Noether symmetry of F(T) cosmology with quintessence and phantom scalar fields
aa r X i v : . [ phy s i c s . g e n - ph ] S e p Noether symmetry of F(T) cosmology with quintessence and phantom scalarfields
Mubasher Jamil ,
1, 2, ∗ D. Momeni , † and R. Myrzakulov ‡ Center for Advanced Mathematics and Physics (CAMP),National University of Sciences and Technology (NUST), H-12, Islamabad, Pakistan Eurasian International Center for Theoretical Physics,Eurasian National University, Astana 010008, Kazakhstan
Abstract:
In this paper, we investigate the Noether symmetries of F ( T ) cosmology in-volving matter and dark energy. In this model, the dark energy is represented by a canonicalscalar field with a potential. Two special cases for dark energy are considered including phan-tom energy and quintessence. We obtain F ( T ) ∼ T / , and the scalar potential V ( φ ) ∼ φ for both models of dark energy and discuss quantum picture of this model. Some astrophys-ical implications are also discussed. Keywords:
Noether symmetries; quintessence; phantom energy; scalar fields; torsion ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected]
I. INTRODUCTION
In the last decade, one of the most active researches in physicist community is investigationof the acceleration of our universe [1], which has confirmed by some observation data such assupernova type Ia [2], baryon acoustic oscillations [3], weak lensing [4] and large scale structure [5].Finding the theoretical explanation of cosmic acceleration has been one of the central problems ofmodern cosmology and theoretical physics [6]. Reviews of some recent and old attempts to resolvethe issue of dark energy and related problems can be found in [7].In order to explain the current accelerated expansion without introducing dark energy, one mayuse a simple generalized version of the so-called teleparallel gravity (TG) [8], namely F ( T ) theory.It is a generalization of the teleparallel gravity by replacing the so-called torsion scalar T with F ( T ).TG was originally developed by Einstein in an attempt of unifying gravity and electromagnetism.The field equations for the F ( T ) gravity are very different from those for f ( R ) gravity, as they aresecond order rather than fourth order. F ( T ) gravity is not locally Lorentz invariant and appear toharbor extra degrees of freedom not present in general relativity [9]. In fact as Li et al [9] pointedout “there are D − F ( T ) gravity in D dimensions, and this impliesthat the extra degrees of freedom correspond to one massive vector field or one massless vectorfield with one scalar field.” In another recent investigation, Li et al [10] studied the cosmologicalperturbation and structure formation in F ( T ) theory and proved that the extra degree of freedomof F ( T ) gravity decays as one goes to smaller scales, and consequently its effects on scales such asgalaxies and galaxies clusters are small. But on large scales, this degree of freedom can producelarge deviations from the standard cosmological model, leading to severe constraints on the F ( T )gravity models as an explanation to the cosmic acceleration.Although teleparallel gravity is not an alternative to general relativity (they are dynamicallyequivalent), but its different formulation allows one to say: gravity is not due to curvature, but totorsion. In other word, using tetrad fields and curvature-less Weitzenbock connection instead oftorsion-less Levi-Civita connection in standard general relativity. We should note that one of themain requirement of F ( T ) gravity is that there exist a class of spin-less connection frames where itstorsion does not vanish [10]. F ( T ) theory leads to interesting cosmological behavior and its variousaspects including thermodynamic laws, phantom crossing and inflation have been examined in theliterature [11].On the other side, we can not ignore the important role of continuous symmetry in the mathe-matical physics. In particular, the well-known Noether’s symmetry theorem is a practical tool intheoretical physics which states that any differentiable symmetry of an action of a physical systemleads to a corresponding