arXiv:0811.4178v1 [gr-qc] 25 Nov 2008
Supersymmetric Cosmology and Dark 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. Tkach† Department of Physics and Astronomy Northwestern University Evanston, IL 60208-3112, USA November 26, 2008 Abstract: Using the superfield approach we construct the n = 2 supersymmetric lagrangian for the FRW Universe with perfect fluid as matter fields. The obtained supersymmetric algebra allowed us to take the square root of the Wheeler-DeWitt equation and solve the corresponding quantum constraint. This model leads to the relation between the vacuum energy density and the energy density of the dust matter. Introduction This paper is for the anniversary volume on the occasion 50th birthday, Sergei Odintsov, our colleague and friend who made an extensive contribution to the cosmological and astrophysics fields. Some time ago we have used the superfield formulation to investigate supersymmetric cosmological models [1]. The main idea is to extend the group of local time reparametrization of the cosmological models to the local time supersymmetry which is a subgroup of the four dimensional space-time supersymmetry. This local supersymmetry procedure has the advantage that, by defining the superfields on superspace, all the component fields in a supermultiplet can be manipulated simultaneously in a manner that automatically preserves supersymmetry. Besides, the fermionic fields are obtained in a clear manner as the supersymmetric partners of the cosmological bosonic variables. ∗ E-mail: † E-mail:
[email protected] [email protected]
1
More recently, using the superfield formulation the canonical procedure quantization for a closed FRW cosmological model filled with pressureless matter (dust) content and the corresponding superpartner was reported [2]. We have obtained the quantization for the energy-like parameter, and it was shown, that this energy is associated with the mass parameter quantization, and that such type 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 supersymmetric lagrangian for the FRW Universe with barotropic perfect fluid as matter field including the cosmological constant. The simplest dark energy candidate is the cosmological constant stemming from energy density of the vacuum [3]. The obtained supersymmetric algebra allowed us to take the square root of the Wheeler-DeWitt equation and solve the corresponding quantum constraint. Classical Action The classical action for a pure gravity system and the corresponding term of matter content, perfect fluid with a constant equation of state parameter γ; p = γρ, and the cosmological term is [2] Z h i c2 R dR 2 N kc4 N c4 Λ 3 + (1) − R+ R + N Mγ c2 R−3γ dt. S= ˜ dt ˜ ˜ 2N G 2G 6G
˜ = 8πG where G is the Newtonian where c is the velocity of light in vacuum, G 6 gravitational constant; k = 1, 0, −1 stands for spherical, plane or hyperspherical three space; N (t), R(t) are the lapse function and the scale factor, respectively; Mγ 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 time reparametrization t′ → t + a(t), (2) if the transformations of R(t) and N (t) are defined as ˙ δR = aR,
δN = (aN ).
(3)
The variation with respect to R(t) and N (t) lead to the classical equation for the scale factor R(t) and the constraint, which generates the local reparametrization of R(t) and N (t). This constraint leads to the Wheeler-DeWitt equation in quantum cosmology. In order to obtain the corresponding supersymmetric action for (1), we follow the superfield approach. For this, we extend the transformation of time reparametrization (2) to the n = 2 local supersymmetry of time (t, η, η¯). Then, we have the following local supersymmetric transformation δt
i = a(t) + [ηβ ′ (t) + η¯β¯′ (t)], 2 2
δη
=
δ η¯ =
1 ¯′ β (t) + 2 1 ′ β (t) + 2
1 [a(t) ˙ + ib(t)]η + 2 1 [a(t) ˙ − ib(t)]¯ η− 2
i ¯˙ ′ β (t)η η¯, 2 i ˙′ β (t)η η¯, 2
(4)
where η is a complex odd parameter (η odd “time” coordinates), β ′ (t) = N −1/2 β(t) is the Grassmann complex parameter of the local “small” n = 2 supersymmetry (SUSY) transformation, and b(t) is the parameter of local U (1) rotations of the complex η. For the closed (k = 1) and plane (k = 0) FRW action we propose the following superfield generalization of the action (1), invariant under the n = 2 local supersymmetric transformation (4) √ Z h c2 −1 c3 k 2 c3 Λ1/2 3 − IN IRDη¯IRDη IR + IR + √ IR − Ssusy = ˜ ˜ ˜ 2G 2G 3 3G √ 1/2 i 3−3γ 2 2Mγ IR 2 dηd¯ η dt, (5) − ˜ 1/2 (3 − 3γ)G where Dη =
∂ ∂ + i¯ η , ∂η ∂t
Dη¯ = −
∂ ∂ − iη , ∂ η¯ ∂t
(6)
are the supercovariant derivatives of the global ”small” supersymmetry of the generalized parameter corresponding to t. The local supercovariant derivatives ˜ η = IN −1/2 Dη , D ˜ η¯ = IN −1/2 Dη¯, and IR(t, η, η¯), IN (t, η, η¯) are have the form D superfields. The Taylor series expansion for the superfields IN (t, η, η¯) and IR(t, η, η¯) are the following IN (t, η, η¯) = IR(t, η, η¯) =
N (t) + iη ψ¯′ (t) + i¯ η ψ ′ (t) + V ′ (t)η η¯, ′ ¯ (t) + i¯ R(t) + iη λ η λ′ (t) + B ′ (t)η η¯.
