# Scalar-Tensor Bianchi VI Models

Authors: J. A. Belinchón
Journal: Journal of Gravity
 (2013)

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## Abstract

We study how may vary the gravitational and the cosmological “constants,” (G and Λ) in several scalar-tensor theories with Bianchi III, VI0, and VIh symmetries. By working under the hypothesis of self-similarity we find exact solutions for two different theoretical models, which are the Jordan-Brans-Dicke (JBD) with Λ(ϕ), and the usual JBD model with potential U(ϕ) (that mimics the behaviour of Λ(ϕ)). We compare both theoretical models, and some physical and geometrical properties of the solutions are also discussed putting special emphasis on the study of the isotropization of the solutions.


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## Body

## 1. Introduction

Current observations of the large scale Cosmic Microwave Background (CMB) suggest to us that our physical universe is expanding in an accelerated way. Such observations [1–3] indicate that the universe is dominated by an unidentified “dark energy” (DE) and suggest that this unidentified dark energy has a negative pressure [4–6]. This last characteristic of the dark energy points to the vacuum energy or cosmological constant Λ, as a possible candidate for dark energy. From the theoretical point of view, it is convenient to consider the cosmological constant as a dynamical quantity in order to solve the so-called coincidence and fine tuning problems. In the same way other observations have pointed out a possible variation of the gravitational constant G [7, 8]. For example, observations of Hulse-Taylor binary pulsar [9, 10], and type Ia supernova observations [11]. For an extensive review see Uzan [12].We have several theoretical models that consider both constants as variable with respect to the cosmic time. Such theories are modified general relativity (MGR), modified scalar cosmological models (MST), and several scalar-tensor theories (STT). The MGR and MST have a drawback, since in them the variations ofG and Λ are introduced in an ad hoc manner. Nevertheless we consider that the STT are the best models to study the variation of G and Λ, since they have been deduced form variational principles and where the time dependence can occur in a natural way, without any new assumption or modification of the theory. This class of models has received a renewed interest in recent times, for two main reasons. Firstly, the new inflationary scenario as the extended inflation has a scalar field that solves several problems present in the old theories. Secondly, string theories and other unified theories contain a scalar field which plays a similar role to the scalar field of the STT. The scalar-tensor theories started with the work of Jordan in 1950 [13]. A prototype of such models was proposed by Brans and Dicke in 1961 [14]. Their aim for presenting this model was to modify Einstein’s theory in such a way as to incorporate the so-called “Mach principle.” These theories have been generalized by Bergmann [15], Nordtvedt [16], and Wagoner [17]. For a recent review of this class of theories we refer to [18, 19].In a recent paper, [20] we have considered a family of scalar-tensor theories under the hypothesis of self-similarity. They are the Jordan-Brans-Dicke model with a dynamical cosmological constant (JBDΛ) [21] and with a potential (JBDU) [19], which is equivalent to a time-dependent cosmological constant. For such theoretical models, we proved how must behave each physical quantity under the hypothesis of self-similarity.We have focused our attention on this class of solutions, since as has been pointed out by Rosquist and Jantzen [22] (for a precise definition of self similar models, see for instance [23]), self-similar models correspond to equilibrium points, playing a dominant role in the dynamics of Bianchi cosmological models. For this reason, Coley [24] has stressed the fact that self-similar models play an important role in describing the asymptotic dynamics of the Bianchi models. A large class of orthogonal spatially homogeneous models (including all class B models) are asymptotically self-similar at the initial singularity and are approximated by exact perfect fluid or vacuum self-similar power-law models. In the same way, exact self-similar power-law models can also approximate general Bianchi models at intermediate stages of their evolution and which is also important, and self-similar solutions can describe the behavior of Bianchi models at late times. Furthermore working under the hypothesis of self-similarity allows us to find exact power-law solutions in such a way that we may compare the obtained solution for each studied model.By applying the theoretical results obtained in [20] we have studied how may vary G and Λ in the described theoretical models using different geometries as Bianchi I [25], Bianchi II [26], Bianchi V, VIIh and IX [27] and Kantowsky-Sach [20]. In this paper, enlarging the program, we would like to study the case of family of Bianchi VI geometry, that is, Bianchi III, VI0, and VIh.Therefore, the paper is organized as follows. In Section2 we start by revising the theoretical results obtained in [20]. Once all the theoretical ingredients have been exposed, in Section 3 we begin describing the main results concerning to the Bianchi VI geometry and its self-similar restrictions. Section 4 is devoted to find exact self-similar solutions for the described theoretical models: JBDΛ and JBDU. We put special emphasis on the study of the isotropization of the solutions. We end in Section 5 with a brief summary of the main results and conclusions.


## 2. Theoretical Models

The first of the scalar-tensor model is described by the field equations [21]: Gij=Tijeff, where Gij is the Einstein tensor
(1)Gij=Rij-12gijR,
and Tijeff is the effective stress-energy tensor defined by
(2)Tijeff=8πc4ϕTijm-Λ(ϕ)gij+Tijϕ,
where Tijm is the stress-energy tensor of the ordinary matter, in this case a perfect fluid model defined by
(3)Tijm=(pm+ρm)uiuj+pmgij,
and where the 4-velocity is defined by ui=(1,0,0,0),  ρm is the energy density, and pm is the pressure. They are related by the equation of state (EoS): pm=γρm, with γ∈(-1,1].  Tijϕ is defined as follows:
(4)Tijϕ=ω(ϕ)ϕ2(ϕ,iϕ,j-12gijϕ,lϕ,l)+1ϕ(ϕ;ij-gij□ϕ),
where ϕ is the scalar function, considering only that ϕ=ϕ(t) and Λ(ϕ) is a potential function and plays the role of a cosmological constant, and ω(ϕ) is the coupling function of the particular theory (the Brans-Dicke parameter), that could be considered as variable.The conservation equations are(5)C□ϕ+dωdϕϕ,lϕ,l=8πT+2ϕ(ϕdΛdϕ-Λ(ϕ)),(Tijm);j=0,
where T=Tii is the trace of the stress-energy tensor and C=3+2ω(ϕ). The gravitational coupling Geff(t) is given by
(6)Geff(t)=(2ω+42ω+3)G*ϕ(t).In a recent paper [20], we have proved that in the framework of self-similar and power-law solutions the main quantities must behave as follow:
(7)ϕ=ϕ0tn~,Λ(t)=Λ0t-(n~+α),ω(ϕ)=const.,ρ=ρ0t-α,
in such a way that the parameters must verify the following relationship: n~+α=2, with α=(1+γ)h, h being a positive constant, that it is defined by the exponents (ai) of the scale factor of the metric (we are only interested in power-law and self-similar solutions), and h=∑i‍ai.The second theoretical model is described by the following field equations (see, for instance, [18, 20]):
(8)FGij=8πTijm-U(ϕ)gij+Tijϕ,
with
(9)Tijϕ=Z[ϕ;iϕ;j-12gijϕ;αϕ;α]+[F;iF;j-gij□F],
and the conservation equations for the scalar and matter fields
(10)2Z□ϕ=FϕR-Zϕϕ′2-2Uϕ,(Tijm);j=0,
where R is the scalar curvature and Fϕ=dF/dϕ. The function U(ϕ) plays the role of a dynamical cosmological constant. The effective gravitational constant Geff between two test masses measured in laboratory Cavendish-type experiments is given by
(11)Geff=G*F[2Z(ϕ)F+4Fϕ22Z(ϕ)F+3Fϕ2].IfF(ϕ)=ϕ,  Z=ω(ϕ)/ϕ, then we obtain
(12)Gij=Tijeff,
with
(13)Tijeff=8πϕTijm-U(ϕ)2ϕgij+Tijϕ,
where Tijm is defined by (3), Tijϕ is defined by (4), and the reduced equation of motion for ϕ yields
(14)C□ϕ=8πT+ϕUϕ-2U(ϕ)-dωdϕϕ,lϕ,l,
with C=3+2ω. Ordinary matter satisfies the conservation equation (Tijm);j=0. Therefore, in this model, the effective gravitational constant Geff behaves as
(15)Geff≈ϕ-1.As we have shown in [20], in the framework of self-similar and power-law solutions, the main quantities must behave as follows:
(16)ϕ=ϕ0tn~,U(t)=U0tn~-2,ω(ϕ)=const.,ρ=ρ0t-α,
where n~+α=2, such that n~<2. Note that as in the Λ model, α=h(γ+1).