conserved quantity, which so called Noether charge [12]. In the literature,applications of the Noether symmetry in generalized theories of gravity have been studied (see [13]and references therein). In additions, Noether symmetry has been used to investigate non-flat [14]and quantum cosmology [15]. The symmetries of the Lagrangian lead to conserved quantities ofthe theory, for instance ‘total energy’ and ‘total angular momentum’. If a generic theory does notpossesses a conserved charge, it implies that this theory has nothing to do with physical reality.Given the fact that F ( T ) theory has no symmetry under Lorentz transformation, it will be in-teresting to know if this theory possesses any other symmetry at all. The application of Noethersymmetries helps in selecting viable models of F ( T ) at a fundamental level. In particular, Weiet al calculated Noether symmetries of F ( T ) cosmology containing matter and found a power-lawsolution F ( T ) ∼ µT n [13]. Further they showed that if n > / F ( T ) ∼ T / . Thus we show that F ( T ) ∼ T / coupled with a canonical scalar field is equivalentto F ( T ) ∼ T n , n > / F ( T ) quantum cosmology. In section-V, we investigatethe cosmography and numerical cosmological implications. Finally we conclude the results insection-VI. II. OUR MODEL
Here, we try to consider Noether symmetry in F ( T ) cosmology in the present of matter andscalar field. The Lagrangian of our model is S = Z d x e ( L F + L m + L φ ) , (1)where e = det( e iµ ) = √− g , where e i ( x µ ) are related to the metric via g µν = η ij e iµ e jν , where allindices run over 0,1,2,3. L F , L m and L φ represent the Lagrangians for gravity model, energy-matter and the scalar field (representing dark energy) respectively. Specifically the total actionreads (in chosen units 16 πG = ~ = c = 1) S = 2 π Z dt a h F ( T ) − ρ m + 12 ǫφ ,µ φ ,µ − V ( φ ) i . (2)where a is the scale factor while H ≡ ˙ a/a is the Hubble parameter. Here ǫ = +1 , − φ has the potentialenergy V ( φ ) (to be determined in the later sections) and ρ m = ρ m a − is the energy density ofmatter with vanishing pressure and ρ m is a constant energy density at some initial time.The Friedmann-Robertson-Walker (FRW) metric representing a spatially flat, homogeneous andisotropic spacetime is given by ds = − dt + a ( t ) ( dx + dy + dz ) . (3)In FRW cosmological background, the Lagrangian is S = 2 π Z dt a h F ( T ) − λ ( T + 6 H ) − ρ m a + 12 ǫφ ,µ φ ,µ − V ( φ ) i . (4)where T = − H is the torsion scalar, λ is the Lagrange multiplier and can be determined byvarying the Lagrangian with respect to torsion scalar T , which yields λ = F T . Integrating by partsin (4), the action converts to S = 2 π Z dt h a ( F − T F T ) − F T a ˙ a − ρ m − a (cid:16) ǫ ˙ φ + V ( φ ) (cid:17)i , (5)and then the point-like Lagrangian reads (ignoring a constant factor 2 π ) L ( a, ˙ a, φ, ˙ φ, T ) = a ( F − T F T ) − F T a ˙ a − ρ m − a (cid:16) ǫ ˙ φ + V ( φ ) (cid:17) . (6)Moreover for a given dynamical system, the Euler-Lagrange equation is ddt (cid:16) ∂ L ∂ ˙ q i (cid:17) − ∂ L ∂q i = 0 , (7)where q i = a, φ, T are the generalized coordinates of the configuration space Q = { a, φ, T } . Using(6) in (7), we obtain the three equations of motion (corresponding to variations of L with respectto T, φ, a respectively) a F T T ( T + 6 H ) = 0 , (8) ǫ ( ¨ φ + 3 H ˙ φ ) − ∂V∂φ = 0 , (9)4 ¨ aa ( F T + 2 T F T ) + 4 H ( F T − T F T ) + F − T F
T T = ρ φ , (10)where ρ φ ≡ ǫ ˙ φ + V ( φ ) . Assuming F T T = 0, from (8) we find T = − H , (11)which is the torsion scalar for FRW model. Using ¨ aa = H + ˙ H and (11) in (10), we find48 H F T T ˙ H − F T (3 H + ˙ H ) − F = p φ , (12)is the modified Raychaudhuri equation, where p φ = − ρ φ (note that p = p φ , p m = 0). We also writedown the Friedmann equation for our model12 H F T + F = ρ φ + ρ m . (13) III. NOETHER SYMMETRY ANALYSIS
