On Dark Energy Models of Single Scalar Field

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On dark energy models of single scalar field

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Please note that terms and conditions apply. JCAP04(2012)003 hysics e P ding based on le ic t logical constant boundary ar linearly. It behaves like an 210093, P.R. China [email protected] es, onal Taiwan University, 10.1088/1475-7516/2012/04/003 , National Taiwan University, an and show that its equation enerally the Lagrangian of this doi: posing a degenerate condition on needs to be generalized. We also he models which are free from the strop and Xinmin Zhang A d [email protected] , Yifu Cai b,c [email protected] , osmology and and osmology C 1 without any instability. Taotao Qiu, − a,f 1112.4255 dark energy theory, cosmological perturbation theory

In this paper we revisit the dynamical dark energy model buil [email protected] rnal of rnal ou An IOP and SISSA journal An IOP and 2012 IOP Publishing Ltd and Sissa Medialab srl Institute of High Energy Physics, Chinese Academy of Scienc Taipei 10617, Taiwan Leung Center for CosmologyTaipei and 106, Particle Taiwan Astrophysics, Nati Department of Physics, ArizonaTempe, State AZ University, 85287, U.S.A. P.O. Box 918-4, Beijing 100049,Joint P.R. Center China for Particle,Nanjing Nuclear University Physics — and Purple Cosmology, Mountain Observatory, Nanjing E-mail: Department of Physics, NanjingNanjing University, 210093, P.R. China Department of Physics and Center for Theoretical Sciences, b c e d a f c Received January 2, 2012 Accepted March 7, 2012 Published April 2, 2012 Abstract.

