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Bianchi II with time varying constants. Self-similar approach. José Antonio Belinchoń Dept. F´ısica. ETS Arquitectura. UPM. Av. Juan de Herrera 4. Madrid 28040. Espanã (Dated: January 24, 2009) We study a perfect fluid Bianchi II models with time varying constants under the self-similarity approach. In the first of the studied model, we consider that only vary G and Λ. The obtained solution is more general that the obtained one for the classical solution since it is valid for an equation of state ω ∈ (−1,∞) while in the classical solution ω ∈ (−1/3,1). Taking into account thecurrentobservations, weconcludethat Gmustbeagrowing timefunctionwhile Λisapositive decreasingfunction. Inthesecondofthestudiedmodelsweconsideravariablespeedoflight(VSL). Weobtainasimilarsolutionasinthefirstmodelarrivingtotheconclusionsthatcmustbeagrowing time function if Λ is a positive decreasing function. 9 0 I. INTRODUCTION. is intrinsic to the underlying model, and what implica- 0 tions this has on our understanding of cosmology. For 2 this reasonBianchimodels areimportantinthe study of n SincethepioneeringworkofDirac([1]),whoproposed, anisotropies. a motivated by the occurrence of large numbers in Uni- Recently , the cosmological implications of a variable J verse,atheorywithatimevariablegravitationalcoupling speed of light during the early evolution of the Universe 4 constant G, cosmological models with variable G and havebeenconsidered. Varyingspeedoflight(VSL)mod- 2 nonvanishing and variable cosmological term, Λ, have been intensively investigated in the physical literature elsproposedbyMoffat([19])andAlbrechtandMagueijo ] ([20]), in which light was travelling faster in the early c (see for example [2]-[14]). periods of the existence of the Universe, might solve the q Inmoderncosmologicaltheories,thecosmologicalcon- same problems as inflation. Hence they could become - r stant remains a focal point of interest (see [15]-[18] for a valuable alternative explanation of the dynamics and g reviews of the problem). A wide range of observations evolutionofourUniverseandprovideanexplanationfor [ now compellingly suggest that the universe possesses a the problem of the variationof the physical“constants”. 1 non-zero cosmological constant. Some of the recent dis- Einstein’s field equations (EFE) for FRW spacetime in v cussions on the cosmological constant “problem” and the VSL theory have been solved by Barrow ([21] and 7 on cosmology with a time-varying cosmologicalconstant [22] for anisotropic models), who also obtained the rate 2 point out that in the absence of any interaction with ofvariationofthespeedoflightrequiredtosolvetheflat- 8 3 matter or radiation, the cosmological constant remains nessandcosmologicalconstantproblems(seeJ.Magueijo . a “constant”. However, in the presence of interactions ([23]) for a review of these theories). 1 with matter or radiation, a solution of Einstein equa- Thismodelisformulatedunderthe strongassumption 0 9 tions and the assumed equation of covariant conserva- that a c variable (where c stands for the speed of light) 0 tionofstress-energywithatime-varyingΛcanbefound. doesnotintroduceanycorrectionsintothecurvatureten- : This entails that energy has to be conserved by a de- sor, furthermore, such formulation does not verify the v crease in the energy density of the vacuum component covariance and the Lorentz invariance as well as the re- i X followed by a corresponding increase in the energy den- sultingfieldequationsdonotverifytheBianchiidentities r sity of matter or radiation Recent observations strongly either (see Bassett et al [24]). a favour a significant and a positive value of Λ with mag- Nevertheless, some authors (T. Harko and M. K. Mak nitude Λ(G~/c3) ≈ 10−123. These observations suggest [25], P.P. Avelino andC.J.A.P. Martins [26] and H. Sho- on accelerating expansion of the universe, q <0. jaieetal[27])haveproposedanewgeneralizationofGen- Our current understanding of the physical universe is eralRelativitywhichalsoallowsarbitrarychangesinthe