## 3. The Metric

Throughout the paperM will denote the usual smooth (connected, Hausdorff, fourdimensional) spacetime manifold with smooth Lorentz metric g of signature (-,+,+,+) (see, for instance, [28]). Thus M is paracompact. A comma, semicolon and the symbol ℒ denote the usual partial, covariant, and Lie derivative, respectively, the covariant derivative being with respect to the Levi-Civita connection on M derived from g. The associated Ricci and stress-energy tensors will be denoted in component form by Rij (≡Rjcdc) and Tij, respectively. We will use a system of units where c=1.A Bianchi VI spacetime is a spatially homogeneous spacetime which admits a group of isometriesG3, acting on spacelike hypersurfaces, generated by the spacelike Killing vectors field (KVs):
(17)ξ1=∂x,ξ2=∂y,ξ3=mx∂x-ny∂y+∂z,
and therefore the nonzero structure constants are C131=m, C232=-n. In synchronous coordinates, the metric is
(18)ds2=-dt2+a2(t)e-2mzdx2+b2(t)e2nzdy2+d2(t)dz2,
where the metric functions a(t),  b(t), and  d(t) are functions of the time coordinate t only. As it is observed, the metric (18) collapses to the following cases:(1)
ifm=-n, then the metric collapses to Bianchi type V
(19)ds2=-dt2+a2(t)e-2mzdx2+b2(t)e-2mzdy2+d2(t)dz2,with(20)ξ1=∂x,ξ2=∂y,ξ3=mx∂x+my∂y+∂z,and thereforeC131=m=C232;(2)
ifn=0, then the metric collapses to Bianchi type III(21)ds2=-dt2+a2(t)e-2mzdx2+b2(t)dy2+d2(t)dz2,with the following KVF:(22)ξ1=∂x,ξ2=∂y,ξ3=mx∂x+∂z,and thereforeC131=m;(3)
ifm=n=0, then the model collapses to Bianchi I model
(23)ds2=-dt2+a2(t)dx2+b2(t)dy2+d2(t)dz2,with(24)ξ1=∂x,ξ2=∂y,ξ3=∂z,andCijk=0.For the metric (18) we may find the following homothetic vector field (HVF) ℋ, defined by ℒℋgij=2αgij (see [22, 23]):
(25)ℋ=t∂t+(1-ta′a)x∂x+(1-tb′b)y∂y+(1-td′d)z∂z,
with the following restrictions on the scale factors:
(26)a(t)=a0ta1,b(t)=b0ta2,d(t)=d0t,
where a1,a2∈ℝ, and without loss of generality we may set d0=1. In this way the HVF may be written in the following form:
(27)ℋ=t∂t+(1-a1)x∂x+(1-a2)y∂y,
in such a way that the metric (18) collapses to
(28)ds2=-dt2+t2a1e-2mzdx2+t2a2e2nzdy2+t2dz2.
The Hubble parameter H, under these restrictions, is defined by
(29)H=(a1+a2+1)3t-1=h3t-1,
with h=a1+a2+1.In this paper we study the following cases.(1)
Model Bianchi typeIII (BIII), where its metric is defined by
(30)ds2=-dt2+t2a1e-2mzdx2+t2dy2+t2dz2,or(31)ds2=-dt2+t2dx2+t2a2e2nzdy2+t2dz2,that is,m=0 and a2=a3=1 or n=0 and a1=a3=1. With these restrictions the deceleration q, anisotropic (𝒜,𝒲2), and gravitational entropy P2 parameters yield
(32)q=ddt(1H)-1=-a1-1a1+2,𝒜=σ23H2=23(a2-1)2(2+a2)2,𝒲2=I324H4=(m-1+a2)2(m+1-a2)236(2+a2)4,P2=I3I2=(m-1+a2)2(m+1-a2)23((a22+1)2+2a2(1-m2)+m2(m2-2)),whereσ2=σijσij is the square shear and I3=CabcdCabcd is the Weyl curvature scalar (see for instance [29]). The second Kretschmann scalar I2 is defined by I2=RijRij; for a precise definition of gravitational entropy, see [30].(2)
Model Bianchi typeVI0 (BVI0), described by the metric
(33)ds2=-dt2+t2a1e-2mzdx2+t2a1e2mzdy2+t2dz2,withm=n,  a1=a2,  and  a3=1. The main kinematical quantities yield
(34)q=-2a1-12a1+1,𝒜=23(a2-1)2(1+2a2)2,𝒲2=m2(3(a1-1)2+m2)9(1+2a1)4,P2=-4m2(3(a1-1)2-m2)3(3a14-2a13+2a12+m2(-2a1+m2)).We also may define aBVI0 metric by following the way suggested by Stephani et al. [28], that is, by considering the KVF:
(35)ξ1=∂x+mz∂y+my∂z,ξ2=∂y,ξ3=∂z,with(36)[ξ1,ξ2]=-mξ3,[ξ2,ξ3]=0,[ξ3,ξ1]=mξ2,and therefore we obtain(37)ds2=-dt2+a2dx2+(b2cosh2mx+d2sinh2mx)dy2-2(b2+d2)coshmxsinhmxdydz+(b2sinh2mx+d2cosh2mx)dz2,but as we have pointed out in [31], with this metric we obtain a very complicate field equation (FE), since they depend on trigonometric functions.(3)
To end, a Bianchi typeVIh (BVIh) is defined by
(38)ds2=-dt2+t2a1e-2mzdx2+t2a2e2nzdy2+t2dz2,withm≠n, a1≠a2, anda3=1. With these restrictions, we obtain
(39)q=-(a1+a2-2)a1+a2+1,𝒜=2(a12+a22+1-a1a2-a1-a2)3(a1+a2+1)2,𝒲2=136(a1+a2+1)4×(-6a1mn-6ma2n-a12mn+a2a1m2-a22mn+a2a1n2+m4+n4-3a13+a14+4n2a22-9m2a1-6a2a1-9n2a2-3a23+a24+6mn+3m2a2+3n2+4m2a12+mn3+3a1n2+m3n+3m2+8a2a1mn-2m2a22-2a12n2+3a1a22+3a12a2-a13a2-a23a1+3a12+3a22),P2=23K(a1-a2+m+n)(a1-a2-m-n)×(a12-3a1+a2a1-n2-m2+3+mn-3a2+a22),where(40)K=-2a1mn-2ma2n+a12mn-a2a1m2+a22mnK=-a2a1n2+2a22a12+m4+n4+2m2n2-a13+a14K=-2n2a22+m2a1+2a2a1+n2a2-a23+a24+2mnK=-m2a2-n2-2m2a12-mn3-a1n2-m3n-m2K=+4a2a1mn-a1a22-a12a2+a13a2+a23a1+a12+a22.For the metric (18) the Einstein tensor yields
(41)G11=a′ab′b+a′ad′d+d′db′b+Ad2,G41=m(a′a-d′d)+n(d′d-b′b),G22=b′′b+d′′d+d′db′b-n2d2,G33=d′′d+a′′a+a′ad′d-m2d2,G44=b′′b+a′′a+a′ab′b+mnd2,whereA=mn-m2-n2.


## 4. Solutions

### 4.1. Model with Cosmological Constant (JBDΛ)

For the metric (18), the effective stress-energy tensor (2) yields
(42)T11=8πρϕ+ω2ϕ′2ϕ2-(a′a+b′b+d′d)ϕ′ϕ+Λ,T41=0,T22=-8πpϕ-ϕ′′ϕ-ω2ϕ′2ϕ2-(b′b+d′d)ϕ′ϕ+Λ,T33=-8πpϕ-ϕ′′ϕ-ω2ϕ′2ϕ2-(a′a+d′d)ϕ′ϕ+Λ,T44=-8πpϕ-ϕ′′ϕ-ω2ϕ′2ϕ2-(a′a+b′b)ϕ′ϕ+Λ.
Once we have defined the field equations, we next find exact self-similar solutions for the different exposed models.


