Dynamical systems analysis of anisotropic cosmologies in gravity
Abstract
In this paper we study the dynamics of orthogonal spatially homogeneous Bianchi cosmologies in gravity. We construct a compact state space by dividing the state space into different sectors. We perform a detailed analysis of the cosmological behaviour in terms of the parameter , determining all the equilibrium points, their stability and corresponding cosmological evolution. In particular, the appropriately compactified state space allows us to investigate static and bouncing solutions. We find no Einstein static solutions, but there do exist cosmologies with bounce behaviours. We also investigate the isotropisation of these models and find that all isotropic points are flat Friedmann like.
pacs:
98.80.JK, 04.50.+h, 05.45.a, and
1 Introduction
Over the past few years, there has been growing interest in higher order theories of gravity (HOTG). This is in part due to the fact that these theories contain extra curvature terms in their equations of motion, resulting in a dynamical behaviour which can be different to General Relativity (GR). In particular these additional terms can mimic cosmological evolution which is usually associated with dark energy [1], dark matter [2, 3] or a cosmological constant [4]. The isotropisation of anisotropic cosmologies can also be significantly altered by these higherorder corrections. In a previous work [5], the existence of an isotropic past attractor within the class of Bianchi type I models was found for a power law Lagrangian of the form . This feature was also found for Bianchi type I, II and IX models in quadratic theories of gravity [6, 7]. In these cases the extra curvature terms can dominate at early times and consequently allow for isotropic initial conditions. This is not possible in GR, where the shear term dominates at early times.
A natural extension of this analysis is to investigate the effect of spatial curvature on the isotropisation in HOTG. In GR, it is wellknown that spatial curvature can source anisotropies for Bianchi models [8, 9]. In this paper we extend the analysis in [5] to the case of orthogonal spatially homogeneous (OSH) Bianchi models [10], in order to investigate the effect of spatial curvature on the isotropisation of models. OSH Bianchi models exhibit local rotational symmetry (LRS), and include the LRS Bianchi types I (BI), III (BIII) and the KantowskiSachs (KS) models. For a review of this class of cosmologies see [11, 12, 10]).
In GR, a cosmological constant or scalar field is required to obtain an Einstein static solution in a closed () FriedmannLemaîtreRobertsonWalker (FLRW) model [13, 14]. The existence of Gödel and Einstein static universes has been investigated for gravitational theories derived from functions of linear and quadratic contractions of the Riemann curvature tensor [15]. Recently, the stability of Einstein static models in some theories of gravity was investigated [16]. It was shown that the modified Einstein static universe is stable under homogeneous perturbations, unlike its GR counterpart [18]. Static solutions are interesting in their own right, but are often an important first step in finding cosmologies that have a “bounce” during their evolution [17].
The existence conditions for a bounce to occur for FLRW universes in gravity have been determined recently [19]. Bouncing cosmological models have been found for FLRW models in gravity [20, 21]. This should in principle be possible for anisotropic models as well, since the higher order corrections can mimic a cosmological constant, and so prevent the model from collapsing to a singularity. In [22], it was shown that bounce conditions for OSH Bianchi models cannot be satisfied in GR with a scalar field, but can be satisfied for KS models in the RandallSundrum type Braneworld scenario.
As in [5], we make use of the dynamical systems approach [23, 14, 24] in this analysis. This approach has been applied to study the dynamics of a range of extended theories of gravity [25, 5, 26, 27, 20, 6, 28, 29, 30, 31, 32]. However, in these works, the dynamical variables were noncompact, i.e. their values did not have finite bounds. This noncompactness of the state space has certain disadvantages (see [33] for detailed discussion of this issue). The standard expansion–normalised variables for example only define a compact state space for simple classes of ever expanding models such as the open and flat FLRW models and the spatially homogeneous Bianchi type I models in GR [24]. As soon as a wider class of models or more complicated underlying gravitational theories are considered, the expansion rate may pass through zero, making the state space non–compact (see e.g. [25, 5, 26, 27]). The points at infinity then correspond to a vanishing Hubble factor, and the non–compact expansion–normalised state space can only contain the expanding (or by time–reversal collapsing) models. In order to obtain the full state space, one would have to carefully attempt to match the expanding and collapsing copies at infinity.
While static solutions correspond to equilibrium points at infinity and can be analysed by performing a Poincaré projection [34, 35], bouncing or recollapsing behaviours on the other hand are very difficult to study in this framework. In both cases ambiguities at infinity can easily occur, since in general only the expanding copy of the state space is studied. A point at infinity may for example appear as an attractor in the expanding non–compact analysis, even though it corresponds to a bounce when also including the collapsing part of the state space.