anchoredon the analysisof expanding, isotropic and ho- speed of light, c, and the gravitational constant, G, but mogeneousmodelswithacosmologicalconstant,andlin- in such a way that variations in the speed of light intro- ear perturbations thereof. This model successfully ac- duces corrections to the curvature tensor in the Einstein counts for the late time universe, as is evidenced by the equations in the cosmological frame. This new formula- observation of large scale cosmic microwave background tion is both covariant and Lorentz invariant and as we observations. Parameter determination from the anal- will show the resulting field equations (FE) verify the ysis of CMB fluctuations appears to confirm this pic- Bianchi identities. ture. However, further analyses seem to suggests some The purpose of this paper is to study a perfect fluid inconsistency. In particular, its appears that the uni- Bianchi II model with time varying constants through verse could have a preferred direction. Followup anal- the self-similarity (SS) hypothesis (see foe example [34]- yses of various sets of WMAP data sets, with different [42]). The study of SS models is quite important since a techniques seem to lead to the same conclusion. It is large class of orthogonal spatially homogeneous models still unclear whether or not the directional preference are asymptotically self-similar at the initial singularity 2 and are approximated by exact perfect fluid or vacuum tively, the covariant derivative being with respect to the self-similar power law models. Exact self-similar power- Levi-Civita connection on M derived from g. The asso- lawmodelscanalsoapproximategeneralBianchimodels ciated Ricci and stress-energy tensors will be denoted in at intermediate stages of their evolution. This last point component form by R (≡Rc ) and T respectively. ij jcd ij is of particular importance in relating Bianchimodels to A Bianchi II space-time is a spatially homogeneous the real Universe. At the same time, self-similar solu- space-time which admits an abelian group of isometries tions can describe the behaviour of Bianchi models at G , acting on spacelike hypersurfaces, generated by the 3 late times i.e. as t→∞. spacelike KVs, We would like to emphasize that in this work we are more interesting in mathematical respects (exact solutions) than in studying strong physical consequences. ξ1 =K∂x, ξ2 =∂y, ξ3 =−Ky∂x+∂z, (1) Nevertheless we consider some observational data in or- and their algebra is: der to rule out some the obtained solutions. Therefore the paper is organized as follows. In the ξ ξ ξ 1 2 3 second section, we begin by defining the metric and cal- ξ 0 0 0 culating its Killing vectors as well as their algebra. We 1 [ξ ,ξ ]=Ckξ , C1 =1. ξ 0 0 ξ i j ij k 23 also calculate the four velocity and the quantities de- 2 1 rivedfromit, i.e. theexpansion,Hubble parameteretc... ξ3 0 −ξ1 0 In section three, we review some basic ideas about self- In synchronous co-ordinates the metric is: similarity. Once we have clarified some concepts then we calculate the homothetic vector field as well as the ds2 =−c2dt2+a2(t)dx2+ b2(t)+K2z2a2(t) dy2+ constrains for the scale factors. In section fourth we review the classicalsolution i.e. a perfect fluid model with +2Ka2(t)zdxdy+d2(t)(cid:0)dz2 (cid:1) (2) “constant” constants. We show that the obtained solution belongs to the LRS Bianchi type II by calculating wherethemetricfunctionsa(t),b(t),d(t)arefunctionsof the fourth Killing vector field. We also calculate all the the time co-ordinate only and K ∈R. The introduction curvature invariants showing that the obtained solution of this constant is essential since if we set K = 1, as it is singular. These results will be valid for the rest of the is the usual way, then there is not SS solution for the studied models. We end this section showing that there outlined field equations. In this paper we are interested is not solution for the vacuum model. In section five, onlyinBianchiIIspace-times,henceallmetricfunctions we study a perfect