Superconducting dark energy Shi-Dong Liang1∗ and Tiberiu Harko2† 1 arXiv:1504.02645v1 [gr-qc] 10 Apr 2015 State Key Laboratory of Optoelectronic Material and Technology, and Guangdong Province Key Laboratory of Display Material and Technology, School of Physics and Engineering, Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China and 2 Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, United Kingdom (Dated: June 26, 2018) Based on the analogy with superconductor physics we consider a scalar-vector-tensor gravitational model, in which the dark energy action is described by a gauge invariant electromagnetic type functional. By assuming that the ground state of the dark energy is in a form of a condensate with the U(1) symmetry spontaneously broken, the gauge invariant electromagnetic dark energy can be described in terms of the combination of a vector and of a scalar field (corresponding to the Goldstone boson), respectively. The gravitational field equations are obtained by also assuming the possibility of a non-minimal coupling between the cosmological mass current and the superconducting dark energy. The cosmological implications of the dark energy model are investigated for a Friedmann-Robertson-Walker homogeneous and isotropic geometry for two particular choices of the electromagnetic type potential, corresponding to a pure electric type field, and to a pure magnetic field, respectively. The time evolutions of the scale factor, matter energy density and deceleration parameter are obtained for both cases, and it is shown that in the presence of the superconducting dark energy the Universe ends its evolution in an exponentially accelerating vacuum de Sitter state. By using the formalism of the irreversible thermodynamic processes for open systems we interpret the generalized conservation equations in the superconducting dark energy model as describing matter creation. The particle production rates, the creation pressure and the entropy evolution are explicitly obtained. PACS numbers: 03.75.Kk, 11.27.+d, 98.80.Cq, 04.20.-q, 04.25.D-, 95.35.+d I. INTRODUCTION The Λ Cold Dark Matter (ΛCDM) model of cosmology is remarkably successful in accounting for most of the observed features of the Universe. The recent Planck satellite data from the 2.7 degree Cosmic Microwave Background full sky survey [1, 2] have generally confirmed again the present day standard cosmological paradigm. However, a number of fundamental questions at the very foundations of cosmology and gravitation still remain open, and unanswered. Perhaps the most important challenge facing modern cosmology is the understanding of the mechanism of the acceleration of the late universe, which is usually attributed to the presence of the mysterious dark energy. [3]. In fact, as fundamental candidates responsible for the cosmic expansion, the standard ΛCDM model of cosmology has favored dark energy models involving time-dependent scalar fields. Scalar fields naturally arise in many particle physics models, including string theory. On the other hand, the underlying dynamics of inflationary models, assumed to be of fundamental importance for the understanding of the early history of the Universe, also depend essentially on a single scalar field, the inflaton, rolling in some underlying potential [4]. The possibility that a single canonical scalar field φ, ∗ Electronic † Electronic address: stlsd@sysu.edu.cn address: t.harko@ucl.ac.uk with a non-zero potential, called quintessence, could be responsible for the late-time cosmic acceleration, was also explored in much detail [5–7]. The well-known action for a scalar field in the presence of gravity is Sφ = Z   √ R 1 α −gd4 x, − ∇ φ∇α φ − V (φ) 16πG 2 (1) where R is the Ricci scalar, G is the gravitational constant, and V (φ) is the self-interaction potential, respectively [8]. In opposition to the behavior of the cosmological constant, the quintessence equation of state changes dynamically with time [9]. In fact, many other exotic fluids have been proposed to explain the accelerated expansion of the Universe. Some of the proposed models are the so-called k−essence models, in which the late-time acceleration is driven by the kinetic energy term of the scalar field [10]. A number of coupled models, where dark energy interacts both quantitatively and qualitatively with dark matter, have also been proposed [11], as well as unified models of dark matter and dark energy [12]. For a review of the dark energy candidates see [13]. An intriguing alternative about the nature of dark energy, which was also intensively investigated in the literature, is the possibility that it could be described by a vector field, which can be at the origin of the present stage of cosmic acceleration. In its simplest formulation 2 the action for the vector field dark energy model is (  Z 3  X √ 1 a aµν R 2 4 − F F + V (A ) + SV = d x −g 16πG a=1 4 µν ) Lm , (2) a where Fµν = ∂µ Aaν − ∂ν Aaµ , A2 = g µν Aaµ Aaν , and Lm is the matter Lagrangian [14]. This vector (or more exactly Yang-Mills) type action for the dark energy thus contains three identical components obtained by generalizing the Lagrangian of a single vector field. The term V (A2 ) is a self-interaction potential that explicitly violates gauge invariance. The cosmological implications of the vector type dark energy models have been investigated in [15]. More general vector field dark energy models, in which the vector field is non-minimally coupled to the gravitational field, have been proposed in [16]. By assuming that the Universe is filled with a massive cosmological vector field, with mass µΛ , which is characterized by a fourpotential Λµ (xν ), µ, ν = 0, 1, 2, 3, which couples nonminimally to gravity, and by introducing, in analogy with electrodynamics, the field tensor Cµν = ∇µ Λν − ∇ν Λµ , the action for the non-minimally coupled vector dark energy theory can be written as Z " 1 R + Cµν C µν + µ2Λ Λµ Λµ + ωΛµ Λµ R + S = − 2 # √ ηΛµ Λν Rµν + 16πG0 Lm −gdΩ, (3) where Rµν is the Ricci tensor and G0 is the gravitational constant. In Eq. (3) ω and η are dimensionless coupling parameters. At first sight the gravitational actions given by Eqs. (1) and (2) look totally different, from both mathematical point of view, as well as from the physical interpretation point of view. However, they can be in fact interpreted and understood as the limiting cases of a single physical model, related to the spontaneous breaking of the electromagnetic U(1) symmetry. Thus an approach is used to describe superconductivity from a fundamental point of view [17–19]. From a general physical point of view in the theory of superconductivity the existence of a quantum condensate (superconducting state) is described by a nonvanishing value of a gauge dependent complex order parameter [17–19]. In bosonic systems superfluid behavior occurs when the expectation value of the bosonic field parameter ψ has a nonzero value, hψi = 6 0. On the other hand the existence of superconductivity is also induced by a nonzero value of the expectation value of the pair field operator. Therefore, in the ground state of a superconducting system a quantum condensate ǫαβ ψ α ψ β forms [17–19]. Since the difermion operator has charge −2e, the important result that the quantum condensate breaks the electromagnetic U(1) symmetry is obtained. Another fundamental quantity in the model is a scalar