Cosmic anisotropic doomsday in Bianchi type I universes

# Cosmic anisotropic doomsday in Bianchi type I universes

Mauricio Cataldo mcataldo@ubiobio.cl    Antonella Cid acidm@ubiobio.cl    Pedro Labraña plabrana@ubiobio.cl Departamento de Física, Universidad del Bío–Bío, Avenida Collao 1202, Casilla 5-C, Concepción, Chile, and
Grupo de Cosmología y Gravitación-UBB
Patricio Mella patricio.mella@uach.cl Centro de Docencia de Ciencias Básicas para Ingeniería, Facultad de Ciencias de la Ingeniería, Universidad Austral de Chile, Casilla 567, Valdivia, Chile, and
Grupo de Cosmología y Gravitación-UBB
July 23, 2019
###### Abstract

Abstract: In this paper we study finite time future singularities in anisotropic Bianchi type I models. It is shown that there exist future singularities similar to Big Rip ones (which appear in the framework of phantom Friedmann-Robertson-Walker cosmologies). Specifically, in an ellipsoidal anisotropic scenario or in a fully anisotropic scenario, the three directional and average scale factors may diverge at a finite future time, together with energy densities and anisotropic pressures. We call these singularities “Anisotropic Big Rip Singularities”. We show that there also exist Bianchi type I models filled with matter, where one or two directional scale factors may diverge. Another type of future anisotropic singularities is shown to be present in vacuum cosmologies, i.e. Kasner spacetimes. These singularities are induced by the shear scalar, which also blows up at a finite time. We call such a singularity “Vacuum Rip”. In this case one directional scale factor blows up, while the other two and average scale factors tend to zero.

###### pacs:
98.80.Cq, 04.30.Nk, 98.70.Vc

## I Introduction

The astrophysical observations howell () give evidence that our Universe is currently in accelerated expansion. In the context of Einstein General Relativity this acceleration is driven by an unknown fluid called dark energy Copeland (), usually described by a state parameter , with . This corresponds to quintessence matter which violates the strong energy condition, and the range to phantom matter, which violates the strong and dominant energy conditions. In this latter case we could have the scale factor, and going to infinity at a finite cosmic time in the future. This type of singularity is dubbed Big Rip Caldwell:2003vq (). This possibility is allowed for isotropic and homogeneous Friedmann-Robertson-Walker (FRW) models by current observational data alam ().

In Big Rip scenarios the curvature invariants , , diverge in the same way as occur in the Big-Bang and Big-Crunch singularities Dabrowskisea (); stoica (). However, in the framework of FRW cosmologies there are different sorts of finite time future singularities. According to Ref. Nojiri (); type5 () the future singularities can be classified in the following types:

Type (“Big Rip”) : For , , and .

Type (“Sudden”) : For , , and .

Type (“Big Freeze”) : For , , and .

Type (“Generalized sudden”): For , , , and higher derivatives of diverge.

Type (“w-singularities”): For , , , , and higher derivatives of are regular.

The quantities , , and are constants.

The type II singularity has been studied by several authors sudden (); suddenb (); type4b (); Dabrowski:2004bz () and includes the subcases of the Big Brake and Big Boost bbbb (). In Ref. fjl () it was shown that for this type of singularity the universe can be extended after the singular event. The type III, IV and V have been studied in Refs. type3 (), Dabrowskisea (); Nojiri () and type5 (), respectively.

There are other types of future singularities that can appear at a finite time, even when the strong energy condition is satisfied fjl (); barrow (); suddenb (); bbbb (). Other interesting types of future singularities, but appearing at an infinite time, are Little Rip lr (), Pseudo-Rip sr () and Little Sibling Rip of Big Rip srbr (). It is noteworthy to mention that an attempt to unify future singular behaviors was made in Ref. grcr (), where the authors introduce the Grand Rip and Grand Bang/Crunch singularities.

It is interesting to note that phantom fields are not the only way to generate scenarios with Big Rip. Such future singularities may be induced, for instance, by fluids with an inhomogeneous equation of state ieos () or interacting coupled fluids cfr (). From the viewpoint of viscous cosmological models, in Ref. cat () it was shown that the bulk viscosity induces a Big Rip singularity, and in Ref. bre () it was studied a Little Rip as a purely viscous effect. Inhomogeneous and spherically symmetric gravitational fields, describing evolving wormholes also may exhibit a Big Rip singularity during its evolution wcat (). Notice that in Ref. Dabrowski:2004bz () an anisotropic and inhomogeneous cosmology of Stephani type was found to possess finite-time sudden singularities, and in Ref. type4b () specific examples of anisotropic sudden singularities in Bianchi type VII universes were constructed.