Mingzhe Li, J On dark energy modelsscalar of field single ArXiv ePrint: Keywords: single scalar field involvingthe higher higher derivative derivatives terms. in curved Byghost spacetime, im mode one can and select thesmoothly, t dynamically equation violate of the statetype null is of energy able dark condition. to energyimperfect cross models fluid, G the thus depends its cosmo study on cosmological such the perturbation a second model theory derivatives of with state explicit may form cross of degenerate Lagrangi JCAP04(2012)003 ] ], ], d 1 2 3 5 7 1, 12 23 13 16 ≤ ] and which w ], show 24 ≤ 1 15 1 − − ], also see [ quintom ] and so on (see 22 6 ]. However, it was 6 ]. However theorists 6 ld [ d more degrees of freedom ian of k-essence form [ rowave background (CMB) ocal string theory [ ] as a “No-Go” theorem for ially flat and accelerated ex- intom model was constructed k energy remains a mystery. alker (FRW) cosmology with 1, dubbed as ], quintom [ dea of dynamical dark energy vative terms [ 21 ne tuning and the coincidence . The first single scalar field 5 n the background EoS crosses − is a small positive cosmological ldly favored [ t is worth noting that a class of intom scenario are difficult to be d to be in the region mination of a component with neg- ], k-essence [ 1. It was proved that in models described 4 ], in combination of the 7-year WMAP [ − ]. A possible approach to stable violations of 12 – 1 – 27 , 26 ], phantom [ ], and the recently released Union2 SNIa data [ 3 – 14 1 ]. However, there is a quantum instability due to an unbounde 25 is always smaller than w ]. This statement was explicitly proven in ref. [ 20 ] for recent reviews). – 11 17 – , 7 Although the recent fits to the data [ To realize a viable quintom scenario, one usually needs to ad 6 1 [ radiation and so onpanding have universe. provided strong InEinstein evidences the gravity, this for context acceleration a of isative spat Friedmann-Robertson-W attributed pressure, to called theTheoretically, do dark the energy. simplest candidateconstant, So for but far, such it a the suffers component problems. nature the of Therefore, difficulties dar many associatedmodels, physicists with such are the as attracted fi quintessence by [ the i 6 Summary 1 Introduction The recent data from type Ia Supernovae (SNIa) and cosmic mic 3 Single scalar dark energy4 model with Degenerate higher higher derivatives derivative model 5 Cosmology with degenerate higher derivative dark energy model Contents 1 Introduction 2 The difficulty of single field dark energy models with EoS crossing dynamically violates the null energy condition (NEC), is mi the cosmological perturbations− encounter a divergence whe decaying tachyonic branes [ vacuum state in such type of models [ refs. [ noticed that consistent singleconstructed. field For models example, the realizing EoS the of qu quintessence is limite dynamical dark energy models. for generalization. It was also obtained in the frame of nonl the Sloan Digitalremarkably Sky the Survey consistence [ ofdynamical the models cosmological with constant, the i equation-of-state (EoS) across are still interestedquintom in model pursuing was realized single by field introducing higher quintom order models deri by a single perfect fluid or a single scalar field with a Lagrang into the dark energy budget.by The a simplest combination and of also the a first canonical qu scalar and a phantom scalar fie while for phantom JCAP04(2012)003 ]. 1 45 ]. – (2.1) 1 re- − ] (see 21 − ]. The 34 10 32 sing e k-essence ]. Later on, various 33 as been studied exten- 1 smoothly. Section 6 is view the difficulty of con- − namical dark energy mod- iminate the ghost mode by ght allow for superluminal is no-go theorem is that in lding as proposed in [ . Thus, this model does not eneral covariant Lagrangian As demonstrated by several ives. We perform numerical perturbations. Our numerical rivative terms do not bring a ifically, we present an explicit derivative terms in the action -essence) cannot cross ant relation with energy and k energy model with degener- ts is that such a single scalar ross rs ago [ pends on the second derivative we revisit quintom dark energy quintom scenario. In section 3, med, namely, see refs. [ pursue how to keep the model o avoid the appearance of ghost tion 5 is devoted to the study of d an explicit dark energy model sing takes place. We understand cal constant boundary smoothly , s were reviewed in ref. [ ve terms, and present the general 2 tion of the DGP model [ V k ˙ ˙ p ρ − 2 s c ]. Various theoretical realizations of quintom H 29 – 2 – + 3 ], in which the negative kinetic modes are bounded ]. It was viewed as a local infrared modification of 28 31 δρ 2 s c = δp ], the EoS of dark energy based on single perfect fluid or singl 21 – 17 , 6 ]). 30 This paper is organized as follows. In section 2, we simply re Recently, a scalar field model which stably violates the NEC h NEC is the ghost condensation of ref. [ maining finite perturbations.both The scenarios key the pointmomentum pressure to density perturbations, perturbation the proof has of a th gauge invari the summary. 