Supersymmetric Barotropic FRW Model and Dark Energy
aa r X i v : . [ g r- q c ] J un Supersymmetric Barotropic FRW Model andDark Energy
J.J. Rosales ∗ Facultad de Ingenier´ıa Mec´anica, El´ectrica y Electr´onica.Campus FIMEE, Universidad de Gto.Carretera Salamanca-Valle de Santiago, km. 3.5 + 1.8 km.Comunidad de Palo Blanco, Salamanca Gto. M´exico.V.I. TkachDepartment of Physics and AstronomyNorthwestern UniversityEvanston, IL 60208-3112, USANovember 25, 2018
Abstract:
Using the superfield approach we construct the n = 2 supersym-metric lagrangian for the FRW Universe with barotropic perfect fluid as matterfield. The obtained supersymmetric algebra allowed us to take the square root ofthe Wheeler-DeWitt equation and solve the corresponding quantum constraint.This model leads to the relation between the vacuum energy density and theenergy density of the dust matter.PACS numbers: 04.20.Fy; 04.60.Ds; 12.60.Jv; 98.70.Dk. Introduction
Einstein’s theory of general relativity is by far the most attractive classicaltheory of gravity today. By describing the gravitational field in terms of thestructure of space-time, Einstein effectively equated the study of gravity withthe study of geometry. In general relativity, space-time is a 4-dimensional man-ifold with Lorentzian metric g µν whose curvature measure the strength of thegravitational field. Given a matter distribution described by a stress-energytensor T µν , the curvature of the metric is determined by Einstein equations G µν = 8 πGT µν . This tensor equation completely describes the classical theory. ∗ E-mail: [email protected] , of a con-sistent treatment for constrained Hamiltonian systems, the way was paved forthe canonical formulation of a quantum theory of gravity. Today, the use ofthe canonical formalism in the reduction of the Einstein gravitational action toHamiltonian form is well known , , . However, the passage from the classical tothe quantum theory using the substitution of dynamical variables by operatorsand Poisson brackets by commutators is complicated by the problem of oper-ator ordering , , , so that one is left with the choice of either abandoning thecanonical approach or studying simplified models called minisuperspace. TheMinisuperspaces are useful toy models for canonical quantum gravity, becausethey capture many of the essential features of general relativity and are at thesame time free of technical difficulties associated with the presence of an in-finite number of degrees of freedom. The Bianchi cosmologies are the primeexample. As is well known, the equation that governs the quantum behaviorof these models is the Wheeler-DeWitt equation, which results in a quadraticHamiltonian leading to an equation of the Klein-Gordon type. The introductionof supersymmetric minisuperspace models has led to the definition and study oflinear ”square root” equations defining the quantum evolution of the Universe.To achieve these Dirac-Type equations one can make use the fact that super-gravity provides a natural square root of gravity − or supersymmetrize themodels − .Some time ago we have used the superfield formulation to investigate super-symmetric cosmological models , . In the previous works − it was shownthat the spatially homogeneous part of the fields in the supergravity theorypreserves the invariance under the local time n = 2 supersymmetry. This