In this paper we extend the study of future singularities by considering anisotropic and homogeneous spacetimes more general than flat FRW ones. Specifically, we analyze Bianchi type I cosmologies, allowing us to show that the anisotropy of spacetime, by means of the shear scalar, may induce future singularities at a finite time, similar to Big Rip ones appearing in the framework of phantom FRW cosmologies. In order to make analytical progress on this topic we shall use some known exact Bianchi type I solutions of the Einstein equations, allowing us to handle exact expressions for directional scale factors , shear scalar , energy density and anisotropic pressures .

Our motivation is based on the fact that several studies on the plausibility of anisotropy in the accelerated expanding universe have been performed in the framework of anisotropic dark energy cosmological models. In Ref. Campanelli () authors found that, in the framework of Bianchi I cosmological models, anisotropy is permitted both in the geometry of the universe and in the dark energy equation of state. Additionally, it is worth to mention that an anisotropic dark energy model can potentially solve the CMB low-quadrupole problem Rodriguez ().

The paper is organized as follows. In Sec. II we write the Einstein equations for Bianchi type I spacetimes. In Sec. III we discuss the Kasner vacuum solution and the future singularities which may appear during its evolution. In Sec. IV we find future singularities in anisotropic Bianchi type I models filled with a stiff fluid. In Sec. V we obtain exact solutions for ellipsoidal universes characterized by a shear scalar proportional to the expansion scalar, and filled with isotropic and anisotropic matter sources. We show that these spacetimes may exhibit anisotropic rip singularities. In Sec. VI we discuss future singularities in fully anisotropic Bianchi I cosmologies filled with an anisotropic, matter source. In Sec. VII we discuss our results.

## Ii Bianchi type I spacetimes and Einstein Field Equations

In this paper we consider models belonging to spatially homogeneous and anisotropic Bianchi type I spacetimes described by the metric

 ds2=dt2−a21(t)dx2−a22(t)dy2−a23(t)dz2, (1)

where are the directional scale factors along the axes, respectively.

This type of cosmologies is particularly interesting because it is the simplest generalization of the homogeneous and isotropic flat FRW models.

The Einstein field equations for this metric may be written in the following form Chimento ():

 3H2=κρ+σ22, (2) −2˙H=κ(ρ+p)+σ2, (3) ˙ρ+3H(ρ+p)=→σ⋅→Σ, (4) ˙→σ+3H→σ=→Σ, (5)

where , we will consider from here on. The average expansion rate , the average pressure , the shear vector , and the transverse pressure vector are respectively defined as

 H=13(H1+H2+H3), (6) p=13(p1+p2+p3), (7) σi=Hi−H, (8) Σi=pi−p, (9)

where . The average expansion rate may be written as , where the average scale factor is defined by

 ¯a=(a1a2a3)1/3, (10)

and the directional expansion rates are given by

 Hi=˙aiai. (11)

From Eqs. (6)-(9) we see that the quantities and satisfy the constraints

 σ1+σ2+σ3=0, (12) Σ1+Σ2+Σ3=0, (13)

respectively.

Additionally we give the definition of some useful anisotropic quantities. The shear tensor is defined by

 σab=hcau(c;d)hdb−13θhab,

where is the expansion scalar, the projection tensor (for the signature ), and the four-velocity. From this expression we obtain the shear scalar given by .

For the considered Bianchi type I metric (1) we have that the expansion scalar, non-zero shear tensor components and the shear scalar are given by

 θ=H1+H2+H3, σ11=−23H1+13(H2+H3), σ22=−23H2+13(H1+H3), σ33=−23H3+13(H1+H2), σ2=23(H21+H22+H23−H1H2−H1H3−H2H3),

respectively.

## Iii Finite-time future anisotropic singularities in vacuum Kasner spacetimes

In this section we study future singularities of anisotropic character by considering Bianchi type I spacetimes without matter, i.e. vacuum solutions for the metric (1) (or Kasner spacetimes).

By putting and into Eqs. (2)-(5) we obtain the following four independent differential equations:

 3H2=σ22, (14) ˙σi+3Hσi=0. (15)

From Eq. (15) we obtain

 σi=σi0¯a3, (16)

then and Eq. (14) gives

 ¯a(t)=(C±12√6σ0t)1/3. (17)

Here and are integration constants, and

 σ0≡√2(σ210+σ220+σ10σ20), (18)

where we have used the relation

 σ30=−σ10−σ20. (19)

From Eqs. (8), (14) and (16) we may write relation

 ˙aiai=(1±√6σi0σ0)˙¯a¯a, (20)

which implies that the directional expansion rates are proportional to the average expansion rate . By using Eq. (17) we obtain for directional scale factors

 ai=a±i0(C±12√6σ0t)13(1±√6σi0σ0), (21)

where are integration constants for branches and respectively, and .

In order to proceed with the analysis, we shall use the initial condition for the directional scale factor . This implies that at we are imposing an expanding scale factor . Then, from Eq. (21) we obtain that , and the scale factor along -direction takes the form

 a1=a±10⎛⎜ ⎜ ⎜⎝1+3H0t1±√6σ10σ0⎞⎟ ⎟ ⎟⎠13(1±√6σ10σ0). (22)

Then, the metric (1) is given by

 ds2=dt2−⎛⎜ ⎜ ⎜⎝1+3H0t1±√6σ10σ0⎞⎟ ⎟ ⎟⎠23(1±√6σ10σ0)dx2− ⎛⎜ ⎜ ⎜⎝1+3H0t1±√6σ10σ0⎞⎟ ⎟ ⎟⎠23(1±√6σ20σ0)dy2− ⎛⎜ ⎜ ⎜⎝1+3H0t1±√6σ10σ0⎞⎟ ⎟ ⎟⎠23(1∓√6(σ10+σ20)σ0)dz2, (23)

where the constants have been absorbed by rescaling the spatial coordinates.

Since we are interested in solutions with finite time future singularities one should require that . This implies that for the positive branch of the considered solution the condition

 σ10<−σ0√6<0 (24)

must be fulfilled by coefficients and , while for the negative branch the condition

 σ10>σ0√6>0 (25)

must be required. Therefore, in metric (23) the scale factor along -direction exhibits a future singularity at finite value of the cosmic time . Notice that Eq. (20) implies that in this scenario we have that and . If we demand that the scale factor along -direction also becomes singular at the finite value we must require additionally that , which implies that for the positive branch, and for the negative branch. However, it can be shown that simultaneously it is not possible to satisfy the conditions and (or and ).

Therefore, only the scale factor becomes singular at finite value of the cosmic time , while the other two scale factors and become zero at this time (see Fig. 1). It is interesting to note that for all directional scale factors (21), the corresponding expansion rates diverge at . From Eq. (20) we conclude that the same is valid for the average expansion rate .

It is worth to mention that if we consider in Eq.(17), then we have in Eq.(21) and the directional scale factor diverges at instead of . The condition for the occurrence of this singularity is the same as before . In this sense, the occurrence of the singularity is independent of the value of the constant : if we must consider for the consistency of Eq.(17), i.e. , if the consistency of Eq.(17) allows us to consider or .

Notice that by defining

 q1=13⎛⎜ ⎜⎝1±√6σ10√2(σ210+σ220+σ10σ20)⎞⎟ ⎟⎠, q2=13⎛⎜ ⎜⎝1±√6σ20√2(σ210+σ220+σ10σ20)⎞⎟ ⎟⎠, (26) q3=13⎛⎜ ⎜⎝1∓√6(σ10+σ20)√2(σ210+σ220+σ10σ20)⎞⎟ ⎟⎠,

for the powers of the scale factors in Eq. (23), we obtain that

 q1+q2+q3=1, (27) q21+q22+q23=1, (28)

where , and are the Kasner parameters. These constraints correspond to the conditions for the well known vacuum Kasner solution.

From the Kasner conditions (27) and (28) we note that if we arrange the Kasner parameters in increasing order , then they change in the ranges Belinski ()

 −13≤q1≤0, (29) 0≤q2≤23, (30) 23≤q3≤1. (31)

From these relations we conclude again that if it is present a future singularity only one of the scale factors may blow up, while the other two tend to zero at a finite value of the cosmic time. For the particular case of an ellipsoidal vacuum cosmology the following parameter values must be required: or . Therefore, only the latter set of parameter values allow us to have a future singularity for an ellipsoidal vacuum universe (see Fig. 2).

