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Accelerated FRW Solutions in Chern-Simons Gravity Juan Crisóstomo, Fernando Gomez, and Patricio Salgado ∗ † ‡ Departamento de F´ısica, Universidad de Concepción, Casilla 160-C, Concepción, Chile Cristian Quinzacara § Departamento de F´ısica, Universidad de Concepción, 4 Casilla 160-C, Concepción, Chile and 1 0 Facultad de Ingenier´ıa y Tecnolog´ıa, Universidad San Sebastián, 2 g Campus Las Tres Pascualas, Lientur 1457, Concepción, Chile u A 9 Mauricio Cataldo¶ 1 Departamento de F´ısica, Universidad del B´ıo-B´ıo, Casilla 5-C, Concepción, Chile. ] c q - Sergio del Campo∗∗ r g Instituto de F´ısica, Pontificia Universidad Católica de Valpara´ıso, [ 3 Av. Universidad 300, Campus Curauma, Valpara´ıso, Chile. v 8 (Dated: August 20, 2014) 2 1 2 . 1 0 4 1 : v i X r a 1 Abstract We consider a five-dimensional Einstein-Chern-Simons action which is composed of a gravitational sector and a sector of matter, where the gravitational sector is given by a Chern-Simons gravity action instead of the Einstein-Hilbert action and where the matter sector is given by the so called perfect fluid. It is shown that (i) the Einstein-Chern-Simons (EChS) field equations subject to suitable conditions can be written in a similar way to the Einstein-Maxwell field equations; (ii) these equations have solutions that describe accelerated expansion for the three possible cosmological models of the universe, namely, spherical expansion, flat expansion and hyperbolic expansion when α, a parameter of theory, is greater than zero. This result allow us to conjeture that this solutions are compatible with the era of Dark Energy and that the energy-momentum tensor for the field ha, a bosonic gauge field from the Chern-Simons gravity action, corresponds to a form of positive cosmological constant. It is also shown that the EChS field equations have solutions compatible with the era of matter: (i) In the case of an open universe, the solutions correspond to an accelerated expansion (α > 0) with a minimum scale factor at initial time that, when the time goes to infinity, the scale factor behaves as a hyperbolic sine function. (ii) In the case of a flat universe, the solutions describing an accelerated expansion whose scale factor behaves as a exponencial function when time grows. (iii) In the case of a closed universe it is found only one solution for a universe in expansion, which behaves as a hyperbolic cosine function when time grows. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] ∗∗ [email protected] 2 I. INTRODUCTION Some time ago was shown that the standard, five-dimensional General Relativity can be obtained fromChern-Simons gravity theory for a certain Lie algebra B[1], whose generators J ,P ,Z ,Z satisfy the commutation relationships ab a ab a { } [J ,J ] = η J η J +η J η J , ab cd ad bc ac bd bc ad bd ac − − [P ,J ] = η P η P , a bc ab c ac b − [J ,Z ] = η Z η Z +η Z η Z , ab cd ad bc ac bd bc ad bd ac − − [Z ,J ] = η Z η Z , a bc ab c ac b − [P ,P ] = Z , a b ab [P ,Z ] = η Z η Z . a bc ab c ac b − This algebra was obtained from the anti de Sitter (AdS) algebra and a particular semigroup S by means of the S-expansion procedure introduced in Refs. [2], [3]. In order to write down a Chern–Simons lagrangian for the B algebra, we start from the one-form gauge connection 1 1 1 1 A = ωabJ + eaP + kabZ + haZ , (1) ab a ab a 2 l 2 l and the two-form curvature 1 1 1 1 F = RabJ + TaP + D kab + eaeb Z 2 ab l a 2 ω l2 ab (cid:18) (cid:19) 1 + D ha +ka eb Z . (2) l ω b a (cid:0) (cid:1) Consistency with the dual procedure of S-expansion in terms of the Maurer-Cartan forms [3] demands that ha inherits units of length from the fu¨nfbein; that is why it is necessary to introduce the l parameter again, this time associated with