fluid model with variable G and Λ. are assumed to be different and the dimension of the We compare the solution with the obtained one in the groupofisometriesactingonthe spacelikehypersurfaces above section showing that we have been able to relax is three. the constrains obtained for the classical model. We rule Once we have defined the metric and we know which outsomeoftheobtainedsolutiontakingintoaccountthe areitskillingvectors,thenwecalculatethefourvelocity. current observations which suggest us that Λ0 > 0 and It must verify, Lξiui = 0, so we may define the four q <0.Insectionsix,we study aperfectfluid modelwith velocity as follows: variablespeedoflight(VSL). We outline the fieldequa- 1 tions taking into account the effects of a c-var into the ui = ,0,0,0 , (3) c curvaturetensor. This pointisessentialinthis approach (cid:18) (cid:19) as we have showed in a previous paper ([28]). We calcu- in such a way that it is verified, g(ui,ui)=−1. late the homothetic vector field for the new metric and From the definition of the 4-velocity we find that: we show the new constrains for the scale factors. If we assume a particular solution for c(t) then we show that 1 a′ b′ d′ 1 it may be a decreasing time function. Only taking into θ =ui = + + = H, ;i c a b d c account observational restriction we are able to rule out (cid:18) (cid:19) some of the obtained solutions as in the above section. H =H1+H2+H3, We end summarizing the conclusions in the last section. d 1 q = −1, dt H (cid:18) (cid:19) 1 II. THE METRIC. σ2 = H2+H2+H2−H H −H H −H H . 3c2 1 2 3 1 2 1 3 2 3 (cid:0) (cid:1) Throughout the paper M will denote the usual III. SELF-SIMILARITY. smooth(connected,Hausdorff,4-dimensional)spacetime manifold with smooth Lorentz metric g of signature (−,+,+,+)(see for example [33]). Thus M is paracom- In general relativity, the term self-similarity can be pact. A comma, semi-colon and the symbol L denote used in two ways. One is for the properties of space- the usual partial, covariant and Lie derivative, respec- times, the other is for the properties of matter fields. 3 Thesearenotequivalentingeneral. Theself-similarityin with a ,a ,a ∈ R, in such a way, that the constants a 1 2 3 i generalrelativity was defined for the first time by Cahill must verify the following restriction: and Taub (see [34], and for general reviews [? ]-[40]). a +a −a =1. (8) Self-similarity is defined by the existence of a homoth- 2 3 1 etic vector V in the spacetime, which satisfies therefore the resulting homothetic vector field is: V =t∂ +(1−a )x∂ +(1−a )y∂ +(1−a )z∂ , (9) L g =2αg , (4) t 1 x 2 y 3 z V ij ij whichisthesameasthefoundoneintheBianchiImodel where gij is the metric tensor,LV denotes Lie differenti- (see for example [43]). ationalong V and α is a constant. This is a specialtype ofconformalKillingvectors. Thisself-similarityis called homothety. If α6=0, then it can be set to be unity by a IV. THE PERFECT FLUID MODEL. THE constantrescaling of V. If α=0, i.e. L g =0, then V CLASSICAL SOLUTION. V ij is a Killing vector. Homothety is a purely geometric property of space- Taking into account the field equations (FE) time so that the physical quantity does not necessarily 1 8πG exhibit self-similarity such as L Z = dZ, where d is a R − Rg = T −Λg , (10) V ij 2 ij c4 ij ij constant and Z is, for example, the pressure, the energy density and so on. From equation (4) it follows that where, in this section, we shall consider that Λ vanish, LVRijkl = 0, and hence LVRij = 0,and LVGij = 0. A and the energy-momentum tensor, Tij, is defined as fol- vector field V that satisfies the above equations is called lows: a curvature collineation, a Ricci collineation and a mat- T =(ρ+p)u u +pg , (11) ij i j ij ter collineation, respectively. It is noted that such equa- tionsdonotnecessarilymeanthatV isahomotheticvec- with the usual equation of state p = ωρ, ω ∈ R, as we tor. We consider the Einstein equations G = 8πGT , have discussed above. ij ij where T is the energy-momentum tensor. If the space- TheresultingFEforthemetric(2)withaperfectfluid