#### 4.1.1. Bianchi III Solution

For this metric, we have found the following solution:(43)a2=-2γ(ω(γ-1)-1)-ω(1-γ2)+2(γ-1)∈(0,1),a1=a3=1,Λ0=0,∀γ∈(-13,0),m=-(a2-1)2+Aω∈(0,1),n=0,ρ0=-(2+a2)Bϕ08πω(ω((a2+1)2-4)-(a2-1)2+A),n~=-2γ(γ-1)(3γ-1)ω(γ+1)+2,q=-ω(3γ+1)(γ-1)+22ω(1-γ)-6γ+4,
where
(44)A=(∫(a2-1)4+ω(a2-1)2(ω(1+a2)2+2(a2+a22+1))∫)1/2,B=2(a2-1)3+ω(a2-1)×(ttttω(2ω(a2+1)2+3(a2+1)2+4)+6+(a2+1)2)+A(ω(2ω(a2+1)+5+a2)+2-2a2tttt).In the first place, we would like to stress that the cosmological constant vanishes, so this model does not describe its possible variation. By fixing the Brans-Dicke parameter,ω=4·104 [32], we may see that the solution is valid  for all γ∈(-1/3,0), since the energy density ρ behaves as a positive decreasing time function for these values of the EoS (we have set ϕ0>0 for physical reasons). The solution is not inflationary for all γ∈(-1/3,0), precisely q<0 only if γ<-1/3. With regard to the anisotropic parameters, we find that the solution cannot be considered as isotropic with regard to the first anisotropic parameter, since 𝒜=(a2-1)2/(4a2-1)2∈[0,5/2), for all γ∈(-1/3,0), that is, 𝒜≪1; note that when γ tends to zero, 𝒜 takes the value 5/2. Nevertheless, we may say that the solution is isotropic with regard to the second anisotropic parameter, since 𝒲2≪0.0004,  for all γ∈(-1/3,0). The gravitational entropy P2∈[0,0.2); it does not show any pathological behaviour and takes values close to zero   for all γ∈(-1/3,0). To end the analysis of this solution, the effective gravitational constant behaves as a time decreasing function   for all γ∈(-1/3,0), since for these values the parameter n~ is positive.


#### 4.1.2. BianchiVI0 Solution

By solving the associated algebraic system to the FE we have found the following solution:(45)a1=a2=-1-γ+ω(γ+1)22(γ2-1)ω+γ-3∈(0,1),m=n=12A-(a2-1)2ω∈(0,12),ρ0=-ϕ0(1+2a2)B8πω(A+4(a22-1)ω-(a2-1)2),n~=8γω(γ+1)-(3γ-1)(γ-1)2ω(γ2-1)+γ-3,q=2(-ω(γ+1)(3γ-1)+2)4ω(γ+1)-3γ+5,Λ0=0,∀γ∈(-13,1),
with
(46)A=(a2-1)2(8a22ω(2ω+3)+(a2-1)2),B=(a2-1)3+A(2ω(2a2+2ωa2+1)+1-a2)+2ω(a2-1)×(2ωa2(7a2+4ωa2+1)+a2(4a2+1)+1).As we may observe, we have again obtainedΛ0=0, which is not in agreement with the current observations and which is worse, and we cannot describe its possible time variation. The solution is only valid   for all γ∈(-1/3,1)=I, since for these values of the EoS, the energy density is positive, that is, ρ0>0. By setting the Brans-Dicke parameter, ω=4·104, we find that the n~ parameter vanishes at γc=1/3, being positive, n~>0,   for all γ∈(-1/3,1/3); therefore, the effective gravitational constant Geff is decreasing when γ<1/3,   for all γ∈I, constant at γc=1/3, and growing when   for all γ∈(1/3,1). The solution is not inflationary, since the deceleration parameter is always positive, q>0   for all γ∈I. Nevertheless we may conclude that the solution is isotropic, since both anisotropic parameters are close to zero, 𝒜∈(0,1), in such a way that 𝒜→1 only when γ→1, while 𝒲2≪0.01,   for all γ∈I. However, the gravitational entropy only takes values close to zero in a small neighborhood of γ=-1/3, tending to -∞ in the rest of the interval.


#### 4.1.3. BianchiVIh Solution

In this case we have obtained the following solution:(47)Λ0=13[12+13ω+2(1+2ω3)((a1+a2)2-a1-a2)],m=33(a2-1),n=33(a1-1),ϕ0=1,ρ0=-148π×(2ω(-4a1-4a2+4(a12+2a22+2a1a2)+1)+2(7a12+7a22+8a1a2)-8(a2+a1)-1),n~=2-(γ+1)(a1+a2+1),q=3(a1+a2+1)-1,
being only valid for γ=1/3; note that for this value the wave equation for the scalar field is easier to solve than in the previously cases, since the trace of the matter stress-energy tensor vanishes. In this solution the cosmological constant takes values different than zero, that is, Λ0≠0, so we are able to describe its behaviour. We have expressed the solution depending on two parameters, a1 and a2; therefore it is necessary to carry out a numerical analysis in order to interpreted it. As usual, we start by fixing the numerical value of the Brans-Dicke parameter, ω=4·104, and we consider that the parameters of the scale factors a1 and a2 vary in the interval (0,1). In the first place we have calculated the set of values where the parameters (ρ0,Λ0,n~) vanish. In Figure 1 we have plotted this situation finding that the solution is only valid within this small tubular neighborhood. In Figure 2, we have detailed how behave each parameter Λ0,  ρ0, and n~. As we can see, the energy density is positive in this small tubular neighborhood where furthermore the cosmological constant is also positive expect in a very small subinterval where it is negative and vanishes (see Figure 2).Figure 1
Solution to BianchiVIh with CC. Plot of the parameters Λ0=0 (blue) and ρ0=0 (red).Figure 2
Solution to BianchiVIh with CC. Tubular neighborhood. Plot of the parameters Λ0 (green), ρ0 (red), and n~ (blue). The plane z=0 (gray).If, for example, we seta1=1/5, then we have obtained the following results (see Figure 3). The energy density parameter ρ0 is only positive, ρ0>0, if and only if a2∈(0.2973605134,0.3026282371)=I1, while the cosmological constant is negative, Λ0<0, if and only if a2∈(0.2999893937,0.3000106065)=I2, note that I2⊂I1, finding that, Λ0≥0,  for all a2∈I1∖I2. In the same way, we may see that the parameter of the scalar field, n~, only vanishes, n~=0, if a2=0.3000106065. In this way we may arrive at the conclusion that the effective gravitational constant Geff is decreasing if a2∈(0.2973605134,0.3000106065), constant at a2=0.3000106065, where furthermore Λ0=0 and G=G0, and if a2∈(0.3000106065,0.3026282371), then Geff is growing and Λ0>0.Figure 3
Solution to BianchiVIh with CC. Plot of the parameters Λ0 (green), ρ0 (red), and n~ (blue) with a1=1/5.The solution is not inflationary, since the deceleration parameterq is positive in I1, and we may say that it is isotropic, since both anisotropic parameters take values close to zero (see Figure 4); only when a1 and a2 tend to zero (which is not physically realistic) we may observe that 𝒜≈0.6, while 𝒲2≈0.12. The gravitational entropy takes the following values: P2≈-5 in I1.Solution to BianchiVIh with CC. Plot of the parameters of anisotropy 𝒜 and 𝒲2.
(a)
(b)


### 4.2. Model with Potential (JBDU)

In this section we consider the theoretical model of the Jordan-Brans-Dicke with potential. For the metric (18), the effective stress-energy tensor (13) yields
(48)T11=8πρϕ+ω2ϕ′2ϕ2-(a′a+b′b+d′d)ϕ′ϕ+Uϕ,T41=0,T22=-8πpϕ-ϕ′′ϕ-ω2ϕ′2ϕ2-(b′b+d′d)ϕ′ϕ+Uϕ,T33=-8πpϕ-ϕ′′ϕ-ω2ϕ′2ϕ2-(a′a+d′d)ϕ′ϕ+Uϕ,T44=-8πpϕ-ϕ′′ϕ-ω2ϕ′2ϕ2-(a′a+b′b)ϕ′ϕ+Uϕ.With these FE and the conservations ones we have obtained the following solution for each of the studied cases.