2 The difficulty of single field dark energy models with EoS cros scalar field (quintessence and phantom are special cases of k structing a single fieldwe dark start energy with model a whichanalysis. single gives rise field We to action also involvingghost discuss higher mode under derivati to what the condition effectivethe field the background description and higher in perturbation de sectionate dynamics 4. higher of Sec derivatives in singlemodel the field frame dar of of quintom flatcomputation dark FRW to universe. energy illustrate Spec with such a degenerate model higher can derivat realize the EoS ac via a spontaneouspropagation Lorentz of symmetry information breaking, in although some cases it [ also mi [ key feature of thesewhile models the is equation of that motionmodes, they remains contain realizing second-order higher in the order orderphenomenological idea t studies pioneered of by thisMotivated Horndeski by type the thirty of feature yea of modelsmodels the were Galileon containing model, perfor higher inof derivative this paper terms. single scalar Wefree field start from involving from extra higher a degreeimposing derivative g of a terms freedom. degenerate and condition Weterms that show linearly. the that Based Lagrangian it on onlyand is the de study degenerate able its Lagrangian, to dynamics we of el calculations buil its show homogeneous that background the and EoS its and is the able perturbation to modes are crossthe well the controlled reason cosmologi when the of cros field realizing model the is single noconflict field longer with quintom be the without able “No-Go” ghos to theorem correspond for quintom to dark a energy perfect bui fluid scenarios and their implications for early universe physic General Relativity, generalizing an effective field descrip sively, which is the so-called Galileon [ We begin byels briefly with reviewing the thegroups EoS difficulty [ across of the constructing cosmological dy constant boundary. JCAP04(2012)003 , ], ˙ 0. ρ 6 we rges p/ < (2.3) (2.2) (2.4) (2.5) X at the . The in this = ˙ X ρ 2 s X Xp π c rent from fterwards, . = 2 δp/δρ 2 p he crossing. To ), and π ρ + ( is the Hubble rate 0 requires p ρ /dt ) > = a/a 2 s p 3 Xφ c . a = ˙ , of which the action is ˙ φp φ φ X 3 ty due to the existence of µ µ H phantom field. Though the 2 a finite to guarantee the Jeans ( ∇ ∇ or a k-essence field is only a √ rges when the quantum effects d φ licitly. For example in ref. [ . Its energy momentum tensor nsity perturbation is defined as µ ity only, = ingle k-essence field, earlier quin- We can see from the action that , − , and g action of second order, crossing point. m the equation of motion or the µν ∇ ν µ g 2 e norm states. ation will lead to arbitrarily large 2 1 , u ) is governed by the equation which π ) µ t ( u φφ = ) φ is defined as the ratio of p p is negative and has a wrong sign. This means 2 s − , u X + φ, X c + ) X π ( X π p ρ L i component of the energy momentum tensor. g i 1, x t x, Xp π∂ +( ( i √ − − φ ∂ x – 3 – and density perturbation. As we learn from the µν 4 + 2 2 X = d . However, from the relation a p p pg V w < π X − − Z − /ρ = is the 0 2 = = will make the equation of perturbation singular and the ˙ X π 0 i S p µν X T X to indicate the partial derivative with respect to ρ T ρ = 3 X , ρ 2 s c L ) we can see that in both cases the pressure perturbation dive and its first derivatives xdta = 3 = 0 with finite 2.1 φ d p ρ should also vanish at the crossing point and change the sign a Z . For k-essence field the sound speed square is generally diffe X 2 1 ρ δρ ) = ˙ ) showed that the kinetic term of ρ 0, vanishes at the point of crossing and changes the sign after t (2) p/ > 2.5 X S 2 s p in Fourier space, here c . The vanishment of = (˙ 0 i is the negative determinant of the metric tensor X T the scale factor. The sound speed square p i g δp a Such kind of instability is classical. Another difficulty eme ik , however, from eq. ( ˙ = ρ ˙ guarantee just as amplitude of the perturbation arbitrarily large around the are considered. At the phantom phase p/ where the dot represents the derivative with respect to time where where with gravitational field equation, a divergent pressure perturb metric perturbations. Thisaction instability for can the alsofunction be perturbations. seen of A fro the generic scalar form of Lagrangian f hence The action ( We have used the subscript the quintom model is constructed by a quintessence field and a phase is anegative ghost energy which states or brings of the violating problem unitarity by of negativ 3 vacuum instabili Single scalar darkTo avoid energy the problem model of with singulartom perturbation higher model possessed derivatives by buildings s introduced multi-degree of freedom exp at the crossing point ˙ know that evolution of the k-essence perturbation has the same form as that of perfect fluid, expressed as, We have neglected thethe metric perturbations sound for speed simplicity. square is For perfect fluid the pressure is a function of the energy dens frame comoving with thestability dark of energy, perturbations it atV should small be scales. positive The de momentum de may be obtained via variational principle from the followin JCAP04(2012)003 . 2. = 2 . So ≥ (3.5) (3.1) (3.4) (3.3) (3.2) = 0 ], and X N N ν ν 47 φ φ degrees of ρσνµ ρµ and φ ρσµ φ ion. To solve N φ φ µν µν ], it was shown µν L ∂φ 22 ∂φ 2 . The equation of L ρ ∂φ 3 ∂ ρσ L µν ∂ 3 ρσ φ ∂ ∂φ iance locally [ grangian generally has erivatives was proposed , we have the following ∂φ∂φ + . crossing the boundary of and νµ ∂φ∂φ ssions simple and without . at the initial surface. This φ + 2 µ ρσν = 0 ν ws. Taking an integration by φ onents. , = sence field are proposed in the with single degree of freedom. φ = 0 µνρ ti-fluid or multi-field models, it ) ) + 2 µ n φ, φ N αβµ , µν φ model with lower derivatives, but , φ ) φ νµρ φ ...