ha. It is interesting to observe that J are still Lorentz generators, but P are no longer ab a AdS boosts; in fact, [P ,P ] = Z . However, ea still transforms as a vector under Lorentz a b ab transformations, as it must be in order to recover gravity in this scheme. A Chern-Simons lagrangian in d = 5 dimensions is defined to be the following local function of a one-form gauge connection A: 1 1 L(5) (A) = k AF2 A3 F+ A5 , (3) ChS − 2 10 (cid:28) (cid:29) 3 where denotes a invariant tensor for the corresponding Lie algebra, F = dA+AA is h···i the corresponding the two-form curvature and k is a constant [4]. Using theorem VII.2 of Ref. [2], it is possible to show that the only non-vanishing com- ponents of a invariant tensor for the B algebra are given by 4l3 J J P = α ǫ , (4) h a1a2 a3a4 a5i 1 3 a1···a5 4l3 J J Z = α ǫ , h a1a2 a3a4 a5i 3 3 a1···a5 4l3 J Z P = α ǫ , h a1a2 a3a4 a5i 3 3 a1···a5 where α1 and α3 are arbitrary independient constants of dimensions [length]−3. Using the extended Cartan’s homotopy formula as in Ref. [5], and integrating by parts, it is possible to write down the Chern-Simons Lagrangian in five dimensions for the algebra B as L(5) = α l2ǫ eaRbcRde EChS 1 abcde 2 +α ǫ Rabecedee +2l2kabRcdTe +l2RabRcdhe 3 abcde 3 (cid:18) (cid:19) (4) +dB (5) EChS (4) where the suface term B is given by EChS 2 1 B(4) = α l2ǫ eaωbc dωde + ωd ωfe EChS 1 abcde 3 2 f (cid:18) (cid:19) 2 1 +α ǫ l2 haωbc +kabec dωde + ωd ωfe 3 abcde 3 2 f " (cid:18) (cid:19) (cid:0) (cid:1) 2 1 1 +l2kabωcd dee + ωd ee + eaebecωde (6) 3 2 f 6 # (cid:18) (cid:19) and where α , α are parameters of the theory, l is a coupling constant, Rab = dωab+ωa ωcb 1 3 c corresponds to the curvature 2-form in the first-order formalism related to the 1-form spin connection [4], [6], [7], and ea, ha and kab are others gauge fields presents in the theory [1]. From (5) we can see that the third term is a surface term and can be removed from this Lagrangian. So that, L(5) = α l2ε RabRcdee EChS 1 abcde 2 +α ǫ Rabecedee +2l2kabRcdTe +l2RabRcdhe (7) 3 abcde 3 (cid:18) (cid:19) 4 is the Einstein-Chern-Simons Lagrangian studied in Ref [1]. It should be noted the absence of kinetic terms for the fields ha and kab in equation (7). The term kinetic for the ha and kab fields are present in the surface term of the Lagrangian (5) given by (6). The Lagrangian (7) show that standard, five-dimensional General Relativity emerges as the l 0 limit of a CS theory for the generalized Poincaré algebra B. Here l is a length → scale, a coupling constant that characterizes different regimes within the theory. The B algebra, on the other hand, is constructed from the AdS algebra and a particular semigroup S by means of the S-expansion procedure. The field content induced by the B algebra includes the vielbein ea, the spin connection ωab and two extra bosonic fields ha and kab, which can be interpreted as boson fields coupled to the field curvature and the parameter l2 can be interpreted as a kind of coupling constant. Recently was found [8] that the standard five-dimensional FRW equations and some of their solutions can be obtained, in a certain limit, from the so-called Chern-Simons-FRW field equations, which are the cosmological field equations corresponding to a Chern-Simons gravity theory. It is the purpose of this paper to show that the Einstein-Chern-Simons (EChS) field equations, subject to (i) the torsion-free condition (Ta = 0) and (ii) the variation of the matter Lagrangian with respect to (w.r.t.) the spin connection is zero (δL /δωab = 0) can M be written in a similar way to the Einstein-Maxwell field equations. The interpretation of theha fieldasa perfect fluid allowus toshow that theEinstein-Chern-Simons fieldequations have an universe in accelerated expansion as a of their solutions. This