su-persymmetry is a subgroup of the four dimensional space-time supersymmetryof the supergravity theory. This local supersymmetry procedure has the advan-tage that, by defining the superfields on superspace, all the component fieldsin a supermultiplet can be manipulated simultaneously in a manner that auto-matically preserves supersymmetry. Besides, the fermionic fields are obtainedin a clear manner as the supersymmetric partners of the cosmological bosonicvariables.More recently, using the superfield formulation the canonical procedure quan-tization for a closed FRW cosmological model filled with pressureless matter(dust) content and the corresponding superpartner was reported − . We haveobtained the quantization for the energy-like parameter, and it was shown, thatthis energy is associated with the mass parameter quantization, and that suchtype of Universe has a quantized masses of the order of the Planck mass.In the present work we are interested in the construction of the n = 2 su-persymmetric lagrangian for the FRW Universe with barotropic perfect fluid asmatter field including the cosmological constant. The obtained supersymmetricalgebra allowed us to take the square root of the Wheeler-DeWitt equation and2olve the corresponding quantum constraint. Classical Action
The classical action for a pure gravity system and the corresponding term ofmatter content, perfect fluid with a constant equation of state parameter γ ; p = γρ , and the cosmological term is − S = Z h − c R N ˜ G (cid:16) dRdt (cid:17) + N kc G R + N c Λ6 ˜
G R + N M γ c R − γ i dt. (1)where c is the velocity of light in vacuum, ˜ G = πG where G is the Newtoniangravitational constant; k = 1 , , − N ( t ) , R ( t ) are the lapse function and the scale factor, respectively; M γ is the mass by unit length − γ .The purpose of this work is the supersymmetrization of the full action (1)using the superfield approach. The action (1) is invariant under the timereparametrization t ′ → t + a ( t ) , (2)if the transformations of R ( t ) and N ( t ) are defined as δR = a ˙ R, δN = ( aN ) . (3)The variation with respect to R ( t ) and N ( t ) lead to the classical equation for thescale factor R ( t ) and the constraint, which generates the local reparametriza-tion of R ( t ) and N ( t ). This constraint leads to the Wheeler-DeWitt equationin quantum cosmology.In order to obtain the corresponding supersymmetric action for (1), we fol-low the superfield approach. For this, we extend the transformation of timereparametrization (2) to the n = 2 local supersymmetry of time ( t, η, ¯ η ). Then,we have the following local supersymmetric transformation δt = a ( t ) + i ηβ ′ ( t ) + ¯ η ¯ β ′ ( t )] ,δη = 12 ¯ β ′ ( t ) + 12 [ ˙ a ( t ) + ib ( t )] η + i β ′ ( t ) η ¯ η, (4) δ ¯ η = 12 β ′ ( t ) + 12 [ ˙ a ( t ) − ib ( t )]¯ η − i β ′ ( t ) η ¯ η, where η is a complex odd parameter ( η odd “time” coordinates), β ′ ( t ) = N − / β ( t ) is the Grassmann complex parameter of the local “small” n = 2supersymmetry (SUSY) transformation, and b ( t ) is the parameter of local U (1)rotations of the complex η . 