Noether Symmetry Approach for Dirac-Born-Infeld Cosmology
Salvatore Capozziello, Mariafelicia De Laurentis, Ratbay Myrzakulov
aa r X i v : . [ g r- q c ] F e b Noether Symmetry Approach for Dirac-Born-Infeld Cosmology
Salvatore Capozziello , , , Mariafelicia De Laurentis ∗ , Ratbay Myrzakulov , , Dipartimento di Fisica, Università di Napoli “Federico II”,Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, ItalyINFN Sezione di Napoli, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy. Gran Sasso Science Institute (INFN), Via F. Crispi 7, I-67100, L’ Aquila, Italy. Tomsk State Pedagogical University, 634061 Tomsk and National Research Tomsk State University, 634050 Tomsk, Russia and Eurasian International Center for Theoretical Physics and Department of General Theoretical Physics,Eurasian National University, Astana 010008, Kazakhstan. (Dated: February 10, 2015)We consider the Noether Symmetry Approach for a cosmological model derived from a tachyonscalar field T with a Dirac-Born-Infeld Lagrangian and a potential V ( T ) . Furthermore, we assumea coupled canonical scalar field φ with an arbitrary interaction potential B ( T, φ ) . Exact solutionsare derived consistent with the accelerated behavior of cosmic fluid. PACS numbers: 04.50.Kd, 98.80.-k, 95.35.+d, 95.36.+x
I. INTRODUCTION
Scalar fields are introduced in cosmology in order togenerate dark energy dynamics and overcome shortcom-ings coming from the standard Λ CDM model. Wideclasses of theories involving scalar fields have been con-sidered in literature by introducing different kinetic termsboth canonical and tachyonic and several interesting re-sults come out in view to address the big puzzle of cos-mic acceleration. Such scalar fields, in general, are ob-tained from particle physics and they can give rise toquintessence or phantom behavior in connection to thefact that they can be canonical or tachyonic [1–7]. Inparticular, there is much interest in the tachyon cos-mology since the presence of tachyon fields is well moti-vated by string theory [8–10]. There are several work onthe interaction between dark energy (tachyon or phan-tom) and dark matter, where different kinds of inter-action terms are phenomenologically introduced [11–16].Specifically, tachyon fields come from the D-brane ac-tion (Dirac-Born-Infeld type Lagrangian) in string theoryand they represent the lowest energy state in unstable D-brane systems [17–22]. Features as classical dust can beproduced from an unstable D-brane through pressurelessgas with finite energy density [23, 24]. Some cosmologicaleffects of a tachyon rolling down to its ground state havebeen widely discussed also using tachyon matter associ-ated with unstable D-branes showing an interesting equa-tion of state as discussed in [25]. Furthermore, tachyonicmatter could provide an explanation for inflation at earlyepochs contributing to some new form of cosmologicaldark matter at late epochs [26]. Inflation derived fromtachyonic fields has been discussed in Ref. [27, 28]. Fur-thermore, Dirac-Born-Infeld type action containing in-variant curvature function, like f ( R, R , ... ) gravity func-tions are considered in Refs. [29–33]. ∗ e-mail address: [email protected] Despite of these interesting results, dynamical equa-tions deriving from this type of Lagrangians are non-linear and general analytic solutions do not exist. Be-cause of this fact, the Noether Symmetry Approach couldbe a suitable method to seek for physically motivatedsolutions [34–37]. The existence of Noether symmetriesleads to a specific form of coupling function and the scalarfield potential. Furthermore, symmetries lead to the ex-istence of conserved quantities that allows to reduce dy-namics thanks to the presence of cyclic variables [38–41].The technique is used to obtain cosmological models inalternative gravity [42–45]. Black holes solutions havealso been obtained by Noether symmetries [46, 47].This paper is organized as follow. In Sec. II we intro-duce the Dirac-Born-Infeld type Lagrangian and derivethe cosmological equations of motion. The Noether Sym-metry Approach is described in Sec. III, and, in Sec. IV,it is applied to the Dirac-Born-Infeld Lagrangian. Someexact solutions are found. Conclusions are drawn in Sec.V.