paper is organized as follows: In Section II we briefly review the Einstein-Chern- Simons field equations. In Section III we study the Einstein-Chern-Simons field equations in the range of validity of general relativity. In Section V we consider accelerated solutions for Einstein-Chern-Simons field equations. We try to find solutions that describes accelerated expansion for cases of open universes, flat universes and closed universes. In Section VI we consider the consistency of the solutions with the ”Era of Matter”. A summary and an appendix conclude this work. 5 II. EINSTEIN-CHERN-SIMONS FIELD EQUATIONS In Ref. [8] was found that in the presence of matter the lagrangian is given by (5) L = L +κL (8) ChS M (5) where L is the five-dimensional Chern-Simons lagrangian given by (7), ChS L = L (ea,ha,ωab) is the matter Lagrangian and κ is a coupling constant related to M M the effective Newton’s constant. The variation of the lagrangian (8) w.r.t. the dynamical fields vielbein ea, spin connection ωab, ha and kab, leads to the following field equations ε 2α Rabeced +α l2RabRcd abcde 3 1 (cid:16) δL +2α l2D kabRcd = κ M, (9) 3 ω δee (cid:17) δL α l2ε RabRcd = κ M, (10) 3 abcde δhe δL 2α l2ε RcdTe = κ M, (11) 3 abcde δkab 2ε α l2RcdT e +α l2D kabTe abcde 1 3 ω (cid:16) +α ecedTe +α l2RcdD he 3 3 ω (cid:17) δL +2α ε l2Rcdke ef = κ M. (12) 3 abcde f δωab For simplicity, we will assume that the torsion vanishes (Ta = 0) and kab = 0. In this case the Eqs.(9 - 12) takes the form δL ε 2α Rabeced +α l2RabRcd = κ M, (13) abcde 3 1 δee (cid:0) α l2ε RabRc(cid:1)d = κδLM, (14) 3 abcde δhe δL M = 0 (15) δkab δL 2α l2ε RcdD he = κ M. (16) 3 abcde ω δωab 6 This field equations system can be written in the form δL δL ε Rabeced = 4κ M +α M , (17) abcde 5 δee δhe (cid:18) (cid:19) δL l2ε RabRcd = 8κ M, (18) abcde 5 δhe δL l2ε RcdD he = 4κ M (19) abcde ω 5 δωab where we introduce κ = κ/8α and α = α /α . 5 3 1 3 − The field equation (9) contains three terms. The first one, proportional to the Einstein tensor. The second one corresponds to a quadratic term in the curvature, and a third one, a term that describes the dynamics of the field kab. Since we asume kab = 0 the last term in left side of Eq. (9) vanishes. InordertowritethisfieldequationmanneranalogoustoEinstein’s equations, onechooses to leave the term proportional to the Einstein tensor on the left side of Eq. (9) κ δL α ǫ Rbcedee = M 1 l2ǫ RbcRde abcde 2α δea − 2α abcde 3 3 and using the Eq. (14) we obtain Eq. (17). This result allows us to interpret δL /δha as the energy momentum tensor for a second M type of matter, not ordinary. Henceforth we will say that δL /δha corresponds to the M energy-momentum tensor for the field ha. The equation of motion for the ha-field is given by Eq.(19). The condition δL /δωab = 0 M (usual in gravity theories), imposed for consistency with the condition Ta = 0, leads to the equation of motion (22) for the ha-field . This means that ha-field is governed by the following field equations δL δL ε Rabeced = 4κ M +α M , (20) abcde 5 δee δhe (cid:18) (cid:19) l2 δL ε RabRcd = M, (21) 8κ abcde δhe 5 ε RcdD he = 0. (22) abcde ω This means that the Einstein-Chern-Simons field equations, subject to the conditions Ta = 0, kab = 0 and δL /δωab = 0, can be re-written in a way similar to the Einstein- M Maxwell field equations. 