Strong Cosmic Censorship in orthogonal Bianchi class B perfect fluids and vacuum models

The Strong Cosmic Censorship conjecture states that for generic initial data to Einstein's field equations, the maximal globally hyperbolic development is inextendible. We prove this conjecture in the class of orthogonal Bianchi class B perfect fluids and vacuum spacetimes, by showing that unboundedness of certain curvature invariants such as the Kretschmann scalar is a generic property. The only spacetimes where this scalar remains bounded exhibit local rotational symmetry or are of plane wave equilibrium type. We further investigate the qualitative behaviour of solutions towards the initial singularity. To this end, we work in the expansion-normalised variables introduced by Hewitt-Wainwright and show that a set of full measure, which is also a countable intersection of open and dense sets in the state space, yields convergence to a specific subarc of the Kasner parabola. We further give an explicit construction enabling the translation between these variables and geometric initial data to Einstein's equations.


Introduction
In her ground-breaking work [FB52], Choquet-Bruhat showed that Einstein's field equations can be formulated as an initial value problem, where the initial data is given on a spacelike Cauchy hypersurface. To given initial data, there is a maximal globally hyperbolic development which is unique up to isometry, as was shown by Choquet-Bruhat and Geroch in [CBG69]. As understood in physics, this implies that this spacetime is uniquely determined by the initial data. The question arises whether one can find a larger development than this when dropping the requirement of global hyperbolicity, and whether this larger development is still unique in a meaningful sense. If two inequivalent developments existed, this would imply that determinism breaks, as the data on the initial Cauchy hypersurface does no longer suffice for determining which of the two developments the universe chooses. The Strong Cosmic Censorship conjecture states that this does not happen, at least not generically, as it conjectures that there is no development larger than the maximal globally hyperbolic development. Remark 1.3. Whenever the spacetime is not vacuum, i. e. the Ricci tensor does not vanish, it may be enough to determine whether the contraction of the Ricci curvature R αβ R αβ is unbounded in the incomplete directions of causal geodesics in the MGHD. Just like the Kretschmann scalar, this is a four-dimensional quantity invariant under isometries, and its unboundedness contradicts the existence of a C 2 extension.
1.1. Bianchi perfect fluid spacetimes. We restrict our discussion to spacetimes (M, g) satisfying Einstein's field equations with R αβ and S the Ricci and scalar curvature of the spacetime and T αβ the stress-energy tensor of a perfect fluid (2) T αβ = µu α u β + p(g αβ + u α u β ), where u is a unit timelike vector field. We assume that the pressure p and the energy density µ satisfy a linear equation of state (3) p = (γ − 1)µ for some constant γ. The case of vacuum is included as µ = 0. We note at this point that Greek indices α, β, . . . range from 0 to 3, while lower case Latin ones i, j, . . . range from 1 to 3. We consider Bianchi perfect fluids, which are perfect fluid spacetimes such that additionally there is a three-dimensional Lie group G acting on the spacetime (M, g) The metric g is invariant under this action, and the unit timelike vector field u appearing in the stress-energy tensor (2) of the perfect fluid is orthogonal to the orbits of this action. This defines non-tilted perfect fluids. The name Bianchi spacetimes stems from the classification of three-dimensional Lie groups by Bianchi from 1903, see [KBS + 03] for a historical overview: Unimodular Lie groups are called class A, non-unimodular ones class B, and depending on the form of the structure constants of the corresponding Lie algebras, they can be further separated into the following types: • Bianchi class A: types I, II, VI 0 , VII 0 , VIII, IX; • Bianchi class B: types IV, V, VI η , VII η , where η ∈ R is a parameter.
In this paper, we focus on Bianchi class B models, but types I and II appear as boundary cases. The models of class A have been the subject of more detailed study in the past, and a number of results have been achieved, including an affirmative answer to the Strong Cosmic Censorship conjecture. We refer to [Rin09] for a detailed exposition, in particular Prop. 22.23 therein, and are going to relate our own findings to a number of other results further down.
In terms of initial data, we arrive at the following setting: The initial data manifold is a threedimensional Lie group G with a metric and second fundamental form which are left-invariant, meaning that they are invariant under the action of G on itself via multiplication from the left. Further, a given constant represents the initial matter configuration. Definition 1.4. Bianchi perfect fluid initial data consists of a Lie group G, a left-invariant Riemannian metric h on G, a left-invariant symmetric covariant two-tensor k on G, and a constant µ 0 ≥ 0, satisfying the constraint equations In these equations, ∇ is the Levi-Civita connection of h, and R the corresponding scalar curvature. Indices are lowered and raised by h.
Here and in the following, we use the Einstein summation convention and sum over indices which occur twice, both as a sub-and as a superindex.
The symmetry of the metric and second fundamental form is preserved under Einstein's equations, and the resulting maximal globally hyperbolic development is isometric to the spatially homogeneous four-dimensional spacetime (4) I × G , g = −dt 2 + t g , with I an open interval and { t g} t∈I a family of left-invariant Riemannian metrics on G ∼ = {t}×G. We give an explicit construction of the maximal globally hyperbolic development in Section 11.
1.2. Results for orthogonal perfect fluid initial data. Let us now state the precise setting we discuss in this paper and give the main results we obtain regarding the Strong Cosmic Censorship conjecture. We focus our attention on Bianchi perfect fluid initial data with a Lie group of Bianchi class B. Type I and II appear as boundary cases. In particular, we exclude Lie groups of type VIII and IX. The Lie algebra g associated to the initial data Lie group G then admits an Abelian subalgebra, see Lemma 11.1. In case of a Lie group of Bianchi class B, this subalgebra can be characterised geometrically as the kernel of a certain one-form, see Lemma 11.2. Using this geometric construction, the subalgebra is unique and denoted by g 2 .
We introduce an orthonormal basis e 1 , e 2 , e 3 of g such that e 2 , e 3 span the Abelian subalgebra g 2 and e 1 is orthogonal to it with respect to the initial metric. By the previous argument, this can be done uniquely up to rotation in the e 2 e 3 -plane and a choice of sign in e 1 , provided the Lie group is of class B.
For Bianchi perfect fluid initial data with a Lie group of class B, one realises that the momentum constraint gives an algebraic relation for the components of the initial symmetric twotensor k with respect to this basis. For all initial data apart from certain cases where the Lie group is of type VI −1/9 , this relation implies that the off-diagonal components k 12 and k 13 of this tensor vanish, see Lemma 11.13. In the remaining special cases, the sets of initial data admit an additional degree of freedom and their maximal globally hyperbolic development is a so-called 'exceptional' Bianchi cosmological spacetime, denoted Bbii in [EM69]. This term should be understood as 'having exceptional behaviour'. We refer to Remark 11.14 for more details on this terminology.
In this paper, we exclude initial data with a Lie group of Bianchi type VI −1/9 and thereby ensure that the resulting spacetime is a 'non-exceptional' or 'orthogonal' Bianchi spacetime. Definition 1.5. Orthogonal perfect fluid Bianchi class B initial data consists of a Lie group G of class B other than type VI −1/9 , a left-invariant Riemannian metric h on G, a left-invariant symmetric covariant two-tensor k on G, and a constant µ 0 ≥ 0, satisfying the constraint equations Orthogonal perfect fluid Bianchi type I and II initial data for Einstein's orthogonal perfect fluid equations is defined similarly, allowing any type I or II Lie group G.
The reason for this restriction is the technique we apply in this paper. We wish to transform initial data sets into a specific set of variables and prove statements in this setting before translating them back. For Lie groups of type VI −1/9 we encounter difficulties as this transformation, described in detail in Section 11, can only be carried out for spacetimes which are 'non-exceptional'.
Certain initial data sets with higher symmetry will be of importance in the discussion of the Strong Cosmic Censorship conjecture. They are defined using a suitably adapted basis of the Lie algebra g: Definition 1.6 (Locally rotationally symmetric initial data). Consider initial data (G, h, k, µ 0 ) as in Def. 1.5, and denote by g the corresponding Lie algebra with two-dimensional Abelian subalgebra g 2 . Let e 1 , e 2 , e 3 an orthonormal basis of g such that e 2 , e 3 span g 2 . The initial data is said to be locally rotationally symmetric (LRS) if the basis can be chosen such that • e 2 commutes with e 1 and e 3 [e 2 , e 1 ] = 0 = [e 2 , e 3 ] , • the commutator [e 1 , e 3 ] is a multiple of e 2 , • the two-tensor k ij is diagonal, with k 11 = k 33 .
Remark 1.7. The notion locally rotationally symmetric in the previous definition stems from the fact that a rotation in the e 1 e 3 -plane is a Lie group isomorphism and an isometry of the initial data. Considering a three-dimensional Lie group G of class B with corresponding Lie algebra g, we find that the following holds: In case [g, g] is two-dimensional, there is no locally rotationally symmetric initial data on this Lie group. If instead [g, g] is one-dimensional, then the vector spanning this set defines a rotation axis contained in g 2 and leaving the Lie algebra invariant. Initial data on the given Lie group is locally rotationally symmetric if and only if the two-tensor k is invariant under this rotation. For more details, we refer to Subsection 11.8. Definition 1.8 (Plane wave equilibrium initial data). Consider initial data (G, h, k, µ 0 ) as in Def. 1.5, and denote by g the corresponding Lie algebra with two-dimensional Abelian subalgebra g 2 . Let e 1 , e 2 , e 3 be an orthonormal basis of g such that e 2 , e 3 span g 2 , and denote by γ k ij the structure constants, i. e.
[e i , e j ] = γ k ij e k . The initial data is said to be of plane wave equilibrium type if the basis can be chosen such that 1A + γ A 1B = −2k AB . We note here that upper case Latin indices A, B, . . . range from 2 to 3, where we assume the frame elements e 2 , e 3 to span g 2 . Definition 1.9. A spacetime which is the maximal globally hyperbolic development of locally rotationally symmetric initial data is called a locally rotationally symmetric spacetime. A spacetime which is the maximal globally hyperbolic development of initial data of plane wave equilibrium type is called a plane wave equilibrium spacetime.
We show in Section 11 how to construct the maximal globally hyperbolic development (4) to given initial data as in Def. 1.5.
To answer the Strong Cosmic Censorship conjecture, we determine whether geometric quantities invariant under isometries of this spacetime remain bounded in the incomplete directions of causal geodesics. For vacuum models we find that a bounded Kretschmann scalar R αβγδ R αβγδ corresponds to a spacetime with local rotational symmetry or of plane wave equilibrium type. Whenever matter is present and γ = 0, we do not have to compute the full Kretschmann scalar, but it is enough to determine whether the contraction of the Ricci curvature R αβ R αβ is unbounded. We show that this is the case for all causal geodesics, there are no exceptions. The arguments for the matter case work similarly as in the Bianchi A case, which was discussed in [Rin00b]. The full statement about curvature blow-up in the incomplete directions of causal geodesics is collected in the following two theorems. Theorem 1.11. Consider orthogonal perfect fluid Bianchi class B initial data (G, h, k, µ 0 ) for vacuum µ 0 = 0, which is neither locally rotationally symmetric of Bianchi type VI −1 nor of plane wave equilibrium type. Let (M, g) be the maximal globally hyperbolic development of the data, solving Einstein's equations for vacuum T αβ = 0. Then the Kretschmann scalar R αβγδ R αβγδ is unbounded in the incomplete directions of causal geodesics. Remark 1.12. We recall that orthogonal perfect fluid Bianchi class B initial data excludes Lie groups of Bianchi type VI −1/9 , see Definition 1.5. In case of such initial data, more precisely for such initial data which is 'exceptional', see Remark 11.14, our statements do not apply. In fact, their behaviour is expected to differ significantly: close to the initial singularity, the corresponding spacetimes are expected to show chaotic oscillatory behaviour, much the same as Bianchi type VIII and IX spacetime in class A.
Due to the additional complications one meets in these models, they have not been studied extensively to this date. An approach similar to the one we choose here is made in [HHW03]. In future works, one can hope that their results and conjectures can be used to find an answer to Strong Cosmic Censorship which also applies to the remaining Bianchi type we do not treat here.
Remark 1.13. Initial data such that both the Kretschmann scalar R αβγδ R αβγδ and the contraction of the Ricci tensor with itself R αβ R αβ remain bounded in the incomplete directions of causal geodesics in the maximal globally hyperbolic development are locally rotationally symmetric or of plane wave equilibrium type. In particular, such initial data has additional symmetry and can therefore be considered non-generic. As a consequence, Strong Cosmic Censorship holds in the class of orthogonal perfect fluid Bianchi class B initial data.
It is interesting to note here that we only in the vacuum case have to exclude certain nongeneric initial data sets. In the matter case, each initial data set has a corresponding development which is inextendible. The presence of matter appears to simplify things, at least in terms of the Strong Cosmic Censorship conjecture.
Remark 1.14. In case µ > 0 and γ = 0, the stress-energy tensor (2) with linear equation of state (3) takes the form p = −µ and can be interpreted as the stress-energy tensor of vacuum with a positive cosmological constant. In this case, we find a statement similar to the previous one: We can show that the maximal globally hyperbolic development (M, g) is of the form (4), with I = (t − , t + ) an interval, and every timeslice {t} × G has positive mean curvature with respect to the normal vector ∂ t . Then the Kretschmann scalar R αβγδ R αβγδ or the contraction of the Ricci tensor with itself R αβ R αβ is unbounded in the incomplete directions of causal geodesics, with the following possible exceptions: • local rotationally symmetric initial data of Bianchi type VI −1 • initial data of Bianchi class B, and in scale free variables the metric and second fundamental form of every slice {t} × G converge to initial data of plane wave equilibrium type, as t → t − . Note that we did not make precise in what sense the initial data converges, and that for initial data on a Lie group of type other than Bianchi VI −1 , we have not clearly stated any property which can be interpreted as non-generic apart from this convergence. Further, we have used the existence of a specific foliation in order to formulate the statements. Precise results are given and proven in scale free variables using a notion of convergence which we introduce in the following. The results, both for the two previous theorems and the special case of positive cosmological constant in vacuum, then follow directly from translating back into the current formulation.