II. DIRAC-BORN-INFELD TYPELAGRANGIAN AND EQUATIONS OF MOTION
Let us consider a system of two interacting scalar fields,a tachyonic field T and a canonical scalar field φ , de-scribed by a Dirac-Born-Infeld type Lagrangian whichcouples the two fields through a potential B ( T, φ ) . Onthe other hand, V ( T ) is the tachyonic potential. TheLagrangian of the system can be written as [48, 49] L = − V ( T ) p − ∂ µ T ∂ µ T + 12 ∂ µ φ∂ µ φ − B ( T, φ ) . (1)For a spatially flat Friedman-Robertson-Walker uni-verse, we obtain the following point-like Lagrangian L = − a V ( T ) p − ˙ T + 12 ˙ φ a − a B ( T, φ ) , (2)where a is the cosmic scale factor of the universe. TheEuler-Lagrange equations give dynamics for φ and T ,that is ¨ φ + 3 H ˙ φ + B φ ( T, φ ) = 0 , (3) ¨ T ˙ T − H ˙ T + V T ( T ) V ( T ) + B T ( T, φ ) p − ˙ T V ( T ) = 0 , (4)where H = ˙ aa is the Hubble parameter. The subscripts‘ T ’ and ‘ φ ’ indicate derivatives of the potentials B ( T, φ ) and V ( T ) with respect to the two fields. Dot indicatesderivatives with respect to the cosmic time. The equationfor the energy is E L = V ( T ) ˙ T p − ˙ T + ˙ φ = 0 (5) III. NOETHER SYMMETRY APPROACH
Solutions for the dynamics given by the Lagrangian(2) can be achieved by selecting cyclic variables relatedto some Noether symmetry [34–37]. In principle, this ap-proach allows to select models compatible with the sym-metry so that the conserved quantities could be consid-ered as a sort of Noether charges. Before going into thedetails of our model, let us introduce the Noether Sym-metry Approach.Let L ( q i , ˙ q i ) be a canonical, independent of time andnon degenerate point-like Lagrangian where ∂ L ∂λ = 0 , detH ij ≡ (cid:13)(cid:13)(cid:13)(cid:13) ∂ L ∂ ˙ q i ∂ ˙ q j (cid:13)(cid:13)(cid:13)(cid:13) = 0 , (6)with H ij the Hessian matrix related to the Lagrangian L and dot the derivative with respect to an affine parameter λ , which, in our case, corresponds to the cosmic time t .In analytic mechanics, L is of the form L = K ( q , ˙ q ) − V ( q ) , (7)where K and V are the kinetic energy and potential en-ergy respectively. We stress that L could assume morecomplicated expressions than Eq. (7) as in [38–40]. K is a positive definite quadratic form in ˙ q . The energyfunction associated with L is: E L ≡ ∂ L ∂ ˙ q i ˙ q i − L , (8)which , when the Lagrangian is in the form (7), reducesto the total energy K + V . In any case, E L is a constantof motion. Since our cosmological problem has a finitenumber of degrees of freedom, we are going to consideronly point transformations. Any invertible transforma-tion of the ‘generalized positions’ Q i = Q i ( q ) induces atransformation of the ‘gene-ralized velocities’ such that: ˙ Q i ( q ) = ∂Q i ∂q j ˙ q j ; (9) the matrix J = (cid:13)(cid:13) ∂Q i /∂q j (cid:13)(cid:13) is the Jacobian of the trans-formation on the positions, and it is assumed to be non-zero. The Jacobian ˜ J of the induced transformation iseasily derived and J 6 = 0 → ˜ J 6 = 0 . In general, thiscondition is not satisfied over the whole space but onlyin the neighborhood of a point. It is a local transfor-mation. A point transformation Q i = Q i ( q ) can dependon a (or more than one) parameter. As starting point,we can assume that a point transformation depends on aparameter ǫ , so that Q i = Q i ( q , ǫ ) , and that it gives riseto a one-parameter Lie group. For infinitesimal values of ǫ , the transformation is then generated by a vector field:for instance, ∂/∂x is a translation along the x axis while x ( ∂/∂y ) − y ( ∂/∂x ) is a rotation around the z axis andso on. In general, an infinitesimal point transformationis re-presented by a generic vector field on Q X = α i ( q ) ∂∂q i . (10)The induced transformation (9) is then represented by X c = α i ∂∂q i + (cid:18) ddλ α i ( q ) (cid:19) ∂∂ ˙ q j . (11) X c is called the complete lift of X . A function F ( q , ˙ q ) isinvariant under the transformation X if L X F ≡ α i ( q ) ∂F∂q i + (cid:18) ddλ α i ( q ) (cid:19) ∂∂ ˙ q j F = 0 , (12)where L X F is the Lie derivative of F . Specifically, if L X L = 0 , X is a symmetry for the dynamics derivedby L . At this point let us consider a Lagrangian L andits Euler-Lagrange equations ddλ ∂ L ∂ ˙ q j − ∂ L ∂q j = 0 . (13)Let us consider also the vector field (11) and, contracting(13) with α i s, it gives α j (cid:18) ddλ ∂ L ∂ ˙ q j − ∂ L ∂q j (cid:19) = 0 . (14)Since we have α j ddλ ∂ L ∂ ˙ q j = ddλ (cid:18) α j ∂ L ∂ ˙ q j (cid:19) − (cid:18) dα j dλ (cid:19) ∂ L ∂ ˙ q j , (15)from Eq. (14), it results that ddλ (cid:18) α j ∂ L ∂ ˙ q j (cid:19) = L X L . (16)The immediate consequence is the Noether theorem whichstates: if L X L = 0 , then the function Σ = α k ∂ L ∂ ˙ q k , (17)is a constant of motion. Some mathematical commentsare necessary now. Eq. (17) can be expressed indepen-dently of coordinates as a contraction of X by a Cartan -form θ L ≡ ∂ L ∂ ˙ q j dq j . (18)For a generic vector field Y = y i ∂/∂x i , and 1-form β = β i dx i , we have, by definition, i Y β = y i β i . Thus Eq.(17) can be expressed as: i X θ L = Σ . (19)Through a point transformation, the vector field X be-comes: ˜ X = (cid:0) i X dQ k (cid:1) ∂∂Q k + (cid:18) ddλ (cid:0) i X dQ k (cid:1)(cid:19) ∂∂ ˙ Q k . (20)We see that ˜ X is still the lift of a vector field defined onthe space of configurations. If X is a symmetry and wechoose a point transformations such that i X dQ = 1; i X dQ i = 0 i = 1 , (21)we get ˜ X = ∂∂Q ; ∂ L ∂Q = 0 . (22)Thus Q is a cyclic coordinate and the dynamics resultsreduced. Furthermore, the change of coordinates givenby (21) is not unique and then a clever choice could bevery important. In general, the solution of Eq. (21) isnot defined on the whole space. It is local in the senseexplained above. Considering the specific case which weare going to analyze, the situation is the following. TheLagrangian is an application L : T Q → R , (23)where R is the set of real numbers and the generator ofsymmetry is X = α ∂∂T + β ∂∂φ + γ ∂∂a + ˙ α ∂∂ ˙ T + ˙ β ∂∂ ˙ φ + ˙ γ ∂∂ ˙ a . (24)As discussed above, a symmetry exists if the equation L X L = 0 has solutions. Then there will be a constantof motion on shell, i.e. for the solutions of the Eulerequations, as stated above Eq. (17). In other words, asymmetry exists if at least one of the functions α , β or γ in Eq.(24) is different from zero. IV. NOETHER SYMMETRY APPROACH FORDIRAC-BORN-INFELD TYPE-LAGRANGIAN
Let us apply now the method described in the previoussection to the Lagrangian in Eq. (2). From the statement L X L = 0 , we obtain: α h − a V T ( T ) p − ˙ T − a B T ( T, φ ) i − a βB φ ( T, φ )+ 3 a γ (cid:20) − V ( T ) p − ˙ T + 12 ˙ φ − B ( T, φ ) (cid:21) + (cid:16) ˙ φ∂ φ α + ˙ T ∂ T α + ˙ a∂ a α (cid:17) a V ( T ) ˙ T p − ˙ T ! + (cid:16) ˙ φ∂ φ β + ˙ T ∂ T β + ˙ a∂ a β (cid:17) (cid:16) ˙ φa (cid:17) = 0 . (25)Setting to zero the coefficients of the terms ˙ φ , ˙ T , ˙ φ ˙ T , ˙ a ˙ φ and ˙ a ˙ T , we obtain respectively: γa + a ∂β∂φ = 0 , (26) ∂α∂T = 0 , (27) a ∂β∂T + a V ( T ) ˙ T p − ˙ T ! ∂α∂φ = 0 , (28) ∂β∂a = 0 , ∂α∂a = 0 . (29)Finally we have to satisfy the constraint aα h V T ( T ) p − ˙ T − B T ( T, φ ) i + aβB φ ( T, φ ) ++ 3 γ h V ( T ) p − ˙ T + B ( T, φ ) i = 0 . (30)Eqs. (26)-(29) are consistent for constant values α ( a, φ, T ) = α , (31) β ( a, φ, T ) = β , (32)and finally γ ( a, φ, T ) = 0 . (33)The existence of non-zero quantities α and β accountsfor the Noether symmetry, provided that they satisfy theconstraint (30), which now becomes α h V T ( T ) p − ˙ T + B T ( T, φ ) i + β B φ ( T, φ ) = 0 . (34)It is evident that this constraint