In order to avoid these ambiguities, we will here construct compact variables that include both expanding and collapsing models, allowing us to study static solutions and bounce behaviour in theories of gravity. This approach is a generalisation of [13], which has been adapted to more complicated models in [36, 37, 39, 38]. We refer to the accompanying work [33] for a detailed comparison between the approach established here and differently constructed non–compact state spaces applied to the class of BI or flat FLRW models in gravity.
We note that we recover the isotropic past attractor found in [5] in this analysis, and we only obtain flat () isotropic equilibrium points. Bounce behaviour is found for BI, BIII and KS cosmologies, but no Einstein static solutions could be found in the phase space. Our analysis also reveals that we can have cosmologies that bounce from expansion to contraction and vice versa, depending on the value of the parameter .
The outline of this paper is as follows: In section 2 we state the field equations and the evolution equations for the OSH Bianchi models. In section 3 we construct a compact state space and then analyse the BIII and KS subspaces separately. Section 4 is devoted to a discussion of the isotropisation of these cosmologies.
The following conventions will be used in this paper: the metric signature is ; Latin indices run from 0 to 3; units are used in which .
2 Preliminaries
The general action for a theory of gravity reads
(1) 
where is the Lagrangian of the matter fields. The fourth order field equations can be obtained by varying (1):
(2) 
where primes denote derivatives with respect to and . The field equations (2) can be rewritten in the standard form
(3) 
(when ), where the effective stress energy momentum tensor is given by
(4) 
It is easy to show that the contracted Bianchi identities give rise to the conservation laws for standard matter [25]. The propagation and constraint equations can be obtained straightforwardly for these field equations (see [40, 5]).
We here consider the case for OSH Bianchi spacetimes, where the Raychaudhuri equation becomes
(5) 
and the trace free GaussCodazzi equation is given by
(6) 
Here is the volume expansion which defines a length scale along the flow lines via the standard relation , and is the standard matter energy density. The magnitude of the shear tensor is given by , and the 3Ricci scalar by (see [10]).
The Friedmann equation is given by
(7) 
Combining the Friedmann and Raychaudhuri equations yields
(8) 
We will assume standard matter to behave like a perfect fluid with barotropic index , so that the conservation equation gives
(9) 
In the following, we assume and .
3 Dynamics of OSH Bianchi cosmologies
3.1 Construction of the compact state space
The overall goal here is to define compact dimensionless expansion–normalised variables and a time variable such that the system of propagation equations above (5)(9) can be converted into a system of autonomous first order differential equations. We choose the expansion normalised time derivative
(10) 
and make the following ansatz for our set of expansion normalised variables ^{1}^{1}1It is important to note that this choice of variables excludes GR, i.e., the case of . See [23, 14] for the dynamical systems analysis of the corresponding cosmologies in GR.:
(11)  
Here is a normalisation of the form
(12) 
where is a linear combination of the terms appearing on the right hand side of the Friedmann equation (7) as discussed below. In order to maintain a monotonically increasing time variable, must be chosen such that the normalisation is real–valued and strictly positive.
Note that we have chosen to define with an opposite sign to that in [5] in order to have a simple form of the Friedmann equation (see below), and can be both positive and negative [10]. We emphasise that the coordinates (11) are strictly speaking only defined for , which means for . Even though the case may not be of physical interest, the limiting case is interesting in the context of the stability analysis, since we obtain equilibrium points with . This means that the system may evolve towards/away from that singular state if these points are attractors or repellers. In the analysis below we will investigate this by taking the limit (by letting ) and find that this puts a constraint on the relation between the coordinates.
We now turn to the issue of compactifying the state space. It is useful to re–write the Friedmann equation (7) as
(13) 
where the quantities with a hat are just the variables defined in (11) without the normalisation . If all the contributions ( and ) to the central term in equation (13) are non–negative, we can simply normalise with (i.e. ), but we have to explicitly make the assumption . We can then conclude that the state space is compact, since all the non–negative terms have to add up to and are consequently bounded between and .
However, while is always positive, and may be positive or negative for the class of models considered here ^{2}^{2}2Note that the sign of is preserved within the open and the closed sectors.. This means that the variables (11) do not in general define a compact state space.
In the following, we will study the class of LRS BIII models with and the class of KS models with separately, as in [13]. While we may in principle normalise with in the Bianchi III subspace, we have to absorb the curvature term into the normalisation in the KS subspace.