Expansion-normalised variables and previous results.
Most of the work in this paper is carried out in a setting different from the one above. Instead of starting with initial data (G, h, k, µ 0 ) to Einstein's equations and proving properties of the resulting maximal globally hyperbolic development, we adopt a different view-point. The information given by initial data (G, h, k, µ 0 ) can be translated into the form of expansion-normalised and dimensionless variables Σ + ,Σ , ∆ ,Ã , N + , which have been introduced in [HW93]. Einstein's field equations translate into an ordinary differential equation in these variables, defined on a compact subset of R 5 and with a changed time coordinate. It is in these variables that we work and obtain the results on curvature blowup. Every initial data set corresponds to a point in the compact subset of R 5 , and once one has determined the solution curve in expansion-normalised variables, the maximal globally hyperbolic development (4) which solves Einstein's field equations with correct initial data can be determined. The detailed construction is carried out in Section 11. Via this construction, our results in expansion-normalised variables can be carried over to the setting of geometric initial data.
Note that the relations (8) and (9) are preserved by the evolution equations (5), see Remark 3.2. We now give a rough explanation as to how the expansion-normalised variables are related to the maximal globally hyperbolic development to given initial data (G, h, k, µ 0 ) as in Def. 1.5. For this, assume that we are given a spacetime as in (4), together with a function µ, which solve Einstein's field equations (1) for a perfect fluid (2) with linear equation of state (3). Assume that ∂ t , the vector field in the I direction which is unit timelike and orthogonal to every {t} × G, coincides with the vector field u from the stress-energy tensor. Assume further that the induced metric and second fundamental form on {0} × G coincide with h and k from the initial data, i. e. (I × G, g, µ) is a development of the data. One obtains consistency between the evolution in expansion-normalised variables and the initial data approach if one chooses the parameter γ from the expansion-normalised variables to coincide with the same-named constant from the linear equation of state (3). Further, if the Lie group G has a group parameter η, then the parameter κ from the expansion-normalised variables is set to satisfy κ = 1/η. If the Lie group has no group parameter, then κ = 0.
For every dimensionless time τ , the point (Σ + ,Σ, ∆,Ã, N + )(τ ) in expansion-normalised variables describes the geometric and dynamical properties of the timeslice {t(τ )} × G. The variables ∆,Ã, N + carry information on the three-dimensional metric of this timeslice, while Σ + ,Σ describe the shear of the vector field ∂ t . A perfect fluid spacetime is described by a curve in R 5 solving the constrained evolution equations (5)-(9), and we sometimes call such a solution an orbit. Transforming the momentum constraint from the point of view of geometric initial data into the expansion-normalised variables yields the constraint equation (8). Similarly, the Hamiltonian constraint justifies the definition of Ω in (10), which is the expansion-normalised version of the energy density µ.
Statements about geometric initial data can be transformed into statements about specific points in the variables (Σ + ,Σ, ∆,Ã, N + ). Conversely, we show in Section 11 how one can, given the solution in expansion-normalised variables through this point, construct the corresponding maximal globally hyperbolic development. Hence, statements about spacetimes solving Einstein's perfect fluid equation on the one hand and orbits solving the evolution equations in expansionnormalised variables (5)-(11) on the other hand carry equivalent information. It is for this reason that we can carry out our analysis wholly in the setting of expansion-normalised variables, and only at the end translate the results back to the point of view of geometric initial data. The setting of expansion-normalised variables is a rather approachable one, as we discuss a polynomial ordinary differential equation on a compact subset of R 5 . For instance, this allows the use of techniques from the theory of dynamical systems.
The system of differential equations (5)-(7) with constraints (8)-(9) was first introduced and studied in [HW93]. In this reference, Hewitt and Wainwright identify invariant subsets representing the different Bianchi types, and we recall these subsets in Tables 1 and 2 below, together  with several other invariant subsets representing models with additional symmetry, Table 3. Additionally, [HW93] discuss several sets of equilibrium points of the evolution equations. Of importance to our work are the Kasner parabola with the two Taub points and the plane wave equilibrium points, as convergence to points in these sets corresponds to the incomplete direction of causal geodesics in the maximal globally hyperbolic development, which is what we need to investigate in light of the Strong Cosmic Censorship conjecture.
Definition 1.15. The Kasner parabola is the subset K defined by .= (Σ + ,Σ, ∆,Ã, N + ) = (1/2, 3/4, 0, 0, 0) . Definition 1.17. The plane wave equilibrium points are the elements of the set L κ defined by Both the Kasner parabola K and the plane wave equilibrium points L k satisfy Ω = 0, are contained in the set defined by the constraint equations (8)-(9) and consist of equilibrium points, i. e. the right-hand side of the evolution equations (5) is zero.
Information about the local stability can be drawn from the linearised evolution equations in the extended five-dimensional space, by which one means the linear approximation of the evolution equations (5)-(7), without restricting to the constraint equations (8)-(9). We give the explicit form of this vector field for points on the Kasner parabola K in Appendix A.1. The eigenvalues of this vector field are (13) 0 2 1 + Σ + ± 3 1 − Σ 2 + 4(1 + Σ + ) 3(2 − γ). Figure 2. The plane wave equilibrium points L κ , projected to the Σ +Σ -plane. For reference, the Kasner parabola K is plotted as a dashed line.
We note at this point that there appear to be typos in the eigenvalues in both [HW93,Sect. 4.4] and [WE97, Sect. 7.2.3]. We give the corrected eigenvalues and state the corresponding eigenvectors in Appendix A.1. The signs of the individual eigenvalues in these two references however are given correctly, and with this information Hewitt-Wainwright are able to identify the points on the Kasner parabola to the right of Taub 2 (with 1/2 < Σ + ≤ 1) as local sources, and the points to the left of Taub 2 (with −1 < Σ + < 1/2) as saddles. For the two Taub points, the linearisation of the evolution equations alone does not determine the local stability, as two, or even three in case of the point Taub 1, of the eigenvalues vanish in these two points.
Similarly, one considers the linearised evolution equations in the extended five-dimensional state space for plane wave equilibrium points. The eigenvalues of this vector field are with N + as in equation (12). See Appendix A.2 for more details. Again, the number of positive, negative, and zero eigenvalues can be used to determine the local stability, and [HW93] identify points with Σ + > −(3γ − 2)/4 as local sources, and points with Σ + < −(3γ − 2)/4 as saddles. In the point Σ + = −(3γ − 2)/4 two eigenvalues vanish, and the local stability cannot be determined this way. Using dynamical systems method, Hewitt-Wainwright show that the Kasner parabola K is of central importance for the asymptotic behaviour of solutions to the evolution equation as τ → −∞, as it contains the α-limit set of (non-constant, generic) solutions. We state and prove a refined version of [HW93,Prop. 5 1.4. New results: convergence behaviour in expansion-normalised variables, curvature blow-up. In the present paper, we obtain more detailed results about the behaviour of solutions to the evolution equations (5)-(11) as τ → −∞. We refine a statement on the αlimit sets of solutions by [HW93]: all α-limit points are contained in the union of the Kasner parabola K, the plane wave equilibrium points L κ and the point {Σ We further prove that every solution has one unique α-limit point, i. e. solutions to the evolution equations converge, to a limit point contained the plane wave equilibrium points L k or the Kasner parabola K, see Prop. 4.4 and Prop. 6.1. Only constant solutions converge to the point Different subsets of this set of limit points have very different qualitative properties: While solutions converging to the plane wave equilibrium points L κ , to the point {Σ + =Σ = ∆ =Ã = N + = 0} and to the subarc of the Kasner parabola K with −1 ≤ Σ + ≤ 1/2 are contained in a 'small' set, namely a countable union of C 1 submanifolds of positive codimension (Prop. 4.2, Prop. 5.1, Thm 7.2, Thm 8.5, Thm 10.3), the remaining arc of the Kasner parabola contains the limit points of all remaining solutions. Considering the set of all possible limit points as a whole, this yields the following statement: Theorem 1.18. Assume either vacuum or inflationary matter, i. e. either Ω = 0 or Ω > 0, γ ∈ [0, 2/3). Then the following holds for solutions to the evolution equations (5)-(11): • Consider the sets describing Bianchi type VI η or VII η , i. e. the sets Then every non-constant solutions converges to the point K ∩ {Σ + = 0}, as τ → −∞.
For Kasner points situated to the right of Taub 2, we further find that they govern the behaviour of all solutions starting close to them. In fact, we show in Prop. 9.1 that for every element of the subarc K ∩ {Σ + > 1/2}, points contained in a sufficiently small neighborhood also converge to this subarc as τ → −∞, to a limit point close to the original one.
As a consequence of this result, for a given parameter κ, a solution converging to a point on the Kasner parabola K situated to the left of Taub 2 has to be either a Bianchi class A (Ã = 0) solution, or it is a Bianchi class B (Ã > 0) solution but can only converge to one of two specific points on K, namely one with Σ + = ± κ/(κ − 3). For this to hold, the parameter κ has to be non-positive. In particular, no Bianchi type VII η solution can converge to a Kasner point situated to the left of Taub 2, as in this Bianchi type one has κ = 1/η > 0.
This statement is very different in spirit than the property of a subarc being a local source or saddle and goes far beyond what can be obtained by considering only the local stability. Instead, we discuss the asymptotic convergence behaviour of the individual expansion-normalised variables one by one, taking into account the full non-linear properties of the evolution equations (5)-(7) in combination with the exact form of the constraint equation (8).
Using the same methods, we also obtain a detailed statement about the Taub points, which due to the multiple zero eigenvalues of the linearised evolution equations in these points cannot be treated by local stability at all. We prove that only the constant orbit converges to the point Taub 1, see Prop. 5.1 and Prop. 5.2. Solutions converging to the point Taub 2 have to be locally  rotationally symmetric, see Definition 3.4 and Thm 7.2, and these locally rotationally symmetric  solutions appear as possible exceptions to an unbounded Kretschmann scalar below in Thm 1.23. For the plane wave equilibrium points with more than one vanishing eigenvalue, we make use of techniques from the theory of dynamical systems and show that solutions are contained in a countable union of submanifolds of dimension at most two and of positive codimension, see Prop. 10.3 and Remark 10.4.
In addition to the statements about asymptotic behaviour in the limit τ → −∞, we also obtain the following result on the late time behaviour.
The main objective of the present paper is to prove the Strong Cosmic Censorship conjecture in the setting of orthogonal Bianchi class B perfect fluid and vacuum spacetimes. This is done by showing that boundedness of the Kretschmann scalar R αβγδ R αβγδ or the contraction of the Ricci curvature with itself R αβ R αβ in the incomplete direction of causal geodesics is a non-generic property. Making use of the transformation between geometric initial data with corresponding maximal globally hyperbolic development (4) on the one hand, and points in the state space of expansion-normalised variables with corresponding solutions to the evolution equations (5)-(9) on the other hand, we can express these two curvature expressions in terms of the expansionnormalised variables. Both quantities then depend only on the initial mean curvature and energy density, the parameters γ and κ, the time τ and the point in expansion-normalised variables, as we explain in the beginning of Section 12. The incomplete direction of causal geodesics corresponds to the limit τ → −∞, see Prop. 11.24. The statements about curvature blow-up in expansion-normalised variables are as follows.
Remark 1.22. We remark at this point that the statement in the matter case is obtained without knowledge on the detailed asymptotic behaviour of the individual variables. In fact, only the evolution equation (11) for Ω is considered to prove this statement, combined with details on how the expansion-normalised variables are obtained from geometric initial data sets. In contrast, the vacuum case treated in the next theorem is rather intricate, and necessitates a detailed discussion of all variables.
It is interesting to see that in the vacuum case, there are certain exceptions to unboundedness of the curvature, while there are none in matter. In a way, one could therefore consider the matter case as the easy case, while it is the vacuum case where interesting-and ultimately more difficult-behaviour becomes visible. In Bianchi class A models, a similar observation has been made, see [Rin00b] and [Rin09].
Theorem 1.23. Consider a solution to equations (5)-(11) with Ω = 0 which is neither the constant solution in the point Taub 1, nor a locally rotationally symmetric Bianchi type I, II or VI −1 solution, nor a plane wave equilibrium solution. Then the Kretschmann scalar R αβγδ R αβγδ is unbounded as τ → −∞.
Remark 1.24. In the previous theorem, locally rotationally symmetric solutions are defined by being contained in the set see Definition 3.4, and consult Tables 1, 2, and 3 for more details on the separation into the  different Bianchi types. In vacuum Ω = 0, a bounded Kretschmann scalar is possible only for solutions converging to one of the Taub points or to the plane wave equilibrium points L k . Only in case of Taub 2 can such a solution be non-constant, and it necessarily has to be locally rotationally symmetric.
Further, there is a countable family of C 1 submanifolds {L m } m∈N of dimension at most two such that For certain Bianchi types, or equivalently certain values of the parameter κ, the following additional restrictions hold: • Bianchi type VI η , which implies κ = 1/η < 0: Every solution in C converges to an element of the plane wave equilibrium points L κ with Σ + = s satisfying • Bianchi type V , which implies κ = 0: Every solution in C is contained in Σ + =Σ = ∆ = N + = 0, andÃ decreases monotonically from 1 to 0.
Remark 1.26. In the state space in expansion-normalised variables, which is the subset of R 5 given by the constraint equations 8-(9), the different Bianchi types are represented by different invariant subsets, see Table 1. In case of Bianchi type VI η , VII η , or IV, this is a subset of dimension four. Consequently, the sets L m defined in the previous theorem are of positive codimension. The invariant subset describing Bianchi type V solutions is of dimension two, and by the additional restriction stated in the theorem the set C is contained in a set of dimension one, which thus is also of positive codimension.
Compared to the statements in terms of initial data to Einstein's equations, Thm 1.10 and Thm 1.11, there is a larger number of exceptions to unboundedness of either geometric scalar in Thm 1.21, Thm 1.23 and Thm 1.25. The reason is that the expansion-normalised formulation allows for every initial data to the evolution equations (5) which satisfies the constraint equations (8) and (9). This in particular includes certain points in R 5 which correspond to Bianchi class A initial data. In Thm 1.10 and Thm 1.11, these are excluded by the assumption on the initial data set.
Part of this material is based upon work supported by the National Science Foundation under Grant No. 0932078 000, while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the winter semester of 2013.
2. Structure of the paper As the scope of this article is rather large, touching a number of different areas and techniques, we wish to give the reader a guide on how to read it, depending on the interests and prerequisites. We also shortly present the main ideas of the proofs.
We have given, in equations (5)-(11), the evolution equations which correspond to Einstein's equations for orthogonal perfect fluid Bianchi B initial data, and stated our main theorems in the introduction. In Sections 3-10, we discuss various properties of this evolution problem. We explain their content in more detail a bit further down.
In Section 11, we move our focus away from a detailed discussion of the evolution equations and towards the relation of these equations to spacetimes solving Einstein's equations. It is here that we prove equivalence between the evolution problem in expansion-normalised variables and the problem of finding, to given geometric initial data, the maximal globally hyperbolic development. We recall the classification of three-dimensional Lie groups and how it relates to Lie groups as initial data sets. This leads, in and after Lemma 11.13, to an explanation of why we excluded Bianchi VI −1/9 Lie groups in the definition of initial data. We recall the construction of expansion-normalised variables for orthogonal Bianchi class B cosmological models, proposed by [HW93], and show how this can be used to construct a spacetime which is shown to by the maximal globally hyperbolic development. To the author's knowledge, these constructions and proofs have not been given before.
In the final section, we give the proofs of the main theorems, which are stated in the introduction. Those which are formulated in expansion-normalised variables, Thm 1.18 Thm 1.21, Thm 1.23, and Thm 1.25 are treated first. Apart from computations of a number of curvature quantities in expansion-normalised variables which we carry out there, most of the work has been done in the sections before, and the proofs of these theorems merely collect the necessary results from Sections 3-10. In order to then prove the remaining main theorems, Thm 1.10 and Thm 1.11, we use the equivalence between the setting of geometric initial data to Einstein's equation and that of the evolution equations in expansion-normalised variables. With this information at hand, proving the remaining main theorem is equivalent to translating the results from Thm 1.21 and Thm 1.23 back to the setting of geometric initial data.
Let us now give a more detailed description of the content of Sections 3-11, which is where the main part of the work is carried out. These sections all discuss more and more detailed properties of the evolution in expansion-normalised variables. In the end, we wish to make a statement about Strong Cosmic Censorship in the C 2 -sense, and one way of doing so is to determine for which initial data both the Kretschmann scalar R αβγδ R αβγδ and the contraction of the Ricci tensor with itself R αβ R αβ remain bounded in the incomplete direction of causal geodesics in the maximal globally hyperbolic development, see Conjecture 1.1, Conjecture 1.2 and Remark 1.3. For spacetimes corresponding to solutions to equations (5)-(11), this direction corresponds to τ → −∞. Consequently, we have to understand the asymptotic behaviour of solutions as τ → −∞, and determine boundedness of the Kretschmann scalar and the contraction of the Ricci tensor with itself along such solutions. If we can show that the set of solutions along which both quantities are bounded at negative times is suitably small and can be considered non-generic, we have proven Strong Cosmic Censorship for this class of initial data.
Section 3 starts with a general discussion of basic properties of equations (5)-(11), such as compactness of the state space, the smoothness of the constraint surface and its division into different invariant subsets corresponding to the different Bianchi types. In Section 4, we examine the so-called α-limit set, which is the set of points where solutions accumulate as τ → −∞. In the case of vacuum and inflationary matter, the α-limit set is contained in the Kasner parabola K and the plane wave equilibrium points L κ , together with one additional point in the case of inflationary matter, see Prop. 4.2. This result about accumulation points is subsequently strengthened to convergence, as τ → −∞, in Prop. 4.4 and Prop. 6.1, where the latter requires additional assumptions. In particular, one needs to argue that the only solution with the point Taub 1 as an α-limit point is the constant orbit, which is the main result we achieve in Section 5. We additionally prove a statement about isotropisation at late times: In case of inflationary matter, all solutions converge to the same point as τ → +∞, see Prop. 4.3.
Before we continue with a summary of the results of Sections 6-11, let us briefly describe the main techniques. In the end, we wish to understand exactly which solutions or sets of solutions converge to specific points, for example to those points which are of interest for the Strong Cosmic Censorship conjecture. To do so, we determine exponential decay or convergence properties of the individual variables using the evolution equations with constraints (5)-(11), for example the variableÃ: For a solution converging to a point ( s, 1 − s 2 , 0, 0, 0), s ∈ [−1, 1], on the Kasner parabola K, as τ → −∞, the evolution equatioñ . This is a consequence of Lemma 4.5, and in this particular case stated as Lemma 6.5. Further in our discussion, we can improve this exponential decay estimate to not only include the slowest, but also the secondslowest exponential term, see also Prop. 6.14.
Similar convergence properties are obtained for the remaining variables upon convergence to the Kasner parabola. We find that the five variables split into two groups, with identical decay rates in each group. The variables Σ + ,Σ andÃ, converge to their respective limit value exponentially to order 4 + 4 s −ε, while ∆ and N + decay to zero exponentially to order 2 + 2 s ±2 3(1 − s 2 ) − ε, where the sign depends on the sign of ∆N + at sufficiently negative times. In some cases, this sign is determined by where the limit point on the Kasner parabola is located with respect to the Taub point 2. This splitting into two different decay rates leads to a tension which we exploit on several occasion: The constraint equation (8) written in the form ΣN 2 + − 3∆ 2 = 3Σ 2 + + κΣ Ã separates the variables according to their decay rate. The fast decay of the left-hand side can only be achieved if the other side vanishes altogether, which is what we use in Lemma 6.8 to show the relation stated in Prop. 1.19 between the parameter κ and the limit value s in Bianchi B solutions. We further use this splitting to obtain lower bounds on the decay rates or even exclude certain subsets of the state space altogether upon convergence to specific subarcs of the Kasner parabola, see the proofs of Lemmata 6.8, 6.11, and 6.12. Once these decay and convergence rates are found, they are used to determine the subsets containing all solutions converging to the point Taub 2, or the subarcs of the Kasner parabola to the left and right of it, respectively.
The most important of the convergence statements achieved in Section 6 are the following: For all variables, the lowest order exponential terms upon convergence to a point on the Kasner parabola ( s, 1 − s 2 , 0, 0, 0) with s ∈ [−1, 1/2] are obtained in Prop. 6.2. For specific cases, this statement is refined in Prop. 6.14, where we determine the lowest order exponential term more precisely and find the second-lowest, again for all variables. Convergence to a point on the Kasner parabola which is situated to the left of the point Taub 2 requires that ∆N + ≥ 0, and we have to treat the case of convergence to the other subarc of the Kasner parabola while ∆N + < 0 separately. In Lemma 6.12, we determine the lowest order exponential terms for all variables for this situation. This section is the foundation of most of the results obtained later.
In Section 7, we determine exactly which solutions converge to the point Taub 2. We rely on the results from the previous section and find that all such solutions have a local rotational symmetry. This result is directly linked to Strong Cosmic Censorship: The only non-constant solutions converging to a point on the Kasner parabola for which both the Kretschmann scalar R αβγδ R αβγδ and the contraction of the Ricci tensor with itself R αβ R αβ possibly remain bounded as τ → −∞ are those which converge to the point Taub 2. As all such solutions are locally rotationally symmetric, convergence to the point Taub 2 and thereby boundedness of these two curvature invariants can be considered non-generic.
Sections 8 and 10 treat the equivalent question for the arc of the Kasner parabola K to the left of Taub 2 and the plane wave equilibrium points L κ . We give qualitative results on the set of solutions with this convergence behaviour, using a theorem from dynamical systems theory which we recall in Appendix B.3. The statements we prove using dynamical systems theory are Thm 8.5 and Thm 10.3, which build upon where the eigenvalues to the linearised evolution equations are located in the complex plane.
Convergence towards one of the remaining points on the Kasner parabola K, located to the right of the point Taub 2, is treated in Section 9. In [HW93], these points have been identified is local sources, and we make this statement more precise. For such perfect fluids where the convergence behaviour at early times is sufficiently well understood, we show that given a solution converging to a Kasner point to the right of Taub 2, all solutions intersecting a sufficiently small neighborhood of its limit point converge to a Kasner point close to this limit point.
In Appendix A, we discuss the linear approximation of the evolution equations in the extended state space, which we investigate both on the Kasner parabola K and on the plane wave equilibrium points L κ .
In the second appendix, Appendix B, we present a result from the theory of dynamical systems. This statement is an important prerequisite for Sections 8 and 10, where we discuss the properties of orbits converging to the Kasner parabola K to the left of the point Taub 2 and to the plane wave equilibrium points L κ .