ij time is homothetic, the energy-momentum tensor of the matter model (11) are: matter fields must satisfy L T = 0. For a perfect V ij fluid case, the energy-momentum tensor takes the form a′b′ a′d′ d′b′ K2a2c2 8πG + + − = ρ, (12) Tij = (p+ρ)uiuj +pgij,where p and ρ are the pressure a b a d d b 4 b2d2 c2 andtheenergydensity,respectively. Then,equations(4) b′′ d′′ d′b′ 3K2a2c2 8πG result in + + − =− ωρ, (13) b d d b 4 b2d2 c2 LVui =−αui, LVρ=−2αρ, LVp=−2αp. (5) d′′ + a′′ + a′d′ + K2a2c2 =−8πGωρ, (14) d a a d 4 b2d2 c2 As shown above, for a perfect fluid, the self-similarity b′′ a′′ a′b′ K2a2c2 8πG + + + =− ωρ, (15) of the spacetime and that of the physical quantity co- b a a b 4 b2d2 c2 incide. However, this fact does not necessarily hold for a′ b′ d′ more general matter fields. Thus the self-similar vari- ρ′+ρ(1+ω) + + =0. (16) a b d ablescanbedeterminedfromdimensionalconsiderations (cid:18) (cid:19) inthecaseofhomothety. Therefore,wecanconcludeho- Now, if we take into account the obtained SS restric- mothety as the general relativistic analogue of complete tions for the scale factors i.e. similarity. a(t)=a ta1, b(t)=b ta2, d(t)=d ta3, 0 0 0 From the constraints (5), we can show that if we con- where a ,a ,a ∈ R, must satisfy a +a −a = 1, we siderthebarotropicequationofstate,i.e.,p=f(ρ),then 1 2 3 2 3 1 get from eq. (16): theequationofstatemusthavetheformp=ωρ,whereω isa constant. Inthis paper,we wouldliketotry toshow ρ=ρ t−γ, (17) 0 how taking into account this class of hypothesis one is where γ =α(1+ω)=(a +a +a )(1+ω). Therefore able to find exact solutions to the field equations within 1 2 3 the system to solve is the following one: the framework of the time varying constants. From equation (4),we find the following homothetic 3K2 a (a −1)+a (a −1)+a a − =−Aω, (18) vector field for the BII metric (2): 2 2 3 3 2 3 4 K2 a′ b′ d′ a (a −1)+a a +a (a −1)+ =−Aω, (19) V =t∂ + 1−t x∂ + 1−t y∂ + 1−t z∂ , 3 3 1 3 1 1 4 t x y z a b d (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) K2 (6) a (a −1)+a a +a (a −1)+ =−Aω, (20) 2 2 1 2 1 1 4 with the following constrains for the scale factors: 2 a +a +a = , (21) a(t)=a ta1, b(t)=b ta2, d(t)=d ta3, (7) 1 2 3 1+ω 0 0 0 4 where we have set the constant A as follows With the obtained results we can see that K2 4a2−1 2 1 A=a a +a a +a a − , H = = , 1 2 1 3 3 2 4 t ω+1 t (cid:18) (cid:19) 4a 3+ω and a +a −a =1. q =− 2 =− <0, ∀ω ∈(−1/3,1), 2 3 1 4a −1 2 The obtained solution is the following one: 2 while the shear behaves as: 1−ω ω+3 a = , a =a = , 1 2(ω+1) 2 3 4(ω+1) (a −1)2 3ω+1 2 1 σ2 = 2 = . 3c2t2 4(ω+1) 3c2t2 therefore the constrain a + a − a = 1 collapses to (cid:18) (cid:19) 2 3 1 2a2−a1 =1, and As it is observed, these quantities fit perfectly with the current observations by High -Z Supernova Team and 1+2ω−3ω2 1 K2 = , ⇐⇒ ω ∈ − ,1 , Supernova Cosmological Project see for example: Gar- 4(ω+1)2 3 navich et al., 1998 [29]; Perlmutter et al., 1997, 1998, (cid:18) (cid:19) 1999 [30]; Riess et al., 1998 [31]; Schmidt et al., 1998 so a2 ∈ 12,1 , a1 ∈ (0,1), K2 ∈ (0,1/4), and A = [32]. 2a a +a2−K2 = (5−ω) .Takinginto accountallthese 1 2 (cid:0)2 (cid:1)4 4(ω+1)2 results the homothetic vector field collapses to A. Curvature behaviour. V =t∂ +(1−a )x∂ +(1−a )(y∂ +z∂ ), t 1 x 2 y z With all these results, we find the following behaviour this is the already solution obtained by Hsu et al ([41]). for the curvature invariants (see for example [45]-[48]). ThissolutionhasbeenalreadyobtainedbyCollins([44]) Ricc Scalar, I , yields 0 Remark 1 As is it observed, we may find the relation 2 K2c2 I = 11a2−10a +2− , (24) a +a −a = 1 from dimensional considerations, only 0 c2t2 2 2 4 2 3 1 (cid:18) (cid:19) looking at the FE. Note that the quantity while Krestchmann scalar, I :=R Rijkl, yields: 1 ijkl a2c2 a2t2a1 ≈ 0 ≈t2a1−2a2−2a3 ≈t−2, 1 b2d2 b2t2a2d2t2a3 I = 432a4−960a3+896a2−384a +64 0 0 1 4c4t4 2 2 2 2 after substitution, it must have dimensions of T−2 as the +K2c2 (cid:0)11K2c2−40a22+80a2−48 . rest of the factors. Therefore we find the relation a + a −a =1. 