field, Φ, which plays the role of the order parameter. Under a gauge transformation Aµ → Aµ + ∂µ Λ, the scalar field transforms like the condensate wave function ψ → eieΛ ψ =⇒ Φ → e2ieΛ Φ. Note that in the zero temperature superconductivity theory one also introduces the Goldstone field φ as the phase of the field Φ, Φ = ρe2ieφ , as well as the gauge invariant Fermi fields, ψ̃ = e−ieφ [17, 18]. Hence from a fundamental point of view a superconducting system can be described by a gauge invariant Lagrangian, depending on the wave function ψ̃, and on the vector potentials Aµ and ∇µ φ. A simplified model is obtained after integrating out the Fermi fields. Thus one obtains a gauge invariant Lagrangian, depending only on Aµ and ∇µ φ, respectively. The important requirement of the gauge invariance of the theory implies that these bosonic fields must appear only in the combinations Fµν = ∇µ Aν −∇ν Aµ and Aµ −∇µ φ, respectively. Therefore the Lagrangian describing a superconductor from a fundamental physical point of view has the form [17, 18] Z 1 L=− Fµν F µν d3~r + Ls (Aµ − ∇µ φ) , (4) 4 where Ls (Aµ − ∇µ φ) is an arbitrary function of the argument Aµ − ∇µ φ. The only physical condition required on the superconductor Lagrangian Ls is that in the absence of Aµ and φ it gives rise to a stable state of the system. In particular, this requires that the point Aµ = ∇µ φ is a local minimum of the theory (this property can fully explain the Meissner effect in superconductivity theory [17]). Therefore, we require that the second derivative of the superconductor Lagrangian Ls with respect to its argument must be nonzero at the point Aµ = ∇µ φ [17, 18]. It is the goal of the present paper to consider a gravitational model in which dark energy is described by a Lagrangian of the form given by Eq. (4), resulting from the breaking of the U(1) symmetry in the ground state dark energy condensate. By analogy with condensed matter physics we call this model the superconducting dark energy model. The gravitational field equations of the model are derived from an action principle, and the cosmological implications are investigated in a background homogeneous and isotropic flat Friedmann-RobertsonWalker geometry. We consider two distinct classes of cosmological models, corresponding to two different choices of the electromagnetic potential Aµ of the dark energy. In the first model Aµ has only a non-vanishing temporal component, while in the second case we assume nonvanishing spatial (magnetic) components of the potential. In both cases we assume that the dark energy selfinteraction potential is constant. In the present dark energy model, due to the coupling between the matter current and the electromagnetic and scalar potentials of the dark energy, the matter energymomentum tensor is not conserved. By using the formalism of the open thermodynamic systems introduced in 3 [22]-[25] (see also [26] for recent investigations of particle creation in cosmology), we interpret the generalized conservation equations in the superconducting dark energy model from a thermodynamic point of view as describing irreversible matter creation processes. Thus in the present model particle creation corresponds to an irreversible energy flow from the superconducting dark energy to the created matter constituents (both normal and dark). We explicitly obtain the equivalent particle number creation rates, the creation pressure and the entropy production rates. The temperature evolution laws of the newly created particles are explicitly derived. We also show that due to the superconducting dark energy - matter current coupling, during the cosmological evolution a large amount of comoving entropy could be produced. The present paper is organized as follows. In Section II the gravitational field equations of the superconducting dark energy model, a scalar-vector-tensor theory with broken U(1) symmetry, are derived from a variational principle. The equations of motion of the scalar and vector fields are also obtained. The cosmological applications of the theory are investigated in Section III. Two distinct dark energy models are considered: an electric type, in which the vector potential has only a time component, and a magnetic type, with the vector potential having only spatial components. The cosmological properties of both models are investigated in detail. The thermodynamic interpretation of the superconducting dark energy model is considered, in the framework of the thermodynamic of open systems and irreversible processes, in Section IV. We discuss and conclude our results in Section V. In this paper we adopt the Landau-Lifshitz [20] metric conventions, and we use the natural system of units with 8πG = c = 1. II. FIELD EQUATIONS OF THE SUPERCONDUCTING DARK ENERGY MODEL In the following we assume that the interaction of the gravitational and of the superconducting dark energy scalar-vector fields is described by a Lagrangian which is required to satisfy the following standard conditions: a) the Lagrangian density is a four-scalar b) the free-field energies are positive-definite for all the metric, scalar and vector fields c) the resulting theory is metric and d) the field equations contain no higher than second order derivatives of the fields [21]. Based on the analogy with superconductor physics we consider a gravitational scalar-vector-tensor action of the form Z " R 1 λ S = − + Fµν F µν − g µν × 2 16π 2
(Aµ − ∇µ φ) (Aν − ∇ν φ) + V A2 , φ − # √ α µν −gdΩ,(5) g jµ (Aν − ∇ν φ) + Lm (gµν , ψ) 2 where λ and α are constants, Lm (gµν , ψ) is the Lagrangian of the total (ordinary baryonic plus dark) matter, and j µ = ρuµ is the total mass current, where ρ is the total matter density (including dark matter), and uµ is the matter four-velocity. We assume that the baryonic and dark matter are comoving. The third term in the action Eq. (5) follows from the assumption that the superconducting dark energy is close to the minimum Aµ = ∇µ φ. In this case the general superconducting Lagrangian (4) can be expanded in power series as [17, 18] 1 δ 2 Ls (Aµ − ∇µ φ)2 +..., 2 δ (Aµ − ∇µ φ)2 (6) where L0 is a constant. Hence the superconducting type Lagrangian Ls (Aµ − ∇µ φ) gives a quadratic contribution in Aµ − ∇µ φ to the gravitational Lagrangian. We have also assumed the possibility of an interaction between the total matter flux j µ and the superconducting dark energy gauge invariant potentials Aµ − ∇µ φ. V (A2 , φ) is the self-interaction potential of the scalar and vector fields, with A2 = Aµ Aµ , in which we have also included the constant L0 . When φ ≡ 0, that is, the scalar field vanishes, the action (5) gives the pure vector model of the dark energy. When the electromagnetic type potential Aµ = 0, we recover the standard action of the minimally coupled scalar-tensor theory. Hence the gravitational action (5) gives a unified framework for the minimal inclusion into the gravitational action of the scalar-vector interactions, under the assumption of the existence of a U(1) broken symmetry. The second and third terms in the gravitational action are also similar to the Stueckelberg Lagrangian [27]. We define the energy-momentum tensor of the matter as [20]  √  √ 2 ∂ ( −gLm ) ∂ ∂ ( −gLm ) Tµν = − √ . (7) − −g ∂g µν ∂xλ ∂ (∂g µν /∂xλ ) Ls (Aµ − ∇µ φ) ≈ L0 + By making the important assumption that the Lagrangian density Lm of the matter depends only on the metric tensor components gµν , and not on its derivatives, we obtain the expression Tµν = Lm gµν − 2∂Lm/∂g µν . By varying the action (5) with respect to the metric tensor we obtain the gravitational field equations for the superconducting dark energy model as   1 1 1 α αβ Rµν − Rgµν = Tµν + −Fµα Fν + Fαβ F gµν + 2 4π 4 λ λ (Aµ − ∇µ φ) (Aν − ∇ν φ) − (Aα − ∇α φ) (Aα − ∇α φ) × 2 α β gµν + αjµ (Aν − ∇ν φ) − j (Aβ − ∇β φ) gµν + 2