For both classes of models, we can construct a compact state space by splitting up the state space into different sectors according to the sign of and . In both the open and the closed subspaces we will have to define sectors, corresponding to the possible signs of the three variables . In the following, we will refer to the spatially open BIII sectors as sector to sector , where the subscript ’o’ stands for ’open’. Similarly, the spatially closed KS sectors will be labeled sectors  , where the ’c’ stands for ’closed’.
After defining the appropriate normalisations for the various sectors, we derive the dynamical equations for the accordingly normalised variables in each sector. For each sector we then analyse the dynamical system in the standard way: we find the equilibrium points and their eigenvalues, which determine their nature for each sector. The overall state space is then obtained by matching the different sectors along their common boundaries.
3.2 The LRS BIII subspace
If , we obtain the class of spatially open LRS BIII cosmologies. This class of models contains the flat LRS BI models as a subclass. In this case enters the Friedmann equation with a non–negative sign and does not have to be absorbed into the normalisation. As can be seen from the Friedmann equation in each sector (see Table 1), and holds in each sector.
3.2.1 Sector
The first open sector denoted is defined to be that part of the state space where . In this case all the contributions to the righthand side of (13) are non–negative, and we can choose . This means we can normalise with , where is the sign function of and for expanding/collapsing phases of the evolution. Note that it is crucial to include in the normalisation: if we were to exclude this factor, time would decrease for the collapsing models, and any results about the dynamical behaviour of collapsing equilibrium points would be timereversed.
It is important to note that we have to exclude in this sector, so we cannot consider static or bouncing solutions here. However, this assumption is not as strong as it first appears: we can see from the Friedmann equation (13) that the only static solution in this sector appears for , because all the quantities enter (13) with a positive sign in this sector by construction. This means that we only have to exclude the static flat isotropic vacuum cosmologies ^{3}^{3}3The same restriction appears in GR, see [13]. Under this restriction, the normalisation above is strictly positive and thus defines a monotonically increasing time variable via (10). Equation (13) now becomes
(14) 
We can directly see from (14) that the appropriately normalised variables (11) define a compact subsector of the total state space:
(15) 
Here is constant and not a dynamical variable.
This sector is different from all the other sectors in both the open BIII and the closed KS subspaces for the following reasons. When gluing together the different sectors to obtain the total state space, we will actually use two copies of : one copy with corresponding to expanding cosmologies and one copy with corresponding to collapsing cosmologies. The two copies are in fact disconnected: The closed sector from the KS subspace separates the expanding and collapsing copies of open sector . Again, this reflects the fact that we cannot study static solutions in sector . In all the other sectors we allow , and the expanding and collapsing sets are connected via the non–invariant subset .
We can now derive the propagation equations for the dynamical systems variables in this sector by using the definitions (10) and (11) and substituting them into the original propagation equations (5)(9). We obtain five equations, one for each of the dynamical variables defined in (11). These variables are constrained by the Friedmann equation (13), which we use to eliminate , resulting in a 4dimensional state space. Note that we have to verify that the constraint is propagated using all five (unconstrained) propagation equations, which we have done for each sector. The effective system^{4}^{4}4If we used the unconstrained 5–dimensional system, we would not constrain the allowed ranges of and for the different equilibrium points correctly. We would also get a fifth zero–valued eigenvalue for all equilibrium points. is given by
(16)  
Only in this sector does the sign of the expansion–rate appears directly in the dynamical equations, and we can see directly that the stability of the collapsing equilibrium points is given by simple timereversal of the stability of the expanding points and vice versa.
The subset (Bianchi I) is a two dimensional invariant sub–manifold, so it is justified to discuss the Bianchi I subspace on its own. This is done in detail in [33]. The vacuum subset and the submanifold are also invariant subspaces. On the other hand, the isotropic subset is not invariant unless . This agrees with GR, where it was found that the spatial curvature can source anisotropies for Bianchi models [8, 9].
We can find the equilibrium points and the corresponding eigenvalues of the dynamical system (3.2.1), and classify the equilibrium points according to the sign of their eigenvalues as attractors, repellers and saddle points (see [35]). Because of the large number of sectors that need to be studied, we do not show the results for each sector. Instead we combine the results from the various sectors in Table 2.
3.2.2 Sectors
Sectors are defined according to the possible signs of as summarised in Table 1. In each sector is defined as the sum of the strictly negative contributions to (13), so that is strictly positive, making strictly positive even for . This means that is a well–defined (nonzero) normalisation, and (10) defines a welldefined monotonously increasing time variable for each sector, even for static or bouncing solutions.
sector  normalisation  Friedmann equation  range of  

^{5}^{5}5We must impose in this sector, see text.  