#### 4.2.1. BianchiIII Solutions

For this model we have get two solutions.(1)
For the first solutions we have obtained the following results:(49)a1=a3=1,n=0,γ=a2+m2-1(a2+2)(a2-1)ϕ0=4(a2-1)(a2+1)2+m2-4,U0=2(a2-1)((a2+1)2+m2-4)2×(∑-ω(-a22+m2+1)(a22+m2-1)2-2m2(a2-1)2(-a2+m2-2)+2(a2+1)(a2+a22+1)(a2-1)3∑),ρ0=-(a2+2)2π((a2+1)2+m2-4)2×(∫ω(a22+m2-1)2+m2(3a22+m2-3)+2(a2+a22+1)(a2-1)∫)2.From the expression forγ we may obtain an exact expression for a2, getting
(50)a2=12γ(1-γ±γ(9γ+4m2-6)+1),γ≠0,a2=1-m2,ifγ=0,and therefore we obtain the following results for the parametersn~ and q:
(51)n~=-12γ(1+3γ2+(γ+1)γ(9γ+4m2-6)+1),q=-(9γ-3γ(9γ+4m2-6)+2m2-3)2(m2-3).Therefore, we begin by studying the set of points where the scale factora2 vanishes, so
(52)a2=0,ifγ=12(1-m2),and, in the same way, we calculate the set of values where the parameters(ρ0,U0,n~) vanish. The numerical analysis performed shows us that the energy density is only positive in a small tubular neighborhood with γ∈(-1/3,0) and m∈(-1,1). In the same neighborhood the potential U and the n~ parameter vanish (see Figure 5).If, for example, we setm=1/5 and the Brans-Dicke parameter, ω=4·104, we may see in Figure 6 that the solution is only valid for all γ∈(-0.3311865733,-0.3264562742)=ℰ(γc), that is, a neighborhood of a critical value of the EoS γc=-0.328, since the energy density only is positive in this neighborhood, ρ0>0, where furthermore the scalar field is also positive, ϕ0>0 (see Figure 6). In this interval the potential function (the cosmological constant) behaves as follow: U0=0 if and only if γ1=-0.3288254637, and γ2=-0.3288130902, in such a way that U0≤0,   for all γ∈(γ1,γ2), being positive in the rest of the interval. The parameter of the scalar field n~ vanishes at γ2, being n~>0 if γ<γ2 and n~<0 if γ>γ2; therefore, the effective gravitational constant decreases if γ<γ2 is constant at γ=γ2 and is growing when γ>γ2,  for all γ∈ℰ(γc).The solution is not inflationary, sinceq>0 for all γ∈(-1/3,0) and m∈(-1,1). To end, we may say that the solution isotropizes since 𝒜→0 and 𝒲2≪0.0001. In the same way we conclude that the gravitational entropy takes values very close to zero in the interval ℰ(γc).(2)
In the second solution we getU0=0. This solution is equal to the obtained one in the case of JBDΛ, so the conclusions are the same as the obtained ones in such case.Figure 5
Solution to Bianchi III withU. Plot of the parameters U0=0 (blue), ρ0=0 (red), and n~=0 (magenta).Figure 6
Solution to Bianchi III withU. Plot of the parameters U0 (blue), ρ0 (red), and n~ (magenta) when m=1/5.


#### 4.2.2. BianchiVI0 Solutions

We have obtained two solutions for this case.(1)
In the first of them we geta1=a2,  a3=1,  m=n, and U0≠0, obtaining
(53)U0=2(a22+n2-a2)(a22+n2-1)2(a2-1)×((a2-1)2((3+2ω)a22-n2)-2ωn4),ρ0=-(1+2a2)4π(a22+n2-1)2×(∫2ω(a22+n2-a2)2+3a22(a2-1)2+n2(5a22-4a2-1+2n2)∫),ϕ0=2(a2-1)a22+n2-a2,γ=2n2+a2-1(1+2a2)(a2-1).From the expression ofγ, we get a2:
(54)a2=γ+1±γ(9γ+16n2-6)+14γifγ≠0,a2=1-2n2ifγ=0,where(55)a2=0ifγ=1-2n2,and therefore(56)q=-(3(3γ-1)-3γ(9γ+16n2-6)+1+8n2)8n2-6,n~=-12γ(3γ2+1+(1+γ)γ(9γ+16n2-6)+1).As mentioned above, we need to perform a numerical analysis to analyze the solution. Withω=4·104, we find that the quantities (ρ,U,n~) only vanish in a tubular neighborhood, being the solution valid if a2∈(0,1) and n∈(-1/2,1/2) as it is shown in Figure 7.If, for example, we setn=1/3, then we may understand how each quantity behaves. The energy density is only positive, ρ0>0, in the interval  for all γ∈(0.12584,0.12879)=ℰ(γc)=I, that is, in a small neighborhood of the critical value of the EoS γc=0.1273. As we can see in Figure 8 in such interval the scalar field is also positive ϕ0>0. The potential U behaves as follows: U0=0 if and only if γ1=0.127322 and γ2=0.127325, so U0≤0,  for all γ∈(γ1,γ2)=I2, and therefore U>0,  for all γ∈I∖I2.The parametern~ vanishes at γ1 in such a way that n~>0 if γ<γ1 and n~<0 if γ>γ1; therefore, the effective gravitational constant Geff behaves as a growing time function if γ<γ1, and it is constant at γ=γ1 and is decreasing if γ>γ1,  for all γ∈ℰ(γc). The solution is not inflationary,  q>0 for all a2∈(0,1) and n∈(-1/2,1/2). With regard to the isotropization of the solution, we reach the conclusion that it isotropizes, since 𝒜→0, and 𝒲2≪0.035 for all a2∈(0,1) and n∈(-1/2,1/2). In the same way, the gravitational entropy takes values very close to zero.(2)
With regard to the second solution, we have obtainedU0=0, and therefore it is similar to the obtained one in the model JBDΛ.Figure 7
Solution to BianchiVI0 with U. Plot of the parameters U0=0 (blue), ρ0=0 (red).Figure 8
Solution to BianchiVI0 with U. Plot of the parameters Λ0 (blue), ρ0 (red), and n~ (magenta), in the interval I.


#### 4.2.3. BianchiVIh Solution

We have only obtained one solution which quite similar to the obtained one in the model JBDΛ. The solution is the following one:
(57)U0=ϕ06(2a1+2a2-1)2(3+2ω),γ=13,m=33(a2-1),n=33(a1-1),ϕ0=1a1+a2+1,ρ0=-ϕ016π×(2ω(2a1+2a2-1)2+2(-4a1-4a2+7a12+7a22+8a1a2)-1),n~=2-(γ+1)(a1+a2+1),q=3(a1+a2+1)-1.
Therefore we arrive at the same conclusions as in the previously model with Λ(ϕ).


## 4.1. Model with Cosmological Constant (JBDΛ)

For the metric (18), the effective stress-energy tensor (2) yields
(42)T11=8πρϕ+ω2ϕ′2ϕ2-(a′a+b′b+d′d)ϕ′ϕ+Λ,T41=0,T22=-8πpϕ-ϕ′′ϕ-ω2ϕ′2ϕ2-(b′b+d′d)ϕ′ϕ+Λ,T33=-8πpϕ-ϕ′′ϕ-ω2ϕ′2ϕ2-(a′a+d′d)ϕ′ϕ+Λ,T44=-8πpϕ-ϕ′′ϕ-ω2ϕ′2ϕ2-(a′a+b′b)ϕ′ϕ+Λ.
Once we have defined the field equations, we next find exact self-similar solutions for the different exposed models.