(7) (8)
In the expressions (7) and (8) we have introduced the redefinitions ψ ′ (t) = ˜ 1/2 ˜ 1/2 N 1/2 ¯ λ and B ′ = G c N B + N 1/2 ψ(t), V ′ = N (t)V (t) + ψ(t)ψ(t), λ′ = G cR1/2 ˜ 1/2 G ¯ ¯ The components of the superfield IN (t, η, η¯) are gauge fields of (ψλ− ψ λ). 2cR1/2 ¯ the 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 component B(t) in (8) is an auxiliary degree of freedom (non-dynamical variable), and ¯ are the fermion partners of the scale factor R(t). After the integration over λ, λ the Grassmann coordinates θ, θ¯ we can rewrite the action (5) in its component form Z ( 2 ˜ 1/2 c R(DR)2 i ¯ ¯ ¯ − N R B 2 − N G B λλ+ − Ssusy = + (λDλ − Dλλ) ˜ 2 2 2cR 2N G √ √ √ c2 kRN c2 kR1/2 ¯ cN k ¯ ¯ + B+ λλ+ (9) (ψλ − ψ λ) + ˜ 1/2 ˜ 1/2 R G 2G 3
1/2 c2 Λ1/2 R3/2 ¯ c2 Λ1/2 ¯ + 2cΛ√ N λλ− ¯ N R2 B + √ (ψλ − ψ λ) +√ ˜ 1/2 ˜ 1/2 3 3G 2 3G √ √ 1−3γ 3γ 2 1/2 ¯ − ψ λ)− ¯ 2 B− cMγ1/2 R− 2 (ψλ − 2cMγ N R 2 o √ ¯ dt. ˜ 1/2 Mγ1/2 N R −3−3γ 2 − 2(1 − 3γ)G λλ
So, the lagrangian for the auxiliary field has the form √ ˜ 1/2 B NR 2 NG c2 Λ1/2 N R2 c2 kRN ¯ B− LB = − B+ √ λλ + B − ˜ 1/2 ˜ 1/2 2 2cR G 3G √ 1−3γ − 2cMγ1/2 N R 2 B.