Basic properties of the expansion-normalised evolution
In this section, we start discussing properties of solutions to the evolution equations with constraints in expansion-normalised variables, equations (5)-(11). We discuss the range of the individual variables and describe several subsets of the state space which remain invariant under the evolution. These correspond to specific Bianchi types, matter models, or families of models with additional symmetry. For a classification of Bianchi Lie groups in terms of their structure constants, we refer to Subsection 11.1.
One sees from the evolution equation of Ω, equation (11), that the sets Ω = 0 and Ω > 0 are invariant, and in the latter case the behaviour of Ω depends on the value of the constant γ. One distinguishes between the following cases: In this paper we also cover the case γ ∈ [2/3, 2). Of particular interest are the values γ = 1 and γ = 4/3, which correspond to dust and radiation respectively. Due to the definition of the variable Ω as the expansion-normalised version of the energy density µ, see equation (68) below, the previous definition coincides with the distinction between vacuum and matter via µ = 0 and µ > 0 resp. In vacuum, one sees that the expression for q, equation (6), simplifies to (14) q = 2 Σ 2 + +Σ = 2 1 −Ã −Ñ , and in the general case the following expressions are of use We note that the differential equations with constraints (5)-(11) as well as the additional expressions for q, equations (14)-(15) are invariant under the symmetry Remark 3.2. The constraint equations (8)-(9) are invariant under the evolution equations (5) with q andÑ as in (6) and (7). For the first constraint, equation (8), this follows from The second and fourth relation in (9) are an immediate consquence of the evolution equations ofÃ in (5) and of Ω in (11), which imply thatÃ = 0 and Ω = 0 are invariant sets.
For the remaining two inequalitiesΣ ≥ 0 andÑ ≥ 0, notice that equality in both cases at some time implies ∆ = 0 = Σ +Ã at that time by the constraint equation (8). The evolution equations forΣ and ∆ in (5) together with those forÑ and Σ +Ã , then reveal thatΣ ≡ 0 ≡Ñ (as well as ∆ ≡ 0 ≡ Σ +Ã ) at all times. Further,Σ andÑ cannot change sign at different times, asΣÑ < 0 is excluded by the constraint equation (8). Hence, the inequalitiesΣ ≥ 0 andÑ ≥ 0 are preserved. The second advantage is that the different Bianchi B types are represented by invariant subsets of the same state space and can thus be discussed simultaneously. Table 1 lists the subsets and their names according to the Bianchi classification of Lie groups. We recall this classification in Subsection 11.1. This table was already given by [HW93], as were Tables 2 and 3 1 . In Table 2, two types of Bianchi A models are given which appear as boundary sets of Bianchi B invariant sets. Notation Restrictions B ± (II)Ã = 0, N + > 0 or N + < 0 B(I)Ã = ∆ = N + = 0 Table 3 lists several invariant subsets which describe Bianchi models with higher symmetry.