V A2 , φ gµν , (8) where Tµν , the energy-momentum tensor of the ordinary matter, is given by Tµν = (ρ + p) uµ uν − pgµν , (9) 4 where p is the total thermodynamic pressure of the matter components (baryonic and dark). By taking the variation of the action Eq. (5) with respect to the scalar field φ we obtain Z " λAµ ∇µ δφ − λg µν ∇µ δφ∇ν φ + δφ S = − #
√ α µ j ∇µ δφ + ∂φ V A2 , φ δφ −gdΩ. (10) 2 With the use of the mathematical identity ∇µ (B µ δφ) = ∇µ B µ δφ + B µ ∇µ δφ, (11) it follows that if the conditions ∇µ Aµ = 0, ∇µ j µ = 0, (12) are imposed on the dark energy vector potential, and on the baryonic matter flow, the variation in the integral (10) of the first term, containing Aµ , and of the last term, containing j µ , vanish identically. Therefore we obtain the result that if the conditions given by Eqs. (12) are satisfied, then the scalar field satisfies the standard KleinGordon equation,
λg µν ∇µ ∇ν φ + ∂φ V A2 , φ = 0. (13) This case corresponds to the minimal coupling of the scalar and vector fields. However, in the following we will use a more general approach, in which no additional constraints are imposed on the fields or on the hydrodynamic flow. Therefore, the variation of the action Eq. (5) gives the following coupled evolution equation for the scalar and vector fields,
α λg µν ∇µ ∇ν φ+∂φ V A2 , φ −λ∇µ Aµ − ∇µ j µ = 0. (14) 2 By varying the superconducting dark energy action Eq. (5) with respect to Aµ , we obtain first Z " 1 µν − δ Aµ S = F ∇ν δAµ − λg µν (Aµ − ∇µ φ) δAµ − 4π #
√ α µ j δAµ + 2∂A2 V A2 , φ Aµ δAµ −gdΩ = 0. 2 (15) By taking into account the identity ∇ν (F µν δAµ ) = ∇ν F µν δAµ + F µν ∇ν δAµ , after partial integration and the use of Gauss’ theorem, it follows that the superconducting dark energy vector field satisfies the equation 1 ∇ν F µν = J µ , 4π (16) where h
i α J µ = λg µν (Aν − ∇ν φ) + j µ − 2∂A2 V A2 , φ Aµ . 2 (17) The divergence of the dark √ energy√field tensor can be obtained as ∇ν F µν = (1/ −g) ∂ν ( −gF µν ). By its definition the dark energy electromagnetic type tensor F µν satisfies the Bianchi identity εαβµν ∇β Fµν = 0, (18) where εαβµν is the complete antisymmetric unit tensor of rank four. Finally, by taking the covariant derivative of the field equations Eqs. (8) we obtain the matter conservation equation in the presence of a superconducting dark energy as α α ∇µ Tνµ + ∇µ [j µ (Aν − ∇ν φ)] − ∇ν j β (Aβ − ∇β φ) + 2

2 ∂φ V A2 , φ Aν + 2∂A2 V A2 , φ Aα ∇α Aν = 0. (19) The derivation of Eq. (19) is presented in Appendix A. By taking into account the explicit form of the energymomentum tensor, given by Eq. (9) we obtain (∇µ ρ + ∇µ p) uµ uν + (ρ + p) uν ∇µ uµ + (ρ + p) uµ ∇µ uν − α α ∇µ pgµν + ∇µ [j µ (Aν − ∇ν φ)] − ∇ν j β (Aβ − ∇β φ) + 2

2 ∂φ V A2 , φ Aν + 2∂A2 V A2 , φ Aα ∇α Aν = 0. (20) By multiplying Eq. (20) with uν , and by taking into account the mathematical identity uν ∇µ uν = 0 we obtain the energy conservation equation in the superconducting dark energy model as α ρ̇ + 3 (ρ + p) H + uν ∇µ [j µ (Aν − ∇ν φ)] − 2 
α d  β j (Aβ − ∇β φ) + ∂φ V A2 , φ uν Aν + 2 ds
2∂A2 V A2 , φ uν Aα ∇α Aν = 0, (21) where we have introduced the Hubble function H = (1/3)∇µ uµ , and we have denoted ˙ = uµ ∇µ = d/ds, respectively, where ds is the line element corresponding to the metric gµν , ds2 = gµν dxµ dxν . By multiplying Eq. (20) with the projection operator hνλ , defined as hνλ = δλν − uλ uν , and satisfying the relation uν hνλ = 0, gives the momentum balance equation for a perfect fluid in the superconducting dark energy model as ( d2 xλ hνλ µ λ λ µ ν ∇ν p − u ∇µ u = + Γµν u u = ds2 ρ+p α α ∇µ [j µ (Aν − ∇ν φ)] + ∇ν j β (Aβ − ∇β φ) − 2 2 )

α 2 2 ∂φ V A , φ Aν − 2∂A2 V A , φ A ∇α Aν . (22) III. COSMOLOGICAL APPLICATIONS We assume that the metric of the Universe is given by the isotropic and homogeneous Friedmann-Robertson- 5 Walker metric,
ds2 = dt2 − a2 (t) dx2 + dy 2 + dz 2 , (23) where a(t) is the scale factor describing the expansion of the Universe. We assume that the cosmological matter is comoving with the cosmological expansion, and therefore we choose the four velocity of the cosmological fluid as uµ = (1, 0, 0, 0). Hence the components of the fourcurrent vector are j µ = (ρ, 0, 0, 0). In the FriedmannRobertson-Walker geometry the Hubble function takes the form H = ȧ/a, since uµ ∇µ = ˙ = d/dt. To describe the decelerating/accelerating nature of the cosmological expansion, we use the deceleration parameter q, with the definition q= Ḣ d 1 − 1 = − 2 − 1. dt H H (24) Moreover, from the homogeneity of the Universe it follows that the scalar and vector fields φ and Aµ can be only functions of the cosmological time t, so that φ = φ(t) and Aµ = Aµ (t) = (A0 (t), A1 (t), A2 (t), A3 (t)), respectively. The non-zero components of the dark energy tensor Fµν are given by Fi0 (t) = −Ȧi (t), i = 1, 2, 3, and F i0 (t) = Ȧi (t)/a2 (t), i = 1, 2, 3. Hence we obtain i2
P3 h Fαβ F αβ = − 2/a2 (t) Ȧ (t) . Then, the cosmoi i=1 logical equations corresponding to the superconducting dark energy model are given by ! 3 h i X α 1 2 2 3H = 1 + A0 − φ̇ ρ + Ȧk + 2 8πa2 (t) k=1 ! 3 2 X
λ λ 2 A0 − φ̇ + 2 Ak + V A2 , φ , (25) 2 2a (t) k=1 1 Ȧ2i 1 2Ḣ + 3H = −p + − 4π a2 (t) 8πa2 (t) 2 3 2 X λ A2 λ A0 − φ̇ + 2 λ 2i − A2k a (t) 2 2a (t) k=1 
α 2 A0 − φ̇ ρ + V A , φ , i = 1, 2, 3, 2 3 X k=1 ! hα   α
λ A0 − φ̇ + ρ − 2∂A2 V A2 , φ A0 = 0, 2 − −  i α ρ + λ A0 − φ̇ H − λȦ0 − ρ̇ + 2
2 ∂φ V A2 , φ = 0, λφ̈ − 3 Ȧ2k ! (26) (27) (28)
Äk + H Ȧk (t) + 4πλAk (t) − 8π∂A2 V A2 , φ Ak (t) = 0, k = 1, 2, 3. (29) As an independent variable we introduce, instead of the cosmological time t, the redshift z, defined as 1+z = 1/a. Therefore dH dz dH dH = = −(1 + z)H . dt dz dt dz (30) As a function of the redshift the deceleration parameter is obtained as q = (1 + z) 1 dH(z) − 1. H(z) dz (31) In the following we will explicitly investigate two distinct superconducting dark energy models. A. Electric dark energy models We assume that the dark energy vector potential has the form Aµ = (A0 (t), 0, 0, 0), that is, the dark energy vector potential has only one, electric type, component. For this choice Fµν ≡ 0, ∀µ, ν ∈ [0, 1, 2, 3]. The gravitational field equations describing the cosmological dynamics in the presence of the superconducting dark energy take the form i 2 h
λ α A0 − φ̇ ρ + A0 − φ̇ + V A2 , φ , 3H 2 = 1 + 2 2 (32) 2Ḣ+3H 2 = −p− 2 α  
λ A0 − φ̇ − A0 − φ̇ ρ+V A2 , φ , 2 2 (33)  α io
d n 3h  a λ A0 − φ̇ + ρ − a3 ∂φ V A2 , φ = 0, (34) dt 2   α
λ A0 − φ̇ + ρ − 2∂A2 V A2 , φ A0 = 0. 2 (35) In order to close the system of equations (32)-(35) the baryonic equation of state p = p (ρ) must also be provided. In the following we will restrict our analysis to the case of a constant self-interaction potential of the super