µ ], the non-minimal coupling to the ...µ L µν φ µν 1 φ es, we only consider the case L 1 µν ( µν ∂ µ ∂ µ 22 − L ∂φ V 3 ∂φ , φ ∂φ are constants. In ref. [ ∂ − µ ν ρσ L νρµ 2 3 n φ ) ∇ µ ∂φ∂φ∂φ ,...,φ ∂ ( ∂φ µ φ 2 φ, φ ∇ µ 2 µν ∇ + αβ A, B 1 ( µ L + ... – 4 – B ∂φ νµ 2 ∂φ ρ 1 ,φ φ ∂ µ 1 + + µ µ ∂φ ∇ L µ and so on are the covariant derivatives of n 1 during the evolution, each component does not cross ∂ αν ∂φ + AX L φ φ,φ φ ∂φ − 1) 2 ( 1 ν ρν = − µ ∂ µ L ( ρσµ φ ∂φ ∇ L φ = 2 =1 N σµ µ X −∇ n − φ µν L + L µν µν ∂φ ∂φ ∂ ≡∇ ) can be rewritten as a function which does not depend on L L L ρσ 2 ∂φ 2 3 ∂φ ∂ L ) is the potential and ∂ σ µ 3.5 ∂ is non-degenerate, this is a fourth order differential equat th order derivative equation, the whole system contains 3 1 φ ∂φ ∂φ∂φ µ ( ∂ α ∂φ N µν ρ V L ∂φ , ∂φ + 2 φ, φ ∂φ µ ∂ 1 ρσ µ µ +2 + ∇ φ ∂φ µ µ ∇ L ], the constrained scalar field which violates Lorentz invar ≡∇ 2 = 1 ∂ 46 µ ∂φ∂φ 2 φ ] where a simple example is taken to illustrate the feature of For a scalar field with higher (but finite) derivatives, its La The first dark energy model of single scalar field with higher d − 22 L 1. The sample model has the Lagrangian of the type, ∂φ ∂ Generally this is a 2 this boundary and has regular perturbation. Besides the mul is still interesting andTo this important end, to some pursue extensionsliterature beyond quintom including the models the perfect higher fluid and derivative k-es field theory [ total EoS of these two fields crosses in curved spacetime, the Lagrangian is a scalar function of where freedom and some ofloss them of are general ghosts. properties of In higher order derivative to fieldmotion keep theori is the discu explicitly that this modelone is field equivalent has to negative theparts kinetic double-field the term. Lagrangian This ( can be shown as follo gravity [ so on. In this paper we only consider the first extension. − where equation, the form, If the matrix Expanding this equation and considering the symmetry this equation we need to impose the values of The equation of motion from this Lagrangian is means such a model essentially possesses two dynamicalin comp [ JCAP04(2012)003 φ = with (3.8) (3.9) (3.6) (4.1) (3.7) BB (3.10) (3.12) (3.11) L νµρ σ R , , the higher σ U φ ) is identically ts kinetic term = 0 = as a function of 3.4 ν φ φ νµρ νµρ µ ond order differential σ 2 φ φ . We first introduce an R , − φ µν σ ) 2 in eq. ( φ 2 νρµ ), we accordingly have the . L 3 φ erate higher derivative field to get ) µν , φ µν lds, ∂ s an arbitrary scalar function ϕ 1 L L ∂φ 2 ∂φ φ ative terms in the equation of φ, ϕ mode. 2 ves antum level, the non-degenerate ρ + ∂ ∂φ∂φ∂φ ∂ . ρσ , s. Non-degeneracy means U( φ ) ∂φ ( ∂φ + − 2 , ) to ( 1 + −L ) 2 νµ √ ) φ, ϕ ) may cross the cosmological constant φ φ . ρν µ ) = φ U( φ, φ 3.6 φ ∇ µ φ, ϕ 2 . 2 2 ), σµ φ, ϕ 2 − . After finding the potential ( φ B L φ U( ∂φ 2 ϕ φ µ φ ν 3.6 µ φ, ∂ − µν 2 ∇ ( ≡L ∇ ∂φ – 5 – φ 1 2 φ L , φ ∂φ and ϕ ϕ ) 2 L σ − µ − = 3 ϕ ϕ φ )= 1 ∂ ) can be removed with the price of introducing extra ∂φ L µν φ − = ρ −∇ L µ φ 2 3.6 L ( φ, ϕ ∇ = ∂φ ∂ 1 2 ′ 1 ∂φ∂φ φ U( + L √ µ ν ∇ φ = + 1 2 1 ρµ µ φ φ φ = represents the derivative with respect to ′ µ µν B L L 2 ∂φ L ∂ ρ 3 ∂φ∂φ ∂ − is a ghost which violates the null energy condition because i ∂φ∂φ L 2 ∂φ ∂ , ) disappear, i.e., φ +2 φ 2 3.4 the Riemann tensor. The equation of motion remains to be a sec . Then change the independent variables ( and ϕ νµρ φ σ it may be considered as a potential of where we have used the commutation of the covariant derivati R This equivalent Lagrangian has a more clear form of double fie derivative term in the Lagrangian ( it belongs to theof more general class in which the Lagrangian i The mode which is equivalent to the following Lagrangian Legendre transform of the Lagrangian in eq. ( zero. Correspondingly, allmotion ( the third and fourth order deriv boundary without divergent perturbation. However at the qu higher derivative model is plagued by the existence of4 ghost Degenerate higher derivativeIn model the degenerate higher derivative model, the matrix field, Now we use thismodel Lagrangian equivalent to to multi-field illustrate model how with lower is derivative the non-degen has a wrong sign. This explains why the model ( through the field redefinitions 0, where the subscript auxiliary field defined as and Due to the non-degeneracy, we may convert the above equation JCAP04(2012)003 . e . − ) φ φ This (4.3) (4.5) (4.9) (4.8) (4.7) (4.6) (4.2) (4.4) µ ) = 0 ] ∇ XG φ X 2 ν X µ ν definitions − ∇ ∇ ∇ φ µ X + µ ∇ φ is totally symmetric ν µν φ. −∇ µ ∇ R model and in infla- , φ. φ ∇ ν ) + X . µ µνρσ µ µ ) φ X ∇ φ ∇ u φ µ ν µ ∇ ν ( X 2 a ∇ ∇ ∇ in the Minkowski space, all the KGB µ y momentum tensor up to F µ φ F F ∇ + C ν field is obtained through the ∇ + ν φ − is Lagrangian can stably cross + ∇ φ u red in it even though we only φ 2 an. This means the degenerate round evolution and properties φ round, ν φ µ ( ν K µ a ∇ X ( ∇ ) becomes µ ∇ XF = F φ F ) 2 = 0 to keep the equation of motion at the µ 2 φ 4.1 + / , −∇ ν − 3 ∇ 2 ], named as µν ) ) φφ ) ∇ n of motion because X µν L φ φ, X F ∂φ µ φ 35 X 2 was considered in the context of Galileon ( ) with constant δS 2 ∂ K 2 δg ρσ 2 ∇ ( F νσ X F φ [( g X ∂φ η 2 X ν + (2 + F √ µρ − ν = +4 ∇ η φ + u φ − – 6 – φ can only be + 2 φ µ 2 G − µ φ µ ) ] for some more discussions about the case in flat spacetime. φ u = L µ ) ρσ ∇ XF µ ∇ φ, 38 ) η p 2 ∇ -inflation. The box term is equivalent to µν µ ∇ 1 X , the energy momentum tensor may be rewritten as µν + X T − µ ρ G φ, X ∇ η X µ ( ( ρ ∇ µ X X ∇ C φ, X G = 0 is the unique way to discriminate higher derivative terms. ( µ ∇ K φX and = +( F ∇ F µν XX p )+ µν L µν F +( ∂φ + K XF 2 ) the equation of motion ( L ∂ )+ implies the Lagrangian only depends on the second derivativ ∂φ pg ρσ + 2 µν − K ∂ g ρσ 4.3 − ∂φ φ ) − K φ, X )+2 µν ), the degenerate Lagrangian may be generally written as ∂φ ( ], named as φ φ 2 = L ρ φ, X ∂φ = = 2 K 2 X ( 43 ∂ ∇ ρσ µν p after integration by parts and dropping a surface term. By re ρ ]. K X K φ,X = T X ( ∂φ φ 33 − ρ L ν = F +2 ∇ ∇ Xφ L φ F µ µ ∇ + ∇ XK φ ) and ] and its generalization was studied in [ 2 K ν X ( µ − 31 ∇ − ∇ φ φ ( φ, X µ With the notations ( Zero matrix φ = K ( ∇ In the flat spacetime it is not necessary to require F 1 K µν X +2 T of Now we will investigate whether thethe dark boundary energy of model cosmological fromof th constant perturbations. by studying its backg G under the interchanges of the indices. See ref. [ third and fourth order derivative terms vanish in the equatio second order, for example if tion model building [ for the degenerate model it is equation, but thereconsider is the a minimal curvature-field coupling couplingmodel to has term the no appea gravity extra in degree the of Lagrangi freedom. terms linearly. With Lorentz invariance, But in the curved spacetime, In terms of the fluid variables We can read offlinear the order pressure of and perturbations energy around the density homogeneous from backg this energ has be shown in ref. [ theory [ The box term at the right hand side with The energy momentum tensorvariation which of sources the the action gravitational with respect to the metric tensor, JCAP04(2012)003 − X = 0. (5.1) (5.2) (4.10) (4.12) (4.11) i K ). The ]: or vise is space 36 , u 4.3 µ ) between a . 2.1 = 1 odel ν . The energy ˙ 0 ˜ u φ cities as u µ H u )˜ p + 3 + ¨ φ ρ is the four acceleration is a second order variable = , µ ) forwardly that ct perturbations of higher +( ν ) u nd linear levels, its spatial nt boundary is that φ ρ a um tensor of the degenerate ˙ µν und evolution and the linear µ φF 2 ch the metric is given by ¨ ∇ φF a e of relativistic imperfect fluid, ρ . This is the very reason why pg e dark energy model ( H tion, we have to check whether u + 3 ecially the relation ( . . ) = 0 is identical to the equation and j ze the quintom idea and avoid the µ − , p ≡ ≃− ˙ φF µ ¨ ] we have defined the four velocity φ φ dx µ ν a + 5 i Xu H a a X ρ ρ 3 F µ dx ( XF 2 a ∇ p / 2 ij − ρ 2 H 3 = 2 δ ) u + φ ) − F = 1, p t φ 3 ρ X ( + + 3 ) X µ 2 µ = 0. We have also used the relation [ + µ XF (2 a ρ u K X ∇ 2 ρ µ – 7 – ( µ u − + (2 u X Xa ¨ − φF µ X µ µ 2 a u − X X ∇ ] or k-essence [ dt = 2 ν , + 2 K only the time component is non-zero, = ˜ u ( 48 = 2 + 2 is homogeneous, so we soon have X µ ) does not have the form of perfect fluid due to the last µ / K µ 2 X µ ˜ 2 u u u φ ˙ − K )˜ φ ∇ ds 4.9 p = = = 2 = + p p ρ ρ X + +( ρ µν could evolve from the positive region to the negative region pg ¨ − φF which is normalized as = + should be first order variables. So if we redefine the four velo X ). For the velocity i 2 µν ˙ a φF T √ 4.4 H φ/ 3 µ − ∇ φ = can be identified as heat flow. Further, one can prove straight µ µ XF like and itscomponents zero-th component vanishes at both background a where by analogy with k-inflation [ This energy momentum tensor ( background universe is a spatially flat FRW spacetime, in whi u a the pressure andthe density degenerate perturbations higher is derivativeproblems model lost possessed is in by possible this single to model k-essence reali field. 5 Cosmology with degenerateIn higher this section, derivative we dark consider energy in m more detail the dynamics of th two terms depending on the four acceleration. In the languag In the last step, we have considered the fact that the product which is orthogonal to the velocity, i.e., versa providing the energythe density perturbations always are positive. stable. In addi the energy-momentum tensor should be At the background level the field The key point for this model to cross the cosmological consta and can be neglectedperturbation if theory. we This restrictmodel equation our has means studies apparently the onorder. the energy the moment same backgro But form it is of essentially perfect different fluid from if perfect we fluid, negle esp where we have considered conservation law in the expanding universe ˙ of motion ( 2 JCAP04(2012)003 ] . 35 φφ (5.3) (5.7) (5.5) (5.6) (5.9) (5.8) K + φφ model [ vity theory , ¨ 2 φF p . . X 0 /M Xφ KGB 2 F > F 2 ) + 2 2 2 p , X Xφ X 4 Xφ X F /M ppressed by the Planck in the 2 2 2 − ) around the background ˙ + 6 ¨ φF φK X F d as φ 2 rbation of the scalar field is 4 Cπ 4.4 X − coupled system based on the Xφ ( 2 X − + XF y other matter. This approxi- F rse for most time and had tiny ence field. Here we will use the 2 φ ) (5.4) fferent sound speed squared, gy perturbations, it is safely to φφ s to get the linear perturbation r this purpose we firstly expand 8 2 rbations of spacetime. However φ − π φ i , − φ e from the action which is second µ ward but tedious calculations, we ˙ X n this paper we will consider the − φF . 