``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）

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

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/0603057

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.,

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

Scaling solutions: example

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

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.

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.

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

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)

(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.

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.

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)

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

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.

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!

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 !

Thanks !