conducting dark energy field, V A2 , φ = V0 = constant. Then from Eqs. (34) and (35) we obtain   α (36) A0 − φ̇ = − ρ. 2λ Hence the generalized Friedmann equations of the cosmological expansion in the presence of the electric type superconducting dark energy become 3H 2 = ρ − α2 2 ρ + V0 = ρ + ρDE , 8λ 2Ḣ + 3H 2 = −p + α2 2 ρ + V0 = −p − pDE , 8λ (37) (38) 6 where we have denoted ρDE = − α2 2 ρ + V0 , 8λ (39) and pDE = − α2 2 ρ − V0 , 8λ (40) respectively. From Eqs. (37) and (38) we obtain 2Ḣ = − (ρ + p) + α2 2 ρ . 4λ (41) The energy conservation equation can be written as      d α2 2 d 3 α2 2 3 ρ− ρ a + p− ρ a = 0. (42) dt 8λ 8λ dt The deceleration parameter can be obtained as
(ρ + 3p) − α2 /2λ ρ2 − 2V0 . q= 2 [ρ − (α2 /8λ) ρ2 + V0 ] (43) For a dust Universe with p = 0, the deceleration parameter takes the form
1 ρ − α2 /2λ ρ2 − 2V0 q= . (44) 2 ρ − (α2 /8λ) ρ2 + V0 From the field equations Eqs. (37)-(41) we obtain the time evolution equation of the baryonic matter density as p  
 √ ρ − (α2 /8λ) ρ2 + V0 p + ρ 1 − α2 /4λ ρ . ρ̇ = − 3 [1 − (α2 /4λ) ρ] (45) In terms of the redshift z, the density evolution equation of the baryon density in the electric type superconducting dark energy models is given by 
 dρ(z) 3 p(z) + ρ(z) 1 − α2 /4λ ρ(z) = . (46) dz 1+z 1 − (α2 /4λ) ρ(z) In order to characterize the dark energy, and its evolution properties, we also introduce the dark energy equation of state parameter wDE , defined as
− α2 /8λ ρ2 + V0 pDE =− . (47) wDE = ρDE (α2 /8λ) ρ2 + V0 1. Dust Universes with electric type superconducting dark energy In the case of the dust Universe, with p = 0, Eq. (45), describing the time dynamics of the matter density in the presence of the superconducting electric type dark energy takes the form √ p ρ̇ = − 3ρ ρ − (α2 /8λ) ρ2 + V0 . (48) By introducing a set of dimensionless variables √
2 (θ, τ,  h, v0 ), defined as ρ = 8λ/α θ, t = α/ 24λτ ,  √ √
2 8λ/ 3α h, v0 = α /8λ V0 , Eq. (48) becomes H= p dθ = −θ θ − θ2 + v0 , dτ (49) while the dimensionless Hubble parameter h is obtained as p (50) h = θ − θ2 + v0 . In the dimensionless time variable τ we have h(τ ) = [3/a(τ )] (da/dτ ). The deceleration parameter of this model is given by q= 1 θ − 4θ2 − 2v0 , 2 θ − θ2 + v0 (51) while the dark energy equation of state parameter wDE can be obtained as wDE = − −θ2 + v0 . θ2 + v0 (52) The time variation of the redshift z can be obtained from the equation 1 da 1 + zp dz = −(1 + z) =− θ − θ2 + v0 . dτ a dτ 3 (53) In terms of the redshift z the evolution of the dust electric type superconducting dark energy Universe is described by the simple relations 3 ρ(z) = ρ0 (1 + z) , (54)  1/2 1 α2 ρ20 H(z) = √ ρ0 (1 + z)3 − (1 + z)6 + V0 , 8λ 3 (55) q(z) =
6 1 ρ0 (1 + z)3 − α2 ρ20 /2λ (1 + z) − 2V0 , 2 ρ0 (1 + z)3 − (α2 ρ20 /8λ) (1 + z)6 + V0 (56) where ρ0 is the matter density of the Universe at the present time z = 0. The variations with respect to the redshift z of the Hubble function h of the Universe, of the matter energy density θ, of the deceleration parameter q, and of the parameter of the dark energy equation of state are represented, for different values of v0 , in Figs. 1-4. The initial values used to numerically integrate the cosmological evolution equations are θ(0) = 0.1, z(0) = 5, and a(0) = 1/6, respectively. As one can see from Fig. 1, the Hubble function h of the Universe is a monotonically increasing function of the redshift (monotonically time decreasing function). In the early stages of evolution, at around z ≈ 5, h is basically 7 0.4 0.30 0.2 0.25 0.0 hHzL qHzL 0.20 -0.2 0.15 -0.4 0.10 -0.6 0.05 -0.8 0 1 2 3 4 0 5 1 2 3 4 5 z z FIG. 1: Variation with respect to the redshift z of the dimensionless Hubble function h of the electric type superconducting dark energy filled Universe with p = 0 for different values of v0 : v0 = 0.001 (solid curve), v0 = 0.002 (dotted curve), v0 = 0.003 (short dashed curve), v0 = 0.004 (dashed curve), and v0 = 0.005 (long dashed curve), respectively. FIG. 3: Variation with respect to the redshift z of the deceleration parameter q of the electric type superconducting dark energy filled Universe with p = 0 for different values of v0 : v0 = 0.001 (solid curve), v0 = 0.002 (dotted curve), v0 = 0.003 (short dashed curve), v0 = 0.004 (dashed curve), and v0 = 0.005 (long dashed curve), respectively. 0.10 0.5 w DE H z L 0.08 ΘHzL 0.06 0.0 0.04 -0.5 0.02 -1.0 0.00 0 1 2 3 4 0 5 1 2 3 4 5 z z FIG. 2: Variation with respect to the redshift z of the dimensionless matter energy density θ of the electric type superconducting dark energy filled Universe with p = 0 for different values of v0 : v0 = 0.001 (solid curve), v0 = 0.002 (dotted curve), v0 = 0.003 (short dashed curve), v0 = 0.004 (dashed curve), and v0 = 0.005 (long dashed curve), respectively. FIG. 4: Variation with respect to the redshift z of the parameter wDE of the dark energy equation of state of the electric type superconducting dark energy filled Universe with p = 0 for different values of v0 : v0 = 0.001 (solid curve), v0 = 0.002 (dotted curve), v0 = 0.003 (short dashed curve), v0 = 0.004 (dashed curve), and v0 = 0.005 (long dashed curve), respectively. independent on the values of v0 . The matter density of the Universe, presented in Fig. 2, is a monotonically increasing function of z, tending in the small z limit to zero, limz→0 θ(z) = 0. Its evolution is basically independent of the range if the numerical values of v0 . The redshift variation of the deceleration parameter q, depicted in Fig. 3, shows that in the present model the Universe starts from a decelerating phase at around z ≈ 5, with q having values of the order of q ≈ 0.2 − 0.3. This initial value increases in the early stages of the cosmological evolution, showing a decelerating expansion. At z ≈ 1 − 2, the Universe starts to accelerate, with the decelerating parameter slightly decreasing and taking negative vales q < 0. The values of the deceleration parameter gradually decrease with decreasing z, and the Universe enters into an accelerating stage, ending its evolution in a de Sitter stage, with q ≈ −1 at z ≈ 0. The parameter wDE of the equation of state of the dark energy, presented in Fig. 4, starts with positive values, and, with decreasing z, it takes negative values. In the small redshift limit it tends to the value wDE = −1. The present day numerical values of the Hubble function H0 and of the deceleration parameter q0 can be obtained as 
1/2 ρ0 − α2 ρ20 /8λ + V0 √ H0 = , (57) 3 and  
1 ρ0 − α2 ρ20 /2λ − 2V0 q0 = , 2 [ρ0 − (α2 ρ20 /8λ) + V0 ] (58) respectively. Therefore the free parameters α and λ of 8 θ(τ ) = 4θ0 v0 eτ √ v0 (θ0 + 2v0 )e e √ 2τ0 v0 θ02 (4v0 2 v0 (−θ02 + θ0 + v0 ) e2τ0 √ τ0 v0 − 4θ0 e + 1)e2τ τ #( √ v0 √ v0  √ v0 0.4 + q √ v0 (−θ02 + θ0 + v0 ) e2τ0 v0 + + 4(θ0 + 2v0 )eτ0 v0 (τ +τ0 ) √ v0 × , (59) 0.3 0.2 0.1  θ02 + 8v02 − 4(θ0 − 2)θ0 v0 × q √ v0 (−θ02 + θ0 + v0 ) e2τ0 v0 − )−1 2θ0 (θ0 + 2v0 )e √ 0.5 hHzL the superconducting electric type dark energy model can be obtained from astronomical observations. Eq. (49) can also be solved exactly, and thus we obtain the density as a function of time in an exact analytical form as given by " q 5 10 15 20 25 z FIG. 5: Variation with respect to the redshift z of the Hubble function h of the radiation fluid Universe with p = ρ/3 in the presence of electric type superconducting dark energy for different values of v0 : v0 = 0.001 (solid curve), v0 = 0.002 (dotted curve), v0 = 0.003 (short dashed curve), v0 = 0.004 (dashed curve), and v0 = 0.005 (long dashed curve), respectively. where we have used the initial condition θ (τ0 ) = θ0 . The radiation fluid Universe with electric type superconducting dark energy 0.4 For a high density radiation fluid Universe, with matter equation of state satisfying the condition p = ρ/3, in the presence of electric type superconducting dark energy the basic evolution equation of the dimensionless matter density θ is given by p dθ θ/3 + θ(1 − 2θ) = − θ − θ2 + v0 . dτ (1 − 2θ) 2θ − 4θ2 − 2v0 . 