2 The full con(cid:0)traction of the Ricci tensor,(cid:1)I(cid:1)2 := RijRij, 3 1 is: Therefore we have obtained a LRS BII model, note 1 that d=b, and hence the metric collapses to this one: I2 = 4c4t4 528a42−1024a32+768a22−256a2+32 ds2 =−c2dt2+a2(t)dx2+ b2(t)+K2z2a2(t) dy2+ K2c2 3K(cid:0)2c2−16a2+8 , +2Ka2(t)zdxdy+b2(t)(cid:0)dz2, (cid:1) (22) this means t(cid:0)hat the model is s(cid:1)in(cid:1)gular. The non-zero components of the Weyl tensor. The which admits the three Killings vectors (1) and this new followingcomponentsoftheWeyltensorrunto±∞when one t→0, C ≈C ≈t4(a2−1), C ≈t2(a2−1), txtx txty tztz z2 y2 ξ =−K − ∂ +z∂ −y∂ . (23) 4 2 2 x y z and (cid:18) (cid:19) Hence their algebra is: Ctyty ≈t2(a2−1)+z2t4(a2−1), these others depend on the value of a , ξ ξ ξ ξ 2 1 2 3 4 ξ1 0 0 0 0 Ctxyz ≈Ctyxz ≈Ctyyz ≈Ctzxy ≈t4a2−3, ξ 0 0 ξ −ξ 2 1 3 ξ3 0 −ξ1 0 ξ2 C ≈C ≈C ≈t2(3a2−2), xzyz xzxz xyxy ξ 0 ξ −ξ 0 4 3 2 i.e. C1 =1, C3 =−1, and C2 =1. C ≈t2(2a2−1)+z2t2(3a2−2). 23 24 34 yzyz 5 i.e. they may run to zero as well as to ±∞ when t→0, Therefore, we conclude that there is no physical so- depending on the value of a . lution for the vacuum model. Note that when we set 2 The Weyl scalar, I = CabcdC = I −2I + 1I2, K = 0, the metric is reduced to Bianchi I one. In the 3 abcd 1 2 3 0 (this definition is only valid when n=4) sameway,it isalsoverifiedthe relationa +a −a =1. 2 3 1 4 I = (a −1)(−2(a −1)−3Kc)+K2c2 3 3c4t4 2 2 V. PERFECT FLUID MODEL WITH G AND × (a2−(cid:0)1(cid:2))(−2(a2−1)+3Kc)+K2c2 . (cid:3) LAMBDA VARIABLE. The e(cid:2)lectric partscalarI4 =EijEij,(see W(cid:3)(cid:1).C. Limet al [49]) TheresultingFEforthemetric(2)withaperfectfluid matter model (11) and with G and Λ time-varying are: 1 2 I = −2(a −1)2+K2c2 , 4 6c4t4 2 (cid:16) (cid:17) while the magnetic part scalar I =H Hij, yields a′b′ a′d′ d′b′ K2a2c2 8πG 5 ij + + − = ρ+Λc2, (29) 3 a b a d d b 4 b2d2 c2 I5 = 2c2t4 (a2−1)2K2. b′′ + d′′ + d′b′ − 3K2a2c2 =−8πGωρ+Λc2, Therefore the Weyl parameter W2 (see W.C. Lim et b d d b 4 b2d2 c2 (30) al [49]) d′′ a′′ a′d′ K2a2c2 8πG 1 + + + =− ωρ+Λc2, W2 = EijEij +HijHij , d a a d 4 b2d2 c2 6 (31) therefore (cid:0) (cid:1) b′′ a′′ a′b′ K2a2c2 8πG 1 2 2 + + + =− ωρ+Λc2, W2 = (a −1)2+K2c2 (2a −2)2+K2c2 , b a a b 4 b2d2 c2 36c4t4 2 2 (32) (cid:16) (cid:17) (cid:16) (cid:17) note that the value of W2 is really small. a′ b′ d′ The gravitationalentropy (see [46]-[47]) ρ′+ρ(1+ω) + + =0, (33) a b d (cid:18) (cid:19) P2 = I3 = I1−2I2+ 13I02 = I1 + 1I02 −2=const., Λ′ =−8πG′ρ. (34) I I I 3I c4 2 2 2 2 since the spacetime is SS (see [50] and [43] for a discus- Now, we shall take into account the obtained SS re- sion). strictions for the scale factors i.e. a(t)=a ta1, b(t)=b ta2, d(t)=d ta3, B. Vacuum model. 0 0 0 wherea ,a ,a ∈R,insucha waythatthey mustverify 1 2 3 Inthissectionweshallshowthatthereisnoself-similar a +a −a =1. 2 3 1 solution for the vacuum model. In this case, after sub- From eq. (33) we get stitution, the resulting FE are: ρ=ρ t−γ, (35) 0 K2 a1a2+a1a3+a3a2− 4 =0, (25) where γ =(ω+1)α and α=(a1+a2+a3). From eq. (29) we obtain: 3K2 a (a −1)+a (a −1)+a a − =0, (26) 2 2 3 3 2 3 4 1 8πG a (a −1)+a a +a (a −1)+ K2 =0, (27) Λ= c2 At−2− c2 ρ0t−(ω+1)α (36) 3 3 1 3 1 1 4 (cid:20) (cid:21) a2(a2−1)+a1a2+a1(a1−1)+ K2 =0, (28) where A=a1a2+a1a3+a2a3− K42. 4 Now,takingintoaccounteq. (34)andeq. (36),algebra so, by solving this system, we only obtain the following brings us to obtain unphysical solutions: G=G tγ−2, (37) a =a =a =K =0, 0 1 2 3 a1 =1, a2 =a3 =K =0, where G = c2A . 1 1 4 0 4πρ0(ω+1)α a =− , a =a = , K2 =− , While the cosmological“constant” behaves as: 1 2 3 3 3 9 1 1 A 2 a1 =−3, a2 =a3 = 3, K =0. Λ= c2 1− (ω+1)α t−2 =Λ0t−2. (38) (cid:18) (cid:19) 6 With all these result we find that the system to solve then Λ = 0, i.e. the model collapses to the standard 0 is the following one: one studied above, and if G is decreasing then Λ < 0. 