3or the closed ( k = 1) and plane ( k = 0) FRW action we propose thefollowing superfield generalization of the action (1), invariant under the n = 2local supersymmetric transformation (4) S susy = Z h − c G IN − IRD ¯ η IRD η IR + c √ k G IR + c Λ / √ G IR −− √ M / γ (3 − γ ) ˜ G / IR − γ i dηd ¯ ηdt, (5)where D η = ∂∂η + i ¯ η ∂∂t , D ¯ η = − ∂∂ ¯ η − iη ∂∂t , (6)are the supercovariant derivatives of the global ”small” supersymmetry of thegeneralized parameter corresponding to t . The local supercovariant derivativeshave the form ˜ D η = IN − / D η , ˜ D ¯ η = IN − / D ¯ η , and IR ( t, η, ¯ η ) , IN ( t, η, ¯ η ) aresuperfields. The supersymmetric action for Λ = 0, γ = 0 was reported in [ ? ].The Taylor series expansion for the superfields IN ( t, η, ¯ η ) and IR ( t, η, ¯ η ) arethe following IN ( t, η, ¯ η ) = N ( t ) + iη ¯ ψ ′ ( t ) + i ¯ ηψ ′ ( t ) + V ′ ( t ) η ¯ η, (7) IR ( t, η, ¯ η ) = R ( t ) + iη ¯ λ ′ ( t ) + i ¯ ηλ ′ ( t ) + B ′ ( t ) η ¯ η. (8)In the expressions (7) and (8) we have introduced the redefinitions ψ ′ ( t ) = N / ψ ( t ), V ′ = N ( t ) V ( t ) + ¯ ψ ( t ) ψ ( t ), λ ′ = ˜ G / N / cR / λ and B ′ = ˜ G / c N B + ˜ G / cR / ( ¯ ψλ − ψ ¯ λ ). The components of the superfield IN ( t, η, ¯ η ) are gauge fields ofthe one-dimensional n = 2 extended supergravity. N ( t ) is the einbein, ψ ( t ) , ¯ ψ ( t )are the complex gravitino fields, and V ( t ) is the U (1) gauge field. The compo-nent B ( t ) in (8) is an auxiliary degree of freedom (non-dynamical variable), and λ, ¯ λ are the fermion partners of the scale factor R ( t ). Thus, we can rewrite theaction (5) in its component form S susy = Z ( − c R ( DR ) N ˜ G + i λDλ − D ¯ λλ ) − N R B − N ˜ G / B cR ¯ λλ ++ c √ kRN ˜ G / B + c √ kR / G / ( ¯ ψλ − ψ ¯ λ ) + cN √ kR ¯ λλ + (9)+ c Λ / √ G / N R B + c Λ / R / √ G / ( ¯ ψλ − ψ ¯ λ ) + 2 c Λ / N √ λλ −−√ cM / γ N R − γ B − √ cM / γ R − γ ( ¯ ψλ − ψ ¯ λ ) −−√ − γ ) ˜ G / M / γ N R − − γ ¯ λλ o dt. So, the lagrangian for the auxiliary field has the form L B = − N R B − N ˜ G / B cR ¯ λλ + c √ kRN ˜ G / B + c Λ / N R √ G / B − √ cM / γ N R − γ B. (10)From the expression (10) we can obtain the equation for the auxiliary fieldvarying the Lagrangian with respect to BB = c √ k ˜ G / − ˜ G / cR ¯ λλ + c Λ / R √ G / − √ cM / γ R − γ − . (11)Then, putting the expression (11) in (9) we have the following supersymmetricaction S susy = Z (cid:26) − c R ( DR ) N ˜ G + c N kR G + c N Λ R G + N c M γ R − γ ++ c √ k Λ / R √ G − √ kc ˜ G / M / γ R − γ − √ c Λ / M / γ √ G / R − γ ++ i λDλ − D ¯ λλ ) + cN √ k R ¯ λλ + √ c Λ / N ¯ λλ + (12)+ ( − γ ) √ N ˜ G / M / γ R − − γ ¯ λλ + c √ kR / G / ( ¯ ψλ − ψ ¯ λ )+ c Λ / √ G / R / ( ¯ ψλ − ψ ¯ λ ) − √ cM / γ R − γ ( ¯ ψλ − ψ ¯ λ ) ) dt, where DR = ˙ R − i ˜ G / cR / ( ψ ¯ λ + ¯ ψλ ) and Dλ = ˙ λ − V λ , D ¯ λ = ˙¯ λ + V ¯ λ . Supersymmetric Quantum Model
In this section we will proceed with the quantization analysis of the system. Theclassical canonical Hamiltonian is calculated in the usual way for the systemswith constraints. It has the form H c = N H + 12 ¯ ψS − ψ ¯ S + 12 V F, (13)where H is the Hamiltonian of the system, S and ¯ S are the supercharges and F is the U (1) rotation generator. The form of the canonical Hamiltonian (13)explains the fact that N, ψ, ¯ ψ and V are Lagrangian multipliers which onlyenforce the first-class constraints H = 0 , S = 0 , ¯ S = 0 and F = 0, whichexpress the invariance under the conformal n = 2 supersymmetric transforma-tions. The first-class constraints may be obtained from the action (12) varying N ( t ) , ψ ( t ), ¯ ψ ( t ) and V ( t ), respectively. The first-class constraints are H = − ˜ G c R