2 (θ − θ2 + v0 ) 0.2 0.1 (60) 0.0 The deceleration parameter of the radiation Universe can be obtained as q= 0.3 ΘHzL 2. (61) The variations with respect to the redshift z of the Hubble function, of the energy density, of the deceleration parameter and of the parameter of the dark energy equation of state of the radiation fluid Universe in the presence of the electric type superconducting dark energy are presented in Figs. 5-8. The initial conditions used to numerically integrate the cosmological evolution equation are θ(0) = 0.45 and z(0) = 25. We assume that the Universe was radiation dominated in the redshift range 5 ≤ z ≤ 25. The dimensionless Hubble function h of the high density Universe, presented in Fig. 5, is a monotonically increasing function of the redshift (monotonically decreasing in time), while the energy density, depicted in Fig. 6, increases monotonically with the redshift z during the cosmological evolution. The deceleration parameter, shown in Fig. 7, has positive values for the redshift interval 9 ≤ z ≤ 25, indicating a decelerating expansion. For large values of v0 the deceleration 5 10 15 20 25 z FIG. 6: Variation with respect to the redshift z of the matter energy density θ of the radiation fluid Universe with p = ρ/3 in the presence of electric type superconducting dark energy for different values of v0 : v0 = 0.001 (solid curve), v0 = 0.002 (dotted curve), v0 = 0.003 (short dashed curve), v0 = 0.004 (dashed curve), and v0 = 0.005 (long dashed curve), respectively. parameter can reach the zero value at redshifts as high as z ≈ 9, limz→9 q|v0 =0.005 ≈ 0. The time variation of the cosmological parameters h and θ is practically independent of the adopted small values of the parameter v0 . The parameter wDE of the dark energy equation of state is represented in Fig. 8. For large redshift values 15 ≤ z ≤ 25, wDE is positive, while for z ≈ 5 it approaches the value wDE = −1, showing that in the present model the Universe becomes dark energy dominated at around z ≈ 5. 9 0.5 0.4 hHzL qHzL 0.5 0.0 0.3 0.2 -0.5 0.1 5 10 15 20 25 0 5 10 z FIG. 7: Variation with respect to the redshift z of the deceleration parameter q of the radiation fluid Universe with p = ρ/3 in the presence of electric type superconducting dark energy for different values of v0 : v0 = 0.001 (solid curve), v0 = 0.002 (dotted curve), v0 = 0.003 (short dashed curve), v0 = 0.004 (dashed curve), and v0 = 0.005 (long dashed curve), respectively. 20 25 FIG. 9: Variation with respect to the redshift z ∈ [0, 25] of the Hubble function h of the Universe filled by electric type superconducting dark energy, for different values of v0 : v0 = 0.001 (solid curve), v0 = 0.002 (dotted curve), v0 = 0.003 (short dashed curve), v0 = 0.004 (dashed curve), and v0 = 0.005 (long dashed curve), respectively. 1.0 0.4 0.5 0.3 ΘHzL w DE H z L 15 z 0.0 0.2 0.1 -0.5 0.0 -1.0 5 5 10 15 20 25 10 15 20 25 z z FIG. 8: Variation with respect to the redshift z of the parameter of the dark energy equation of state wDE of the radiation fluid Universe with p = ρ/3 in the presence of electric type superconducting dark energy for different values of v0 : v0 = 0.001 (solid curve), v0 = 0.002 (dotted curve), v0 = 0.003 (short dashed curve), v0 = 0.004 (dashed curve), and v0 = 0.005 (long dashed curve), respectively. 3. The unified picture of the evolution of the Universe in the electric type superconducting dark energy model Finally, to conclude the investigation of the electric type superconducting dark energy model, we present a unified picture of the evolution of the Universe for the redshift range z ∈ [0, 25]. The variations of the Hubble function, matter energy density, deceleration parameter, and the parameter of the dark energy equation of state are plotted in Figs. 9-12. To study the evolution of the electric type superconducting dark energy we adopt for the redshift z the range from 0 to 25. We assume that in the range z ∈ [5, 25] the matter content of the Universe can be (at least approximately) described by a radiation type equation of state FIG. 10: Variation with respect to the redshift z ∈ [0, 25] of the matter energy density θ of the Universe filled with electric type superconducting dark energy, for different values of v0 : v0 = 0.001 (solid curve), v0 = 0.002 (dotted curve), v0 = 0.003 (short dashed curve), v0 = 0.004 (dashed curve), and v0 = 0.005 (long dashed curve), respectively. p = ρ/3. At z = 5 the Universe enters in the matter dominated era, with p ≈ 0. In this simplified model the transition from the radiation dominated era to the matter dominated phase is smooth, with all physical and thermodynamical quantities continue at the transition point. Therefore the Hubble function and the matter density, represented in Figs. 9 and 10, are monotonically increasing functions of the redshift for the entire period. The deceleration parameter, shown in Fig. 11, has a complex behavior. The Universe starts at z = 25 with a deceleration parameter having a value of q ≈ 0.10, and its early evolution is strongly decelerating, with q reaching the value q ≈ 0.9 at z ≈ 12, for v0 = 0.001. For larger values of v0 the expansion of the Universe is faster, with q reaching the value q = 0.6 at z ≈ 14. After q has reached its maximum, it starts to decrease with decreasing z, and, depending on the numerical value of v0 , reaches the value 10 4. Electric type superconducting dark energy Universe with conserved electric field and matter current qHzL 0.5 Finally, we investigate the case in which the electric potential A0 and the matter current satisfy the conservation equations given by Eqs. (12). The time variation of A0 can be immediately obtained from the Lorentz gauge equation imposed on Aµ , which gives 0.0 -0.5 -1.0 0 5 10 15 20 25 z FIG. 11: Variation with respect to the redshift z ∈ [0, 25] of the deceleration parameter q of the Universe filled with electric type superconducting dark energy, for different values of v0 : v0 = 0.001 (solid curve), v0 = 0.002 (dotted curve), v0 = 0.003 (short dashed curve), v0 = 0.004 (dashed curve), and v0 = 0.005 (long dashed curve), respectively. A0 (t) = C0 , a3 (63) respectively, where C0 is an arbitrary integration constant. The continuity equation of the matter hydrodynamic flow, ∇µ (ρuµ ) = 0 gives a similar dependence for the matter density ρ, ρ0 ρ(t) = 3 , (64) a where ρ0 is an arbitrary integration constant. The evolution of the scalar field φ is decoupled from the electric and matter component, and in the presence of a constant
potential, V A2 , φ = constant, follows a similar law as the electric and the matter fields, 1.0 0.5 w DE H z L
1 ∂ √ 1 d 3 0