In conclusion, due to the anisotropic character of Bianchi type I metrics, in the Kasner vacuum solution all three scale factors do not exhibit simultaneously a future singularity: just one of the scale factors may exhibit such a singularity at , while the other two do not. In this case the average scale factor (17) does not exhibit a singular behavior, becoming zero at . We note that this scenario necessarily corresponds to an average contracting universe. However, all directional expansion rates as well as the average expansion rate diverge at , and due to the vacuum character of the Kasner solutions, the scalar curvature is always zero.

Because of the absence of matter content, the discussed anisotropic future singularities are not similar to any of the finite-time singularities listed in the introduction section. We shall call such a singularity a Vacuum Rip.

It should be emphasized that these vacuum rips are not produced by fluids violating the dominant energy conditions (DEC) Hawking (), i.e. and , as stated for FRW cosmologies filled with a phantom fluid. The Kasner vacuum solution satisfies DEC, and by writing the Kasner metric in the form where the shear is explicitly included, we have shown that future singularities may be induced by the anisotropy of the spacetime, by making a suitable choice of the model parameters and .

## Iv Finite-time future anisotropic singularities with a stiff fluid

To elucidate the role of the shear, in the occurrence of future singularities, in the presence of matter fields we will consider the “toy model” of fully anisotropic Bianchi type I spacetimes (1), filled with a stiff fluid, for which the condition for the powers of scale factors (27) is still valid. This “toy model” is interesting because it allows us to consider finite time future singularities in a cosmology, fulfilling energy conditions (, ). The other aspect to be considered is that this cosmological model allows us to handle exact analytical expressions for studying relevant quantities.

We use the field equations in the form given by Eqs. (2)-(5). Since the pressure is isotropic, in this case we have that , and from Eq. (4) we have for the equation of state that the energy density is given by . On the other hand, from Eq. (5) we obtain the solution (16), and then Eq. (2) implies that the average scale factor is given by

 ¯a(t)=(C±√3ρ0+32σ20t)1/3, (32)

where is an integration constant and is given by Eq. (18). Note that by making we obtain the average scale factor (17) discussed in the previous section.

From Eq. (8) we may write for directional expansion rates the following equation:

 ˙aiai∓13√12ρ0+6σ022C±√12ρ0+6σ02t= (33) σi0C±√3ρ0+32σ02t,

which implies that the directional scale factors are given by

 ai=a±i0⎛⎜ ⎜⎝C±√12ρ0+6σ202t⎞⎟ ⎟⎠13±2σi0√12ρ0+6σ20, (34)

where are integration constants.

By using the initial condition for the directional scale factor we obtain from Eq. (34) that , and then the scale factor along -direction takes the form

 a1=a±10⎛⎜ ⎜⎝1+3√12ρ0+6σ20√12ρ0+6σ20±6σ10H0t⎞⎟ ⎟⎠13±2σ10√12ρ0+6σ20.

Hence, the resulting metric may be written as

 ds2=dt2−(1+H0γt)23±4σ10√12ρ0+6σ20dx2− (1+H0γt)23±4σ20√12ρ0+6σ20dy2− (1+H0γt)23∓4(σ10+σ20)√12ρ0+6σ20dz2, (36)

and the energy density and the pressure are given by

 ρ=p=36H20ρ0(√12ρ0+6σ20±6σ10)2(1+H0γt)2, (37)

where

 γ=13±2σ10√12ρ0+6σ20, (38)

i.e. the power of the scale factor along -direction.

In order to induce a future singularity, and considering that , we must require that . This implies that

 σ10<−16√12ρ0+6σ20<0, (39)

for the positive branch, and

 σ10>16√12ρ0+6σ20>0, (40)

for the negative branch.

By taking into account Eq. (18) we conclude that

 σ10<14(σ20−√9σ220+8ρ0)<0, (41)

for the positive branch, and

 σ10>14(σ20+√9σ220+8ρ0)>0, (42)

for the negative branch.

Notice that by defining the powers of the scale factors in the metric (IV) as

 q1=13±2σ10√12ρ0+6σ20, q2=13±2σ20√12ρ0+6σ20, (43) q3=13∓2(σ10+σ20)√12ρ0+6σ20,

the parameters satisfy the condition (27), independently of the values of , and . Then, the average scale factor takes the form

 ¯a=⎛⎜ ⎜⎝1+3√12ρ0+6σ20√12ρ0+6σ20±6σ10H0t⎞⎟ ⎟⎠13. (44)

However, now we have that

 q21+q22+q23≠1. (45)

By putting the parameters in Eqs. (IV) become the Kasner parameters of relations (III), implying that the condition (28) is fulfilled for vanishing matter.