π R − c kR G − c Λ R G − M γ c R − γ + √ c Λ / M / γ √ G / R − γ −− c √ k Λ / R √ G + √ kc ˜ G / M / γ R − γ − c √ k R ¯ λλ − √ c Λ / ¯ λλ − (6 γ − √ G / M / γ R − − γ ¯ λλ, (14) S = (cid:16) i ˜ G / cR / π R − c √ kR / ˜ G / − c Λ / R / √ G / + √ cM / γ R − γ (cid:17) λ, (15)¯ S = (cid:16) − i ˜ G / cR / π R − c √ kR / ˜ G / − c Λ / R / √ G / + √ cM / γ R − γ (cid:17) ¯ λ, (16) F = − ¯ λλ, (17)where π R = − c R ˜ GN ˙ R + icR / N ˜ G / ( ¯ ψλ + ψ ¯ λ ) is the canonical momentum associatedto R . The canonical Dirac brackets are defined as { R, π R } = 1 , { λ, ¯ λ } = i. (18)With respect to these brackets the super-algebra for the generators H, S, ¯ S and F becomes { S, ¯ S } = − iH, { S, H } = { ¯ S, H } = 0 , { F, S } = iS, { F, ¯ S } = i ¯ S. (19)In a quantum theory the brackets (18) must be replaced by anticommutatorsand commutators, they can be considered as generators of the Clifford algebra.We have { λ, ¯ λ } = − ¯ h, [ R, π R ] = i ¯ h with π R = − i ¯ h ∂∂R (20)¯ λ = ξ − λ † ξ = − λ † , { λ, λ † } = ¯ h, λ † ξ = ξλ † and ξ † = ξ. Then, for the operator ¯ S the following equation is satisfied¯ S = ξ − S † ξ. (21)Therefore, the anticommutator of supercharges S and their conjugated operator¯ S under our defined conjugation has the form (cid:8) S, ¯ S (cid:9) = ξ − (cid:8) S, ¯ S (cid:9) ξ = (cid:8) S, ¯ S (cid:9) , (22)and the Hamiltonian operator is self-conjugated under the operation ¯ H = ξ − H † ξ . We can choose the matrix representation for the fermionic parameters λ, ¯ λ and ξ as λ = √ ¯ hσ − , ¯ λ = −√ ¯ hσ + , ξ = σ , (23)with σ ± = ( σ ± iσ ), where σ , σ , σ are the Pauli matrices.In the quantum level we must consider the nature of the Grassmann variables λ and ¯ λ , with respect to these we perform the antisymmetrization, then we canwrite the bilinear combination in the form of the commutators, ¯ λ, λ → [¯ λ, λ ],and this leads to the following quantum Hamiltonian H . H quantum = − ˜ G c R − / π R R − / π R − c kR G − c Λ R G − M γ c R − γ √ c Λ / M / γ √ G / R − γ − c √ k Λ / R √ G ++ √ kc ˜ G / M / γ R − γ − c √ k R [¯ λ, λ ] − √ c Λ / [¯ λ, λ ] −− (6 γ − √ G / M / γ R − − γ [¯ λ, λ ] . (24)The supercharges S , ¯ S and the fermion number F have the following structures: S = Aλ, S † = A † λ † (25)where A = i ˜ G / c R − / π R − c √ k ˜ G / R / − c Λ / R / √ G / + √ cM / γ R − γ , (26)and F = −
12 [¯ λ, λ ] . (27)An ambiguity exist in the factor ordering of these operators, such ambiguities al-ways arise, when the operator expression contains the product of non-commutingoperator R and π R , as in our case. It is then necessary to find some criteria toknow which factor ordering should be selected. We propose the following ruleto integrate with the inner product of two states − . This inner product iscalculated performing the integration with the measure R / dR . With this mea-sure the conjugate momentum π R is non-Hermitian with π † R = R − / π R R / .However, the combination ( R − / π R ) † = π † R R − / = R − / π R is a Hermitianone, and ( R − / π R R / π R ) † = R − / π R R / π R is Hermitian too. This choicein our supersymmetric quantum approach n = 2 eliminates the factor orderingambiguity by fixing the ordering parameter p =
12 30 − , , . Superquantum Solutions