Notation
Class of models Restrictions We want to give particular attention to the locally rotationally symmetric (LRS) models. The property of local rotational symmetry in expansion-normalised variables is defined as follows: Comparison with the three LRS subsets given in Table 3 shows that the union of these subsets equals the union of all locally rotationally symmetric solutions. We have now defined the notion of local rotational symmetry twice, in terms of expansion-normalised variables in Def. 3.4 and in terms of initial data to Einstein's orthogonal perfect fluid equations in Def. 1.6. In Subsection 11.8, we clarify why these two definitions are equivalent under suitable transformation between the initial data setting and the expansion-normalised variables.
Considering the derivatives 1 Note that only one of the three LRS subsets in Table 3 was mentioned in [HW93], namely LRS Bianchi II.
LRS Bianchi VI −1 first appeared in [WE97], it is also called LRS Bianchi III in the literature. In the literature, the subset LRS Bianchi II in Table 3 is defined with ∆ 2 = Σ 2 + N 2 + . Considering the evolution equations of 3Σ 2 + −Σ and Σ + N + ± ∆, we realise however that it is the case with negative sign Σ + N + − ∆ = 0 which is preserved, not the one with positive sign, see also equations (17).
one sees that the three LRS subsets are invariant under the evolution equations (5). For the remaining sets given in Tables 1, 2 and 3, invariance follows by direct computation, see also Remark 3.2 for Bianchi VII η FLRW.
Remark 3.5. The evolution equations (5)-(7) define a dynamical system in R 5 , and we will on several occasions apply dynamical systems methods to this. Whenever we restrict our attention to only these evolution equations, i. e. without assuming the constraint equations (8)-(9), we call this the evolution equations in the extended state space.
The evolution in the physical, i. e. constrained, state space, is obtained via restriction to the set defined by the constraint equations (8)-(9). The first equation is invariant under the evolution and describes a submanifold as long as the gradient of its left-hand side does not vanish, implying that the hypersurface is non-singular. The only exceptions to a nonvanishing gradient are: • If κ > 0: Σ + =Σ = ∆ = 0, N 2 + = κÃ. This defines an invariant set of dimension one. • If κ = 0: Σ + = N + = ∆ = 0. This defines an invariant set of dimension two.
Remark 3.6. There is a similar set of expansion-normalised coordinates which is used to describe Bianchi class A models, but does not apply to class B models. It was introduced by Wainwright and Hsu in [WH89] and motivated the definition of the present coordinates. In the cases of Bianchi I and II, models can be decribed in both sets of variables.
Certain Bianchi perfect fluid spacetimes with a Lie group of type VI −1/9 cannot be described by the evolution equations (5)-(11). This is the case for the so-called 'exceptional' Bianchi B perfect fluids, a notion which we explain in Remark 11.14. Initial data sets in these spacetimes admit an additional degree of freedom compared to the 'non-exceptional' ones given in Definition 1.5. In these 'exceptional' cases as well, it is possible to introduce a set of expansion-normalised variables, this has been done in [HHW03]. Due to the additional freedom, these spacetimes are described by evolution equations in six dimensions instead of five.
4. The Kasner parabola and the plane wave equilibrium points The dynamical system for the expansion-normalised variables described by equations (5)-(11) possesses a number of equilibrium points and sets, i. e. points where the right-hand side of the evolution equations (5) becomes zero. These equilibrium sets have been studied in [HW93], to whom we refer for more details. For our present discussion, the Kasner parabola K and the plane wave equilibrium points L k are of importance, see Definition 1.15 and Definition 1.17 On the Kasner parabola K, one finds Ω = 0. Furthermore, in case γ < 2, the set K is characterised by q = 2. A closer look at the evolution equations reveals that the Kasner parabola is a curve in the Σ +Σ -plane consisting of individual equilibrium points. Information about the local stability can be drawn from the linearised evolution equations in the extended five-dimensional space. We give the explicit form of this vector field for points on the Kasner parabola K in Appendix A.1. The eigenvalues of this vector field are given in (13). The number of positive, negative and zero eigenvalues corresponds to the qualitative behaviour of orbits close to the Kasner parabola. This is a result from dynamical systems theory, which we state in Appendix B, but do not make use of in this section. The eigenvalues to the linearised evolution equations will appear further down as the exponential decay rates of certain linear combination of the variables.
We notice that for two special points on the Kasner parabola K the number of zero eigenvalues is greater than one, namely the points Taub 1 and Taub 2, see Definition 1.16. Especially the latter point will play a dominant role in our discussion, as it is the limit point of locally rotationally symmetric solutions, which constitute exceptions to the Strong Cosmic Censorship conjecture. To the right of the point Taub 2 (1/2 < Σ + ≤ 1) all non-vanishing eigenvalues are positive, while to the left of Taub 2 (−1 < Σ + < 1/2) exactly one of the four non-vanishing eigenvalues is negative. This difference reflects a difference in qualitative behaviour which we explore in more detail in this paper.
The plane wave equilibrium points L κ form a curve consisting of individual equilibrium points with Ω = 0, as was the case for the Kasner parabola. Using the function which is discussed in [HW93], the set L κ can be characterised as follows: Direct computation shows that the constraint equation (8) is equivalent to which due to the non-negativity ofΣ,Ñ and Ω, see (9), means that the function Z is non-negative, and vanishes if and only if Note that the identity −Σ + (1 + Σ + ) =Σ is a reformulation of Ω = Z, using the identities (21). Consequently, Z = 0 characterises the plane wave equilibrium points together with the Kasner point Taub 1. In caseÃ = 0, Z = 0 characterises exactly the point Taub 1. The derivative of the function Z is due to the evolution equations (5).
Remark 4.1. Both the Kasner parabola K and the plane wave equilibrium points L κ can be interpreted in terms of initial data to Einstein's orthogonal perfect fluid equations. This is done in detail in Subsection 11.8, where we explain how to establish a relation between initial data and solutions to Einstein's equations on the one hand and initial data and solutions to the evolution equations (5)-(11) on the other. We find that • The Kasner parabola K corresponds to vacuum Bianchi type I initial data.
• The set of plane wave equilibrium points L κ together with the point Taub 1 correspond to plane wave equilibrium initial data as in Def. 1.8. • The point Taub 1 corresponds to initial data of Bianchi type I which is of plane wave equilibrium type. • The point Taub 2 corresponds to initial data of Bianchi type I which is locally rotationally symmetric and the symmetric two-tensor k additionally satisfies k 11 > k 22 .
Alongside with the proof of this proposition, we find a statement about asymptotic behaviour in the future time direction in case of inflationary matter: for τ → +∞, all non-constant solutions converge to one point. Note that this is the only time that we consider this time direction, all other statements treat the case τ → −∞.

Proof of Prop. 4.2.
We start with the case of a vacuum solution Ω = 0. If q < 2 and Z > 0, then the derivative of Z defined in (19) is negative, which means that the function Z is strictly monotone decreasing. The α-limit set is therefore contained in the union of {q = 2} and {Z = 0}. The first set is the Kasner parabola K, while the second describes the plane wave equilibrium points L κ together with the point Taub 1. Suppose there is a solution with an α-limit point in L κ ∪ T1. Then there is a sequence of times τ k → −∞ such that In combination with Z ≥ 0 and monotonicity, this implies that Z is vanishing identically along the whole orbit. The orbit is therefore contained in the zero set of Z, which are the plane wave equilibrium points together with the point Taub 1. Hence the orbit is the constant orbit. For inflationary matter solutions Ω > 0, γ ∈ [0, 2/3), one reformulates the evolution equation for Ω into the form (1 − Ω) Ω using (15). Due to the restrictions on the individual variables (16), this shows that Ω is monotone increasing and implies that the α-limit set is contained in the union of {Ω = 0} and {Ω = 1}, as the latter is equivalent to the bracket vanishing. It follows immediately from monotonicity that if there is an α-limit point in {Ω = 1}, then the whole solution is contained in this set. The condition Ω = 1 characterises the point Σ + =Σ = ∆ =Ã = N + = 0, which proves the last statement.
For 0 < Ω < 1, one computes and concludes from and the fact that both Ω and Z are non-negative that the quantity e (6−3γ)τ Z/Ω decreases monotonically to some non-negative constant B 1 as τ → −∞. This has to be understood in the sense that the expression decreases when going backwards in time, here and on similar occurences of this formulation further down. Monotone convergence to B 1 implies where the integral is well-defined and bounded by 2 due to the bound on Σ + from (16). There are two cases to consider. If B 1 > 0, one can reformulate the previous equality into and sees that the right-hand side is bounded. The evolution equation for the left-hand side is . In our case, one concludes that q − 2 is integrable on the interval (−∞, 0). One further knows from the evolution equations that the derivative of q − 2 is a polynomial in the expansion-normalised variables and consequently bounded, and thus can conclude that q → 2 as τ → −∞. The α-limit set is therefore contained in {q = 2}, which characterises the Kasner parabola.
If B 1 = 0, then we have already argued above that Z Ω ≤ 2.
As Ω → 0 for τ → −∞ due to monotonicity and the fact that the α-limit set is contained in {Ω = 0}, this implies Z → 0. One concludes that the α-limit set is contained in the set {Z = 0}, which characterises the plane wave equilibrium points L κ together with the point Taub 1. This proves the first statement for inflationary matter models. The two cases for B 1 are mutually exclusive, and the only point satisfying both q = 2 and Z = 0 is the point Taub 1, which is contained in the Kasner parabola K. This concludes the proof.

Proof of Prop. 4.3.
We have found in the previous proof that Ω is monotone increasing with Due to the restrictions on the individual variables (16), this shows that all solutions with Ω > 0 have to satisfy Ω → 1 as τ → +∞, because the condition Ω = 1 is equivalent to the bracket vanishing. Due to the definition of Ω in equation (10) and the restriction of the individual variables in (16), this concludes the proof.
In the next proposition, we strengthen the result about the α-limit set: In inflationary matter models, there cannot be more than one α-limit point in the plane wave equilibrium points L κ , i. e. we find convergence.
Proposition 4.4 (Convergence to plane wave equilibrium points). Assume inflationary matter, i. e. Ω > 0, γ ∈ [0, 2/3), and consider a solution to equations (5)-(11) whose α-limit set is Proof. If the point Taub 1 is the only α-limit point, there is nothing to show. We assume therefore without loss of generality that there is at least one α-limit point contained in L κ . Due to the assumption on the α-limit set, the solution satisfies Z → 0 as τ → −∞. The zero set of Z is characterised by expressions (21), and inserting these into the third expression for q in (15), we conclude that the solution satisfies All plane wave equilibrium points L κ satisfy −1 < Σ + ≤ 0, which means that q converges to a non-negative value. In combination with the fact that 3γ − 2 is a strictly negative constant, this implies that the factor 2q−(3γ −2) in the evolution equation (11) is strictly positive for τ ≤ τ 0 sufficiently negative. Consequently, Ω decays to zero exponentially as τ → −∞.
We have seen in the proof of Prop. 4.2, that the solutions under consideration satisfy B 1 = 0, and consequently 0 ≤ Z ≤ 2Ω. Therefore, Z decays to zero exponentially as well. Boundedness of the state space, see Remark 3.3, therefore reveals that the right-hand side of equation (20) decays to zero exponentially, and therefore the same has to hold for all terms appearing on the left-hand side. Using the identities (21) to rewrite Ω − Z → 0, we find exponential decay for and as a consequence also for q + 2Σ + , applying the third expression in equation (15). Inserting this into the evolution equation (5) shows the same decay for Σ ′ + , therefore Σ + converges to some s ∈ [−1, 0]. The limiting values for the remaining variables follows from the definition of the plane wave equilibrium points L κ , Def. 1.17.
Consider now a non-constant solution to (5)-(11) in vacuum or inflationary matter whose α-limit set has a non-empty intersection with K \ T1 instead. Due to Prop. 4.2, this implies that the whole α-limit set is contained in the Kasner parabola K. Then as otherwise compactness of the state space, see Remark 3.3, would yield an α-limit point which does not lie on the Kasner parabola. A convergence result similar to the previous statement is achieved further down in Prop. 6.1. Its proof needs some additional work.
For the discussion of more detailed properties of solutions close to the α-limit points, we frequently make use of the following lemma which appears with slightly different notation as [Rin00a, Lemma 8].

Lemma 4.5. Consider a positive function
This lemma can be applied to the density parameter Ω without any further assumptions: In vacuum Ω = 0 holds, while inflationary matter models satisfy Ω > 0 with γ ∈ [0, 2/3). If the α-limit set is contained in the Kasner parabola K, one uses (22) to find convergence to zero at rate for τ ≤ τ ε sufficiently negative.
Notation 4.6. When we are only interested in the upper bound as τ → −∞, we also make use of the big O notation, i. e. we write as τ → −∞, if we want to say that there is a τ 0 > −∞ and a C Ω > 0 such that τ ≤ τ 0 implies Ω(τ ) ≤ C Ω e (6−3γ)τ . As we are in the present paper not interested in any other limit than τ → −∞, we frequently omit the range of τ .
If the function ζ in the previous lemma converges exponentially, the statement can be improved as follows.
Proof. Integration of The last two factors are contained in some interval [c M , C M ] ⊂ (0, ∞) for τ smaller than a fixed number τ 0 .

Convergence to Taub 1
In this section, we show that in vacuum and inflationary matter the only orbit with Taub 1 as an α-limit point is the constant orbit. As a consequence, this special Kasner point can be neglected when we determine in more detail the asymptotic behaviour close to the Kasner parabola. Remark 5.3. As the point Taub 1 is contained in the vacuum set Ω = 0, we conclude from these propositions that if Ω > 0, then the point Taub 1 is not allowed as an α-limit point.
Proof of Prop. 5.1. The proof revolves around the function Z defined by equation (19), and we start with the case of a vacuum solution Ω = 0. As the point Taub 1 is an α-limit point but does not lie in L κ , Prop. 4.2 yields that the α-limit set is contained in the Kasner parabola K, which is characterised by q = 2. In the point Taub 1, the function Z vanishes. This point being an α-limit point therefore implies that Z → 0 along a time sequence τ k → −∞. In the proof of Prop. 4.2, we have used the existence of such a time sequence together with monotonicity of the function Z to conclude that Z is vanishing constantly along the whole orbit. This argument applies to the present case, and we conclude that the orbit is contained in the zero set of Z. The set Z = 0 equals the union of L k and the point Taub 1 and consists solely of equilibrium points. As a consequence, the solution has to be the constant one in the point Taub 1, as this is the only point which satisfies both Z = 0 and q = 2.
In the case of matter Ω > 0, one first realises that due to its evolution equation (5), ifÃ vanishes at one time, then it vanishes along the whole orbit. One then reformulates the evolution of Σ + using the definition ofÑ from (7) together with equation (15) for q to find IfÃ = 0, then Σ ′ + < 0, as Σ + is contained in the interval [−1, 1] by Remark 3.3. Hence, Σ + is monotone decreasing, and an argument similar to the one for Z in the vacuum case applies: The assumption on the α-limit set gives a sequence of times τ k → −∞ such that In combination with Σ + ≥ −1 and monotonicity, this shows that Σ + = −1 along the whole orbit. This in turn implies that Z vanishes along the orbit and concludes the proof forÃ = 0, as then Ω = 0 due to (21), a contradiction. Assume thereforeÃ > 0 and Ω > 0. We first prove convergence, i. e. that Taub 1 is the unique α-limit point, then show that the only orbit converging to this point is the constant one. From Prop. 4.2 we know that the α-limit set is either contained in K or in L κ ∪ T1, and together with Prop. 4.4 this implies that we only have to show convergence for solutions where the α-limit set is contained in the Kasner parabola K.
As shown in (23), Ω then decays as for τ ≤ τ ε , since all factors apart from Ω are at least bounded. Therefore, there exists a nonnegative constant B 2 such that In case B 2 > 0, the function Z is bounded away from zero, which excludes Taub 1 as an α-limit point. Consequently B 2 = 0, which means The convergence of q to 2, see (22), implies |q − 2| < ε for sufficiently negative times, hence As Z → 0, and additionallyÃ → 0 from the assumption that the α-limit set is contained in the Kasner parabola K, one concludes that Σ + → −1. The only point in K with this property is the point Taub 1, which implies convergence to this point. It remains to exclude non-constant solutions withÃ > 0 and Ω > 0 which converge to the point Taub 1 as τ → −∞. Knowing that Σ + → −1 and q → 2, one can apply Lemma 4.5 to the evolution equation ofÃ, equation (5), to obtain for τ ≤ τ ε . Using the decay of the function Z together with boundedness of the state space, Remark 3.3, in equation (20) yields ∆ 2 = O(e (6−3γ−ε)τ ). With this, one computes as N + is at least bounded, and finds and therefore However, due to (24)Ã decays at most as e ετ , and the bracket on the left-hand side converges to −1, a contradiction. ThusÃ > 0 is not possible, which concludes the proof.