Therefore, in the case of Bianchi type I cosmologies filled with stiff matter the singularity appears at . As in the vacuum case, from Eqs. (IV) we note that we can have only one of the directional scale factors blowing up together with the energy density and pressure. It becomes clear from expressions (IV)-(41) that this rip singularity is induced by the anisotropy of the spacetime.

Note that, if we require that two of powers are negative, as it is allowed by the condition (27), then , implying that the energy density becomes negative, violating the weak energy condition (see Figs. 3 and 4).

The average scale factor (44) does not exhibit a singularity at , where it vanishes. This anisotropic rip singularity appears at only for a contracting average scale factor.

## V Future anisotropic rip singularities in ellipsoidal universes

In this section we consider the evolution of Bianchi type I cosmologies with a matter content characterized by isotropic and anisotropic pressure.

Specifically, we consider particular cases of Bianchi type I models described by the condition . Thus the line element (1) takes the form

 ds2=dt2−a21(t)(dx2+dy2)−a23(t)dz2, (46)

which possesses spatial sections with planar symmetry and an axis of symmetry directed along the -axis. The functions of the cosmic time and are the directional scale factors along x, y and z directions respectively. The metric (46) describes a space that has an ellipsoidal rate of expansion at any moment of the cosmological time, dubbed also Locally Rotationally Symmetric Bianchi I.

In this case the Einstein field equations (2)-(5) may be written in the form

 ρ=˙a21a21+2˙a1˙a3a1a3, (47) p1=−(¨a1a1+˙a1˙a3a1a3+¨a3a3), (48) p3=−(2¨a1a1+˙a21a21), (49)

where and are the transversal and longitudinal pressures respectively. For the metric (46) the average scale factor is given by , and the average expansion rate takes the form

 H=˙¯a¯a=13(2˙a1a1+˙a3a3). (50)

In order to handle exact solutions to the metric (46), we further make the assumption that the scale factors and are constrained to be given by

 a1(t)=aα3(t), (51)

where is a constant. Thus the metric (46) takes the following form:

 ds2=dt2−a2α3(t)(dx2+dy2)−a23(t)dz2. (52)

This metric becomes isotropic for . It is interesting to note that the metric (52) is characterized by the condition that expansion scalar is proportional to the shear scalar .

The measure of anisotropy is constant in a number of Bianchi-type spacetimes representing perfect fluid cosmologies with barotropic equations of state (see Roy () and references therein). Given that these models may allow nearly isotropic scenarios, they can be used for studying the effects of anisotropy in our universe by confronting them with observational data. In our case, this condition will allow us to work with analytical ellipsoidal cosmologies exhibiting anisotropic rip singularities.

### v.1 Anisotropic rip singularities with isotropic pressure: p=ρ

Let us suppose that . Thus from Eqs. (47)-(49) and (51), the relevant metric function is given by

 a3(t)=c1(1+c2t)1(2α+1), (53)

where and are integration constants.

We shall rewrite this scale factor by using for the directional Hubble parameter the condition

 H3(t=0)=H0. (54)

Thus, the scale factor (53) takes the form

 a3(t)=c1(1+(2α+1)H0t)1(2α+1), (55)

and the metric (52) is given by

 ds2=dt2−(1+(2α+1)H0t)2α2α+1(dx2+dy2) −(1+(2α+1)H0t)22α+1dz2, (56)

where the constant has been absorbed by rescaling the spatial coordinates.

In this case the energy density and pressure result to be:

 ρ=p=α(α+2)H20(1+(2α+1)H0t)2, (57)

which means that the isotropic requirement for the pressure implies that the matter filling the universe is characterized by a stiff equation of state.

From the expression (55) we see that a future rip singularity appears for when , or for . On the other hand, in order to have a positive energy density we must also require that or , which excludes the case . For , the scale factor (55), energy density and pressure blow up at the finite value of the cosmic time , while the scale factor becomes zero at this time. In this case the average scale factor is given by

 ¯a(t)=(1+(2α+1)H0t)13, (58)

and does not exhibit a singularity at .

In conclusion, for the metric (52) the requirement of isotropic pressure implies that the matter content behaves as a stiff fluid. The evolution of this cosmology exhibits a future singularity for () at . Due to the functions , and blow up at this time, this singularity is similar to the FRW Big Rip one but of anisotropic character, since at the scale factor along and directions becomes zero, as well as the average scale factor . From Eq. (58) we note that this scenario corresponds to a contracting universe.