4 ) XF π∂ X ∇ ) i Xφ X 8 X XF − ∂ X F ¨ 4 φF 2 2 µ − φ x, t B + 2 a ) ( X XF − ∇ X φ 2 π X 4 − XF F + F 8 ) − 2 − ˙ + ˙ φ F π )+ XF φ − φ ( t ˙ A K HXF ( H ) φ + , ( , φ – 8 – X XF (6 g XF H π 3 X F 4 0. Furthermore, the absence of ghost mode corre- 8 6 + 2 ( H √ ˙ ¨ φ ¨ )= XF φ − x φF − − > 4 xdta ) H 2( 2 s + X F d 3 6 ) c 2 X ) + 3 d x, t ˙ XX − φ F ( Z − ( − φ Xφ ˙ X Z H φ XF φ = XK 1 2 K H ˙ XX + S φK 6 + 2 XF ], a full treatment of the gravity- F ¨ + 2 − φ 4 − )= ( XK ˙ 43 X φ π 2( − φφ ( K H XX − ) to express the propagating velocity of the perturbations. + 2 F ˙ 6 (2) φF ) ˙ = X S X φ − 5.7 X XK K K . This sound speed depends on the gravity theory, here the gra H A B + 2 = XX + 2 = πG + 2 φ 8 2 s A X ¨ φ / c XK K 2( = 1 HXF = − + 2 -inflation model [ 2 p 2 (6 ′ s G X X c M d If the universe is dominated by the scalar field, as discussed Similar to the analysis in single k-essence model, the pertu The classical stability requires dt K K = = = A C B For complete consideration wefor should dark also energy, include it the iscontribution pertu believed to to the be curvature.neglect subdominant So the in for the metric studying unive perturbationsmation the which will dark greatly are ener simplify sourcedequations. the mainly analysis. b One There is are to two directly way perturb the equation of motion is Einstein’s general relativity.sound speed For in dark eq. energy ( studied i sponds to with to the second orderobtain of the the desired perturbations. second After order straightfor action for We can see from the action that the sound speed square is define or the evolution. Another one isorder to of adapt perturbations the as variationalsecond we principl have way to shown in discuss section thethe 2 perturbation action for of k-ess the scalar field. Fo a small deviation from the homogenous background Arnowitt-Deser-Misner (ADM) method will give a slightly di where both numeratormass and denominator are modified by terms su JCAP04(2012)003 ] s ), A 49 = 0 well 4.4 2 (5.10) (5.11) (5.12) (5.13) (5.14) c 61 fits well − 1 violating DE 167 for the . − = 10 , hence when 1 p . i − ˙ φ + ) and the sound . The choices of and Ω 725, also compat- ρ w < . In both figures, . = 0 2 5.6 122 2 DE ˙ φ egular at the crossing 10 ), one may find that ) 2 and w . = 0 and in the second = 0, i.e., ˙ and 158 and volution and the coeffi- 1 × = 0, both coefficients H . 2 these two cases give rise 4.3 p 1 and corresponding model e the analysis in ref. [ 1 1 c − 0 p c . re considering is simple: ion of state of dark energy − + F = ns of dark energy consistent + tion of state of dark energy ρ xplicitly, we study an explicit + 3 i ussed in section 2. So in this = ρ = 3 , nd part has d, which comes from eq. ( he perturbation is regular and defined in ( φ For the first case where 2 2 , eld has the right kinetic term to 1 and th the current constraint by the figures c t values of H A DE 1 . Xφ K c w is proportional to X, 2 1 ) c ˙ X. φ B − + 9 ) + 2 ¨ H φ 1, we have φ = 0, φ φ 1 1 2 2 nor − c c µ c a c 2 ( ∇ in both cases are around 0 A − − X )= ˙ 1 + 6 1 µ DE φ ˙ φ ) – 9 – ∇ 2 − − 1 H φ 2 2 c 3 2 φ φ c remain finite and positive as we will show explicitly 2 2 − − c c ), we get the pressure and energy density of this model − ¨ φ A X ( 1 (1 4.8 − = ( = ( c H p ρ = and . Note that the third term can also be viewed as an “effective 1 ]. Moreover, the parameter 2 s L ) are both positive during evolutions in these two cases. Thi c + 3 c = 1 and chose the parameter 13 − ¨ ) and ( φ 5.7 ) 2 = 4.7 πG 160. The values of Ω φ . 2 F 0 c ) one can find that neither ± − and ˙ 5.2 φ 2 1 but both 143 . H − 1 1 are different for these two cases. In the first figure c Xφ − 6 2 2 are constants. Compared with the notations in eq. ( c 05. In both cases we set the initial conditions as c . = − 2 + , 1 (1 DE are not vanished in general. The equation of motion would be r c = 0 X w 2 − In order to illustrate the realization of quintom scenario e Finally we should emphasize that with different parameters, We performed the numerical calculations of the background e B c = point. This issingle different field from model, the forparameters, case particular of choices it of k-essence is the modelevolves possible functions across disc to find solutions in which the equa below. Within these solutionscircumvent the the fluctuation pathology of of the scalar ghost, fi even though its backgrou and the null energy condition. form of the degenerate dark energy model. The Lagrangian we a the dark energy crosses the cosmological constant boundary In addition, we have the equation of motion for the scalar fiel ible with the observations [ where Compared to eq. ( we have used the unit 8 gives means that at theclassically crossing stable. there is no ghost instability and t respectively as: speed squared defined in ( K At the point where the equation of state crossing mass term”. From eqs. ( to predictions of different fate of dark energy in the future. cients for perturbations for different parameter choices in parameter one within the matter dominatedwith era the in observations. ordercrosses to We the can make cosmological the see constant evolutio that boundary, and in the both presen cases the equat with the observational data. At present time ln first and second casesobservational respectively. data These from are CMB, consistent LSS wi and Supernovae. For exampl JCAP04(2012)003 , 122 10 . The (5.17) (5.18) (5.15) (5.16) × C 1 forever 0 . 2 at early