With this choice of normalisation, only positive contributions remain in the Friedmann equation, and the appropriately normalised variables define a compact sub–sector of the total state space, as can be seen from the respective versions of the Friedmann equation in Table 1. Note that the Friedmann equation looks different in each sector, which is of course due to the different normalisation for each sector. We also gain a second constraint equation which arises from the definition of :
(17) 
which can be written in terms of the variables (11) in each sector.
It is straightforward to derive the dynamical equations for each sector, and again we analyse them as outlined in the previous subsection. We confirm in each sector that the flat LRS BI subset is indeed an invariant submanifold.
3.2.3 Equilibrium points of the full LRS BIII state space
The equilibrium points of the entire BIII state space are obtained by combining the equilibrium points in each sector. We summarise them in Table 2. Note that not all the points occur in all of the sectors, and some points only occur in a given sector for certain ranges of or a specific equation of state . For this reason, we cannot express all the equilibrium points in terms of the same variables. When possible we state the coordinates in terms of the dimensionless variables defined for sector , i.e if the given point occurs in this sector. This is true for all the points except the line , whose coordinates are described in terms of the variables defined in sector (see below for more details on the relation between and ).
We emphasise that if the same point occurs in different sectors, it will have different coordinates in each of these sectors. In particular, can be a function of or in sectors even if is a constant in sector . This simply reflects the fact that we have to exclude the static solutions in sector but not in the other sectors. This issue will be of importance when looking for static solutions in section 3.4.3. In order to ensure that equilibrium points obtained in different sectors correspond to the same solution, we have to look at the exact solution at these points. This is outlined in section 3.4.
Note that each of the isolated equilibrium points has an expanding () and a collapsing () version as indicated in the labeling of the points via the subscript in Table 2. Similarly, the lines each have an expanding and a contracting branch (see below). We will however drop the subscript in the following unless we explicitly address an expanding or contracting solution.
We find the three equilibrium points , and corresponding to spatially flat Friedmann cosmologies. The expanding versions of these points correspond to the equally labeled points in the BI analysis [5] (see [33] for detailed comparison). These points were also found in the Friedmann analysis [25]. and are vacuum Friedmann points, while represents a nonvacuum Friedmann point whose scale factor evolution resembles the well known FriedmannGR perfect fluid solution with .
We now address the two lines of equilibrium points denoted by and . Both these lines correspond to the spatially flat anisotropic BI cosmologies. The ratio of shear and curvature component changes as we move along both lines. We note that in [5] a single line of equilibrium points denoted was found. In section 3.6 we will discuss in more detail how and are related to .
We emphasise that for the two expanding and contracting branches are disconnected and appear as two copies of the line labeled by in Table 2. Each of these two branches range from purely shear dominated () to isotropic (), to purely shear dominated with opposite orientation (). For on the other hand the expanding and contracting branches are connected: each and ranges from expanding () and static () to collapsing (). The two disconnected copies and correspond to positive and negative values of the shear respectively. Note that there is no isotropic subset of in analogy to the fact that there is no static subset of .
A closer look shows that and are actually the same object in different sectors: has hence occurs in sectors , , and , while is the analog with occurring in sectors , , and . This statement is confirmed by looking at the exact solutions corresponding to the points on both lines; we find that both these lines have the same parametric solution of scale factor and shear (see section below). For this reason, we could in fact give the two lines the same label. However, it is useful to treat them separately, since we obtain different bifurcations in the sectors with and respectively. Furthermore, the subset of the line denoted by allows for static solutions unlike the subset labeled . This is due to the fact that a negative curvature contribution can effectively act as a cosmological constant by counter–balancing other contributions in the Friedmann equation. This is explored in section 3.4.3 below.
Finally, we find the equilibrium points and corresponding to spatially open models. Point is independent of and , while depends on the value of . The points and can be spatially open, flat or closed depending on the value of and/or , i.e. they move through the different sectors of the total state space as are varied. This is reflected in Tables 9 and 10, where we summarise the stability properties of the equilibrium points of the closed and open subspaces separately, and observe that these two points occur in each subspace for certain ranges of only.