Proof of Prop. 5.2.
The proof is similar to the one of Prop. 5.1 for inflationary matter, i. e. Ω > 0 with γ ∈ [0, 2/3). In that setting, we first had to show that the point Taub 1 is the unique αlimit point, but for the current statement, this holds by assumption. Showing that there are no non-constant solution converging to the point Taub 1 then hinged on the fact that In the case γ ∈ [2/3, 2), it is still possible to choose ε > 0 sufficiently small that this holds. The argument at the end of the proof of Prop.5.1 excluding non-constant solutions then applies without any change.

Convergence properties and asymptotic decay towards the Kasner parabola
This section is the longest and most technical in our discussion of the evolution equations (5)-(11), and it is here that we prove the main statements we build upon in the following.
We focus our attention on solutions whose α-limit set is contained in the Kasner parabola K. For vacuum models, all non-constant solutions satisfy this property, while in the case of inflationary matter we have to additionally assume that the α-limit set does not intersect the plane wave equilibrium points L κ , see Prop. 4.2.
In a first step we show that these non-constant solutions with α-limit set in K converge, i. e. every such solution has a unique accumulation point. This is done in Prop. 6.1. An equivalent convergence result for inflationary matter solutions with an α-limit point in L κ has been obtained in Prop. 4.4, and we therefore find convergence for all inflationary matter and vacuum solutions.
The main aim of this section is to now obtain decay and convergence rates of the individual variables under the assumption of convergence to a limit point on the Kasner parabola K. The behaviour we discover is exponential decay or convergence, and the different exponents coincide with specific eigenvalues to the linearised evolution equations in the extended state space, see (13). We further find that the rates of convergence depend on where the limit point is situated relative to the point Taub 2. For solutions converging to the point Taub 2 or to a limit point to the left of this point, the lowest order exponential terms are determined in Prop. 6.2. In certain situations, we can refine this statement to even include the second-lowest term, see Prop. 6.14. Solutions which converge to such a point on the Kasner parabola necessarily have to satisfy that ∆N + > 0 for sufficiently negative times or ∆ ≡ 0 ≡ N + , as we conclude from Lemma 6.4 and Lemma 6.11.
In case ∆N + < 0 for sufficiently negative times, we determine the exponential convergence rates in Lemma 6.12.
We remark that even though we prove convergence only in the case of vacuum and inflationary matter, we then drop this restriction on the matter in all the following statements and only assume convergence to a limit point on the Kasner parabola. The results we show hold for all matter models apart from, for some statements, the stiff fluid case γ = 2. Proposition 6.1 (Convergence to the Kasner parabola). Assume either vacuum or inflationary matter, i. e. either Ω = 0 or Ω > 0, γ ∈ [0, 2/3), and consider a non-constant solution to equations (5)-(11). In the inflationary matter case, assume additionally that the α-limit does not intersect the plane wave equilibrium points L κ . Then there is an s ∈ (−1, 1] such that We prove this statement below. Proposition 6.2. Let γ ∈ [0, 2) and consider a solution to equations (5)-(11) converging to along the whole orbit, and .= min (6 − 3γ, 4 + 4 s ) if Ω > 0, and Π . .= 4 + 4 s if Ω = 0. Furthermore, the following properties hold: • If ∆ and N + do not both vanish identically, then ∆N + > 0 along the whole orbit and for The proof is divided into several steps which will have additional individual use later. The arguments revolve around the constraint equation (8) written in the form (25)ΣN 2 + − 3∆ 2 = 3Σ 2 + + κΣ Ã which is then used to determine the asymptotic decay properties of the individual variables.

Lemma 6.3. Consider a solution to equations
Proof. This follows from inspection of the evolution equation (5) for Σ ′ + , using the range of the variables given by the constraints (9) and Remark 3.3.
Lemma 6.4. Consider a solution to equations (5)-(11) whose α-limit set is contained in K and such thatΣ(τ ) > δ > 0 for τ ≤ τ 0 . Then one of the following statements holds: Note that this statement does not require any assumption on the matter model but holds for all values of γ ∈ [0, 2].
Proof of Prop. 6.1. Using Prop. 4.2, we can conclude that under the given assumptions, the αlimit points are contained in the Kasner parabola K, both for the vacuum and the inflationary case. We show in the following that no solution can have α-limit points with different Σ +values. As points on the Kasner parabola are uniquely identified by their Σ + -value, this implies convergence.
According to Prop. 5.1, either the orbit is the constant one in the point Taub 1, or this Kasner point is not contained in the α-limit set. The first case is excluded by the assumption. In the latter case, one can assume that for sufficiently negative times the orbit is bounded away from the point Taub 1, which implies that Σ + is bounded from below by some constant greater than −1 for sufficiently negative times τ . Consequently,Σ is bounded away from 0, as all possible α-limit points are contained in K ∩ {Σ + ≤ 0} and the point Taub 1 is excluded. Using additionally that q → 2 due to (22), the term 2(q + 2Σ + ) is bounded from below by some suitable constant D 1 > 0 for sufficiently negative times, and the evolution equation (5) forÃ reads One computes from the evolution equations (5) that 12∆N + A and sees that according to Lemma 6.4 this derivative does not change sign for τ sufficiently negative. As a consequence, the term N 2 + /Ã either converges to a non-negative real number or diverges to ∞ as τ → −∞. In the latter case, the evolution of Σ + is dominated by N 2 + alone, in the sense that Σ ′ + as in (26) has negative sign for sufficiently negative times. One concludes as for Σ + > 0 that there is a unique α-limit point. If the limit of N 2 + /Ã is finite, this means N 2 In combination with the decay estimate (23) on Ω, this yields that Σ ′ + is integrable and implies convergence.
As a direct consequence of convergence which we have shown in Prop. 6.1, we can apply Lemma 4.5 toÃ.
The next lemma is of a technical nature and will be used in the following.
andÃ and Ω decay as in Lemma 6.5 and equation (23), integration yields The assumption on γ ensures that Π > 0. Due to convergence to the Kasner parabola which implies 2 − q ≤ ε for sufficiently negative times, one finds that for some suitably chosen constant D 2 > 0 and sufficiently negative times τR Therefore, one concludes which is a contradiction to Σ + → s > −1. As a consequence, the functionR is bounded on (−∞, 0), and Having found detailed decay properties forÃ and Ω, the next step is to determine the asymptotic behaviour of ∆ and N + . Their decay rates are intertwined: One searches for a linear combination of ∆ and N + such that the evolution equation has a form suitable for Lemma 4.5, i. e.
and ζ converging as τ → −∞. It turns out that the limit of ζ not only depends on the value of s but also on the sign of ∆N + , and one recovers exactly the eigenvalues 2 + 2 s ±2 3(1 − s 2 ) of the linearised evolution equations on the Kasner parabola, see (13) and Appendix A.1.
If on the other hand ∆N + (τ ) < 0 for all τ ≤ τ 0 , then for every ε > 0 there existsε > 0 and τ ε > −∞ such that τ ≤ τ ε implies We remark at this point that for this statement, no restriction on γ is imposed.
In case ∆N + > 0 and s ∈ (−1, 1), one computes with two functions f 1 , f 2 converging to 0 as τ → −∞. We do not need the explicit form of these two functions here, but use them in the proof of Prop. 6.14. Note that this computation makes use of the fact that ∆ and N + have the same sign. As the last term in the bracket vanishes asymptotically, Lemma 4.5 yields the decay of ∆ +rN + in case both ∆ and N + are positive. If both are negative, the statement follows due to the invariance of the evolution equations (5)-(11) under a change of sign in these two variables.
In case s = 1, we findr = 0 and can no longer conclude that the quotient in the last line in equation (29) vanishes asymptotically. We assume ∆N + > 0 and compute forε > 0. By similar argument as above, we obtain the requested statement.
In order to treat the cases where ∆N + < 0 it is enough to replace every occurence ofr andε by −r and −ε, respectively.
The decay of ∆ and N + , depending on whether they have the same or opposite sign, determines the decay of the remaining variables.
holds along the whole orbit, andÃ > 0 implies that Remark 6.9. As a consequence of this result, one finds that in the setting of this lemma Proof of Lemma 6.8. If s = −1, then due to Prop. 5.1 and Prop. 5.2 the solution is the constant orbit for which all the required properties hold. Let us therefore assume s ∈ (−1, 1). It follows from Lemma 6.7 that for sufficiently negative τ withr as in equation (28). In particularr > 0, which implies that ∆ andrN + have the same sign and yields the upper bound of ∆ and N + in the statement. In case s = 1, we can use the same argument withr replaced byε. Comparing the decay of N + with the decayÃ = O(e (4+4 s −ε)τ ) from Lemma 6.5, we find Due to its definition (15), the quantity q inherits its convergence rate either from the decay of Ω as in equation (23) or from the one ofÑ +Ã, whichever is slower: for Π . .= min (6 − 3γ, 4 + 4 s ). Vacuum is defined by Ω = 0, and one sees that the statement holds if one sets Π . .= 4 + 4 s in this case. Using the above information in the evolution equations (5) for Σ + andΣ yields and thus gives the convergence rates for Σ + andΣ.
Writing the constraint equation as in (25) and applying the convergence rates of Σ + andΣ, one sees that The left-hand side is of order O(e (4+4 s +4 √ 3(1−s 2 )−ε)τ ) due to the above. The right-hand side-if non-vanishing-consists of the bracket with its explicitly given decay and the factorÃ which-if non-vanishing-decays at most as e (4+4 s +ε)τ , see Lemma 6.5. In order for equation (30) to be consistent, eitherÃ = 0 or 3 s 2 +κ(1 − s 2 ) = 0 has to hold.
For the term in the bracket, one further computes from the evolution equations (5) and . In Lemma 6.6, the function and consequently due to the above. IfÃ > 0, then the constant term on the right-hand side vanishes due to our previous arguments, and one can conclude that This yields that the right-hand side of equation (25) 1 decays exponentially to order at least 8 + 8 s +4 3(1 − s 2 ) − ε, both forÃ = 0 andÃ > 0: In the first case, the right-hand side vanishes identically, while in the latter case we have determined the decay properties for both factors on the right-hand side individually. It remains to show the lower bound for ∆ and N + in case s ∈ (−1, 1). From the last argument we conclude that 1 3Σ as τ → −∞, due to the lower bound on the denominator found in Lemma 6.7. As 1 − s 2 > 0, the variableΣ is bounded away from zero for sufficiently negative times. If one of the two squared terms tends to zero along a time sequence, so does the other one (along the same time sequence), which would imply that a contradiction. Consequently, both ∆ ∆ +rN + and N + ∆ +rN + are bounded from below by a positive constant for τ ≤ τ ε sufficiently negative. This gives the lower bounds on ∆ and N + and concludes the proof. Lemma 6.10. Let γ ∈ [0, 2) and consider a solution to equations (5)-(11) converging to ( s, 1 − s 2 , 0, 0, 0) with s ∈ [−1, 1]. Assume that ∆ = N + = 0 along the orbit. Theñ For the decay properties, there is nothing to show for s = −1, as this is the constant orbit due to Prop. 5.1 and Prop. 5.2. In the other cases, one simplifies equation (7) forÑ , equation (15) for q and the evolution equations (5) for Σ + andΣ using ∆ = N + = 0 to find then inserts the decay ofÃ and Ω from Lemma 6.5 and equation (23) respectively to conclude the proof.
For an orbit with ∆N + ≥ 0 for sufficiently negative times, there is no a priori restriction on where its limit point on the Kasner parabola is located with respect to Taub 2. In the case of opposite signs, the limit point has to be situated to the right of Taub 2, as the next lemma shows.
Proof. As Σ + ∈ [−1, 1] due to Remark 3.3, the limit value s is contained in the same interval. The case s = −1 can be excluded by Prop. 5.1 and Prop. 5.2. Due to Lemma 6.4 and under the assumption that s ∈ (−1, 1), we know that ∆N + has a fixed sign for sufficiently negative τ , hence it is enough to show that orbits with ∆N + (τ ) < 0 for τ ≤ τ 0 cannot converge to a Kasner point with s ∈ (−1, 1/2].
Consider first the case −1 < s < 1/2 and recall from Lemma 6.7 for τ sufficiently negative, wherer is as in equation (28). The bracket in the exponent is strictly negative for sufficiently small ε, which means that |∆ −rN + | grows exponentially as τ → −∞. This contradicts the fact that ∆ and N + converge to zero.
We can therefore assume that s = 1/2. One finds the special valuer = 1/2 and notices 2 + 2 s −2 3(1 − s 2 ) = 0. Hence for τ sufficiently negative. This estimate holds for ∆ and N + individually, as the following argument shows: The constraint equation (25) can be reformulated into The right-hand side divided by 3( with an asymptotically vanishing function f 3 , while the left-hand side divided by the same expression decays exponentially to order O(e (4+4 s −2ε)τ ) due to Lemma 6.5. This yields Using the definition ofÑ , the decay ofÃ from Lemma (6.5), and equation (15) for q, one additionally findsÑ The rest of the proof aims at constructing a contradiction to the slow decay behaviour of N + . This becomes possible by considering the evolution equations in more detail and relating the decay of several quantities. One starts by noting that Taylor expansion of the square root applied to the constraint equation (25) yields where we used Lemma 6.5, and the sign stems from our assumption ∆N + < 0. We then multiply the constraint equation (8) by N 2 + and reformulate the resulting expression using equation (7) definingÑ and equation (15) in the form The last line follows from convergence to the point Taub 2, i. e. s = 1/2, in combination with Lemma 6.5:Ã decays as O(e (6−ε)τ ), which is faster than the decay of Ω, given by O(e (6−3γ−ε)τ ).
Using once more Taylor expansion of the square root yields In the next step one determines the behaviour of Σ + andΣ. The evolution of Σ + + 1 has been computed in (27). With the function R defined as in Lemma 6.6, integration of (e R (Σ + + 1)) ′ yields and thus ForΣ, one findsΣ Hence Integration yields One can eliminate R using a linear combination of equations (32) and (33) and finds Now, we consider the evolution of N + . Due to equations (31) and (34) and the slow decay of N + , one finds which, recalling the discussion of R in the proof of Lemma 6.6, implies that there is a function f 4 which is integrable on (−∞, 0) and satisfies As Σ ′ + ≤ 0 asymptotically due to Lemma 6.3, every solution converging to the point Taub 2 satisfies 4Σ + ≤ 2, which means that the function |e −F4 N + | increases as τ → −∞. It follows that N + does not converge to 0, a contradiction.
With the results found above, we are finally in a position to prove Prop. 6.2.
Proof of Prop. 6.2. In case s = −1, the solution is the constant solution (Prop. 5.1 and Prop. 5.2), for which the statement trivially holds. Otherwise, the decay of Ω andÃ follows from equation (23) and Lemma 6.5. According to Lemma 6.4, there are three cases to be considered regarding the sign of ∆N + . As s ∈ (−1, 1/2], the case ∆N + < 0 is excluded by Lemma 6.11, and the remaining two cases are discussed in Lemma 6.8 and 6.10. In the proof of the previous lemma, we have discussed the asymptotic behaviour of orbits converging to a Kasner point to the left of Taub 2, as τ → −∞. In particular, we could exclude ∆N + < 0 asymptotically. For orbits converging to a Kasner point to the right of the point Taub 2, i. e. with Σ + converging to s ∈ (1/2, 1], we cannot exclude the negative sign. This stems from the fact that the non-positive eigenvalue 2 + 2 s −2 3(1 − s 2 ) which was used to construct a contradiction in the proof of Lemma 6.11 becomes positive when changing from the Kasner arc to the left of Taub 2 to the one on the right. Orbits which satisfy ∆N + < 0 asymptotically and converge to a point on the Kasner parabola to the right of Taub 2 exist, and as in the case of ∆N + > 0 asymptotically, one finds that the rates of convergence are related to eigenvalues of the linearised evolution equation: For ∆N + > 0 asymptotically, we found exponential decay of order 2 + 2 s +2 3(1 − s 2 ) (see Prop. 6.2), while for ∆N + < 0 asymptotically, we obtain exponential decay of order 2 + 2 s −2 3(1 − s 2 ).
Lemma 6.12. Let γ ∈ [0, 2) and consider a solution to equations (5)-(11) converging to ( s, 1 − Proof. If s = −1, then due to Prop. 5.1 and Prop. 5.2 the solution is the constant orbit for which all the required properties hold. Let us therefore assume s ∈ (−1, 1]. As in the beginning of the proof of Lemma 6.8, we conclude the upper bounds for ∆ and N + from Lemma 6.7. The decay for Ω andÃ has been determined in equation (23) and Lemma 6.5 independently of the sign of ∆N + . In particular,Ã decays faster than ∆ 2 and N 2 + , and consequently the evolution is no longer dominated byÃ, but rather by those two variables. More preciselỹ with Π defined as in the statement of the lemma. Using these properties in the evolution equations for Σ + andΣ yields and hence gives the convergence rates for Σ + andΣ.
In order to show the lower bounds for ∆ and N + in case s ∈ (−1, 1), one considers the expression 1 3Σ which is a reformulation of the constraint equation (25). Due to the previous argument, the numerator of the right-hand side decays as O(e (4+4 s −ε)τ ), while the denominator is bounded from below by e (2+2 s −2 √ 3(1−s 2 )+ε)τ due to Lemma 6.7. The left-hand side consequently converges to 0 as τ → −∞. We conclude the same way we did when proving the lower bounds of Lemma 6.8.
Proof. In case s = −1, nothing has to be shown, as this is the constant orbit (Prop. 5.1 and Prop. 5.2). For the remaining values, all that has to be shown to apply Lemma 4.7 is that the expressions appearing in the evolution equations ofÃ and Ω converge at exponential rates. From Lemma 6.4, one knows that ∆N + has constant sign for sufficiently negative times. The case ∆N + > 0 is covered in Lemma 6.8. If ∆ = N + = 0 along the whole orbit, then Lemma 6.10 yields the result. In case ∆N + < 0 exponential convergence is shown in Lemma 6.12. Note however that this case of opposite sign can be excluded for s ∈ (−1, 1/2] by Lemma 6.11.
These more detailed estimates can now be used to determine the convergence of the remaining variables in more detail. So far, we have found the constant term and the slowest order of exponential convergence. This is improved in the following way: We can relate the slowest order of exponential convergence of several variables, and determine which is the next non-vanishing order.
We carry this out for orbits converging to the left of the point Taub 2, as it is in this case that we make use of the more detailed convergence properties in Section 8. We point out, however, that the same approach can also be used on the remaining Kasner limit points.
ForÃ, the decay established in Lemma 6.13 implies that e −(4+4 s )τÃ is bounded for sufficiently negative τ . As e −(4+4 s)τÃ ′ = (2(q − 2) + 4(Σ + − s))e −(4+4 s)τÃ and 2(q − 2) + 4(Σ + − s ) = O(e (Π−ε)τ ), the left-hand side decays as O(e (Π−ε)τ ), and one finds The exact value of α is not known, only that it is positive due toÃ > 0. But this form shows that there are no terms of exponential order between 4 + 4 s and 4 + 4 s +Π. By the same method one finds that with a constant ω > 0, if not Ω ≡ 0 along the whole orbit. The improved decay properties ofÃ and Ω together with the ones from Lemma 6.8 implỹ Integrating these expressions and making use of the relation s 2 = κ/(κ − 3) yields that for s = 0 (or equivalently κ = 0) one finds (35) and (36). In case s = 0 and κ = 0, one obtains equation (38) forΣ, and Note that 2Π ≤ 8 < 4 + 4 √ 3. We are going to obtain a stronger estimate further down. To find the improved decay for the remaining variables ∆ and N + , we recall from the proof of Lemma 6.7 that which there appeared as equation (29). Here, one hasr = (1 − s 2 )/3, and a closer look at the computation carried out in that proof reveals that the two functions f 1 , f 2 which vanish asymptotically as τ → −∞ have the form Due to the previous results, both functions decay as O(e (Π−ε)τ ). As ∆ andrN + have the same sign, we see that the quotient with the functions f 1 , f 2 in equation (40) inherits the asymptotic behaviour of these two functions, i. e. decays exponentially to order O(e (Π−ε)τ ). Hence, we have and find e − 2+2 s +2 for some constant β = 0 having the same sign as ∆ and N + . Consequently, we obtain We further see from Remark 6.9 that  The very first bracket is of order e (2+2 s +2 √ 3(1−s 2 ))τ and not faster. Thus, Together with equation (41), this implies the decay expressions for ∆ and N + in the statement, with β ∆ =rβ N+ .
It remains to show equation (37). We know from the above thatÑ = N 2 3)τ ). Using the function R defined as in Lemma 6.6, we integrate 3)τ and obtain As the function R is non-negative, this gives the decay for Σ + and concludes the proof.