(10)
From the expression (10) we can obtain the equation for the auxiliary field varying the Lagrangian with respect to B √ 2 1/2 ˜ 1/2 √ G c2 k ¯ + c√Λ R − 2cM 1/2 R −3γ−1 2 . (11) − B= λλ γ 2 ˜ 1/2 ˜ 1/2 2cR G 3G Then, putting the expression (11) in (9) we have the following supersymmetric action Z 2 c R(DR)2 c4 N kR c4 N ΛR3 − + + + N c2 Mγ R−3γ + Ssusy = ˜ ˜ ˜ 2N G 2G 6G √ √ 3 √ 3 1/2 1/2 3−3γ c4 kΛ1/2 R2 2kc 2c Λ Mγ 1/2 1−3γ √ √ Mγ R 2 − R 2 + + − ˜ 1/2 ˜ ˜ 1/2 G 3G 3G √ √ i ¯ ¯ + 3 cΛ1/2 N λλ+ ¯ + cN k λλ ¯ + (λDλ − Dλλ) (12) 2 2R 2 √ (−1 + 6γ) ˜ 1/2 1/2 −3−3γ ¯ c2 kR1/2 ¯ ¯ √ N G Mγ R 2 λλ + (ψλ − ψ λ) + ˜ 1/2 2 2G ) √ c2 Λ1/2 3/2 ¯ 2 1/2 − 3γ ¯ ¯ ¯ + √ R (ψλ − ψ λ) − cMγ R 2 (ψλ − ψ λ) dt, ˜ 1/2 2 2 3G where DR = R˙ −
˜ 1/2 iG ¯ (ψ λ 2cR1/2
¯ and Dλ = λ˙ − 1 V λ, Dλ ¯˙ + 1 V λ. ¯=λ ¯ + ψλ) 2 2
Supersymmetric Quantum Model In this section we will proceed with the quantization analysis of the system. The classical canonical Hamiltonian is calculated in the usual way for the systems with constraints. It has the form 1 1 1¯ − ψ S¯ + V F, Hc = N H + ψS 2 2 2
(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) 4
explains the fact that N, ψ, ψ¯ and V are Lagrangian multipliers which only enforce the first-class constraints H = 0, S = 0, S¯ = 0 and F = 0, which express the invariance under the conformal n = 2 supersymmetric transformations. The first-class constraints may be obtained from the action (12) varying ¯ and V (t), respectively. The first-class constraints are N (t), ψ(t),ψ(t) √ 3 1/2 1/2 ˜ 3−3γ G c4 kR c4 ΛR3 2c Λ Mγ 2 2 −3γ √ H = − 2 πR − − − Mγ c R + R 2 − 1/2 ˜ ˜ ˜ 2c R 2G 6G 3G √ 1/2 2 √ √ 3 √ 4 1−3γ c kΛ R c k¯ 2kc 3 1/2 ¯ √ − Mγ1/2 R 2 − λλ − cΛ λλ − + ˜ 1/2 ˜ 2R 2 G 3G (6γ − 1) ˜ 1/2 1/2 −3−3γ ¯ √ (14) G Mγ R 2 λλ, − 2 √ iG ˜ 1/2 c2 Λ1/2 R3/2 √ c2 kR1/2 1/2 − 3γ 2 √ λ, (15) − S = + π − 2cM R R γ ˜ 1/2 ˜ 1/2 cR1/2 G 3G √ iG ˜ 1/2 c2 Λ1/2 R3/2 √ c2 kR1/2 1/2 − 3γ ¯ 2 √ S¯ = − λ, (16) − + π − 2cM R R γ ˜ 1/2 ˜ 1/2 cR1/2 G 3G ¯ F = −λλ, (17) 2 R ˙ icR1/2 ¯ ¯ where πR = − cGN ˜ R + 2N G ˜ 1/2 (ψλ + ψ λ) is the canonical momentum associated to R. The canonical Dirac brackets are defined as
{R, πR } = 1,
¯ = i. {λ, λ}
(18)
With respect to these brackets the super-algebra for the generators H, S, S¯ and F becomes ¯ = −2iH, {S, S}
¯ H} = 0, {S, H} = {S,
{F, S} = iS,
¯ = iS. ¯ {F, S}
(19)
In a quantum theory the brackets (18) must be replaced by anticommutators and commutators, they can be considered as generators of the Clifford algebra. We have ∂ ¯ {λ, λ} = −¯ h, [R, πR ] = i¯h with πR = −i¯h (20) ∂R ¯ = ξ −1 λ† ξ = −λ† , λ {λ, λ† } = h ¯, λ† ξ = ξλ† and ξ † = ξ.
Then, for the operator S¯ the following equation is satisfied S¯ = ξ −1 S † ξ.