### 4.1.1. Bianchi III Solution

For this metric, we have found the following solution:(43)a2=-2γ(ω(γ-1)-1)-ω(1-γ2)+2(γ-1)∈(0,1),a1=a3=1,Λ0=0,∀γ∈(-13,0),m=-(a2-1)2+Aω∈(0,1),n=0,ρ0=-(2+a2)Bϕ08πω(ω((a2+1)2-4)-(a2-1)2+A),n~=-2γ(γ-1)(3γ-1)ω(γ+1)+2,q=-ω(3γ+1)(γ-1)+22ω(1-γ)-6γ+4,
where
(44)A=(∫(a2-1)4+ω(a2-1)2(ω(1+a2)2+2(a2+a22+1))∫)1/2,B=2(a2-1)3+ω(a2-1)×(ttttω(2ω(a2+1)2+3(a2+1)2+4)+6+(a2+1)2)+A(ω(2ω(a2+1)+5+a2)+2-2a2tttt).In the first place, we would like to stress that the cosmological constant vanishes, so this model does not describe its possible variation. By fixing the Brans-Dicke parameter,ω=4·104 [32], we may see that the solution is valid  for all γ∈(-1/3,0), since the energy density ρ behaves as a positive decreasing time function for these values of the EoS (we have set ϕ0>0 for physical reasons). The solution is not inflationary for all γ∈(-1/3,0), precisely q<0 only if γ<-1/3. With regard to the anisotropic parameters, we find that the solution cannot be considered as isotropic with regard to the first anisotropic parameter, since 𝒜=(a2-1)2/(4a2-1)2∈[0,5/2), for all γ∈(-1/3,0), that is, 𝒜≪1; note that when γ tends to zero, 𝒜 takes the value 5/2. Nevertheless, we may say that the solution is isotropic with regard to the second anisotropic parameter, since 𝒲2≪0.0004,  for all γ∈(-1/3,0). The gravitational entropy P2∈[0,0.2); it does not show any pathological behaviour and takes values close to zero   for all γ∈(-1/3,0). To end the analysis of this solution, the effective gravitational constant behaves as a time decreasing function   for all γ∈(-1/3,0), since for these values the parameter n~ is positive.


### 4.1.2. BianchiVI0 Solution

By solving the associated algebraic system to the FE we have found the following solution:(45)a1=a2=-1-γ+ω(γ+1)22(γ2-1)ω+γ-3∈(0,1),m=n=12A-(a2-1)2ω∈(0,12),ρ0=-ϕ0(1+2a2)B8πω(A+4(a22-1)ω-(a2-1)2),n~=8γω(γ+1)-(3γ-1)(γ-1)2ω(γ2-1)+γ-3,q=2(-ω(γ+1)(3γ-1)+2)4ω(γ+1)-3γ+5,Λ0=0,∀γ∈(-13,1),
with
(46)A=(a2-1)2(8a22ω(2ω+3)+(a2-1)2),B=(a2-1)3+A(2ω(2a2+2ωa2+1)+1-a2)+2ω(a2-1)×(2ωa2(7a2+4ωa2+1)+a2(4a2+1)+1).As we may observe, we have again obtainedΛ0=0, which is not in agreement with the current observations and which is worse, and we cannot describe its possible time variation. The solution is only valid   for all γ∈(-1/3,1)=I, since for these values of the EoS, the energy density is positive, that is, ρ0>0. By setting the Brans-Dicke parameter, ω=4·104, we find that the n~ parameter vanishes at γc=1/3, being positive, n~>0,   for all γ∈(-1/3,1/3); therefore, the effective gravitational constant Geff is decreasing when γ<1/3,   for all γ∈I, constant at γc=1/3, and growing when   for all γ∈(1/3,1). The solution is not inflationary, since the deceleration parameter is always positive, q>0   for all γ∈I. Nevertheless we may conclude that the solution is isotropic, since both anisotropic parameters are close to zero, 𝒜∈(0,1), in such a way that 𝒜→1 only when γ→1, while 𝒲2≪0.01,   for all γ∈I. However, the gravitational entropy only takes values close to zero in a small neighborhood of γ=-1/3, tending to -∞ in the rest of the interval.


### 4.1.3. BianchiVIh Solution

In this case we have obtained the following solution:(47)Λ0=13[12+13ω+2(1+2ω3)((a1+a2)2-a1-a2)],m=33(a2-1),n=33(a1-1),ϕ0=1,ρ0=-148π×(2ω(-4a1-4a2+4(a12+2a22+2a1a2)+1)+2(7a12+7a22+8a1a2)-8(a2+a1)-1),n~=2-(γ+1)(a1+a2+1),q=3(a1+a2+1)-1,
being only valid for γ=1/3; note that for this value the wave equation for the scalar field is easier to solve than in the previously cases, since the trace of the matter stress-energy tensor vanishes. In this solution the cosmological constant takes values different than zero, that is, Λ0≠0, so we are able to describe its behaviour. We have expressed the solution depending on two parameters, a1 and a2; therefore it is necessary to carry out a numerical analysis in order to interpreted it. As usual, we start by fixing the numerical value of the Brans-Dicke parameter, ω=4·104, and we consider that the parameters of the scale factors a1 and a2 vary in the interval (0,1). In the first place we have calculated the set of values where the parameters (ρ0,Λ0,n~) vanish. In Figure 1 we have plotted this situation finding that the solution is only valid within this small tubular neighborhood. In Figure 2, we have detailed how behave each parameter Λ0,  ρ0, and n~. As we can see, the energy density is positive in this small tubular neighborhood where furthermore the cosmological constant is also positive expect in a very small subinterval where it is negative and vanishes (see Figure 2).Figure 1
Solution to BianchiVIh with CC. Plot of the parameters Λ0=0 (blue) and ρ0=0 (red).Figure 2
Solution to BianchiVIh with CC. Tubular neighborhood. Plot of the parameters Λ0 (green), ρ0 (red), and n~ (blue). The plane z=0 (gray).If, for example, we seta1=1/5, then we have obtained the following results (see Figure 3). The energy density parameter ρ0 is only positive, ρ0>0, if and only if a2∈(0.2973605134,0.3026282371)=I1, while the cosmological constant is negative, Λ0<0, if and only if a2∈(0.2999893937,0.3000106065)=I2, note that I2⊂I1, finding that, Λ0≥0,  for all a2∈I1∖I2. In the same way, we may see that the parameter of the scalar field, n~, only vanishes, n~=0, if a2=0.3000106065. In this way we may arrive at the conclusion that the effective gravitational constant Geff is decreasing if a2∈(0.2973605134,0.3000106065), constant at a2=0.3000106065, where furthermore Λ0=0 and G=G0, and if a2∈(0.3000106065,0.3026282371), then Geff is growing and Λ0>0.Figure 3
Solution to BianchiVIh with CC. Plot of the parameters Λ0 (green), ρ0 (red), and n~ (blue) with a1=1/5.The solution is not inflationary, since the deceleration parameterq is positive in I1, and we may say that it is isotropic, since both anisotropic parameters take values close to zero (see Figure 4); only when a1 and a2 tend to zero (which is not physically realistic) we may observe that 𝒜≈0.6, while 𝒲2≈0.12. The gravitational entropy takes the following values: P2≈-5 in I1.Solution to BianchiVIh with CC. Plot of the parameters of anisotropy 𝒜 and 𝒲2.
(a)
(b)


## 4.1.1. Bianchi III Solution

For this metric, we have found the following solution:(43)a2=-2γ(ω(γ-1)-1)-ω(1-γ2)+2(γ-1)∈(0,1),a1=a3=1,Λ0=0,∀γ∈(-13,0),m=-(a2-1)2+Aω∈(0,1),n=0,ρ0=-(2+a2)Bϕ08πω(ω((a2+1)2-4)-(a2-1)2+A),n~=-2γ(γ-1)(3γ-1)ω(γ+1)+2,q=-ω(3γ+1)(γ-1)+22ω(1-γ)-6γ+4,
where
(44)A=(∫(a2-1)4+ω(a2-1)2(ω(1+a2)2+2(a2+a22+1))∫)1/2,B=2(a2-1)3+ω(a2-1)×(ttttω(2ω(a2+1)2+3(a2+1)2+4)+6+(a2+1)2)+A(ω(2ω(a2+1)+5+a2)+2-2a2tttt).In the first place, we would like to stress that the cosmological constant vanishes, so this model does not describe its possible variation. By fixing the Brans-Dicke parameter,ω=4·104 [32], we may see that the solution is valid  for all γ∈(-1/3,0), since the energy density ρ behaves as a positive decreasing time function for these values of the EoS (we have set ϕ0>0 for physical reasons). The solution is not inflationary for all γ∈(-1/3,0), precisely q<0 only if γ<-1/3. With regard to the anisotropic parameters, we find that the solution cannot be considered as isotropic with regard to the first anisotropic parameter, since 𝒜=(a2-1)2/(4a2-1)2∈[0,5/2), for all γ∈(-1/3,0), that is, 𝒜≪1; note that when γ tends to zero, 𝒜 takes the value 5/2. Nevertheless, we may say that the solution is isotropic with regard to the second anisotropic parameter, since 𝒲2≪0.0004,  for all γ∈(-1/3,0). The gravitational entropy P2∈[0,0.2); it does not show any pathological behaviour and takes values close to zero   for all γ∈(-1/3,0). To end the analysis of this solution, the effective gravitational constant behaves as a time decreasing function   for all γ∈(-1/3,0), since for these values the parameter n~ is positive.