0 Therefore if we consider the recent observations which 3K2 α−2 suggestusthatΛ >0,wemustruleouttheothercases, a (a −1)+a (a −1)+a a − =A , 0 2 2 3 3 2 3 4 α concluding that G is a growing time function and Λ is a (cid:18) (cid:19) (39) positive decreasing function. K2 α−2 The energy-density behaves as a (a −1)+a a +a (a −1)+ =A , 3 3 1 3 1 1 4 (cid:18) α ((cid:19)40) ρ=ρot−γ, with γ ∈(0,6) (44) K2 α−2 so it is always a time decreasing function if ω >−1, and a (a −1)+a a +a (a −1)+ =A , 2 2 1 2 1 1 4 α to end, we find the following behaviour for the Hubble, (cid:18) ((cid:19)41) deceleration and shear parameters: whose solution is: 4a −1 4a H = 2 , q =− 2 <0, a =2a −1, a =a , K2 =−4a2+6a −2, (42) t 4a −1 1 2 2 3 2 2 2 thereforethissolutionhasonlysenseifa ∈ 1,1 .Note 2 2 that for these values, a >0, a =2a −1∈(0,1). This (a −1)2 1 1 2 (cid:0) (cid:1) σ2 = 2 −→0. (45) result is valid for all equation of state. We would like 3c2t2 to point out that this solution is more general that the obtained one in the perfect fluid case with “constant” Asintheabovemodel,the metricofthis onecollapses constants, since it is valid for ω ∈(−1,∞), although we to a LRSBII type (22). are only interested in ω ∈(−1,a], a≥1. It is important to emphasize that ω 6=−1. VI. PERFECT FLUID MODEL WITH VSL. With this solution we may see that all the quantities dependontwovariables,a andω.Newtongravitational 2 constant behaves as follows: We start by defining the new metric as follows: G= c2A tγ−2 =G tg, G >0, (43) ds2 =−c(t)2dt2+a2(t)dx2+ b2(t)+K2z2a2(t) dy2+ 0 0 4πρ (ω+1)α 0 +2Ka2(t)zdxdy+d2(t)dz(cid:0)2, (cid:1) (46) with g =(ω+1)(4a −1)−2∈(−2,4), and A=5a2− 2 2 2a − K2 = 6a2 − 7a + 1 ∈ 1,3 . So depending of note that we have simply replaced c by c(t), so we shall 2 4 2 2 2 2 4 consider the four velocity: the different combinations between a and ω, G may be 2 (cid:0) (cid:1) a growing or a decreasing time function. 1 With regardto the behaviour ofthe cosmologicalcon- u= ,0,0,0 , / uiui =−1. (47) c(t) stant, as we can see, it is a decreasing time function, as (cid:18) (cid:19) itisexpected. Onlyrestto knowwhichisthe signofΛ . 0 The time derivatives of G,c and Λ are related by the We may express its behaviour in the following tables Bianchi identities ω a2 Λ0 ω a2 Λ0 1 ;j 8πG ;j R − Rg = T −Λg , (48) (−1,1) ց1/2 ≤0 (−1,−1/2) ր1 <0 ij 2 ij c4 ij ij (cid:18) (cid:19) (cid:18) (cid:19) 1 ց1/2 0 −1/2 ր1 0 in our case we obtain >1 ց1/2 ր1 >−1/2 ր1 ր1 Λ′c4 G′ c′ this means, for example, that when we fix a ց 1/2 i.e. ρ′+ρ(1+ω)H + +ρ −4 =0, (49) 2 8πG G c that a2 tends to 1/2, for all value of ω between -1 and (cid:18) (cid:19) 1, Λ0 < 0, Λ0 = 0, if ω = 1 and only Λ0 > 0 when where H = a′ + b′ + d′, but if we take into account tωhe>n1∀.ωIn∈th(e−1sa,y−1w/a2y)w, weemfianydsatyhatthaΛt0if<w0e.fiΛx0a=2 ր0,1if, the condition,aTi;jj =b 0, itdis obtained the following set of ω = −1/2 and only it is found Λ > 0 when ω > −1/2. equations: 0 So may only say that the sign of Λ , depends on the 0 ρ′+ρ(1+ω)H =0, (50) parametersa andω.If wetakeinto accountthe current 2 observations which suggest that Λ > 0, then we may Λ′c4 G′ c′ 0 + −4 =0. (51) rule out the values which make Λ0 <0. 8πGρ G c AsitisobservedthebehaviourofGandthesignofΛ , 0 are related. If G is growing then Λ > 0, if G = const. Therefore, the resulting FE are: 0 7 a′b′ a′d′ d′b′ K2a2c2 8πG + + − = ρ+Λc2, (52) a b a d d b 4 b2d2 c2 b′′ d′′ d′b′ c′ b′ d′ 3K2a2c2 8πG + + − + − =− p+Λc2, (53) b d d b c b d 4 b2d2 c2 (cid:18) (cid:19) d′′ a′d′ a′′ c′ a′ d′ K2a2c2 8πG + + − + + =− p+Λc2, (54) d a d a c a d 4 b2d2 c2 (cid:18) (cid:19) b′′ a′b′ a′′ c′ a′ b′ K2a2c2 8πG + + − + + =− p+Λc2, (55) b a b a c a b 4 b2d2 c2 (cid:18) (cid:19) a′ b′ d′ ρ′+ρ(1+ω) + + =0. (56) a b d (cid:18) (cid:19) Λ′c4 G′ c′ + −4 =0. (57) 8πGρ G c Note that we have taken into account the effects of a where γ =(1+ω)α. c−var into the curvature tensor (see [28] for a discussion In the same way it is easily calculated the shear about the issue). 