The Noether symmetry approach is useful in obtaining exact solution to the given Lagrangiani.e. unknown functions in a given Lagrangian can be determined up to some arbitrary constants.The Noether symmetry generator is a vector field defined by X = α ∂∂a + β ∂∂φ + η ∂∂T + ˙ α ∂∂ ˙ a + ˙ β ∂∂ ˙ φ + ˙ η ∂∂ ˙ T , (14)where dot represents the total derivative given by ddt ≡ ˙ φ ∂∂φ + ˙ a ∂∂a + ˙ T ∂∂T . (15)The vector field X can be thought of as a vector field on T Q = ( a, ˙ a, φ, ˙ φ, T, ˙ T ) is the relatedtangent bundle on which L is defined.A Noether symmetry X of a Lagrangian L exists if the Lie derivative of L along the vector field X vanishes i.e. L X L = X L = α ∂ L ∂a + β ∂ L ∂φ + η ∂ L ∂T + ˙ α ∂ L ∂ ˙ a + ˙ β ∂ L ∂ ˙ φ + ˙ η ∂ L ∂ ˙ T = 0 . (16)By requiring the coefficients of ˙ a , ˙ φ , ˙ T , ˙ a ˙ φ , ˙ a ˙ T and ˙ φ ˙ T to be zero in Eq. (16), we find3 αF − αT F T − αV ( φ ) − ηaT F T T − βaV ′ ( φ ) = 0 , (17) αF T + ηaF T T + 2 aF T ∂α∂a = 0 , (18)32 α + a ∂β∂φ = 0 , (19)12 F T ∂α∂φ + ǫa ∂β∂a = 0 , (20)12 aF T ∂α∂T = 0 , (21) ǫa ∂β∂T = 0 . (22)By assuming F T = 0, from Eqs. (21) and (22), we conclude α = α ( a, φ ) , β = β ( a, φ ) . (23)Now we must solve the system of equations (17)-(20). The non-trivial solution for this systemreads as the following form (Model-I) F ( T ) = 43 c T + c , (24) V ( φ ) = c + c ( c φ + c ) , (25) α ( a, φ ) = − c a, (26) β ( a, φ, T ) = c φ + c , (27) η ( a, φ, T ) = 83 c T . (28)Here c . . . c are arbitrary constants. It is interesting to note that the form of potential V andtorsion function F is the same for both models of dark energy. The quadratic potential (25) hasbeen used to model cosmic inflation including chaotic inflation in super-gravity models [19].Using (26)-(28), the Noether symmetries are X = − a ∂∂a + 83 T ∂∂T + φ ∂∂φ , X = ∂∂T . (29)The symmetry X represents the scaling i.e. the Lagrangian remains invariant under scaling trans-formation while the second symmetry X shows that Lagrangian is invariant under T translation.These NS generators form a two dimensional closed algebra[ X , X ] = − X . (30)The conjugate momenta for the variables of configuration space Q can be defined as p a = ∂ L ∂ ˙ a = − a ˙ aF T , (31) p φ = ∂ L ∂ ˙ φ = ǫ ˙ a ˙ φ, (32) p T = ∂ L ∂ ˙ T = 0 . (33)Notice that p T = 0 on account of symmetry X . The Noether charge of the system reads Q = αp a + βp φ + ηp T = 8 c a ˙ aF T + ( c φ + c ) ǫ ˙ a ˙ φ. (34)Using (24) in (34), we get Q = − / c ( ˙ aa ) / − a ǫ ˙ φ ( c φ + c ) . (35) • Remark-1: If ǫ = 0, the Noether charge coincides with the results reported in [18] afteridentifying the parameters µ = c , n = . • Remark-2:
One obvious solution of the system (17)-(22) is α = η = 0 , β = constant , V ( φ ) = constant (36)In this case a, T are cyclic coordinates and we have the following constant charge Q = βp φ = − βǫa ˙ φ. (37)This is the same as the Noether symmetry analysis of purely scalar fields in general relativity[17]. For Q = 0, φ =constant, which corresponds to ‘cosmological constant’. But if Q = 0then we have ˙ φ ∝ a − and the scalar field dilutes with the expansion of the Universe. • Remark-3:
Another interesting solution of the system (17)-(22) is ǫ = V ( φ ) = 0 which inthis case has been discussed in [18]. For the perfect fluids with EoS w = 0, the system ofNoether symmetry condition is non-integrable. • Remark-4:
We find another interesting solution of a teleparallel gravity with a scalar field(Model-II) F ( T ) = − ǫc T + c , (38) V ( φ ) = c e − φc (1 + e φ + c c ) , (39) α ( a, φ ) = 23 r − c a sinh (cid:16) φ + c c (cid:17) , (40) β ( a, φ ) = r − c a cosh (cid:16) φ + c c (cid:17) , (41) η ( a, φ, t ) = arbitrary. (42)The corresponding Noether charge is given by Q = r − c a h − ǫc a ˙ a sinh (cid:16) φ + c c (cid:17) + ǫa ˙ φ cosh (cid:16) φ + c c (cid:17)i . (43) IV. F ( T ) QUANTUM COSMOLOGY
The Hamiltonian for a given Lagrangian reads H = X i p i ˙ q i − L . (44)Using Eqs. (31)-(33) in (44), we get H = − p a aF T − p φ ǫa − a ( F − T F T ) + ρ m + a V ( φ ) . (45)The Hamiltonian equations are ˙ q i = ∂ H ∂p i , ˙ p i = − ∂ H ∂q i . ˙ a = { a, H} , ˙ p a = { p a , H} , (46)˙ T = { T, H} , ˙ p T = { p T , H} ≡ , (47)˙ φ = { φ, H} , ˙ p φ = { p φ , H} . (48)The Hamiltonian constraint equation H ≡ − aF T ∂ S∂a − ǫa ∂ S∂φ − a ( F − T F T ) + ρ m + a V ( φ ) = 0 . (49)For the quantum picture of our model, we define a wave function ψ and ∂∂a i → − ι ~ ∂∂q i . Then thewave equation (which is the Hamiltonian constraint) reads: − ~ aF T ∂ ψ∂a − ~ ǫa ∂ ψ∂φ + U ( a, T, φ ) ψ = 0 , (50)where U ( a, T, φ ) = a ( F − T F T ) − ρ m − a V ( φ ) . Solution of the above wave equation for F ( T ) − scalar field model is not our main purpose here. V. NUMERICAL RESULTS AND COSMOGRAPHY