∇µ Aµ = √ −gAµ = 3 a A = 0, µ −g ∂x a dt (62) and 0.0 φ(t) = -0.5 -1.0 0 5 10 15 20 25 z FIG. 12: Variation with respect to the redshift z ∈ [0, 25] of the parameter of the dark energy equation of state wDE for an Universe filled with electric type superconducting dark energy for different values of v0 : v0 = 0.001 (solid curve), v0 = 0.002 (dotted curve), v0 = 0.003 (short dashed curve), v0 = 0.004 (dashed curve), and v0 = 0.005 (long dashed curve), respectively. q = 0 for z ≈ 5 − 7. Then the deceleration parameter enters the negative range, with the Universe starting to accelerate at a higher rate, and reaching the value q ≈ −1 (the de Sitter phase) at around z = 0. The parameter of the dark energy equation of state wDE , represented in Fig. 12, is slowly decreasing from its maximum value 1 in the redshift range z ≈ 17−25. For z < 15, wDE decreases rapidly with decreasing z, and, depending on the numerical value of v0 , reaches the limiting value wDE ≈ −1 at z ≈ 7.5 − 10. φ0 , a3 (65) where φ0 is an arbitrary integration constant. In the large time limit, all these fields tend to zero, and the Universe enters in an exponential, de Sitter type expansionary phase. However, the presence of the electric type superconducting dark energy modifies the cosmological dynamics of the Universe before it enters in the de Sitter stage. B. Magnetic dark energy models As a second superconducting vector type dark energy model we consider the case in which dark energy has a magnetic type structure, with its vector potential given by Aµ = (0, A1 (t), A2 (t), A3 (t)). In order to have an isotropic expansion the components of the superconducting magnetic vector potential must satisfy the condition A1 (t) = A2 (t) = A3 (t) = A(t). Hence for this choice of Aµ we obtain Fi0 = −Ȧi , F i0 = Ȧi /a2 , i = 1, 2, 3, (1/16π) Fαβ F αβ = − (3/8π) Ȧ2 /a2 , (1/4π) Fα0 F0α = − (3/4π) Ȧ2 /a2 , and (1/4π)Fαi Fiα = − (1/4π) Ȧ/a2 (no summation upon the index i). Therefore the gravitational field equations describing the isotropic and homogeneous Universe in the presence of superconducting dark energy take the form  λ 3 Ȧ2 3λ A2 α  + + V0 , (66) 3H 2 = 1 − φ̇ ρ + φ̇2 + 2 2 8π a2 2 a2 11 1 Ȧ2 λ A2 λ α + +V0 , (67) 2Ḣ +3H 2 = −p+ ρφ̇− φ̇2 − 2 2 8π a2 2 a2 λφ̇ − α ρ = 0, 2 Therefore the system of gravitational field equations describing the superconducting magnetic type cosmological dark energy model takes the form (68) 1 d   aȦ = −4πλA, a dt (69) where for simplicity we have adopted a constant value V0 for the self-interaction potential of the scalar and vector
fields, V A2 , φ = V0 = constant. The energy conservation equation takes the form d 3
α 3 15 2 7 d 3
a ρ +p a = a φ̇ρ̇ + ȧȦ + λȧA2 − dt dt 2 8π 3 1 2 2 3 (70) λφ̇ + V0 a ȧ. 2 With the use of Eq. (68) we can substitute the derivative of the scalar field in terms of the matter density ρ. 3 α2 2 3λ A2 3 Ȧ2 ȧ2 =ρ− + + V0 , ρ + 2 2 a 8λ 8π a 2 a2 ä ȧ2 α2 2 λ A2 1 Ȧ2 2 + 2 = −p + + + V0 , ρ − a a 8λ 8π a2 2 a2 ȧ Ä + Ȧ + 4πλA = 0. a wDE =  

− α2 /8λ ρ2 + (3/8π) Ȧ2 /a2 + (3λ/2) A2 /a2 + V0   − . (α2 /8λ) ρ2 − (1/8π) Ȧ2 /a2 + (λ/2) (A2 /a2 ) + V0 From Eqs. (71) and (72) we obtain 2Ḣ = −(ρ + p) + α2 2 A2 1 Ȧ2 − λ . ρ − 4λ 2π a2 a2 ρ̇ = 1. 15 Ȧ2 8π a2 + 7 A2 3 λ a2 1 2 2 − 3α 8λ ρ +
2 − α4λ a3 ρ 3ρ − 3V0 H. (77) Dust Universes with magnetic type superconducting dark energy For the case of dust matter, with negligible thermodynamic pressure, we can take p = 0 in the gravitational field equations. Then by introducing a setn of dimensionless variables (θ, Σ,τ, h, v0 ) dep √
fined as ρ = 8λ/α2 θ, A = 8π/3Σ, t = α/ 8λ τ, (74) The deceleration parameter of the magnetic type superconducting dark energy model can be represented as H= (75) √ 
o 8λ/α h, V0 = 8λ/α2 V0 , and by denoting σ = πα2 /2, the system of Eqs. (71)-(73) can be written in a dimensionless form as  2 Σ2 1 dΣ 2 2 + σ 2 + v0 , (78) 3h = θ − θ + 2 a dτ a (76) In the case of a dust Universe, with p = 0, from the conservation equation Eq. (70) we obtain for the time derivative of the energy density ρ the equation (72) (73)  
ρ + 3p − α2 /2λ ρ2 + (3/4π) Ȧ2 /a2 − 2V0 i.   q= h 2 ρ − ((α2 /8λ) ρ2 ) + (3/8π) Ȧ2 /a2 + (3λ/2) (A2 /a2 ) + V0 while the parameter of the equation of state of the dark energy is given by (71) 2 dh 1 1 + 3h2 = θ2 − dτ 3 a2  4 1 dh = θ(2θ − 1) − 2 dτ 3 a2 dΣ dτ  2 dΣ dτ + 2 σ Σ2 + v0 , 3 a2 (79) 2 Σ2 − σ 2, 3 a (80) d2 Σ dΣ +h + σΣ = 0, dτ 2 dτ s 2 (81) Σ2 + v0 . a2 (82) The deceleration parameter and the parameter of the equation of state of the dark energy of the Universe filled 1 dz = −(1 + z) √ dτ 3 θ − θ2 + 1 a2  dΣ dτ +σ 12 with magnetic type superconducting dark energy are obtained as 0.030 0.025 wDE = − ′2 2 2 2 −θ + Σ /a + σΣ /a + v0 , θ2 − Σ′2 /3a2 + σΣ2 /3a2 + v0 θ0 , a3 0.015 0.010 (84) 0.005 where a prime denotes the derivative with respect to the dimensionless time τ . By taking the derivative with respect to τ of Eq. (78), and with the use of Eqs. (80) and (81) we obtain for the time variation of the matter density the equation θ= 0.020 (85) where θ0 is an arbitrary constant of integration. The variations with respect of the redshift z ∈ [0, 3] of the Hubble function, of the matter energy density θ, and of the deceleration parameter q of the Universe filled with magnetic type superconducting dark energy, obtained by numerically integrating Eqs. (79), (81), (82), and (85), are presented, for a fixed value of σ = 0.0001, and for different values of v0 , in Figs. 7-13. The initial conditions use to integrate the system of cosmological evolution equations are θ(0) = θ0 = 0.25, a(0) = 1, Σ(0) = 0.01, z(0) = 5, and Σ′ (0) = 0.01, respectively. 0.000 0 1 2 3 4 5 z FIG. 14: Matter energy density of the dust Universe in the presence of magnetic type superconducting dark energy as a function of the redshift for σ = 0.0001, and for different values of v0 : v0 = 0.001 (solid curve), v0 = 0.002 (dotted curve), v0 = 0.003 (short dashed curve), v0 = 0.004 (dashed curve), and v0 = 0.005 (long dashed curve), respectively. 0.4 0.2 0.0 qHzL 2 (83) ΘHzL θ − 4θ2 + 2Σ′2 /a2 − 2v0 , q= 2 (θ − θ2 + Σ′2 /a2 + σΣ2 /a2 + v0 ) -0.2 -0.4 -0.6 -0.8 0.10 -1.0 0 0.08 1 2 3 4 5 hHzL z 0.06 0.04 0.02 0 1 2 3 4 5 FIG. 15: Redshift evolution of the deceleration parameter of the dust Universe in the presence of magnetic type superconducting dark energy for σ = 0.0001, and for different values of v0 : v0 = 0.001 (solid curve), v0 = 0.002 (dotted curve), v0 = 0.003 (short dashed curve), v0 = 0.004 (dashed curve), and v0 = 0.005 (long dashed curve), respectively. z FIG. 13: The Hubble function of the Universe in the presence of magnetic type superconducting dark energy as a function of redshift for σ = 0.0001, and for different values of v0 : v0 = 0.001 (solid curve), v0 = 0.002 (dotted curve), v0 = 0.003 (short dashed curve), v0 = 0.004 (dashed curve), and v0 = 0.005 (long dashed curve), respectively. As one can see from Fig. 13, the Hubble function of the Universe filled with magnetic type superconducting dark energy is a monotonically increasing function of z (time decreasing function), indicating an expansionary evolution. The matter energy density θ, represented in Fig. 14, monotonically increases with the redshift, and tends to zero in the limit of small z. Its dynamics is basically independent on the adopted numerical values of the parameters σ and v0 . The dust magnetic Universe starts from a decelerating state at z = 5, with positive values of the deceleration parameter q > 0, shown in Fig. 15. The cosmological evolution is generally decelerating for 2 ≤ z ≤ 5, with q reaching the value zero at z ≈ 2. Then the Universe begins to accelerate, with q < 0, and in the large time (small z) limit we have limz→0 q(z) = −1. Thus, in the final stages of evolution of the Universe fillet with magnetic type superconducting dark energy the cosmological expansion is of de Sitter type, with the dark energy driving the Universe’s expansion. The parameter of the dark energy equation of state, represented in Fig. 16, is smaller than zero in the entire redshift range 0 ≤ z ≤ 5, and it tends to -1 in the limit of small redshifts. 13 0.0 system the second law of thermodynamics, in its most general form, is given by [23] -0.2