## 4.1.2. BianchiVI0 Solution

By solving the associated algebraic system to the FE we have found the following solution:(45)a1=a2=-1-γ+ω(γ+1)22(γ2-1)ω+γ-3∈(0,1),m=n=12A-(a2-1)2ω∈(0,12),ρ0=-ϕ0(1+2a2)B8πω(A+4(a22-1)ω-(a2-1)2),n~=8γω(γ+1)-(3γ-1)(γ-1)2ω(γ2-1)+γ-3,q=2(-ω(γ+1)(3γ-1)+2)4ω(γ+1)-3γ+5,Λ0=0,∀γ∈(-13,1),
with
(46)A=(a2-1)2(8a22ω(2ω+3)+(a2-1)2),B=(a2-1)3+A(2ω(2a2+2ωa2+1)+1-a2)+2ω(a2-1)×(2ωa2(7a2+4ωa2+1)+a2(4a2+1)+1).As we may observe, we have again obtainedΛ0=0, which is not in agreement with the current observations and which is worse, and we cannot describe its possible time variation. The solution is only valid   for all γ∈(-1/3,1)=I, since for these values of the EoS, the energy density is positive, that is, ρ0>0. By setting the Brans-Dicke parameter, ω=4·104, we find that the n~ parameter vanishes at γc=1/3, being positive, n~>0,   for all γ∈(-1/3,1/3); therefore, the effective gravitational constant Geff is decreasing when γ<1/3,   for all γ∈I, constant at γc=1/3, and growing when   for all γ∈(1/3,1). The solution is not inflationary, since the deceleration parameter is always positive, q>0   for all γ∈I. Nevertheless we may conclude that the solution is isotropic, since both anisotropic parameters are close to zero, 𝒜∈(0,1), in such a way that 𝒜→1 only when γ→1, while 𝒲2≪0.01,   for all γ∈I. However, the gravitational entropy only takes values close to zero in a small neighborhood of γ=-1/3, tending to -∞ in the rest of the interval.


## 4.1.3. BianchiVIh Solution

In this case we have obtained the following solution:(47)Λ0=13[12+13ω+2(1+2ω3)((a1+a2)2-a1-a2)],m=33(a2-1),n=33(a1-1),ϕ0=1,ρ0=-148π×(2ω(-4a1-4a2+4(a12+2a22+2a1a2)+1)+2(7a12+7a22+8a1a2)-8(a2+a1)-1),n~=2-(γ+1)(a1+a2+1),q=3(a1+a2+1)-1,
being only valid for γ=1/3; note that for this value the wave equation for the scalar field is easier to solve than in the previously cases, since the trace of the matter stress-energy tensor vanishes. In this solution the cosmological constant takes values different than zero, that is, Λ0≠0, so we are able to describe its behaviour. We have expressed the solution depending on two parameters, a1 and a2; therefore it is necessary to carry out a numerical analysis in order to interpreted it. As usual, we start by fixing the numerical value of the Brans-Dicke parameter, ω=4·104, and we consider that the parameters of the scale factors a1 and a2 vary in the interval (0,1). In the first place we have calculated the set of values where the parameters (ρ0,Λ0,n~) vanish. In Figure 1 we have plotted this situation finding that the solution is only valid within this small tubular neighborhood. In Figure 2, we have detailed how behave each parameter Λ0,  ρ0, and n~. As we can see, the energy density is positive in this small tubular neighborhood where furthermore the cosmological constant is also positive expect in a very small subinterval where it is negative and vanishes (see Figure 2).Figure 1
Solution to BianchiVIh with CC. Plot of the parameters Λ0=0 (blue) and ρ0=0 (red).Figure 2
Solution to BianchiVIh with CC. Tubular neighborhood. Plot of the parameters Λ0 (green), ρ0 (red), and n~ (blue). The plane z=0 (gray).If, for example, we seta1=1/5, then we have obtained the following results (see Figure 3). The energy density parameter ρ0 is only positive, ρ0>0, if and only if a2∈(0.2973605134,0.3026282371)=I1, while the cosmological constant is negative, Λ0<0, if and only if a2∈(0.2999893937,0.3000106065)=I2, note that I2⊂I1, finding that, Λ0≥0,  for all a2∈I1∖I2. In the same way, we may see that the parameter of the scalar field, n~, only vanishes, n~=0, if a2=0.3000106065. In this way we may arrive at the conclusion that the effective gravitational constant Geff is decreasing if a2∈(0.2973605134,0.3000106065), constant at a2=0.3000106065, where furthermore Λ0=0 and G=G0, and if a2∈(0.3000106065,0.3026282371), then Geff is growing and Λ0>0.Figure 3
Solution to BianchiVIh with CC. Plot of the parameters Λ0 (green), ρ0 (red), and n~ (blue) with a1=1/5.The solution is not inflationary, since the deceleration parameterq is positive in I1, and we may say that it is isotropic, since both anisotropic parameters take values close to zero (see Figure 4); only when a1 and a2 tend to zero (which is not physically realistic) we may observe that 𝒜≈0.6, while 𝒲2≈0.12. The gravitational entropy takes the following values: P2≈-5 in I1.Solution to BianchiVIh with CC. Plot of the parameters of anisotropy 𝒜 and 𝒲2.
(a)
(b)


## 4.2. Model with Potential (JBDU)

In this section we consider the theoretical model of the Jordan-Brans-Dicke with potential. For the metric (18), the effective stress-energy tensor (13) yields
(48)T11=8πρϕ+ω2ϕ′2ϕ2-(a′a+b′b+d′d)ϕ′ϕ+Uϕ,T41=0,T22=-8πpϕ-ϕ′′ϕ-ω2ϕ′2ϕ2-(b′b+d′d)ϕ′ϕ+Uϕ,T33=-8πpϕ-ϕ′′ϕ-ω2ϕ′2ϕ2-(a′a+d′d)ϕ′ϕ+Uϕ,T44=-8πpϕ-ϕ′′ϕ-ω2ϕ′2ϕ2-(a′a+b′b)ϕ′ϕ+Uϕ.With these FE and the conservations ones we have obtained the following solution for each of the studied cases.