Asymptotics at Taub 2
In this section, we determine precisely which orbits converge to the point Taub 2, i. e. which solutions to equations (5)-(11) satisfy We have to restrict ourselves to γ ∈ [0, 2) in order to apply our results from the previous section. From Prop. 6.2, we know that along every solution converging to the point Taub 2 A(κ + 1) = 0 has to hold, and we found precise decay conditions for all variables. We deduce in the present section that only locally rotationally symmetric models of Bianchi type I, II and VI −1 are possible. In order to prove this statement, we show that the invariant set consisting of these three Bianchi types, compare Table 3 and Definition 3.4, contains all solution converging to the point Taub 2.
Proposition 7.1. Let γ ∈ [0, 2) and consider a solution to (5)-(11) which satisfies Then the solution is contained in the invariant set The proof revolves around the function It is constructed in such a way that its zero set characterises the invariant set from Prop. 7.1. The proof consists of showing that the function vanishes along all orbits converging to the point Taub 2. This is an adaptation of an approach which has already been used successfully in [Rin00a,Sect. 4] in the case of Bianchi A vacuum models. However, our method of showing that the function f indeed vanishes is slightly different.

Proof. The derivatives of the constituents of f have been computed in (17). Combining this with
A(κ + 1) = 0 which holds due to Prop. 6.2, this immediately yields that the set f = 0 is invariant. From the definition of f as the sum of two squares we see that either f ≡ 0 or f > 0. In the first case, the statement follows. Assume therefore that f > 0 holds. Let us have a closer look at the derivatives of the two terms which compose f, see equation (17). As due to the assumption on the limit point, we conclude that for some function ζ which is integrable on (−∞, T ] for some sufficiently negative time T . This follows from the convergence rates obtained in Prop. 6.2. Consequently, integration yields As a direct consequence of Prop. 7.1, the following theorem characterises the models whose orbits converge to the Kasner point Taub 2. Theorem 7.2. Let γ ∈ [0, 2) and consider a solution to equations (5)-(11) converging to (1/2, 3/4, 0, 0, 0) as τ → −∞. Then this solution is one of the following:

Asymptotics to the left of Taub 2
In this section, we discuss the set of solutions converging to a Kasner point to the left of Taub 2 as τ → −∞, i. e. with a limit point ( s, 1 − s 2 , 0, 0, 0) satisfying −1 < s < 1/2. We have shown in Prop. 6.2 that for such orbits eitherÃ = 0 has to hold, or the limit value s and the parameter κ have to satisfy the relation 3 s 2 +κ(1 − s 2 ) = 0. This is the same restriction we appealed to in the previous section, when discussing convergence to the point Taub 2. In other words, for a given κ, any solution to equations (5)-(11) converging to a point ( s, 1 − s 2 , 0, 0, 0) on the Kasner parabola with −1 < s < 1/2 is either a Bianchi A solution or has to converge to the limit point with s = ± κ/(κ − 3). This last special case, which can only occur if κ ≤ 0, is discussed in more detail in this section, using concepts, notation and results from dynamical systems theory which we recall in Appendix B.
The object we are interested in is the submanifold called centre-unstable manifold C u , as it contains the maximal negatively invariant set A − (U ) of a suitable open neighborhood U of the point ( s, 1−s 2 , 0, 0, 0). This latter set consists of all points which remain in U under the evolution in the negative time direction. In particular, all solutions converging to a point in U as τ → −∞ are contained in this set A − (U ) for sufficiently negative times.
To determine the properties of the centre-unstable manifold C u , we consider the linearised evolution equations in the extended five-dimensional state space, by which we mean the linear approximation of the evolution equations (5), with q andÑ defined as in equations (6) and (7), but without assuming the constraint equations (8) and (9). The corresponding matrix is a linear transformation of the five-dimensional tangent space to R 5 at equilibrium points of the evolution. We give the explicit form of the linear mapping and its eigenvectors and -values in Appendix A.1. For every Kasner point to the left of Taub 2, there are exactly four eigenvectors such that the corresponding eigenvalues are non-negative.
The centre-unstable manifold C u which we want to understand is tangent to the set E m c ⊕E m u which in its turn is spanned by the eigenvectors to eigenvalues with non-negative real part. The information on the number of such eigenvalues therefore translates into properties of the centre-unstable manifold, and thus into information on the set which solutions converging to a Kasner point to the left of Taub 2 have to eventually be contained in. We find the following statement: There is a four-dimensional manifold in R 5 such that every solution converging to a point ( s, 1 − s 2 , 0, 0, 0) on the Kasner parabola with −1 < s < 1/2 is contained in this manifold for sufficiently negative times. leaves U as τ → −∞.
Remark 8.2. We see in the proof that we could change the regularity of the manifold M K to C r , for some r < ∞. Further, as all eigenvalues on K are real, the manifold M K is tangent to those eigenvectors which correspond to non-negative eigenvalues.
Proof. Every point on the Kasner parabola is a zero-dimensional manifold consisting of equilibrium points of the evolution equations. Using the notation of Appendix B, we can therefore apply Thm B.3 to this zero-dimensional submanifold of equilibrium points to find a centre-unstable manifold C u near this point, and a neighborhood U of ( s, 1 − s 2 , 0, 0, 0) such that the maximal negatively invariant set A − (U ) is contained in C u . The point ( s, 1 − s 2 , 0, 0, 0) is contained in the manifold C u by Def. B.2. Without loss of generality, we therefore restrict the manifold to U . By definition, the manifold C u is tangential to E m c ⊕ E m u , which are the subspaces of the tangent space at ( s, 1 − s 2 , 0, 0, 0) associated with eigenvalues on the imaginary axis and in the right half-plane: The evolution equations (5) are polynomial and consequently C ∞ . We can therefore apply Thm B.3 with some finite r, for example r = 1. Consequently, C u is a four-dimensional submanifold, and it has the requested properties.
Remark 8.4. This statement could have been achieved using results from the theory of dynamical systems which are less powerful than the one we used here, Thm B.3, as we only apply it to zero-dimensional manifolds and the individual points they contain. However, in Section 10, we make use of this theorem again, this time using it to a fuller extent.
The previous statement gives information on the centre-unstable manifold corresponding to the evolution in the extended state space. In order to understand the evolution in the nonextended state space, i. e. restricted to the set of points such that the constraint equations (8) and (9) are satisfied, we need to understand the relation between the manifold we found and the constraint surface defined by equation (8). More precisely, we are interested in the codimension of their intersection, which is what we determine in Thm 8.5. As the constraint equation (8) defines a set which becomes singular in certain points, see Remark 3.5, we cannot deduce this codimension solely from the knowledge about normal and tangent directions derived above. In addition, we use the convergence behaviour of the individual variables from Prop. 6.14, where we determined the slowest order exponential term and the next non-vanishing one, for convergence to Kasner points to the left of the point Taub 2.
Consider a solution as in the statement which satisfies that ∆ and N + do not both vanish identically. Due to convergence, the solution is contained in the neighborhood U from the previous proposition for sufficiently negative times, and therefore the solution has to lie in the submanifold M K for sufficiently negative times. At the same time, the solution satisfies the constraint equation (8). We therefore have to show that the constraint surface and the submanifold M K either have an empty intersection or intersect transversally in a set of the correct dimension, when additionally restricted toÃ > 0 and ∆, N + not both vanishing identically. To do this, we compare the normal direction of the constraint surface to the vectors spanning M K . By counting the number of spanning vectors which are orthogonal to the gradient direction, we find the dimension and properties of the set in question.
As we consider solutions withÃ > 0 and ∆, N + not both vanishing, and κ = 3 s 2 s 2 −1 < 0 by Prop. 6.2, we conclude from Remark 3.5 that the gradient of the constraint equation does not vanish along such solutions. Consequently, the set of points defined by equation (8) is smooth along the solutions under consideration. However, due to the special value of κ the constraint surface becomes singular in the limit point ( s, 1−s 2 , 0, 0, 0). We therefore cannot simply compute the scalar product between the (vanishing) gradient of the constraint equation and the vectors spanning the submanifold M K to show transversality, at least not in the limit point itself. Instead, we consider the gradient in its general form (18), but replace the fourth component using the constraint equation (25) to obtain Applying the improved convergence properties found in Prop. 6.14 as well as Remark 6.9 shows that the decay behaviour of this gradient is where we used the special value of κ. The last term here denotes a vector in R 5 whose every component has the denoted decay. This decay is faster than the one of the first vector, which decays to order 4 + 4 s. Consequently, we can normalise the gradient vector by multiplying with e −(4+4 s )τ /α to eliminate the highest order of decay. Rescaling in this manner gives a well-defined non-vanishing gradient direction even up to the singular point ( s, 1 − s 2 , 0, 0, 0). Let us now turn to M K . In the limit point ( s, 1 − s 2 , 0, 0, 0), this four-dimensional submanifold is spanned by the eigenvectors to non-negative eigenvalues, see Appendix A.1 for the explicit form of these eigenvectors. Direct computation of the scalar product shows that the rescaled gradient of the constraint equation, i. e. the vector (−2 s, −κ/3, 0, 0, 0), is orthogonal to the eigenvectors to eigenvalues 2 + 2 s ±2 3(1 − s 2 ), 4(1 + s ) and 3(2 − γ), but not the eigenvector to 0.
Similarly, we can compare the spanning directions of M K to the normal direction of the set Ω = 0, i. e. the gradient of Ω. Due to equation (10), this gradient is for the point on the Kasner parabola K with Σ + = s. Direct computation of the scalar product shows that this gradient is orthogonal to the eigenvectors to eigenvalues 2 + 2 s ±2 3(1 − s 2 ), 4(1 + s ) and 0, but not the eigenvector to 3(2 − γ). As the manifold M K is C 1 , the spanning vectors depend continuously on the point. Consequently, in a sufficiently small neighborhood of the point ( s, 1 − s 2 , 0, 0, 0), the manifold M K and the constraint surface intersect transversally in a submanifold of dimension at most three if they intersect at all. The solutions for non-vacuum, i. e. Ω > 0, form a set of dimension four in the constraint surface, see also Table 1, as κ < 0 by assumption. Hence, in a sufficiently small neighborhood the intersection of the manifold M K and the constraint surface is of codimension at least one in the set of all non-vacuum solutions.
For vacuum solutions, we realise that one of the four eigenvectors in question is non-orthogonal to the gradient of Ω, and another one is non-orthogonal to the rescaled gradient of the constraint equation. Consequently, restricting the manifold M K first to the set Ω = 0 and then additionally to the constraint surface by an argument similar to the one for Ω > 0 yields that in a sufficiently small neighborhood of the point ( s, 1 − s 2 , 0, 0, 0), the intersection of the manifold M K , the constraint surface and the set Ω = 0 is a submanifold of dimension at most two. As all vacuum solutions form a set of dimension three, in a sufficiently small neighborhood this intersection is of codimension at least one in the set of all vacuum solutions.
We now apply the flow corresponding to the evolution equations and integer times to this intersection manifold. As the flow is a diffeomorphism coming from a polynomial evolution equation, the resulting set is a countable union of C 1 submanifolds of codimension at least one in the respective set of solutions, and by construction contains all solutions satisfying the properties listed in the statement. Further, the set Ω = 0 is invariant under the flow.
Remark 8.6. In case κ = 0, we cannot apply the same reasoning, as the improved convergence properties from Prop. 6.14 and the relation between β ∆ and β N+ found in the same proposition yield that the gradient (18) of the constraint equation decays as where we used Remark 6.9 for the fourth component. One can normalise this vector by multiplication with e −(2+2 √ 3)τ /(2β ∆ ) but the resulting direction in the limit point (0, 1, 0, 0, 0) is orthogonal to all four eigenvectors to non-negative eigenvalues. This means that the submanifold M K and the constraint surface do not intersect transversally, but are tangent in the limit point. This does not give any additional information on the set containing possible solutions.