makes a direct connec-tion between the tachyon potential V ( T ) and the poten-tial B ( T, φ ) . Finally, we find the constant of motion,namely the Noether charge, as Σ = α a V ( T ) ˙ T p − ˙ T ! + β a ˙ φ. (35)As we can see the constant of motion connects directlythe dynamical evolution of the scalar field φ and thetachyon field T only through the tachyonic potential V ( T ) , and not the coupled potential B ( T, φ ) . Consid-ering the system of Eqs. (3), (4) and (34), we can findsome particular solutions making appropriate positions.In fact supposing that the tachyon scalar field does notdepend on time, being T = T with T a constant, thesystem of equations is solved for φ ( t ) = φ e nt ,a ( t ) = a e − nt ,B φ = 0 ,B T ( T ) = − V T ( T ) , (36)where φ and n are arbitrary constants while a = (cid:18) Σ β φ n (cid:19) . A second solution can be achieved con-sidering the tachyon scalar field to be constant, being T = T , and Σ = β = 0 . In this case the system ofequations is solved for φ ( t ) = φ e nt ,a ( t ) = a e − (cid:16) n n (cid:17) t ,B T = 0 ,B ( t ) = B e nt ,V = V , (37)where φ , n , a , B and V are arbitrary constants.Clearly, the expansion explicitly depends on n . In Figs.1 and 2 are showed the trends of the potential and of thescale factor for different values of n . We are adopting adimensionless time parameter where t = 1 correspondsto a Hubble time H − . V. CONCLUSIONS
In this paper, we have obtained the dynamics equa-tions and investigated the conditions for the existenceof a Noether symmetry in a Dirac-Born-Infeld type La-grangian with a tachyonic potential V ( T ) coupled toa canonical scalar field φ through an arbitrary interac-tion potential B ( T, φ ) . We have shown that a Noethersymmetry exists and is related to a constant of motion.Then, we have considered two particular cases where thetachyon scalar field T is a constant and solved the sys-tem of equations. In the first case, it is possible to findparticular solutions for the system of Eqs. (3), (4) and(34), obtaining the explicit behavior of the scale factor a ( t ) and of the canonical scalar field φ ( t ) . As it is clear inthe solutions (36), both of them have exponential behav-ior, while the coupling potential B is independent of φ .From the condition existing between the derivatives of B and V with respect T , calculated in T = T , it is possibleto obtain a constant value for the tachyonic field. In the ph i ( t ) a ( t ) n=−2n=−1n=1n=2n=−2n=−1n=1n=2 Figure 1: In the top figure is plotted the behavior of φ ( t ) andin the bottom figure the behavior of a ( t ) for different values of n respectively for the solution (36). The solid lines representthe behavior for n = − , the dashed for n = − , the dotterfor n = 1 and dashed-dot for n = 2 . ph i ( t ) a ( t ) n=−4n=−1n=1n=4n=−4n=−1n=1n=4 Figure 2: Plots of φ ( t ) and a ( t ) for the solution (37) for dif-ferent values of n . The solid lines represent the behavior for n = − , the dashed for n = − , the dotter for n = 1 anddashed-dot for n = 4 second case we find the solutions (37), where a ( t ) , φ ( t ) and B ( t ) are exponential functions. In a forthcoming pa-per, we will develop these considerations to more generalpotentials V ( T ) and B ( T, φ ) . [1] A. de la Macorra and G. Piccinelli, General scalar fieldsas quintessence, Phys Rev. D , (2000) 123503 .[2] E. Copeland, M.R. Garousi, M.Sami and S.Tsujikawa,What is needed of a tachyon if it is to be the dark energy? Phys. Rev.D , (2005) 043003.[3] A.de la Macorra, U.Filobello, G.Germán, The Mass, nor-malization and late time behavior of the tachyon field , Physics Letters B , (2006) 355 .[4] A.V.Frolov, L. Kofman, A.A Starobinsky, Prospects andproblems of tachion matter cosmology,
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