(21)
Therefore, the anticommutator of supercharges S and their conjugated operator S¯ under our defined conjugation has the form (22) S, S¯ = ξ −1 S, S¯ ξ = S, S¯ ,
¯ = and the Hamiltonian operator is self-conjugated under the operation H −1 † ξ H ξ. We can choose the matrix representation for the fermionic parameters ¯ and ξ as λ, λ √ √ ¯ = − ¯hσ+ , hσ− , λ ξ = σ3 , (23) λ= ¯ 5
with σ± = 21 (σ1 ± iσ2 ), where σ1 , σ2 , σ3 are the Pauli matrices. In the quantum level we must consider the nature of the Grassmann variables ¯ with respect to these we perform the antisymmetrization, then we can λ and λ, ¯ λ], ¯ λ → 1 [λ, write the bilinear combination in the form of the commutators, λ, 2 and this leads to the following quantum Hamiltonian H. Hquantum
˜ c4 kR c4 ΛR3 G − − Mγ c2 R−3γ = − 2 R−1/2 πR R−1/2 πR − ˜ ˜ 2c 2G 6G √ √ 3 1/2 1/2 3−3γ 2c Λ Mγ c4 kΛ1/2 R2 2 √ √ R + + − ˜ 1/2 ˜ 3G 3G √ 3 √ √ 1−3γ 2kc 3 1/2 ¯ c k ¯ + Mγ1/2 R 2 − [λ, λ] − cΛ [λ, λ] − 1/2 ˜ 4R 4 G (6γ − 1) ˜ 1/2 1/2 −3−3γ ¯ √ G Mγ R 2 [λ, λ]. (24) − 2 2
The supercharges S, S¯ and the fermion number F have the following structures: S † = A† λ†
S = Aλ,
(25)
where A=
√ ˜ 1/2 3γ iG c2 k 1/2 c2 Λ1/2 R3/2 √ + 2cMγ1/2 R− 2 , R − √ R−1/2 πR − ˜ 1/2 ˜ 1/2 c 3G G
(26)
and
1 ¯ λ]. (27) F = − [λ, 2 An ambiguity exist in the factor ordering of these operators, such ambiguities always arise, when the operator expression contains the product of non-commuting operator R and πR , as in our case. It is then necessary to find some criteria to know which factor ordering should be selected. The inner product is calculated performing the integration with the measure R1/2 dR. With this measure the † conjugate momentum πR is non-Hermitian with πR = R−1/2 πR R1/2 . However, † −1/2 −1/2 † −1/2 the combination (R πR ) = πR R = R πR is a Hermitian one, and (R−1/2 πR R1/2 πR )† = R−1/2 πR R1/2 πR is Hermitian too. This choice in our supersymmetric quantum approach n = 2 eliminates the factor ordering ambiguity by fixing the ordering parameter p = 21 . Superquantum Solutions
In the quantum theory, the first-class constraints H = 0, S = 0, S¯ = 0 and F = 0 become conditions on the wave function Ψ(R). Furthermore, any physical state must be satisfied the quantum constraints HΨ(R) = 0,
¯ SΨ(R) = 0,
SΨ(R) = 0,
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F Ψ(R) = 0,
(28)
where the first equation is the Wheeler-DeWitt equation for the minisuperspace model. The eigenstates of the Hamiltonian (24) have two components in the matrix representation (23) Ψ1 Ψ= . (29) Ψ2
However, the supersymmetric physical states are obtained applying the super¯ = 0. With the conformal algebra given by (19), charges operators SΨ = 0, SΨ these are rewritten in the following form ¯ (λS¯ − λS)Ψ = 0.