In this section, using (24) and (25) in (9) in (10), we obtain the Euler-Lagrange equations: ǫ ¨ φ + 3 ǫH ˙ φ = 2 c c ( c φ + c ) , (51)4 ¨ aa ( F T + 2 T F T ) + 4 H ( F T − T F T ) + F − T F
T T = 12 ǫ ˙ φ + c + c ( c φ + c ) . (52)We numerically solve (51) and (52) and display our results in the figures 1 to 4. From figure-1, thee-folding parameter for quintessence increases almost exponentially while for phantom, its staysflat. In figure-2, the quintessence scalar field increases while phantom field oscillates and decaywith time. From figure-3, the Hubble parameter decreases from its current value to nearly unityand stays close to zero for phantom while it starts increasing for quintessence. In figure-4, the stateparameter for phantom decreases to sub-negative values while for quintessence, it stays near thecosmological constant boundary.Our model (24) must be checked by observational parameters from cosmographical view, fol-lowing the methodology presented in [20] we must check the following equations f ( T ) = 6 H (Ω m − , (53) f ′ ( T ) = 1 . (54)Explicity we have c = 2 H (3Ω m − , (55) c = − / H / . (56)Using (55) and (56) in (24) it is possible to find the values of the present value of F ( T ) and thefirst derivatives of it using the cosmographic parameters set with a given value of Ω m . A. Reduction of the equations to a single equation for scale factor a ( t ) In this section, using (24) and (25) we want to find a single equation for scale factor a ( t ).From (51) we get ǫ ( a ˙ φ ) t = 2 c c a ψ. (57)where ψ = c φ + c . On the other hand, from (35) we get: ǫa ˙ φ = − Q + / c ( ˙ aa ) / ψ = − Uψ (58)where U = Q + / c ( ˙ aa ) / . Differentation of this equation gives ǫ ( a ˙ φ ) t = − ˙ U ψ − U ˙ ψψ (59)So that finally we have 2 c c a ψ = − ˙ U ψ − U ˙ ψψ (60)0From (41) follows ˙ ψ = − c Uǫa ψ (61)At least, Eqs. (43) and (44) gives2 ǫc c a ψ + ǫa ˙ U ψ − c U = 0 (62)Hence we obtain ψ = ± − ǫa ˙ U ± √ D ǫc c a (63)where D = a ˙ U + 8 ǫc c a U (64)For ψ we get the form ψ = s √ D ∓ ǫa ˙ U ǫc c a (65)or φ = − c + 1 c s √ D ∓ ǫa ˙ U ǫc c a (66)Now let us rewrite the equation (52) as[4 ˙ H (1 + 2 T ) + 0 . c T − . + c − c = 2 c c U √ D ∓ ǫa ˙ U + √ D ∓ ǫa ˙ U ǫc a . (67)This equation is very complicated in the form of the Y ( a, ˙ a, ¨ a ) = 0. We will not solve this equation.From analytical view, there is no simple method for converting this equation to a simpler model.The remaining job is the numerical analysis which we done in previous section. VI. CONCLUSION
Noether symmetry analysis is a useful tool to find unknown parameters involved in the La-grangian. As is observed in the literature, this approach has been used to find explicit forms of f ( R ) and recently F ( T ) gravities. In this Letter, we considered the F ( T ) cosmology with matterand (phantom or quintessence) scalar fields with potential function. Although phantom DE is theleast desirable candidate of DE as it violates relativistic energy conditions and leads to future timesingularities, we consider it for the sake of completeness of our model since some astrophysical1observations support it. Although in literature, one can find numerous forms of F ( T ) written in anad hoc way, while the advantage of Noether symmetry is that it helps in calculating a viable form ofthis arbitrary function. We obtained F ( T ) ∼ T / , and the scalar potential V ( φ ) ∼ φ as a viablecandidate of dark energy. In comparison with [18] dealing with F ( T ) ∼ T n , n > / n < / V ( φ ) ∼ φ for the scalar field. Besides DE,this potential has applications in ‘chaotic’ inflation model. We also wrote the Schrodinger waveequation for our model and discussed cosmological implications of our model. Our model can beused for the construction of the F ( T ) Quantum Cosmology. Acknowledgment
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