1 In this model, the homothetic vector field is: σ2 = H2+H2+H2−H H −H H −H H , 3c2(t) 1 2 3 1 2 1 3 2 3 cdt cdt (cid:0) (cid:1) X = ∂ + 1− H x∂ + i.e. t 1 x c(t) c(t) (cid:18)R (cid:19) (cid:18) R (cid:19) cdt cdt 1 3 −2 + 1− c(t) H2 y∂y+ 1− c(t) H3 z∂z, (58) σ2 = 3 α2i − αiαj cdt . (64) (cid:18) R (cid:19) (cid:18) R (cid:19) i=1 i6=j (cid:18)Z (cid:19) X X with the following restrictions   As it is observed all the quantities depend on c(t)dt. α1 α2 α3 Now only rest to calculate G and Λ. a=a cdt , b=b cdt , d=d cdt , R 0 0 0 From eqs. (52 and 63) we get: (cid:18)Z (cid:19) (cid:18)Z (cid:19) (cid:18)Z (cid:19) (59) 2 −γ c 8πG with (α )3 ∈ R, in such a way that they must satisfy A = ρ c +Λc2, (65) i i=1 c c2 0 the relation α2 +α3 −α1 = 1. Since we expect to get (cid:18) (cid:19) (cid:18)Z (cid:19) a growing scale factors, we shall consider that cdt wherewehaveRwritten,forsimplicity, cinsteadof cdt, monulystnbeeedasptoosbiteivientgergorwabinlegbtuimteitfumnacytiobne.deNcoretea(cid:0)stRinhgata(cid:1)cs andwehavesetA=α1α2+α1α3+αR2α3−K42,therRefore we shall show below. 2Ac 8πρ G G′ c′ c Λ′ =− − 0 −4 −γ . (66) Remark 2 As it is observed, if c=const.,we regain the c 3 c4 c γ G c c (cid:20) (cid:21) usual homothetic vector field. i.e. Now, tak(cid:0)iRng(cid:1)into acc(cid:0)oRun(cid:1)t eq. (57), we getRthat X =t∂ +(1−tH )x∂ +(1−tH )y∂ +(1−tH )z∂ , t 1 x 2 y 3 z (60) − c4 2Ac + 8πρ0G G′ −4c′ −γ c + while the scale factors behave as 8πGρ" c 3 c4 c γ (cid:20)G c c(cid:21)# a=a0(t)α1, b=b0(t)α2, d=d0(t)α3, (61) (cid:0)R (cid:1) (cid:0)R (cid:1) R G′ c′ with α +α −α =1, as in the above studied cases. + −4 =0, (67) 2 3 1 G c By defining and hence we obtain c c γ−2 H =α =⇒ H =α , (62) A i i cdt cdt G= c4 c , (68) 4πρ γ 0 (cid:18)Z (cid:19) where α= 3i=R1αi, we find, from eq. (5R6), the behavior and in this way we find that the cosmological constant of the energy density i.e. behaves as P −γ 2 −2 ρ=ρ cdt , (63) Λ=A 1− c . (69) 0 γ (cid:18)Z (cid:19) (cid:18) (cid:19)(cid:18)Z (cid:19) 8 Aswecansee,fromeqs. (68and69),wehavethatare The cosmological constant behaves as verified the following relationships −2 −2 2 Gρ ≈ c −2, Λ c 2 =const. (70) Λ=A(cid:18)1− γ(cid:19)(cid:18)Z cdt(cid:19) =Λ0(cid:18)Z cdt(cid:19) , c4 (cid:18)Z (cid:19) (cid:18)Z (cid:19) where the sign of Λ depends on the different values of 0 Now, we will try to find the value of the constants γ, (α )3 . Taking into account the field eqs. (52-55) we i i=1 find that, obviously eq. (52) vanish, but from eqs. (53- <0 if γ ∈(0,2) 55) we get Λ0 ==0 if γ =2 . >0 if γ ∈(2,6) 3K2 α−2 a (a −1)+a (a −1)+a a − =A , 2 2 3 3 2 3 4 α Sinceourmodelisformallyself-similar,then([37]-[36]) (cid:18) (cid:19) (71) have shown, that all the quantities must follow a power K2 α−2 law, so, we may assume that for example, c takes the a3(a3−1)+a1a3+a1(a1−1)+ =A , following form: c(t) =c tǫ, with ǫ∈ R. Hence, the scale 4 α 0 (cid:18) (cid:19) factors behaves as: (72) K2 α−2 a=a tα1(ǫ+1), b=b tα2(ǫ+1) =d, (77) a (a −1)+a a +a (a −1)+ =A , 0 0 2 2 1 2 1 1 4 α (cid:18) (cid:19) (73) insucha waythat they behave asgrowingtime function if ǫ ∈ (−1,∞), α > 0. This means that c may be a i findingthereforethesamesolutionasintheabovemodel, decreasing time function. The homothetic vector field i.e. collapses to this one a =2a −1, a =a , K2 =−4a2+6a −2, (74) t 1 2 2 2 2 2 V = ∂ +(1−a )(2x∂ +y∂ +z∂ ) (78) t 2 x y z ǫ+1 this solution has only sense if a ∈ 1,1 , note that for these values, a1 >0. 2 2 The Hubble parameter behaves as: H = (4α2 − Wewouldliketoemphasizethatto(cid:0) out(cid:1)linethesystem 1)(ǫ+1)t−1 = α˜t−1, so we find again the restriction (71-73) it is essential to take into account the effects of for ǫ, ǫ ∈ (−1,∞). In the same way, we find from eq. −2 a c-var into the curvature tensor. If we do not consider Λ=Λ c , that, since Λ must be a decreasing time 0 sucheffectsthenthesystemdependson candtherefore function, this is only possible if ǫ∈(−1,∞), the special it is not algebraic (see [28] for a detailed discussion). case, ǫ =(cid:0)R−(cid:1)1, is forbidden, note that cdt = c0 tǫ+1 > R ǫ+1 Therefore the behaviour of the main quantities is the 0, ∀ǫ∈(−1,∞). R following one. The energy density behaves as So, we find that −γ ρ≈t−γ(ǫ+1), G≈tγ(ǫ+1)+2ǫ−2, Λ≈t−2(ǫ+1), (79) ρ=ρ c(t)dt , (75) 0 (cid:18)Z (cid:19) withǫ∈(−1,∞),andγ =(ω+1)α=(ω+1)(4a −1). 