7 From (20-22) we can see that if L = 0, then in five dimensions there is no solution of M Schwarzschild type [1], [9]. III. EINSTEIN-CHERN-SIMONS EQUATIONS IN THE RANGE OF VALIDITY OF GENERAL RELATIVITY From (20-21) we can see that general relativity is valid when (i) the curvature Rab takes values not excessively large (ii) the parameter l takes small values (l 0) [1]; (iii) the −→ constant α takes values not excessively large. In fact, in this case we have that (21) takes the form δL M 0. (23) δhe ≈ Introducing (23) into (20) we obtain the Einstein’s field equation δL ε Rabeced 4κ M. (24) abcde 5 ≈ δee If Rab is not large then δL /δea is also not large. This means that General Relativity M can be seen as a low energy limit of Einstein-Chern-Simons gravity. So that, in the range of validity of the General Relativity, the equations (20-22) are given by δL ε Rabeced = 4κ M, (25) abcde 5 δee ε RcdD he = 0. (26) abcde ω On the another hand, if Rab is large enough, so that when it is multiplied by l2 (which is very small) will have a non-negligible results, then we will find that δL /δha is not M negligible. This means that, in this case, we must consider the entire system of equations (20-22). IV. EINSTEIN-CHERN-SIMONS FIELD EQUATIONS FOR A FRIEDMANN- ROBERTSON-WALKER-LIKE SPACETIME Theshapeofthefieldea isobtainedfromoftheapplicationofthecosmologicalprincipleto the metric tensor of spacetime: it is considered a splitting of the 5D-manifoldin a maximally symmetric four-dimensional manifoldandonetemporaldimension (M = R Σ ). This leads 4 × 8 to five dimensional Friedmann-Robertson-Walker (FRW) metric. So that, the vielbein can be chosen like in [8]: e0 = dt, a(t) e1 = dr, √1 kr2 − e2 = a(t)rdθ , 2 e3 = a(t)rsinθ dθ , 2 3 e4 = a(t)rsinθ sinθ dθ (27) 2 3 4 where a(t) is the scale factor of the universe and k is the sign of the curvature of space (Σ ): 4 (i) +1 for a closed space (S4), (ii) 0 for a flat space (E4) and (iii) 1 for an open space − (hyperbolic). The application of the cosmological principle to the metric tensor of the spacetime also constrains the shape of the field ha (see for example [8]). A detailed discussion can be also found in Ref. [10]. The bosonic field ha is given by h0 = h(0)dt = h(0)e0, a(t) h1 = h(t) dr = h(t)e1, √1 kr2 − h2 = h(t)a(t)rdθ = h(t)e2, 2 h3 = h(t)a(t)rsinθ dθ = h(t)e3, 2 3 h4 = h(t)a(t)rsinθ sinθ dθ = h(t)e4 (28) 2 3 4 where h(0) is a constant and h(t) is a function of time t that must be determined. Sub- stituting (28) into Eq. (22) we obtain the explicit form of the equations of motion for the ha-field, which will be displayed in Eq.(39). In accordance with the equation (20), we will consider a fluid composed of two perfect fluids, thefirst onerelatedtoordinaryenergy-momentum tensor (T δLM)andthesecond µν ∼ δea one related to field ha (T(h) δLM). The energy-momentum tensors in the comoving frame, µν ∼ δha are given by T = diag(ρ,p,p,p,p), (29) µν T(h) = diag ρ(h),p(h),p(h),p(h),p(h) , (30) µν (cid:16) (cid:17) 9 where ρ is the matter density and p is the pressure of fluid. Then, the energy-momentum tensor for the composed fluid is T˜ = T +αT(h) (31) µν µν µν = diag ρ+αρ(h),p+αp(h), (cid:16) p+αp(h),p+αp(h),p+αp(h) (32) (cid:17) = diag(ρ˜,p˜,p˜,p˜,p˜). (33) In the torsion-free case, the energy momentum tensor of ordinary matter satisfies a conservation equation and the Einstein tensor has also zero divergence. In this case the energy momentum tensor for the non-ordinary matter must also satisfy a conservation equation. In fact, from Eq. (20) we find Tµ = 0 , T(h)µ = 0 (34) ∇µ ν ∇µ ν Introducing (27 - 33) into eqs. (20 - 22) we find the following field equations (see Ref. [8] and Appendix A) a˙2 +k 6 = κ ρ˜, (35) a2 5 (cid:18) (cid:19) a¨ a˙2 +k 3 + = κ p˜, (36) a a2 − 5 (cid:20) (cid:18) (cid:19)(cid:21) 3l2 a˙2 +k 2 = ρ(h), (37) κ a2 5 (cid:18) (cid:19) 3l2a¨ a˙2 +k = p(h), (38) κ a a2 − 5 (cid:18) (cid:19) a˙2 +k a˙ ˙ (h h(0)) +h = 0. (39) a2 − a (cid:18) (cid:19)(cid:20) (cid:21) We should note that equation (35) was studied in Ref. [11] in the context of inflationary cosmology . The Equations (35) and (36) are very similar to the Friedmann equations in five dimensions. However now ρ and p are subject to restrictions imposed by the remaining equations. 10