are positive

− 2 s − 5 5 c = 3 1 and sound speed 4 4 c and A 3 3 A 2 2 with constant 1 1 . From it we can solve C 3 − +

0 0 nderstood as follows. With a will approach to φ lna , parameter lna ively -1 -1 1 and approaches DE DE φ, → he universe was matter dominated. − µ w -2 -2 φ . ∇ ant in the future. 0 φ -3 -3 2 . 2 ˙ φ 2 1 in the future. Both , 2 1 ) -4 -4 2 ˙ HJ ˙ − c φ φ crosses 1 ) c − ¨ -5 -5 H φ = 0 1 1 1.0 0.8 0.6 0.4 0.2 0.0 1.4 1.2 1.0 0.8 0.6 0.4 0.2 H 2 s 12 DE φ 0

c c c 1 µ c w − 5 5 , which scales as 6 ∇ 2 HJ 1 ˙ ], φ – 10 – 4 4 + 1 + 6 √ ). The parameters are chosen to be (1+2 35 H + 3 − 1 X ± − 3 3 0 c µ ˙ 1 5.10 3 J = ( = ∇ 2 2 1 = − p ρ , the energy density ratio Ω c ˙ ˙ 1 1 φ φ = 0 is nothing but the equation of motion. For the back- = DE

w 0 0 µ µ = J J lna lna 0 µ -1 -1 1 recently, and approach to J ∇ − -2 -2 -3 -3 -4 -4 -5 -5 5 4 3 2 1 0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 DE A w , the model is symmetric under the ﬁeld shift of dark energy of the model ( is homogeneous the conservation law simpliﬁes as 2 c . The equation of state 2 s φ c that ˙ φ = 0 and the initial conditions are chosen in the far past when t 2 On the other hand the energy density and pressure are respect The equation of sate crosses and the universe entersvanished into the de Sitter phase. This can be u time, then when the dark energy dominates the universe during the evolution. The universe will be dark energy domin which reduces to the case discussed in [ and the “charge density” Figure 1 for c corresponding Noether current is squared and the conservation law ground, JCAP04(2012)003 , H → 1 1 are 122 c . So 3 2 s 10 c 3 (5.19) (5.21) (5.20) DE − × → w a 0 ˙ and . φ

ubble rate

A 5 5 = 3 1 and sound speed 4 4 c . So the equation A 1 3 3 c 1 ), we should choose 2 2 2 5.17 1 1 . , 2 iod, the energy density of 2

0 0 − H , parameter lna than in the matter dominated lna 2 1 1 -1 -1 c he universe was matter dominated. DE H )= sits to the phase of dark energy 1 1 ¨ 1, in eq. ( 54 φ c -2 -2 1 3 . 0. The equation of state c 0 → -3 -3 ˙ approaches to zero so that φ> 2 dominant in the future. → 6 ˙ 0 -4 -4 φ H HJ 1 J ¨ 1 φ (1+2 c c -5 -5 → 1.0 0.8 0.6 0.4 0.2 0.0 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 DE H

2 s

2 c 1 2 ˙ and →− φ c − 5 5 ) 6 ˙ approaches to the constant 4 ˙ 1 φ φ / – 11 – 4 4 ¨ 1 ¨ φ is more close to √ φ H ). The parameters are chosen to be H − 1 ˙ 1 1 φ ) c c 3 3 c and the matter energy density decreases as 2 1 c 1+ 5.10 3 2 2 a = 1 + 2 , the energy density ratio Ω 1 + 6 1 + 6 ˙ 1 1 φ − − DE ( −