Point  Description  

Friedmann flat  
Friedmann flat  
Friedmann flat  
Line  flat LRS Bianchi I  
Line  flat LRS Bianchi I  
open LRS BIII  
open LRS BIII  
vacuum BI, BIII or KS  
BI, BIII or KS 
3.3 The KantowskiSachs subspace
When , we obtain the class of spatially closed KS cosmologies. Here is positive and needs to be absorbed into the normalisation in all sectors. This means that in this subspace, is strictly positive in all closed subsectors  , hence is strictly positive even for . We can therefore consider static and bouncing solutions in all sectors that make up the closed subspace. The flat subspace is obtained in the limit . As explained in the previous subsection, we have to exclude static flat isotropic vacuum cosmologies in this limit.
The closed sectors can be defined as in the BIII case, except that no longer appears in the Friedmann equation (see Table 3). Similar to the BIII case, the first sector labeled is defined as the subset of the state space where . In this case we choose , so that equation (13) becomes
(18) 
The curvature can be obtained from (17), which in this sector becomes
(19) 
From (18) and (19) it is clear that the appropriately normalised variables (11) define a compact subsector of the total state space with
(20) 
Note that the variable will not be used explicitly in any of the closed sectors.
As in the BIII case, we derive the propagation equations for the dynamical systems variables in this sector and reduce the dimensionality of the state space to four by eliminating via the Friedmann constraint (18). Again we have verified that the constraint is preserved using all five propagation equations. We obtain the following dynamical system:
(21)  
We recover the following features from the BIII subspace: The flat subset (here corresponding to ) is invariant, as can be seen from the equation together with the Friedmann equation (18). Other invariant subspaces are the hypersurfaces and . The isotropic subset is not invariant unless .
sector  normalisation  Friedmann equation  range of  

The sectors  are defined according to the possible signs of as summarised in Table 3: In each sector is defined as the sum of the strictly negative contributions to (13). The dynamical equations analogous to (3.3) can be derived straightforwardly for each sector. We then solve these equations in each sector for their respective equilibrium points and the corresponding eigenvalues, and classify the equilibrium points according to their dynamical properties. The results are combined with the results from the open sectors and summarised in Tables 6–10.
3.4 Exact solutions corresponding to the equilibrium points
We now derive the solutions corresponding to the various equilibrium points. Special attention has to be paid to the points with , since these correspond to the limit , which may makes the coordinate singular. We will study this issue in detail below. Note that it is legitimate to take the limit in the original field equations as long as , which results in the constraint . Consequently it is only possible to study the limit for and when solving for the solutions corresponding to the equilibrium points with .
It is important to emphasise that the dynamical system by itself is welldefined for ; only when going back to the original equations to solve for the exact solutions corresponding to the equilibrium points with do we notice that there may not be an exact solution corresponding to these coordinates.
We now proceed to find the exact solutions corresponding to the non–static () equilibrium points. As usual, we can solve the energy conservation equation (9) for the non–vacuum solutions to obtain
(22) 
where is determined by the coordinate of the given equilibrium point. We require , which constrains the allowed range of or for a given equilibrium point (see below).
In order to determine the scale factor evolution at each equilibrium point, we rewrite the Raychaudhuri equation (5) as
(23) 
where we express the deceleration parameter at each point in terms of the dimensionless variables (11):
(24) 
Note that this equation is invariant in different sectors: for a given equilibrium point, each coordinate divided by is the same in all sectors. This ensures that the corresponding solution is invariant, no matter with which coordinates we describe the equilibrium point.
Similarly, we re–write the trace free Gauss Codazzi equation (6) as
(25) 
and the curvature constraint (8) as
(26) 
for a given equilibrium point with coordinates and deceleration parameter .
3.4.1 Powerlaw solutions
We first study the non–stationary () cosmologies, for which (23) has the solution
(27) 
We have set the Big Bang time . Given , we can solve for all the other dynamical quantities for a given equilibrium point to obtain the scale factor evolution
(28) 
the shear
(29) 
and the curvature scalar
(30) 
Again, we point out that even though a given equilibrium point formally has different coordinates in the different sectors, the exact solutions corresponding to the point are invariant, since the coordinates only enter the solutions (28)–(30) with a factor . The solutions for each point are summarised in Table 4, where constants of integration were obtained by substituting the solutions into the original equations.
When substituting the points with into the original field equations, we find that these are only satisfied for special values of . This is reflected in Table 4. Point only has a solution for and . The solutions for points and only satisfy the original equations for , which has been excluded from the start. These points therefore do not have any physical power–law solutions. The points on the lines only have corresponding solutions for special coordinate values, making only two points on each line physical (see below).