Asymptotics to the right of Taub 2
In this section, we turn our attention to solutions with a limit point on the Kasner parabola to the right of Taub 2, i. e. a limit point in K ∩ {Σ + > 1/2}. For such points, all eigenvalues but one are positive as soon as γ < 2. One can therefore expect that this arc of equilibrium points acts as a source, even in the extended state space, as mentioned by Hewitt-Wainwright in [HW93]. As one considers an arc of equilibrium points, it is desirable to make a more thorough analysis. We carry out this analysis, using not the signs of the eigenvalues but the explicit evolution equations. In our understanding, an arc of the Kasner parabola can be considered a source if for any point on this arc, every orbit which enters a sufficiently small neighborhood of that point also converges to the Kasner parabola as τ → −∞, and the limiting point is close to that particular point on the arc. Corollary 9.2 states that this holds for the arc of the Kasner parabola which lies to the right of the point Taub 2, for vacuum and inflationary matter models. In general, we are able to show that the Σ + coordinate cannot differ too much.
Proof. It follows from the previous proposition in combination with Prop. 4.2 and the fact that that the α-limit set of every such solution is contained in the Kasner parabola K and does not intersect the plane wave equilibrium points L κ . Due to Prop. 6.1, the solution converges to a Kasner point ( s, 1 − s 2 , 0, 0, 0), and the estimate on s follows from applying Prop. 9.1 again.
To prove the previous proposition, the main idea is to use the fact that to the right of Taub 2, the Kasner parabola has a slope which is steeper than −1. Then, one shows that one can bound solutions from below by some straight line with this slope. This is the statement of Lemma 9.3 below and we provide a visualisation in Figure 3. As the convergence point has to lie above this line but below the Kasner parabola, one gains control over where the convergence point has to be situated exactly.
To prove the remaining estimate, we fix a time τ 1 ≤ τ 0 and distinguish between the two cases that ∆N + (τ 1 ) ≥ 0 or ∆N + (τ 1 ) < 0. We prove that in both cases for some constant M > 0 independent of τ 1 . This estimate is then used to conclude the proof.
We start with the case ∆N + (τ 1 ) ≥ 0. As Σ + (τ ) > 1/2 holds for all τ ≤ τ 0 due to the estimate on Σ + , we find which gives the desired statement. Next, we consider the case ∆N + (τ 1 ) < 0. The constraint equation (8) in the form using Σ + ∈ [−1, 1] andΣ ∈ [0, 1]. Due to the assumption on the sign of ∆N + , this implies at time τ 1 . With this and equation (15) for q one computes, suppressing the time τ 1 for readibility, that for a function f 1 which is bounded due to the compactness on the state space, say for some constant M > 0. The term containing Ω is non-positive. For the first bracket, one easily sees that hence the term containing N 2 + is non-positive as well. In total, this implies that We now integrate inequality (42) from τ < τ 0 to τ 0 . Using the estimate onÃ yields and setting δ . .= ε 2 1 (2M ) −2 concludes the proof.
One now constructs an even smaller neighborhood U whose closure is contained inŨ and such that all orbits starting in this smaller neighborhood are contained inŨ . The construction proceeds as follows: Note that due to the restriction on the state space, equation (9), one finds that That is, the orbit projected to the (Σ + ,Σ)-plane lies below the Kasner parabola, which is the graph of a function with slope −2Σ + . For Σ + in the interval (1/2, 1), this slope is strictly less than −1 and decaying. One can therefore choose a constant d < 1 −s 2 such that the straight line with slope −1 through (s, d) intersects the Kasner parabola at somes < Σ + <s + ε. Let 0 < ε 1 < min ((1 −s 2 − d)/2, ε), choose δ = δ(ε 1 ) as in Lemma 9.3 and let U be the set defined by |Σ + −s| < ε 1 , Σ + +Σ >s + d + ε 1 ,Ã < δ.
By Lemma 9.3, any orbit which is contained in U at time τ 0 satisfies for all τ ≤ τ 0 . From the first inequality we conclude that Σ + (τ ) >s − ε at all times τ ≤ τ 0 . The second inequality implies that the graph of the solution lies above the straight line from equation (43). Both inequalities are visualised in Figure 3. Because the graph also has to lie below the Kasner parabola, but we have chosen the constant d such that the straight line and the Kasner parabola intersect at somes < Σ + <s + ε, this implies that Σ + (τ ) <s + ε for all times τ ≤ τ 0 and every α-limit point. This concludes the proof.

Asymptotics towards the plane wave equilibrium solutions
In this section, we use the theory of dynamical systems to determine the qualitative behaviour of solutions converging to the plane wave equilibrium points L κ as τ → −∞. The statements are not qualitatively new, as certain parts of L κ have already been identified as "saddles" or "sinks" in [HW93]. Here, we state and prove more detailed properties of solutions converging to L κ . The approach we use here is similar to the one in Section 8, where we discussed the behaviour of solutions converging to a Kasner point situated to the left of the point Taub 2.
Remark 10.2. We see in the proof that we could change the regularity of the submanifolds to C r , for some r < ∞. Further, the eigenvalues on L k which are not real have real part −2(1 + Σ + ).
The only possibility where this is non-negative for a point in one of the compact sets is if γ = 2 and Σ + = −1. This implies that the solution converges to the point Taub 1 and hence is constant, see Prop. 5.2. For the current statement, this situation is not of interest, and excluding the case K 4 = L k ∩ {Σ + = −1}, the submanifolds are tangent to those eigenvectors which correspond to non-negative eigenvalues.
For the proof, we make use of the concepts and notation introduced in Appendix B, see also the explanation in the beginning of Section 8. The set of points which remain in U i under the evolution in negative time direction is the maximal negatively invariant set A − (U i ), and we prove the proposition using properties of the centre-unstable manifold C u .
Proof. The arc L κ ∩ {−1 < Σ + < −(3γ − 2)/4} is a manifold in R 5 consisting of equilibrium points of the evolution equation, with three eigenvalues of the linearised evolution equations lying in the left half-plane, one vanishing, and one lying in the right half-plane, see Def. 1.17 and the adjacent text.
According to Thm B.3, there is a centre-unstable manifold C u near K 1 and a neighborhood U 1 of K 1 such that the maximal negatively invariant set A − (U 1 ) is contained in C u . Without loss of generality, we can restrict the manifold C u to the open set U 1 .
The manifold C u is by definition tangential to E m c ⊕ E m u , which are the subspaces of the tangent spaces at points on K 1 associated with eigenvalues on the imaginary axis and in the right half-plane. These are the eigenvalues 0 and −4Σ + − (3γ − 2) whose eigenvectors span a twodimensional subspace, which implies that C u is a submanifold of dimension two. The evolution equations (5) are polynomial and consequently C ∞ , and the plane wave equilibrium points form a smooth curve. We can therefore apply Thm B.3 with some finite r, for example r = 1.
The proof for the arc L κ ∩{−(3γ − 2)/4 < Σ + < 0} proceeds in the same way, with 0 being the only eigenvalue with non-negative real part. The two individual points on the arc L κ constitute (zero-dimensional) manifolds of equilibrium points on their own, to which we can also apply Thm B.3. In case Σ + = 0, the value of γ determines where the eigenvalue −4Σ + − (3γ − 2) is situated in the complex plane.
As in Section 8, we now have to restrict the submanifolds to the constraint surface. There are two main differences between the situation in that section and the present one: The dimension of all centre-unstable manifolds is now at most two, and the constraint surface is singular only in the point Σ + = 0, see Remark 3.5.
On the other hand, intersecting the centre-unstable manifolds for plane wave equilibrium points L κ with the constraint surface does not necessarily result in a lower dimension: We know from the proof of the previous proposition that the centre-unstable manifolds are tangent to the eigenvectors to eigenvalues with positive or zero real parts, which are 0 and possibly −4Σ + − (3γ − 2), depending on the relation between Σ + and γ, see Appendix A.2. It follows by direct computation that these eigenvectors are orthogonal to the gradient of equation (8). Consequently, the centre-unstable manifolds and the constraint surface do not intersect transversally. We can only conclude that restricting the submanifolds to the constraint surface yields submanifolds of at most the same dimension, that is dimension at most one or two.  The relation between the parameter κ and the limiting value s follows from the definition of the plane wave equilibrium points, Def. 1.17, as N 2 + ≥ 0. We start with the case −(3γ − 2)/4 < s < 0. Fix a compact subarc K 2 of the arc L κ ∩ {−(3γ − 2)/4 < Σ + < 0} containing the limiting point of the solution. As the solution converges to a point in K 2 , it is contained in the neighborhood U 2 from the previous statement for sufficiently negative times, and therefore the solution has to lie in the one-dimensional submanifold M Lκ,right for sufficiently negative times. The set of plane wave equilibrium points L κ itself forms a onedimensional submanifold as well. It consists of constant solutions, and K 2 consequently must be contained in the centre-unstable manifold M Lκ,right . Due to the dimension, both submanifolds coincide in a sufficiently small neighborhood of the limiting point, and the first part of the statement follows. The solution satisfies Ω = 0 due to the definition of L κ , see Def. 1.17 and below.
For the second case, choose a countable family of compact subarcs K m , m ∈ N, which exhaust the arc L κ ∩ {−1 < Σ + < −(3γ − 2)/4}. For every K m as well as for the point on L κ with Σ + = −(3γ − 2)/4, consider the submanifolds found in Prop. 10.1. Convergence implies that for sufficiently negative times the solution cannot escape the corresponding open neighborhoods, and it consequently lies in one of these submanifolds for sufficiently negative times.
Still without restricting to the constraint equations we apply the flow corresponding to the evolution equations and integer times to these submanifolds. The flow resulting from a polynomial evolution equation implies that the regularity of the submanifolds is preserved. As the graph of any solution is invariant under this flow, solutions to the evolution equations in the extended state space are fully contained in the resulting family of submanifolds L ′ m . By construction, the family is countable. In the extended state space, the statement about the dimension follows from Prop. 10.1 and the fact that the dimension of a submanifold is invariant under diffeomorphisms, consequently invariant under the flow. Restricting to the constraint equations cannot increase the dimension, which concludes the proof in this case. The remaining case where s = 0 is treated in the same way.
Remark 10.4. Consider the element of the plane wave equilibrium points L κ which satisfies Σ + = s ∈ (−1, 0]. We have found in the previous theorem that the question whether there is a solution to equations (5)-(11) converging to this point, and whether it is constant or not, depends on the relation between s and the group parameter κ, as well as the relation between s and the matter parameter γ.
In case of vacuum Ω = 0, the only solutions converging to L κ are the constant solutions, see Prop. 4.2. Assume that Ω > 0. All solutions have to be non-constant and in particular cannot converge to limit points with −(3γ − 2)/4 < s < 0. We analyse the possible combinations of s, κ and γ, depending on the Bianchi type.
• Bianchi class A solutions cannot converge to the plane wave equilibrium points L κ , due toÃ = 0 for class A but non-vanishing on L κ . • Bianchi type VI η : As κ < 0 in this Bianchi type, solutions can only converge to limit points satisfying on Bianchi type V solutions. As q ≤ 2 andΣ ≥ 0, due to (16), this implies thatΣ is monotone decreasing or constant. As the only element of the plane wave equilibrium points L κ with s = 0 satisfiesΣ = 0, this shows thatΣ vanishes at all times. Further, the element in the plane wave equilibrium point with s = 0 satisfiesÃ = 1. With the information on the other variables, the evolution equation ofÃ reads Its range, see (16) then implies that for γ ≥ 2/3, only the constant orbit is possible. In case 0 ≤ γ < 2/3, the solution is contained in Σ + =Σ = ∆ = N + = 0 andÃ decreases mononotically from 1 to 0.