In the quantum theory, the first-class constraints H = 0 , S = 0 , ¯ S = 0 and F = 0become conditions on the wave function Ψ( R ). Furthermore, any physical statemust be satisfied the quantum constraints H Ψ( R ) = 0 , S Ψ( R ) = 0 , ¯ S Ψ( R ) = 0 , F Ψ( R ) = 0 , (28)where the first equation is the Wheeler-DeWitt equation for the minisuperspacemodel. The eigenstates of the Hamiltonian (24) have two components in thematrix representation (23) Ψ = (cid:18) Ψ Ψ (cid:19) . (29)However, the supersymmetric physical states are obtained applying the super-charges operators S Ψ = 0 , ¯ S Ψ = 0. With the conformal algebra given by (19),these are rewritten in the following form( λ ¯ S − ¯ λS )Ψ = 0 . (30)7sing the matrix representation for λ and ¯ λ we obtain the following differentialequations for Ψ ( R ) and Ψ ( R ) components (cid:16) ¯ h ˜ G / c R − / ∂∂R − c √ kR / ˜ G / − c Λ / R / √ G / + √ cM / γ R − γ (cid:17) Ψ ( R ) = 0 . (31) (cid:16) ¯ h ˜ G / c R − / ∂∂R + c √ kR / ˜ G / + c Λ / R / √ G / − √ cM / γ R − γ (cid:17) Ψ ( R ) = 0 . (32)Solving these equation, we have the following wave functions solutionsΨ ( R ) = C exp h √ kc R h ˜ G + c Λ / √ h ˜ G R − √ c M / γ (3 − γ )¯ h ˜ G / R − γ i , (33)Ψ ( R ) = ˜ C exp h − √ kc R h ˜ G − c Λ / √ h ˜ G R + 2 √ c M / γ (3 − γ )¯ h ˜ G / R − γ i . (34)In the case of the flat universe ( k = 0) and for the dust-like matter ( γ = 0) wehave the following solutions (using the relation M γ =0 = R ρ γ =0 )Ψ ( R ) = C exp h √ π (cid:16) ρ Λ ρ pl (cid:17) / (cid:16) Rl pl (cid:17) − √ π (cid:16) ρ γ =0 ρ pl (cid:17) / (cid:16) Rl pl (cid:17) i , (35)Ψ ( R ) = C exp h − √ π (cid:16) ρ Λ ρ pl (cid:17) / (cid:16) Rl pl (cid:17) + 2 √ π (cid:16) ρ γ =0 ρ pl (cid:17) / (cid:16) Rl pl (cid:17) i , (36)where ρ pl = c ¯ hG is the Planck density and l pl = (cid:16) ¯ hGc (cid:17) / is the Planck length.We can see, that the function Ψ in (35) has good behavior when R → ∞ under the condition ρ Λ < ρ γ =0 , while Ψ does not. On the other hand, thewave function Ψ in (36) has good behavior when R → ∞ under the condi-tion ρ Λ > ρ γ =0 , because the principal contribution comes from the first termof the exponent, while Ψ does not have good behavior. However, only thescalar product for the second wave function Ψ is normalizable in the measure R / dR under the condition ρ Λ > ρ γ =0 . This condition does not contradictthe astrophysical observation at ρ Λ ≥ ρ M , due to the fact that the dust matterintroduces the main contribution to the total energy density of matter ρ M .On the other hand, according to recent astrophysical data, our universe isdominated by a mysterious form of the dark energy , which counts to about75 −
80 per cent of the total energy density. As a result, the universe expansionis accelerating , . Vacuum energy density ρ Λ = c Λ8 πG is a concrete example ofthe dark energy. Conclusion γ = − . +0 . − . . However, in the literature we can find differ-ent theoretical models for the dark energy with state parameter γ > − γ < −
1. In the present work we have discussed the case for γ = 0 correspondingto the FRW universe with barotropic perfect fluid as matter field. In the case ofthe flat universe ( k = 0) and the dust-like matter γ = 0 we have obtained twowave functions. However, only the second wave function is normalizable underthe condition ρ Λ > ρ γ =0 , which leads to the cosmological value Λ > πGc ρ γ =0 . Acknowledgments
We thanks M. Gu´ıa, J. Torres, D.A. Rosales and Carlos Montoro for severaluseful remarks.