### 4.2.1. BianchiIII Solutions

For this model we have get two solutions.(1)
For the first solutions we have obtained the following results:(49)a1=a3=1,n=0,γ=a2+m2-1(a2+2)(a2-1)ϕ0=4(a2-1)(a2+1)2+m2-4,U0=2(a2-1)((a2+1)2+m2-4)2×(∑-ω(-a22+m2+1)(a22+m2-1)2-2m2(a2-1)2(-a2+m2-2)+2(a2+1)(a2+a22+1)(a2-1)3∑),ρ0=-(a2+2)2π((a2+1)2+m2-4)2×(∫ω(a22+m2-1)2+m2(3a22+m2-3)+2(a2+a22+1)(a2-1)∫)2.From the expression forγ we may obtain an exact expression for a2, getting
(50)a2=12γ(1-γ±γ(9γ+4m2-6)+1),γ≠0,a2=1-m2,ifγ=0,and therefore we obtain the following results for the parametersn~ and q:
(51)n~=-12γ(1+3γ2+(γ+1)γ(9γ+4m2-6)+1),q=-(9γ-3γ(9γ+4m2-6)+2m2-3)2(m2-3).Therefore, we begin by studying the set of points where the scale factora2 vanishes, so
(52)a2=0,ifγ=12(1-m2),and, in the same way, we calculate the set of values where the parameters(ρ0,U0,n~) vanish. The numerical analysis performed shows us that the energy density is only positive in a small tubular neighborhood with γ∈(-1/3,0) and m∈(-1,1). In the same neighborhood the potential U and the n~ parameter vanish (see Figure 5).If, for example, we setm=1/5 and the Brans-Dicke parameter, ω=4·104, we may see in Figure 6 that the solution is only valid for all γ∈(-0.3311865733,-0.3264562742)=ℰ(γc), that is, a neighborhood of a critical value of the EoS γc=-0.328, since the energy density only is positive in this neighborhood, ρ0>0, where furthermore the scalar field is also positive, ϕ0>0 (see Figure 6). In this interval the potential function (the cosmological constant) behaves as follow: U0=0 if and only if γ1=-0.3288254637, and γ2=-0.3288130902, in such a way that U0≤0,   for all γ∈(γ1,γ2), being positive in the rest of the interval. The parameter of the scalar field n~ vanishes at γ2, being n~>0 if γ<γ2 and n~<0 if γ>γ2; therefore, the effective gravitational constant decreases if γ<γ2 is constant at γ=γ2 and is growing when γ>γ2,  for all γ∈ℰ(γc).The solution is not inflationary, sinceq>0 for all γ∈(-1/3,0) and m∈(-1,1). To end, we may say that the solution isotropizes since 𝒜→0 and 𝒲2≪0.0001. In the same way we conclude that the gravitational entropy takes values very close to zero in the interval ℰ(γc).(2)
In the second solution we getU0=0. This solution is equal to the obtained one in the case of JBDΛ, so the conclusions are the same as the obtained ones in such case.Figure 5
Solution to Bianchi III withU. Plot of the parameters U0=0 (blue), ρ0=0 (red), and n~=0 (magenta).Figure 6
Solution to Bianchi III withU. Plot of the parameters U0 (blue), ρ0 (red), and n~ (magenta) when m=1/5.


### 4.2.2. BianchiVI0 Solutions

We have obtained two solutions for this case.(1)
In the first of them we geta1=a2,  a3=1,  m=n, and U0≠0, obtaining
(53)U0=2(a22+n2-a2)(a22+n2-1)2(a2-1)×((a2-1)2((3+2ω)a22-n2)-2ωn4),ρ0=-(1+2a2)4π(a22+n2-1)2×(∫2ω(a22+n2-a2)2+3a22(a2-1)2+n2(5a22-4a2-1+2n2)∫),ϕ0=2(a2-1)a22+n2-a2,γ=2n2+a2-1(1+2a2)(a2-1).From the expression ofγ, we get a2:
(54)a2=γ+1±γ(9γ+16n2-6)+14γifγ≠0,a2=1-2n2ifγ=0,where(55)a2=0ifγ=1-2n2,and therefore(56)q=-(3(3γ-1)-3γ(9γ+16n2-6)+1+8n2)8n2-6,n~=-12γ(3γ2+1+(1+γ)γ(9γ+16n2-6)+1).As mentioned above, we need to perform a numerical analysis to analyze the solution. Withω=4·104, we find that the quantities (ρ,U,n~) only vanish in a tubular neighborhood, being the solution valid if a2∈(0,1) and n∈(-1/2,1/2) as it is shown in Figure 7.If, for example, we setn=1/3, then we may understand how each quantity behaves. The energy density is only positive, ρ0>0, in the interval  for all γ∈(0.12584,0.12879)=ℰ(γc)=I, that is, in a small neighborhood of the critical value of the EoS γc=0.1273. As we can see in Figure 8 in such interval the scalar field is also positive ϕ0>0. The potential U behaves as follows: U0=0 if and only if γ1=0.127322 and γ2=0.127325, so U0≤0,  for all γ∈(γ1,γ2)=I2, and therefore U>0,  for all γ∈I∖I2.The parametern~ vanishes at γ1 in such a way that n~>0 if γ<γ1 and n~<0 if γ>γ1; therefore, the effective gravitational constant Geff behaves as a growing time function if γ<γ1, and it is constant at γ=γ1 and is decreasing if γ>γ1,  for all γ∈ℰ(γc). The solution is not inflationary,  q>0 for all a2∈(0,1) and n∈(-1/2,1/2). With regard to the isotropization of the solution, we reach the conclusion that it isotropizes, since 𝒜→0, and 𝒲2≪0.035 for all a2∈(0,1) and n∈(-1/2,1/2). In the same way, the gravitational entropy takes values very close to zero.(2)
With regard to the second solution, we have obtainedU0=0, and therefore it is similar to the obtained one in the model JBDΛ.Figure 7
Solution to BianchiVI0 with U. Plot of the parameters U0=0 (blue), ρ0=0 (red).Figure 8
Solution to BianchiVI0 with U. Plot of the parameters Λ0 (blue), ρ0 (red), and n~ (magenta), in the interval I.


### 4.2.3. BianchiVIh Solution

We have only obtained one solution which quite similar to the obtained one in the model JBDΛ. The solution is the following one:
(57)U0=ϕ06(2a1+2a2-1)2(3+2ω),γ=13,m=33(a2-1),n=33(a1-1),ϕ0=1a1+a2+1,ρ0=-ϕ016π×(2ω(2a1+2a2-1)2+2(-4a1-4a2+7a12+7a22+8a1a2)-1),n~=2-(γ+1)(a1+a2+1),q=3(a1+a2+1)-1.
Therefore we arrive at the same conclusions as in the previously model with Λ(ϕ).


## 4.2.1. BianchiIII Solutions

For this model we have get two solutions.(1)
For the first solutions we have obtained the following results:(49)a1=a3=1,n=0,γ=a2+m2-1(a2+2)(a2-1)ϕ0=4(a2-1)(a2+1)2+m2-4,U0=2(a2-1)((a2+1)2+m2-4)2×(∑-ω(-a22+m2+1)(a22+m2-1)2-2m2(a2-1)2(-a2+m2-2)+2(a2+1)(a2+a22+1)(a2-1)3∑),ρ0=-(a2+2)2π((a2+1)2+m2-4)2×(∫ω(a22+m2-1)2+m2(3a22+m2-3)+2(a2+a22+1)(a2-1)∫)2.From the expression forγ we may obtain an exact expression for a2, getting
(50)a2=12γ(1-γ±γ(9γ+4m2-6)+1),γ≠0,a2=1-m2,ifγ=0,and therefore we obtain the following results for the parametersn~ and q:
(51)n~=-12γ(1+3γ2+(γ+1)γ(9γ+4m2-6)+1),q=-(9γ-3γ(9γ+4m2-6)+2m2-3)2(m2-3).Therefore, we begin by studying the set of points where the scale factora2 vanishes, so
(52)a2=0,ifγ=12(1-m2),and, in the same way, we calculate the set of values where the parameters(ρ0,U0,n~) vanish. The numerical analysis performed shows us that the energy density is only positive in a small tubular neighborhood with γ∈(-1/3,0) and m∈(-1,1). In the same neighborhood the potential U and the n~ parameter vanish (see Figure 5).If, for example, we setm=1/5 and the Brans-Dicke parameter, ω=4·104, we may see in Figure 6 that the solution is only valid for all γ∈(-0.3311865733,-0.3264562742)=ℰ(γc), that is, a neighborhood of a critical value of the EoS γc=-0.328, since the energy density only is positive in this neighborhood, ρ0>0, where furthermore the scalar field is also positive, ϕ0>0 (see Figure 6). In this interval the potential function (the cosmological constant) behaves as follow: U0=0 if and only if γ1=-0.3288254637, and γ2=-0.3288130902, in such a way that U0≤0,   for all γ∈(γ1,γ2), being positive in the rest of the interval. The parameter of the scalar field n~ vanishes at γ2, being n~>0 if γ<γ2 and n~<0 if γ>γ2; therefore, the effective gravitational constant decreases if γ<γ2 is constant at γ=γ2 and is growing when γ>γ2,  for all γ∈ℰ(γc).The solution is not inflationary, sinceq>0 for all γ∈(-1/3,0) and m∈(-1,1). To end, we may say that the solution isotropizes since 𝒜→0 and 𝒲2≪0.0001. In the same way we conclude that the gravitational entropy takes values very close to zero in the interval ℰ(γc).(2)
In the second solution we getU0=0. This solution is equal to the obtained one in the case of JBDΛ, so the conclusions are the same as the obtained ones in such case.Figure 5
Solution to Bianchi III withU. Plot of the parameters U0=0 (blue), ρ0=0 (red), and n~=0 (magenta).Figure 6
Solution to Bianchi III withU. Plot of the parameters U0 (blue), ρ0 (red), and n~ (magenta) when m=1/5.