d d dQ ρ + p d ρa3 + p a3 = + na3 , dt dt dt n dt w DE H z L -0.4 -0.6 -0.8 -1.0 0 1 2 3 4 5 z FIG. 16: Redshift evolution of the parameter of the dark energy equation of state of the dust Universe in the presence of magnetic type superconducting dark energy for σ = 0.0001, and for different values of v0 : v0 = 0.001 (solid curve), v0 = 0.002 (dotted curve), v0 = 0.003 (short dashed curve), v0 = 0.004 (dashed curve), and v0 = 0.005 (long dashed curve), respectively. IV. THERMODYNAMIC INTERPRETATION OF THE SUPERCONDUCTING DARK ENERGY MODELS In the present Section we analyze the physical interpretation of the superconducting dark energy model by adopting the point of view of the thermodynamics of the matter creation irreversible processes [22]-[25]. As we have already seen, the energy conservation equation of the superconducting dark energy models, Eq. (21), contain, as compared to the standard adiabatic conservation equation, an extra term, which can be interpreted thermodynamically as a matter creation rate. According to irreversible thermodynamics, matter creation represents an entropy source, generating an entropy flux, and thus modifying the temperature evolution of the considered gravitational system. On the other hand, due to our choice of the geometry of the Universe, all the nondiagonal components of the total energy–momentum tensor of the superconducting dark energy model are equal (total) to zero, so that Tµν = 0, µ 6= ν. In particular, from the point of view of the thermodynamics of the irreversible processes and open systems, this condition implies the impossibility of heat transfer in the Friedmann– Robertson–Walker models of superconducting dark en(total) ≡ 0, i = 1, 2, 3 must ergy, since the condition T0i always hold. A. Matter creation rates and the creation pressure To analyze the thermodynamical implications of the superconducting dark energy models at the cosmological scale we start with an open system containing N particles in a volume V = a3 , and characterized by an energy density ρ and a thermodynamic pressure p. For such a (86) where dQ is the heat received by the system during time dt, and n = N/V is the particle number density, respectively. Due to our choice of the geometry of the Universe, in a homogeneous and isotropic system filled with superconducting dark energy only adiabatic transformations, defined by the condition dQ = 0, are possible. Therefore in the following we ignore proper heat transfer processes in the superconductor type cosmological system. However, under the assumption of adiabatic transformations, Eq. (86), representing the general formulation of the second law
of thermodynamics, contains the term [(ρ+ p)/n]d na3 /dt, which explicitly takes into account the variation of the number of particles in a given volume. Hence, in the general thermodynamic approach of open systems, even for the case of adiabatic transformations with dQ = 0, there is a ”heat” (internal energy), received/lost by the system, which is entirely due to the change in the particle number n. From the cosmological perspective of the superconducting dark energy models, the change in the particle number is due to the transfer of energy from dark energy to matter. Thus in this class of cosmological models matter creation acts as a source of internal energy, as well as of entropy. For adiabatic transformations dQ/dt = 0, Eq. (86) can be written in an equivalent form as ρ̇ + 3(ρ + p)H = ρ+p (ṅ + 3Hn) . n (87) Therefore, from the point of view of the thermodynamics of open systems, Eq. (21), giving the energy conservation equation in the superconducting dark energy models, can be interpreted as describing particle creation in an homogeneous and isotropic geometry, with the time variation of the particle number obtained from the equation ṅ + 3nH = Γn, (88) where the particle creation rate Γ is defined as ( 1 α Γ = − uν ∇µ [j µ (Aν − ∇ν φ)] + ρ+p 2 
α d  β j (Aβ − ∇β φ) − ∂φ V A2 , φ uν Aν − 2 ds )
(89) 2∂A2 V A2 , φ uν Aα ∇α Aν . Therefore the energy conservation equation in the superconducting dark energy model can be written in the alternative form ρ̇ + 3(ρ + p)H = (ρ + p)Γ. (90) 14 As shown initially in [23], for adiabatic transformations Eq. (86), describing irreversible particle creation in an open thermodynamic systems, can be formulated as an effective energy conservation equation of the form
d d ρa3 + (p + pc ) a3 = 0, dt dt (91) which can be written in an equivalent form as, ρ̇ + 3 (ρ + p + pc ) H = 0, Therefore in the superconducting dark energy model the creation pressure can be obtained as ( α 1 − uν ∇µ [j µ (Aν − ∇ν φ)] + pc = − 3H 2 
α d  β j (Aβ − ∇β φ) − ∂φ V A2 , φ uν Aν − 2 ds )
ν α 2 (94) 2∂A2 V A , φ u A ∇α Aν . Particle creation rates and creation pressure in the electric type superconducting dark energy model dS = de S + di S, ρ̇ + 3(ρ + p)H = α2 ρ (ρ̇ + 3Hρ) . 4λ (95) From Eq. (95) it follows that for p = 0 the matter energy is conserved, ρ̇ + 3(ρ + p)H = 0, and there is no particle creation from the superconducting dark energy. However, for p 6= 0, matter and energy transfer processes take place in the presence of the superconducting electric type dark energy, with the particle creation rate Γ given by Γ= α2 ρ (ρ̇ + 3Hρ) . 4λ ρ + p (96) The creation pressure for this model can be obtained as pc = − α2 ρ (ρ̇ + 3Hρ) . 12λ H (97) (98) where we assume that di S > 0. Both the entropy flow and the entropy production rate in the superconducting dark energy model can be evaluated by starting from the total differential of the entropy, given by [23],


(99) T d s̄a3 = d ρa3 + pda3 − µd na3 , where T is the temperature of the open thermodynamic system with superconducting particle creation, s̄ = S/a3 is the entropy per unit volume, and µ is the chemical potential, defined in the usual way as µn = (ρ + p) − T s̄. (100) For closed systems and adiabatic transformations dS = 0 and di S = 0. However, in the presence of matter creation there is a non-zero contribution to the entropy. For homogeneous systems the entropy flow term de S cancels, so that de S = 0. But matter creation from superconducting dark energy acts as a source for entropy production, with the corresponding entropy time variation obtained as [23] T As an example of the thermodynamic description of the superconducting dark energy models we consider the electric type superconducting dark energy case, for which the energy conservation equation is given by Eq. (42), can be formulated as Entropy and temperature evolution In order to formulate the second law of thermodynamics for open systems, and to apply it to the superconducting dark energy model, we must decompose the entropy change in the cosmological fluid into two components: the entropy flow term de S, and the entropy creation term di S. Hence the total entropy S of an open thermodynamic system can be written as [22, 23] (92) where we have introduced the term pc , called the creation pressure, and which is defined as [23]
ρ+p ρ + p d na3 =− (ṅ + 3nH) = pc = − 3 n da 3nH ρ+p Γ − . (93) 3 H 1. B.