(30)
¯ we obtain the following differential Using the matrix representation for λ and λ equations for Ψ1 (R) and Ψ2 (R) components √ ¯ ˜ 1/2 3γ hG c2 Λ1/2 R3/2 √ c2 kR1/2 −1/2 ∂ − √ R − + 2cMγ1/2 R− 2 Ψ1 (R) = 0. (31) ˜ 1/2 ˜ 1/2 c ∂R G 3G √ 1/2 ¯ 1/2 2 1/2 2 ˜ 3γ hG c Λ R3/2 √ ∂ c kR + √ R−1/2 + − 2cMγ1/2 R− 2 Ψ2 (R) = 0. (32) ˜ 1/2 ˜ 1/2 c ∂R G 3G Solving these equation, we have the following wave functions solutions √ 1/2 i h √kc3 R2 3−3γ c3 Λ1/2 3 2 2c2 Mγ √ Ψ1 (R) = C exp R 2 , + R − ˜ ˜ ˜ 1/2 2¯ hG 3 3¯ hG (3 − 3γ)¯ hG √ √ 1/2 i h 3−3γ c3 Λ1/2 3 2 2c2 Mγ kc3 R2 Ψ2 (R) = C˜ exp − R 2 . − √ R + ˜ ˜ ˜ 1/2 2¯ hG 3 3¯hG (3 − 3γ)¯ hG
(33) (34)
In the case of the flat universe (k = 0) and for the dust-like matter (γ = 0) we have the following solutions (using the relation Mγ=0 = 21 R3 ργ=0 ) √ h 1 ρ 1/2 R 3 2 ργ=0 1/2 R 3 i Λ √ √ , (35) − Ψ1 (R) = C1 exp lpl lpl 6π ρpl 6π ρpl √ h 1 ρΛ 1/2 R 3 2 ργ=0 1/2 R 3 i Ψ2 (R) = C2 exp − √ , (36) +√ lpl lpl 6π ρpl 6π ρpl 1/2 5 where ρpl = h¯cG2 is the Planck density and lpl = h¯cG is the Planck length. 3
We can see, that the function Ψ1 in (35) has good behavior when R → ∞ under the condition ρΛ < 2ργ=0 , while Ψ2 does not. On the other hand, the wave function Ψ2 in (36) has good behavior when R → ∞ under the condition ρΛ > 2ργ=0 , because the principal contribution comes from the first term of the exponent, while Ψ1 does not have good behavior. However, only the scalar product for the second wave function Ψ2 is normalizable in the measure R1/2 dR under the condition ρΛ > 2ργ=0 . This condition does not contradict the astrophysical observation at ρΛ ≈ (2 − 3)ρM , due to the fact that the dust matter 7
introduces the main contribution to the total energy density of matter ρM . On the other hand, according to recent astrophysical data, our universe is dominated by a mysterious form of the dark energy [4], which counts to about 70 per cent of the total energy density. As a result, the universe expansion is c2 Λ accelerating [5, 6]. Vacuum energy density ρΛ = 8πG is a concrete example of the dark energy. Conclusion The recent cosmological data give us the following range for the dark energy state parameter γ = −0.96+0.08 −0.09 . However, in the literature we can find different theoretical models for the dark energy with state parameter γ > −1 and γ < −1, see reviews [7, 8] and the articles [9, 10]. In the present work we have discussed the case for γ = 0 corresponding to the FRW universe with barotropic perfect fluid as matter field. In the case of the flat universe (k = 0) and the dust-like matter γ = 0 we have obtained two wave functions. However, only the second wave function is normalizable under the condition ρΛ > 2ργ=0 , which leads to the cosmological value Λ > 16πG c2 ργ=0 .
References [1] O. Obreg´on, J.J. Rosales and V.I. Tkach, Phys. Rev. D 53, R1750 (1996); V.I. Tkach, J.J. Rosales and O. Obreg´on, Class. Quantum Grav. 13, 2349 (1996). [2] C. Ortiz, J.J. Rosales, J. Socorro, J. Torres and V.I. Tkach. Phys. Lett. A 340, 51-58, (2005); O. Obreg´on, J.J. Rosales, J. Socorro and V.I. Tkach, Class. Quantum Gravity, 16, 2861 (1999); V.I. Tkach, J.J. Rosales and J. Socorro, Class. Quantum Gravity, 16, 797 (1999); M. Ryan Jr, Hamiltonian Cosmology, Springer Verlag (1972); J. Socorro, M.A. Reyes and F.A. Gelbert, Phys. Lett. A 313, 338 (2003). [3] V.I. Tkach, arXiv:0808.3429; C. Beck, arXiv:0810.0752. [4] T. Padmanabhan, Phys. Rept. 380, 235 (2003). [5] S. Perlmutter, et al; Astrophysics J., 517, 565 (1999). [6] A.G. Riess, et al; Astrophysics J., 607, 665 (2004). [7] S. Nojiri, S.D. Odintsov, Int. J. Geom. Math. Mod. Phys. 4, 115 (2007) and references therein. [8] E.J. Copeland, M. Sami, S. Tsujikawa, Int. J. Mod. Phys. D 15, 1753 (2006) and references therein.
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[9] S. Nojiri, S.D. Odintsov, arXiv: 0801.4843; arXiv: 0807.0685; arxiv: 0810.1557. [10] K. Bamba, C. Geng, S. Nojiri, S.D, Odinsov, arXiv:0810.4296
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