w 0 0 3 1 lna = lna ≃ -1 -1 1 recently, and becomes larger than 0 in the future. Both 2 − DE , soon after the beginning -2 -2 H 3 w − -3 -3 a -4 -4 1. The universe enters into the de Sitter phase in the future. , it is easy to obtain that . This time the universe is still dominated by matter and the H 2 -5 -5 2 5 4 3 2 1 0 / 1.5 1.0 0.5 0.0 3 -0.5 -1.0 -1.5 -2.0 -2.5 H ˙ A H 1 DE − →− c w scales as a 3 ) of dark energy of the model ( 0 ¨ φ . The equation of state J 2 s 1 c c → − approaches to the constant (54 05 and the initial conditions are chosen in the far past when t . ¨ φ H scales as = 0 (1+2 2 Because This is athe phantom dark attractor energy solution increases to as the fast model. as At this per with the initial conditionsdomination we at low have redshift. chosen In the this universe phase tran Figure 2 and positive during the evolution. The universe will be matter- To guarantee the positivity of the energy density, 6 c era. However, the Hubble rate is of state H the plus sign, i.e., − The equation of sate crosses so squared JCAP04(2012)003 (5.22) (5.23) re and r model increases e effective DE 1. However, ], the authors w − 51 leading to quantum insta- he cosmological constant . Hence in the future dark er derivative operators are widely existing in standard 6 increases to the value much ter phase and − n, the effective mass term is matter-dominant stage. This 2 y when the universe shifts to we considered the degenerate alysis. Particularly, we chose ion of dark energy in this case a utation. Our numerical results φ its generalizations. We started ies. n eliminate the classical insta- we have chosen is negative and f attention in the literature (for 2 e pressure of the dark energy as cosmological constant boundary ian cannot give rise to the sta- a degrees of freedom brought by c higher derivative operator, and c . initial singularity can be replaced ndition in the curved spacetime, 2 and approaches φ 2 The past studies have shown that 1 and approaches to the phantom ˙ e physics. In refs. [ φ − − H φ, 1 2 ˙ 1, but this phase is not stable because c φ 3 2 c → − − ). We studied the cosmological application ˙ φ ) DE = 2 4.3 w 0 φ 2 – 12 – c increases from HJ − 05, the shift symmetry is violated by the “effective” . DE + 3 0 w ˙ = (1 = 0 J 0 2 c J 2. Because the initial value of − increases continually and the effective mass term becomes mo = gets a minor modification compared with the first case, φ ]). A very interesting question is how to build a single scala DE 0 ). The equation of state increases from the negative region to the positive one, so th 50 J w φ 5.11 0, > H 1 1 c 3 A quintom model which successfully violates the NEC without For the second case where ≃ ˙ larger than unity, it willseen dominate from the eq. energy ( density and th the value of the field more important. So the universe will not enter into the de Sit bility also has important implications to the early univers mass term remains smallthe during dark a energy fairly dominated long phase, period. Recentl continually and gets back to larger than zero again. When φ attractor solution With the parameters andvery small the at initial the conditions timeis of we similar matter have to domination. chose the The first early case, evolut its equation of state crosses of this model,an including explicit the example background and and performedverified that a perturbation this detailed model an numerical is comp and able the to realize perturbation the evolves EoS smoothly to cross without the any patholog found that a quintomand model inflationary cosmologies. can In avoid this the picture, big the moment bang of singularity which gives rise toa the crossing single without scalar anyble field instabilities. violation satisfying of aintroduced. NEC. generic k-essence The However, Lagrang non-degenerate itbilities higher becomes but derivative is possible model stillhigher when ca plagued derivative high model by inspired thewith by ghost a the mode. single Galileon scalar theory In ofthen which and this suggested the a paper Lagrangian degenerate containsthese condition generi higher to derivative eliminate operators. thethe extr Lagrangian Using can the be degenerate reduced co to the form ( the energy density of the dark energy will decrease as fast as mass term. The “charge density” satisfies the equation where the density energy might exit dominationphenomenon and of the the universe model will has return not been to discussed before. 6 Summary The model buildingboundary, i.e., of the dark quintom darkexample energy energy see which has refs. attracted is a [ lot possible o to cross t JCAP04(2012)003 , ], en- field ]. 55 ] ]. (1988) 668 SPIRE , (2011) 525 rametrizations: IN SPIRE ] [ 56 IN ]. ] [ B 302 , ]. , , ]. SPIRE ]. l (NSC) under Project IN 2. , 1120054 and by SRF for undation of China under SPIRE astro-ph/9908168 ] [ [ C is supported by funds of IN ]. rturbation theory has been ]. Recently, it was observed 5142 and 11033005, and by ], non-Gaussianities [ SPIRE earch Fund for the Doctoral SPIRE rted in part by the National ] [ IN ]. 57 ]; IN 54 Nucl. Phys. r derivative operators. – c acceleration , ] [ ] [ A dynamical solution to the SPIRE astro-ph/9708069 52 [ arXiv:0909.2776 IN [ s[ SPIRE (2002) 23 SPIRE ] [ IN Quintom cosmology: theoretical IN Commun. Theor. Phys. , ] [ ] [ B 545 arXiv:1106.5658 [ (2010) 1 1 exactly leads to a bouncing solution in the (1998) 1582 arXiv:0803.0982 [ Cosmological imprint of an energy component with − Dynamics of dark energy 493 arXiv:0903.0866 80 [ – 13 – astro-ph/0404224 Kinetically driven quintessence [ Dark energy Dark energy constraints from the cosmic age and Dark energy and the accelerating universe ]. The physics of cosmic acceleration hep-th/0603057 [ Phys. Lett. (2011) 119 , (2008) 385 astro-ph/0004134 astro-ph/9912463 [ SPIRE ]. Consequently, we expect that bouncing cosmologies can [ Phys. Rept. (2009) 397 IN (2005) 35 46 [ 58 Cosmological consequences of a rolling homogeneous scalar , B 703 ]. 59 Phys. Rev. Lett. (2006) 1753 , B 607 ], and the related preheating phase [ SPIRE Probing the dynamics of dark energy with divergence-free pa IN (2000) 4438 56 D 15 ] [ , Phys. Lett. (2000) 023511 (1988) 3406 85 , 53 A phantom menace? Cosmology and the fate of dilatation symmetry D 62 D 37 Phys. Lett. , ]. ]. SPIRE SPIRE arXiv:1103.5870 IN IN a global fit study [ [ C. Armendariz-Picon, V.F. Mukhanovproblem and of P.J. a Steinhardt, small cosmological constant and late time cosmi Int. J. Mod. Phys. Ann. Rev. Astron. Astrophys. Ann. Rev. Nucl. Part. Sci. implications and observations Phys. Rev. Lett. Phys. Rev. supernova Phys. Rev. [ general equation of state [4] R.R. Caldwell, [6] B. Feng, X.-L. Wang and X.-M. Zhang, [7] E.J. Copeland, M. Sami and S. Tsujikawa, [8] J. Frieman, M. Turner and D. Huterer, [9] R.R. Caldwell and M. Kamionkowski, [2] C. Wetterich, [1] B. Ratra and P.J.E. Peebles, [5] T. Chiba, T. Okabe and M. Yamaguchi, [3] R.R. Caldwell, R. Dave and P.J. Steinhardt, [10] Y.-F. Cai, E.N. Saridakis, M.R. Setare and J.-Q. Xia, [12] H. Li and X. Zhang, [11] M. Li, X.-D. Li, S. Wang and Y. Wang, by a big bounce.extensively developed, namely, Based on adiabatic on perturbation this scenario, the corresponding pe tropy fluctuations [ that the Galileon model with the EoS across Acknowledgments The research ofGrants ML No. is 11075074 supported andProgram in No. of 11065004, part Higher by by the EducationROCS, National Specialized (SRFDP) SEM. Science Res under Fo TQ GrantNo. No. is NSC98-2811-M-002-501 2009009 supported and by No.physics Taiwan NSC98-2119-M-002-001. department National at Y Science ArizonaScience Counci State Foundation of University. 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