As in Sec. III, the anisotropic future singularities are not produced by fluids violating the DEC, since in this case . This type of singularities is induced by the anisotropy of the spacetime, since if shear vanishes, i.e. , then the solution becomes the standard isotropic FRW cosmology filled with a stiff fluid, which does not exhibit any future singularity at a finite value of the cosmic time, and only presents the initial singularity or Big Bang.

### v.2 Big Rip singularities with anisotropic pressure

Let us suppose that the transversal and longitudinal pressures are given by

 p1=ω1ρ, (59) p3=ω3ρ, (60)

respectively, where and are constant state parameters. Thus, from Eqs. (47), (48) and (51) we obtain that

 a3(t)=c1(1+c2t)α+1α2ω1+α2+2αω1+α+1, (61)

where and are integration constants. By using the initial condition (54) the scale factor (61) takes the form

 a3(t)=c1(1+H0γt)γ, (62)

where

 γ=α+1α2ω1+α2+2αω1+α+1. (63)

In this case the metric takes the form:

 ds2=dt2−(1+H0γt)αγ(dx2+dy2)− (1+H0γt)γdz2, (64)

where the constant has been absorbed by rescaling the spatial coordinates and we have considered , and the energy density and the pressure take the form

 ρ=α(α+2)H20(1+H0γt)2, (65)
 p3=1+2αω1−α1+αρ, (66)

respectively. Eq. (66) implies that the state parameter is given by

 ω3=1+2αω1−α1+α. (67)

It becomes clear that, in order to have a positive energy density we must require or and for the scale factor (62) exhibits a future singularity at . At this value of the cosmic time the energy density and pressures also blow up. In this case for we can have one of the scale factors blowing up at , while for all three scale factor may blow up at . It is interesting to note that the average scale factor, given by

 ¯a(t)=(1+H0γt)γ(2α+1)3, (68)

also may exhibit a singular behavior at for and . These inequalities imply, with the help of Eq. (63), that , hence we have that the state parameter can not be greater than for any . For the power of Eq. (68) is always positive and the average scale factor tends to zero at the time , while the scale factors and go to zero and blows up at this time. This singularity corresponds to an axisymmetric pancake rip defined in TABLE 1. Note that for we have , and for we have .

In conclusion, in the case of ellipsoidal universes we may have one, or two, or all three scale factors blowing up at . For the latter case, it is crucial to require and . This condition will be realized by requiring

 α>1forω1=−1, (69) α−<α<α+for−1<ω1<−√32, α>α−forω1<−1. (70)

where

 α±=12−(1+2ω1)±√4ω21−31+ω1. (71)

Therefore, we can have future singularities of cigar and pancake rip types (see TABLE 1). The cigar singularities may be of anisotropic () or symmetric types (), while the pancake singularities may be anisotropic and infinite ( and ) or of axisymmetric (infinite) type ().

In Figs. 5 and 6 we show the qualitative behaviors of the three scale factors, average scale factor, energy density and pressures for ellipsoidal universes (V.2). In this example the future singularity is of Big Rip type, and the universe rips itself apart in all directions at a finite time, with diverging energy density and pressures.

## Vi Big rip in fully anisotropic Bianchi type I cosmologies

It is possible to construct a Bianchi type I generalization of the ellipsoidal cosmology, exhibiting a future singularity, with three different scale factors and barotropic anisotropic pressures. For instance, let us choose the scale factors in the form

 ai=(1+H0tγ)si, (72)

where , and , and are constants. In this case, from Einstein equations the energy density and pressures are given by

 ρ=(α+β+αβ)H20(1+H0tγ)2, (73)

and (), where

 ω1 = 1+β−γ(1+β2+β)γ(α+β+αβ), (74) ω2 = 1+α−γ(1+α+α2)γ(α+β+αβ), (75) ω3 = α+β−γ(α2+β2+αβ)γ(α+β+αβ). (76)

In this case the average scale factor is given by

 ¯a=(1+H0tγ)(α+β+1)γ3. (77)

We notice that for we recover the ellipsoidal cosmology of Subsection VB.

Now we are interested in studying scenarios with and a positive energy density. This means that, we must require

 α+β+αβ>0. (78)

Simultaneously we require that the power of the average scale factor in Eq. (77) be negative, i.e.

 (α+β+1)γ<0. (79)

From Eqs. (78) and (77) we obtain

 α>−1,β>−α1+α. (80)

It becomes clear that for