di S d dS ρ+p d na3 − µ na3 = = T = dt dt n dt dt
s̄ d 3 (101) na ≥ 0, T n dt From Eq. (101) we obtain for the time variation of the entropy the equation dS S = (ṅ + 3Hn) = ΓS ≥ 0, dt n (102) giving for the entropy increase due to particle creation the expression S(t) = S0 e Rt 0 Γ(t′ )dt′ , (103) where S0 = S(0) is a constant. With the use of Eq. (96), we obtain for the entropy production in the superconducting dark energy models the equation ( 1 dS α 1 − uν ∇µ [j µ (Aν − ∇ν φ)] + = S dt ρ+p 2 
α d  β j (Aβ − ∇β φ) − ∂φ V A2 , φ uν Aν − 2 ds )
(104) 2∂A2 V A2 , φ uν Aα ∇α Aν ≥ 0. 15 Equivalently, the above equation can be written as 1 dS α2 ρ = (ρ̇ + 3Hρ) ≥ 0. S dt 4λ ρ + p (105) An important thermodynamic quantity, the entropy flux vector S µ of the particles created from the superconducting dark energy, is defined according to [24] S µ = nσuµ , ρ+p dn, n ρ+p − T σ, n ∂ρ ∂ρ Ṫ + 3(ρ + p)H = (ρ + p)Γ. ṅ + ∂n ∂T ∂ρ ρ + p T ∂p = − , ∂n n n ∂T ρ+p ṅ = 0, n (115) we obtain for the temperature evolution of the newly created particles in the superconducting dark energy model the expression (108) ṅ Ṫ = c2s = c2s (Γ − 3H) . T n (109) where we have taken into account the important relation nT σ̇ = ρ̇ − (114) With the use of the general thermodynamic relation [24] we obtain ∇µ S µ = (ṅ + 3nH) σ + nuµ ∇µ σ =   ρ+p 1 (ṅ + 3Hn) −µ , T n (113) Therefore the energy conservation equation Eq. (90) can be written in the expanded form (107) and by using the definition of the chemical potential µ of the superconducting thermodynamic system as given by µ= ρ = ρ(n, T ), p = p(n, T ). (106) where σ = S/N is the specific entropy per particle. The entropy flux vector S µ must satisfy during the entire cosmological evolution the second law of thermodynamics, which requires that the constraint ∇µ S µ ≥ 0 be satisfied for all times. By taking into account the fundamental Gibbs relation [24], nT dσ = dρ − thermodynamic variables, the particle number density n, and the temperatures T , respectively. Hence the energy density ρ and the thermodynamic pressure p can be obtained, in terms of the particle number n and temperature T , by using the standard form of the equilibrium equations of state of the matter created by the superconducting dark energy, where the speed of sound cs is defined as c2s = ∂p/∂ρ. If the matter newly created from the superconducting dark energy satisfies a barotropic equation of state p = (γ − 1) ρ, 1 ≤ γ ≤ 2, the matter temperature evolution can be obtained as T = T0 nγ−1 . (110) which follows immediately from Eq. (87). With the use of Eq. (96) we obtain for the entropy production rate due to the particle creation processes in the superconducting dark energy model the expression ( α n µ − uν ∇µ [j µ (Aν − ∇ν φ)] + ∇µ S = (ρ + p) T 2 
α d  β j (Aβ − ∇β φ) − ∂φ V A2 , φ uν Aν − 2 ds ) 
ρ+p 2 ν α 2∂A2 V A , φ u A ∇α Aν −µ . n (111) The entropy production rate via superconducting particle creation processes given by Eq. (111) can be written in a simpler form as   α2 1 ρ+p ρ ∇µ S µ = n −µ (ρ̇ + 3Hρ) . (112) 4λ T n ρ+p In the general case the thermodynamic state of a perfect comoving fluid is described by only two essential (116) V. (117) DISCUSSIONS AND FINAL REMARKS In the present paper we have considered an electromagnetic type dark energy model, in which the electromagnetic gauge invariance is spontaneously broken. The action for such a system must be invariant under gauge transformations, Aµ (x) → Aµ (x) + ∂µ Λ(x), ψn (x) → exp (iqn Λ(x)/~) ψn (x), where qn are the charges destroyed by the field ψn [17]. These phase changes lead to the formation of an ordered state. By writing all charged fields as a function of a scalar field φ(x), when the matter fields are integrated out we obtain a Lagrangian that is a gauge invariant functional of the fields Aµ and φ. Such a physical model can explain easily all the observed properties of superconductors [17]-[19]. Tentatively, we also propose it as a dark energy model with a broken electromagnetic gauge invariance. From a physical point of view the superconducting dark energy model is a two-field model, leading to a scalar-vector-tensor cosmological theory. It can also be viewed as a unified scalar - vector field dark energy model, in which the scalar field φ and the vector field Aµ appear in the gauge invariant combination Aµ − ∇µ φ. 16 Moreover, similarly the standard electrodynamic case, we have assumed the possible existence of a generalized coupling between the matter current j µ and the gauge invariant combination of the potentials Aµ − ∇µ φ. We have investigated the cosmological implications of this model, by restricting our analysis to the case of a homogeneous and isotropic geometry. We have considered two distinct classes of models, whose main properties are determined by the form of the electromagnetic potential Aµ . The first model corresponds to an electric type choice for the dark energy potential, with A0 the only non-zero component. For this case the general solution of the gravitational field equations was obtained numerically for the dust and the radiation filled Universe, respectively. In both cases we have assumed that the self-interaction potential of the electromagnetic type dark energy is a constant. In both cases in the long time limit the Universe ends in a decelerating phase. A similar result is obtained for the magnetic type dark energy model, in which the electromagnetic potential is restricted to the form Aµ = (0, A(t), A(t), A(t)). For this case we have investigated, by numerically solving the gravitational field equations, the zero thermodynamic pressure (dust) cosmological model, in the presence of a constant self-interaction potential. Similarly to the electric type dark energy model, the magnetic superconducting dark energy model drives the Universe, in the long time limit, in an accelerated, de Sitter type, expansionary phase. Due to the coupling between the dark energy potentials and the matter current, the matter energy-momentum tensor is not conserve in the present approach. We have interpreted, in the framework of the thermodynamics of open systems and irreversible processes [22]-[25], the nonconservation of the matter energy-momentum tensor as describing particle creation, and energy transfer from the superconducting dark energy to ordinary matter. We have explicitly obtained the particle creation rates, as well as the effective creation pressure generated by the irreversible transformation of the field energy into matter. The entropy production rate and the overall entropy evolution of the Universe was also obtained, with the total entropy being given by the exponential of the time integral of the particle production rate Γ. The possible observational study in a cosmological context of the irreversible matter–creation processes in the homogeneous and isotropic flat Friedmann–Robertson– Walker geometry in the superconducting dark energy models may represent one of the possibilities of consider- ing the viability of this dark energy model. However, in order to confirm the validity of the superconducting dark energy model developed in the present paper, it is necessary to carefully consider a much wider range of cosmological and astrophysical tests for this type of models. In particular, an essential test of the superconducting dark energy model would be the investigation of the classical macroscopic predictions of the model in large-scale structure formation with linear perturbations, and with the consideration of the Newtonian limit for small scales. Supernovae observations fitting, and the study of the effects of the matter creation on the Cosmic Microwave Background anisotropies could lead to other important tests and parameter constraints of the model. Essentially the superconducting dark energy model introduced in the present paper is a simple toy model, whose main goal is to stimulate the study of alternative or more general electromagnetic type dark energy models. The superconducting dark energy model introduced in the present paper leads to the possibility that matter creation, associated with matter current - dark energy electromagnetic potential coupling may also happen in the present - day universe, as initially considered by Dirac [28]. 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(A1) 4π 4 Hence for the divergence of the electromagnetic component we find first   1 1 αβ (em)µ µα µα . ∇µ Tν = F ∇ν Fαβ − ∇µ Fνα F − Fνα ∇µ F 4π 2 (A2) By taking into account that (1/4π) ∇µ F µα = J α , and ∇ν Fαβ = −∇α Fβν − ∇β Fνα , it follows that ∇µ Tν(em)µ = 1 4π 1 1 − F αβ ∇α Fβν − F αβ ∇β Fνα − 2 2 18 F µα ∇µ Fνα ! − Fνα J α . (A3) Then for the divergence of the matter energymomentum tensor we obtain α µ (A6) and The terms in the bracket vanish, and therefore we find " α (em)µ ∇µ Tν = Fνα λg αβ (Aβ − ∇β φ) + j α − 2 #
(A4) 2∂A2 V A2 , φ Aα . ∇µ Tνµ µ λFµν (Aµ − ∇µ φ) , µ α β j (∇ν Aβ − ∇ν ∇β φ) = 2 α µ α j (∇µ Aν − ∇µ ∇ν φ) + j µ Fµν , (A7) 2 2 αj µ (∇µ Aν − ∇µ ∇ν φ) − respectively. With the use of the evolution equation of the scalar field we find the relation, λ (∇µ Aµ − ∇µ ∇µ φ) (Aν − ∇ν φ) =
α ∂φ V A2 , φ (Aν − ∇ν φ) − ∇µ j µ (Aν − ∇ν φ) . (A8) 2 − Fνα J + λ (∇µ A − ∇µ ∇ φ) (Aν − ∇ν φ) + λ (A − ∇µ φ) Fµν + α∇µ j µ (Aν − ∇ν φ) + α α α µ j (∇µ Aν − ∇µ ∇ν φ) + j µ Fµν − ∇ν j β (Aβ − ∇β φ) + By substituting the above relation and the expression of 2 2

2 the divergence of the electromagnetic part of the energy∂φ V A2 , φ ∇ν φ + 2Aα ∂φ V A2 , φ ∇ν Aα = 0, (A5) momentum tensor in Eq. (A8), we finally obtain where we have used the identities λ λ (Aµ − ∇µ φ) (∇µ Aν − ∇µ ∇ν φ) − (∇ν Aα − ∇ν ∇α φ) × 2 λ α (Aα − ∇α φ) − (A − ∇α φ) (∇ν Aα − ∇ν ∇α φ) = 2 α α ∇µ [j µ (Aν − ∇ν φ)] − ∇ν j β (Aβ − ∇β φ) + 2

2 ∂φ V A2 , φ Aν + 2∂A2 V A2 , φ Aα ∇α Aν = 0. (A9) ∇µ Tνµ +