2 where γ = (ω+1)α, and α = 2α +α , while Hubble Therefore the energy density is alwaysa decreasing time 2 1 and the deceleration parameters behave as function if ω ∈ (−1,1] and ǫ ∈ (−1,∞). As above, the behaviour of G depends on (a ,ω,ǫ), hence it may be a 2 c 1 c′ c growingfunctionaswellasadecreasingone. Withregard H =α , q = 1− −1, (76) to the cosmological constant, the most important thing cdt α c c (cid:18) R (cid:19) is to know its sign. Recalculating all the computations and the shear yRields we find that Λ = A˜c−2 1− 2 , where γ˜ = γ(ǫ+1) = 0 0 γ˜ −2 (ω+1)(4a2−1)(ǫ+1),(cid:16)in suc(cid:17)h a way that this sign is 1 σ2 = (α −1)2 cdt . affected by the perturbing parameter (ǫ+1). We have 3 2 (cid:18)Z (cid:19) set A˜= 5α22−2α2 (ǫ+1)2−K42,rememberthat,α1 = 2α −1. On the other hand, G behaves as, G = 2 (cid:0) (cid:1) To clarify this results we may fix ω =1, a =2/3 and G0c4 cdt γ−2,whereG0 >0.Hence,itsbehaviourwill ǫ=±1/2, finding 2 depend on the different values of γ. If γ ∈ (0,2) we find (cid:0)R (cid:1) thatGisadecreasingtimefunction. Ifγ =2,G=const. ǫ ρ G Λ 0 and all the model collapses to the standard one studied intheabovesections(G=const.,c=const.andΛ=0). −1/2 t−5/3 t−4/3 <0. While if γ ∈(2,6) then G is a growing time function. 1/2 t−5 t4 >0 9 Hence, if we take into account the current observation, With regard to the perfect fluid model with G and Λ then we must rule out those values which make Λ < 0. variable we have arrived to the conclusion that its solu- 0 Therefore we conclude that c and G are growing time tion is quite similar to the obtained one for the classical functionswhileΛisapositivetimedecreasingtimefunc- model. Butwehavebeenabletoenlargethesetofvalues tion. for the equation of state, in this case is ω ∈(−1,∞). As Withregardtothedecelerationparameterwefindthat we have mentioned above the solution is similar to the it has the following behavior: classical model so it also belongs to the LRS BII kind. WithregardtothebehaviourofGandΛwehaveshowed 1 q =−1+ , (80) that G is a growing time function while Λ is a positive α˜ decreasing time function. We have ruled out the rest of the solution taking into account the recent observation whereα˜ =(4a −1)(ǫ+1),indicating asthatitis quite 2 which suggest us that Λ >0 and q <0. unlikely that ǫ→−1. We find that q <0, if α˜ >1. 0 In the VSL model we have arrived to similar conclusions as in the above model i.e. the solution is valid VII. CONCLUSIONS. for ω ∈ (−1,∞). We have started outlining the FE but taking into account the effects of a c−var into the cur- We have studied several perfect fluid Bianchi II mod- vature tensor. In this approach is essential to take into els under the self-similarity hypothesis. We have started accountsucheffects, inotherway,itisimpossibletoget, our study reviewing the classical solution and showing aftersubstitution,andalgebraicsystemofequations. We that it belongs to the LRS BII kind by calculating the have also calculated the homothetic vector field for the fourth Killing vector. The self-similar solution brings us newmetric aswellasthe constrainsforthe scalefactors. to get a power law solution for the scale factors i.e. they Inthiscaseallthequantitiesdependon c(t)dt.Bysolv- behave as a = a ta1, b = b ta2 = d, in such a way the ingtheassociatedalgebraicsystemwehaveobtainedthe 0 0 R the constants a must satisfy the relation 2a −a = 1. same solution as in the above studied case. Hence this i 2 1 The only drawback is that this solution is only valid for solution belongs to the LRS BII type. In a generic way an equation of state ω ∈ (−1/3,1). We have also shown we have discussed the behaviour of the main quantities that this solutionfits perfectly with the currentobserva- ruling out solutions like Λ ≤ 0. If we assume a power 0 tionsforH andq.Withregardtoitscurvaturebehaviour law for c(t), c = tǫ say, then we arrive to the conclu- we may conclude that the model is singular. The Weyl sionthatitmaybe adecreasingtime function. 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