Excluding these nonphysical points, we find that the only non–vacuum solutions are given by and . Substituting the solution (22) into the definition of , we find that the constant must satisfy
In order for these solutions to be physical, we require that and therefore . For we find that this condition is satisfied for
(31) 
while for it is only valid for
(32) 
where .
For points and the solutions only depend on , while the solutions at and depend on both and . We can see from these solutions that points and are the isotropic analogs of points and respectively.
The lines and have the same solutions for shear and energy density. As noted above, they are the same line but for different ranges of and hence . contains the isotropic subset of solutions () while contains the static subset ().
In Table 5 we summarise the behaviour of the deceleration parameter . By studying the deceleration parameter, we can determine whether the power law solutions above correspond to accelerated () or decelerated () expansion or contraction. The expansion (or contraction) of point is decelerating for or and accelerating for . Point and lines only admit decelerating behaviours. Point has a decelerated behaviour for and an accelerated behaviour for . The equilibrium points and for , have decelerated behaviours when and accelerated behaviours when .
3.4.2 Stationary solutions
If , we obtain stationary solutions (), which have an exponentially increasing scale factor. As reflected in Table 5, the vacuum points and correspond to de Sitter solutions for the bifurcation value for all equations of state, while the matter points and are de Sitter–like for all but only, and appears to be de Sitter–like for for all values of . Since has , we will have to study this case in more detail below.
For a constant expansion rate
(33) 
the scale factor has the following solution
(34) 
The energy conservation equation becomes
(35) 
The trace free Gauss Codazzi equation (6) can be rewritten as
(36) 
which on integration yields
(37) 
where is an integration constant. The evolution of the Ricci scalar can be obtained by substituting the solutions above into (8), to find
(38) 
As before, we substitute the solutions at each equilibrium point into the definition of the coordinates, which constrains the constants of integration for each point. In particular, holds for all stationary equilibrium points, which means that we only have constant or vanishing shear.
As in the power–law case, we see that all the equilibrium points except for and correspond to vacuum solutions . For point the energy density is given by
(39) 
and for the energy density is given by
(40) 
Both of these solutions only hold for with .
Again, we substitute the generic solutions into the original field equations for each point, and find that the original equations are satisfied for all points with . It is however not possible to find a stationary solution at point (which has ), even after carefully considering the limit .
3.4.3 Static solutions
The static equilibrium points are characterised by . These points satisfy , where the second identity comes from the fact that if , then we require that from the definition of the variables, as discussed below. ^{6}^{6}6Note that unlike in the bouncing or recollapsing case below, we do not consider here, since this corresponds to the limit . While we may want to study a bounce where the Ricci scalar approaches zero and then grows again, we are not interested in static solutions that have vanishing Ricci curvature at all times.
We will now explore which of the equilibrium points obtained above correspond to static solutions. As indicated above, even though holds in the first sector as stated in Table 2, can be a function of and/or in the other sectors. In order to find the static equilibrium points, we have to look at the coordinates that each equilibrium point takes in each sector, and find the values of and/or for which in the given sector.
An obvious static solution appears to be the subset on line for all values of and . We can however not find a solution corresponding to this limit, since implies , which contradicts the value of the shear coordinate of this equilibrium point. We can study the eigenvalues associated with the line in the limit and find that the static subset is an unstable saddle point for all values of for both and .
The point appears to admit a static solution for the bifurcation value . This bifurcation only occurs in sectors 2, 3, 6 and 7 of the open and the closed sectors. However, it is not possible to find a solution satisfying the coordinates of the static equilibrium point that satisfies the original field equations. For this reason, this static equilibrium point is unphysical. We explore the stability of the static solution in the limit from the appropriate sides: for example, point only lies in the open sector 2 for or , making only the limit welldefined. We find that this bifurcation represents a saddle point in the state space since two of the eigenvalues approach from the left and from the right, making the point unstable.
Even though the –coordinate of point is a function of in sectors 2, 4, 5 and 7, cannot be zero for any values of . This means point does not admit any static solutions.
Point can only be static in the limit in sector 6 for and in sector 3 and 5 for . Again, we cannot find a solution for this special case, but this case is physically not interesting either way.
The –coordinate of point is zero in sectors 68 for , but again there is no solution corresponding to this limit.
Even though point has as a function of in open sectors 2 and 6, is non–zero for the allowed ranges of n.
Point becomes static in the limit in sectors 4, 6 and 8, which again is not physically relevant.
Point  Scale factor ()  Shear ()  Ricci Scalar ()  