References
1. P.A.M. Dirac, Can. J. Math. 2, 129 (1950).2. P.A.M. Dirac, Lectures on Quantum Mechanics (Academic, New York,1965).3. P.A.M. Dirac, Proc. R. Soc. London A 246, 333 (1958).4. P.A.M. Dirac, Phys. Rev. , 924 (1959).5. R. Arnowitt, S. Deser and C.W. Misner, in Gravitation: An Introductionto Current Research, edited by L. Witten (Wiley, New York, 1962).6. J. Anderson, Phys. Rev. , 1182 (1959).7. J. Anderson, In Proceedings of the First Eastern Theoretical Physics Con-ferences, edited by M.E. Rose (Gordon and Breach, New York, 1963).8. A. Komar, Phys. Rev.
D 20 , 830, (1979).9. C. Teitelboim, Phys. Lett.
B 69 , 240 (1977).10. C. Teitelboim, Phys. Rev. Lett. , 1106 (1977).11. R. Tabensky and C. Teitelboim, Phys. Lett. B 69 , 453 (1977).12. A. Mac´ıas, O. Obreg´on and M.P. Ryan, Jr., Class. Quantum Grav. ,1477 (1987).13. P.D. D’Eath and D.I. Hughes, Phys. Lett. B 214 , 498 (1988).14. P.D. D’Eath and D.I. Hughes, Nucl. Phys.
B378 , 381 (1991).15. J. Socorro, O. Obreg´on and A. Mac´ıas, Phys. Rev.
D 45 , 2026 (1992).96. P.D. D’Eath, S.W. Hawking and O. Obreg´on, Phys. Lett.
B 300 , 44(1993).17. P.D. D’Eath, Phys. Rev.
D 48 , 713 (1993).18. M. Asano, M. Tanimoto and N. Noshino, Phys. Lett.
B 314 , 303 (1993).19. R. Capovilla and J. Guven, Class. Quantum Grav. , 1961, (1994).20. P.D. D’Eath, Phys. Lett. B 320 , 12 (1994).21. A.D.Y. Cheng, P.D. D’Eath and P.R.L.V. Moniz, Phys. Rev.
D 49 , 5246,(1994).22. R. Capovilla and O. Obreg´on, Phys. Rev.
D 49 , 6562, (1994).23. A. Csord´as and R. Graham, Phys. Rev. Lett. , 4129 (1994).24. R. Graham, Phys. Rev. Lett. , 1381, (1991).25. R. Graham, Phys. Lett. B 277 , 393, (1992).26. J. Bene and R. Graham, Phys. Rev.
D 49 , 799, (1994).27. O. Obreg´on, J. Socorro and J. Ben´ıtez, Phys. Rev.
D 47 , 4471 (1993).28. O. Obreg´on, J.J. Rosales and V.I. Tkach, Phys. Rev.
D 53 , R1750 (1996).29. V.I. Tkach, J.J. Rosales and O. Obreg´on, Class. Quantum Grav. , 2349(1996).30. C. Ortiz, J.J. Rosales, J. Socorro, J. Torres and V.I. Tkach. Phys. Lett. A 340 , 51-58, (2005).31. O. Obreg´on, J.J. Rosales, J. Socorro and V.I. Tkach, Class. QuantumGravity, , 2861 (1999).32. V.I. Tkach, J.J. Rosales and J. Socorro, Class. Quantum Gravity, , 797(1999).33. M. Ryan Jr, Hamiltonian Cosmology, Springer Verlag (1972).34. J. Socorro, M.A. Reyes and F.A. Gelbert, Phys. Lett. A 313 , 338 (2003).35. P.D. D’Eath, D.I. Hughes, Phys. Lett.
B 214 , 498 (1988).36. P.D. D’Eath, D.I. Hughes, Nucl. Phys.
B 378
381 (1991).37. T. Christodoulakis, J. Zanelli, Phys. Rev.
D 29
B 102
237 (1984).39. S.W Hawking, D.N. Page, Nucl. Phys.
B 264
185 (1986).100. J. Halliwel, in: S. Coleman, et al.(Eds.), Introductory Lectures on Quan-tum Cosmology (Jerusalem Winter School), vol. 7, World Scientific, Sin-gapore, pp. 159-243 1991.41. T. Padmanabhan, Phys. Rept. , 235 (2003).42. S. Perlmutter, et al; Astrophysics J., , 565 (1999).43. A.G. Riess, et al; Astrophysics J.,607