The vacuum is stable at quantum level for .
when w =0 m
Accelerated expansion occurs for
when w =0 m
The coupling Q can lead to a viable scaling solution.

14. Generalization to scalar-field dependent coupling
When Q depends on the field , we can also derive
the scaling Lagrangian:
Luca Amendola
and junior
where
Amendola, Quartin, S.T., Waga, Physical Review D74 (2006)
!!??
Redefining a new variable:
The same Lagrangian as the case of constant Q
One can always transform to the Lagrangian for
a constant coupling.

15. Can we have a viable scaling cosmology?
Is it possible to have the following sequence ? (for n=1)
Radiation era
(輻射）
Matter era
(物質）
Accelerated scaling era
with p>>1
（加速)
We showed that this sequence does not occur for a vast class of
scaling models proposed in current literature.
Amendola, Quartin, S.T., Waga (2006)

16. Quintessence with an exponential potential (Amendola, 1999)
radiation
dark energy
dark matter
If we require the scaling
accelerated attractor,
Matter phase is absent!

17. We showed that the scaling models
where
with integer m
are excluded to get a sequence of radiation, matter,
accelerated scaling solution. This includes
(i) Quintessence with an exponential potential
g=1 - c/Y
(ii) Dilatonic ghost condensate
g=-1 +cY
The evolution of matter era to accelerated scaling era is
forbidden because of singularities in phase space.

18. Conclusion （結論) We derived the condition for scalar-field Lagangian
for the existence of scaling solutions:
This includes a wide variety of dark energy models:
(coupled)-quintessence, phantom, ghost condensate,
tachyon, k-essence, …
We studied a possibility to obtain a sequence of radiation, ,
matter, accelerated scaling solution and showed that
a vast class of generalized coupled scalar-field Largangian
does not give rise to this sequence.
Challenges for scaling cosmologies!

19. Let’s hope to know what dark energy is in foreseeable future !
Chaplin-gas ?
Quintessence ?
Ghost
Condensate ?
Phantom ?
???
Let’s hope to know what dark energy is
in foreseeable future !

20. Thanks !