## 4.2.2. BianchiVI0 Solutions

We have obtained two solutions for this case.(1)
In the first of them we geta1=a2,  a3=1,  m=n, and U0≠0, obtaining
(53)U0=2(a22+n2-a2)(a22+n2-1)2(a2-1)×((a2-1)2((3+2ω)a22-n2)-2ωn4),ρ0=-(1+2a2)4π(a22+n2-1)2×(∫2ω(a22+n2-a2)2+3a22(a2-1)2+n2(5a22-4a2-1+2n2)∫),ϕ0=2(a2-1)a22+n2-a2,γ=2n2+a2-1(1+2a2)(a2-1).From the expression ofγ, we get a2:
(54)a2=γ+1±γ(9γ+16n2-6)+14γifγ≠0,a2=1-2n2ifγ=0,where(55)a2=0ifγ=1-2n2,and therefore(56)q=-(3(3γ-1)-3γ(9γ+16n2-6)+1+8n2)8n2-6,n~=-12γ(3γ2+1+(1+γ)γ(9γ+16n2-6)+1).As mentioned above, we need to perform a numerical analysis to analyze the solution. Withω=4·104, we find that the quantities (ρ,U,n~) only vanish in a tubular neighborhood, being the solution valid if a2∈(0,1) and n∈(-1/2,1/2) as it is shown in Figure 7.If, for example, we setn=1/3, then we may understand how each quantity behaves. The energy density is only positive, ρ0>0, in the interval  for all γ∈(0.12584,0.12879)=ℰ(γc)=I, that is, in a small neighborhood of the critical value of the EoS γc=0.1273. As we can see in Figure 8 in such interval the scalar field is also positive ϕ0>0. The potential U behaves as follows: U0=0 if and only if γ1=0.127322 and γ2=0.127325, so U0≤0,  for all γ∈(γ1,γ2)=I2, and therefore U>0,  for all γ∈I∖I2.The parametern~ vanishes at γ1 in such a way that n~>0 if γ<γ1 and n~<0 if γ>γ1; therefore, the effective gravitational constant Geff behaves as a growing time function if γ<γ1, and it is constant at γ=γ1 and is decreasing if γ>γ1,  for all γ∈ℰ(γc). The solution is not inflationary,  q>0 for all a2∈(0,1) and n∈(-1/2,1/2). With regard to the isotropization of the solution, we reach the conclusion that it isotropizes, since 𝒜→0, and 𝒲2≪0.035 for all a2∈(0,1) and n∈(-1/2,1/2). In the same way, the gravitational entropy takes values very close to zero.(2)
With regard to the second solution, we have obtainedU0=0, and therefore it is similar to the obtained one in the model JBDΛ.Figure 7
Solution to BianchiVI0 with U. Plot of the parameters U0=0 (blue), ρ0=0 (red).Figure 8
Solution to BianchiVI0 with U. Plot of the parameters Λ0 (blue), ρ0 (red), and n~ (magenta), in the interval I.


## 4.2.3. BianchiVIh Solution

We have only obtained one solution which quite similar to the obtained one in the model JBDΛ. The solution is the following one:
(57)U0=ϕ06(2a1+2a2-1)2(3+2ω),γ=13,m=33(a2-1),n=33(a1-1),ϕ0=1a1+a2+1,ρ0=-ϕ016π×(2ω(2a1+2a2-1)2+2(-4a1-4a2+7a12+7a22+8a1a2)-1),n~=2-(γ+1)(a1+a2+1),q=3(a1+a2+1)-1.
Therefore we arrive at the same conclusions as in the previously model with Λ(ϕ).


## 5. Conclusions

We have studied how may vary the “constants”G and Λ (or U) in two different scalar-tensor models: the JBD model with Λ and the JBD model with potential U. Under the hypothesis of self-similarity we have been able to obtain exact solutions for the Bianchi VI geometry (including types III, VI0, and VIh), which allow us to compare them. We have reached the conclusion that the possible variation of the cosmological constant is better described within the JBDU theoretical framework, since with the JBDΛ model Λ usually vanishes. We have summarized in Table 1 the set of solutions that we have found for both models, JBDΛ and JBDU.Table 1
Summary of the solutions.
Model
Λ
U
III
Λ
=
0, G↘, γ∈(-1/3,0)
⇔
U
=
0, (case Λ)
III
U
>
0, G~G(γc,m)γ∈(-1/3,0)𝒜
V
I
0
Λ
=
0, G~G(γ), γ∈(-1/3,1)𝒜
⇔
U
=
0, (case Λ)
V
I
0
U
>
0, G~G(a2,n)γ∈(-1/3,1)𝒜
V
I
h
Λ
>
0, G~G(a1,a2), γ=1/3𝒜
⇔
U
>
0, G~G(a1,a2), γ=1/3𝒜With regard to the first of the studied models, the JBDΛ, we have been able to obtain a solution for each geometry. The first of them (Bianchi III) is only valid for all γ∈(-1/3,0) and predicts that the cosmological constant vanishes, Λ=0, which is in contradiction with the current observations. Nevertheless we have studied this solution reaching the conclusion that it is noninflationary and does not isotropize with respect to the first anisotropic parameter 𝒜. The effective gravitational constant Geff is always decreasing for all γ∈(-1/3,0). In the case of the Bianchi type VI0 we have found that the cosmological constant is also zero, Λ=0, but the solution is valid for all γ∈(-1/3,1). The effective gravitational constant Geff shows a different behaviour depending on the EoS γ. It may be growing if γ>1/3, constant at γ=1/3, and it behaves as a decreasing time function if γ<1/3, where γ∈(-1/3,1). The solution isotropizes, and it is noninflationary. The solution obtained for the model Bianchi type VIh has a nonzero cosmological constant, Λ≠0, but the solution is only valid for a particular value of the EoS, γ=1/3. It is also noninflationary, and it isotropizes. The numerical analysis carryout shows us that the solution is only valid in a small tubular neighborhood, where a1,a2∈(0,1/2) in fact the amplitude of this neighborhood depends on the value of the Brans-Dicke parameter ω in such a way that if future observations fix an upper limit for this constant then the diameter of this tubular neighborhood will be small. In this tubular neighborhood, the energy density is positive as well as the cosmological constant Λ. However, there is a really small subtubular neighborhood where Λ vanishes, or it is negative. In this way we have concluded that the solution predicts a Λ>0 (except within this subtubular neighborhood) in agreement with the observations, while the effective gravitational constant may be growing, constant at a critical point γc (where the parameter n~ vanishes), and decreasing.With regard to the model with potential, JBDU, we have obtained solutions similar to the obtained ones in the model JBDΛ, and therefore we have reached the same conclusions. Nevertheless we have also obtained other solutions which predict a nonvanishing cosmological constant, in this case U≠0, for the geometries Bianchi types III and VI0. In the case of the Bianchi type III, the solution is only valid in a small tubular neighborhood, where γ∈(-1/3,0) and m∈(-1,1). In this tubular neighborhood the energy density is positive, and the solution predicts that the potential function U (that mimics the behaviour of the cosmological constant) is also positive except in a very small subtubular neighborhood where it is negative or vanishes. Within this neighborhood the effective gravitational constant may be growing, constant at a critical point γc (where the parameter n~ vanishes), or decreasing depending of the values of γ and n. To end, the solution found for the model Bianchi type VI0 with U≠0 is only valid in a small tubular neighborhood, where a2∈(0,1) and n∈(-1/2,1/2), it is noninflationary and isotropizes. In this neighborhood the quantities ρ,  U, and Geff show similar behaviour as the described ones in this kind of solutions, that is, ρ>0,  U>0 (except in a small subtubular neighborhood), and Geff is growing, constant, or decreasing depending on the parameters a2 and n.



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*Source: 491905-2013-06-05.xml*