Equivalence of the initial data perspective and the expansion-normalised variables
The goal of this section is to justify the use of expansion-normalised variables, and show that under the correct transformation the evolution of these variables is equivalent to solving Einstein's equation for orthogonal Bianchi B perfect fluid initial data. At the same time, we show how to construct, for given initial data (G, h, k, µ 0 ) as in Def. 1.5, the maximal globally hyperbolic development and prove properties of this spacetime. This is done via the expansion-normalised variables (Σ + ,Σ, ∆,Ã, N + ) for Bianchi class B models.
The expansion-normalised variables in Bianchi class B models were developed in [HW93], motivated by a similar set of coordinates for Bianchi A models introduced in [WH89]. Their deduction starts out with given structure constants γ δ αβ of a suitably chosen four-dimensional orthonormal frame of the spacetime in question. To connect this to the initial data perspective, we have to understand how such a frame can be constructed from the knowledge of the metric h and the two-tensor k on the Lie group G. In particular, this means choosing a suitable threedimensional basis of the Lie algebra associated with the Lie group. For this reason, we collect the necessary background on three-dimensional Lie groups in Subsection 11.1.
We recall the deduction of the expansion-normalised variables from given structure constants γ δ αβ in Subsection 11.2. Note that this deduction already starts with the full spacetime, in particular with structure constants γ δ αβ in four dimensions. For initial data, this information is not available, only three-dimensional spacelike structure constants γ k ij make sense. The structure constants γ j 0i require the existence of a timelike vector field e 0 . However, to begin with there is no such vector field. Nonetheless, we can define objectsγ j 0i using the metric h and the two-tensor k and need to make sure that they have the form required for the construction by [HW93], which means that the only non-vanishing ones arẽ In order to see that in our setting we can indeed choose a suitable basis such that the commutators have the required form, we discuss in more detail initial data sets where the three-dimensional Riemannian manifold is a Lie group with left-invariant metric. This is done in Subsection 11.3, and it is here that we explain the terms 'exceptional' and 'orthogonal' as well as the reason for excluding Lie groups of type VI −1/9 . The objectsγ 1 01 ,γ B 0A have to be understood as merely numbers, devoid of any geometric meaning. It is only a posteriori that we can interpret these numbers as structure constants of a suitable four-dimensional frame. With these objects at our disposal, we can follow through with the transformation explained in Subsection 11.2. We wish to point out however that we make use only of the algebraic relations, not their geometric interpretation. This yields initial data (Σ + ,Σ, ∆,Ã, N + )(0) for the evolution equations in expansion-normalised variables, equations (5)-(11).
In order to avoid confusion and shorten notation, we are going to denote initial data to Einstein's field equation, i. e. initial data as in Def. 1.5, by geometric initial data, and initial data to the evolution equations in expansion-normalised variables, equations (5)-(11), by dynamical initial data. The details of how to translate geometric initial data into dynamical initial data are given in the first part of the construction of the maximal globally hyperbolic development, Subsection 11.4.
Once geometric initial data is translated into the expansion-normalised variables setting, i. e. into dynamical initial data, and we have obtained a solution in these variables, the main work lies in the construction of a global four-dimensional frame with structure constants behaving correctly over time, and such that our initially defined objectsγ j 0i are consistent with the geometric objects γ j 0i at the starting time t = 0, i. e. on the initial Cauchy hypersurface. Finally, we use this four-dimensional frame to construct a spacetime metric. This is done in the second and third part of the construction, Subsections 11.5 and 11.6. Having obtained through this construction a spacetime into which the initial data is embedded in the correct way, we investigate its properties in Subsection 11.7. We can show that this spacetime is in fact the maximal globally hyperbolic development of the geometric initial data.
In Subsection 11.8, we then consider Bianchi spacetimes with additional symmetries, namely local rotational symmetry and plane wave equilibrium solutions. We compare their definitions from the point of view of geometric initial data with their definitions in expansion-normalised variables and show that these definitions coincide, respectively.
The construction of expansion-normalised variables by [HW93] which we explain in Subsection 11.2 is related to the study of orthogonally transitively G 2 cosmologies. Even though [HW93] appeal to results having been obtained in this context, our construction does not make use of these additional statements but is self-contained. We do nonetheless give several remarks explaining the relations, but these are logically independent from the construction presented here.
In the Bianchi class A setting, results equivalent to what we achieve in this section have been obtained by [Rin01], however with a somewhat different approach. 11.1. Bianchi classification of three-dimensional Lie groups. We give a brief introduction to the classification of three-dimensional Lie groups proposed by Bianchi in 1903. The article [KBS + 03] gives a historical overview and provides insights into how our modern understanding of this classification came to be. In [EM69], the details are laid out, and for the first part of this subsection we refer to [Rin13, App. E] for the details.
For a three-dimensional Lie group G, let {e i }, i = 1, 2, 3, be a basis of the associated Lie algebra g, and let γ k ij denote the structure constants, i. e.
[e i , e j ] = γ k ij e k . The equivalent information is encoded in the symmetric matrix n and the vector a given by with ǫ ijk = ǫ ijk the permutation symbol which satisfies ǫ 123 = 1 and is antisymmetric in all indices. The brackets () denote symmetrisation, in fact γ The structure constants have to satisfy the Jacobi identity, which is equivalent to the condition Applying a suitable change of basis yields n and a in a specific simplified form and gives the following classification of three-dimensional Lie groups, of which we sketch a proof further down in Lemma 11.6. As there is a one-to-one correspondence between simply connected Lie groups and their Lie algebras, see [War71,Thm 3.28], we formulate this classification in terms of (simply connected) Lie groups. i) In the case of simply connected unimodular Lie groups (Bianchi class A) which are defined by a = 0, one can choose n diagonal, i. e. n = diag (ν 1 , ν 2 , ν 3 ), such that it falls in exactly one of the categories given in Table 4. Given a left-invariant metric in G, the basis e 1 , e 2 , e 3 producing a and n of this form can be chosen orthonormal. ii) In the case of simply connected non-unimodular Lie groups (Bianchi class B) which are defined by a = 0, one can choose a 1 = 0, a 2 = a 3 = 0, and n diagonal, i. e. n = diag (ν 1 , ν 2 , ν 3 ). The Jacobi identity then implies ν 1 = 0. All possible types and their resp. names are listed in Table 5. Again, given a left-invariant metric in G, the basis e 1 , e 2 , e 3 can be chosen orthonormal.
Lie groups of Bianchi type VI and VII have an additional degree of freedom which is captured in the quantity η. In the chosen basis this parameter satisfies (47) ην 2 ν 3 = a 2 1 and is invariant under scaling. We give a more geometric definition further down, see Lemma 11.5. Proof. Choose a basis e 1 , e 2 , e 3 of the Lie algebra such that n and a are of one of the forms given in i) and ii). If the Lie group is not of Bianchi type II, VIII or IX, then ν 1 = 0. One computes that [e 2 , e 3 ] = γ i 23 e i = ǫ 23l n li + a 2 δ i 3 − a 3 δ i 2 e i = ǫ 231 n 1i e i = ν 1 e 1 = 0, which implies that e 2 , e 3 span an Abelian subalgebra. For a Lie group of Bianchi type II one finds ν 2 = 0, and e 1 , e 3 commute.
Assume now that the Lie group is of type VIII or IX and suppose that there are two linearly independent vectors b, c ∈ g which commute, i. e.
This implies hence for all k = 1, 2, 3 As the two Bianchi types in question are of class A, one finds from equation (46) that = 1, 2, 3, are non-vanishing, and equation (48) is therefore equivalent to the system of equations or in other words b × c = 0, where × denotes the cross product. Hence, b and c are parallel, a contradiction.
In case the Lie group is of class B, there is also a geometric way of defining the Abelian subalgebra. For this, we consider the adjoint ad : g → End (g), ad x y = [x, y], see [War71,Prop. 3.47].

Lemma 11.2. Let G a three-dimensional Lie group of class B and H the one-form
Denote by g 2 the kernel of H. Then g 2 is an Abelian subalgebra and coincides with the subalgebra identified in Lemma 11.1.
Proof. We choose a basis e 1 , e 2 , e 3 of g and denote by e k , k = 1, 2, 3, the dual basis. Setting a k , k = 1, 2, 3, as in (45), we find As the Lie group is of class B, we can choose the basis e 1 , e 2 , e 3 as above in ii), meaning that a 1 = 0, a 2 = a 3 = 0, and n is diagonal, i. e. n = diag (0, ν 2 , ν 3 ). With this, we find that the kernel g 2 is spanned by the two basis elements e 2 and e 3 . Comparison with the proof of Lemma 11.1 shows that it is the same subalgebra we identified there.
Remark 11.3. Given a class B Lie group G with a left-invariant metric, we obtain the following useful interpretation of the kernel g 2 : it provides a splitting of the Lie algebra g 2 into a two-dimensional Abelian subalgebra and the direction orthogonal to it. For any orthonormal basis e 1 , e 2 , e 3 of g such that e 2 , e 3 span g 2 , the basis element e 1 is the one with the non-vanishing a i .
Remark 11.4. Given a Lie group of class B with Lie algebra g and a left-invariant metric on G, we have at our disposal the uniquely defined Abelian subalgebra g 2 . If instead the Lie group is of class A but not of type VIII or IX, we can fix an Abelian subalgebra, which exists due to Lemma 11.1. In both cases, we can then introduce an orthonormal basis e 1 , e 2 , e 3 of g such that e 2 , e 3 span the Abelian subalgebra and e 1 is orthogonal to this span. Given g 2 , this basis is uniquely defined up to a rotation in the e 2 e 3 -plane and a choice of sign in e 1 . One easily sees that it is equivalent to say that one uses the choice of orthonormal basis from i) and ii), and then allows for a rotation in the e 2 e 3 -plane. This holds in all cases apart from the case of Bianchi type II, where it is necessary to first switch the basis elements e 1 and e 2 . In the later subsections, in particular for the deduction of the different variables in Subsection 11.2, this rotation in the e 2 e 3 -plane is left as a gauge freedom.
With this knowledge, we are now in a position to give an invariant definition of the parameter η appearing in the Lie groups of type VI η and VII η , which are of class B.
Lemma 11.5. Let G a three-dimensional Lie group of type VI η or VII η , η = 0 in either case, with associated Lie algebra g, and g 2 the uniquely defined Abelian subalgebra which is the kernel of the one-form H. Let v 1 ∈ g \ g 2 . Then .= ad v1 | g2 : g 2 → g 2 , and a different choicev 1 ∈ g \ g 2 results into a mapÂ 2 differing from A 2 by a constant non-zero multiple. Further, setting is well-defined independently of the choice of v 1 and consistent with equation (47), i. e. under a choice of basis as in ii) one finds ην 2 ν 3 = a 2 1 . Proof. Consider a basis e 1 , e 2 , e 3 as in ii), i. e. satisfying a 1 = 0, a 2 = a 3 = 0, and n = diag (0, ν 2 , ν 3 ). By Remark 11.4, the Abelian subalgebra g 2 is spanned by e 2 , e 3 . Consequently, we can express the given unit vector v 1 as v 1 = b 1 e 1 + A b A e A and find as [e 2 , e 3 ] = 0 and γ 1 1A = 0. This proves the first statement. For the second statement, we realise that v 1 , e 2 , e 3 also forms a basis of g, From the last computation, it even follows immediately that η is independent of the choice of v 1 , and we can for all purposes assume that v 1 = e 1 . Doing so, the linear mapping A 2 in the chosen basis e 2 , e 3 is described by the matrix which implies tr A 2 = 2a 1 , det A 2 = a 2 1 + ν 2 ν 3 , and yields This is equivalent to the requested relation and in addition shows that η is well-defined, as both ν 2 and ν 3 are non-vanishing for the Bianchi types in question.
We conclude this subsection by proving that the types we listed in Tables 4 and 5, together with the parameter η, indeed provide a classification of all three-dimensional Lie algebras.
Lemma 11.6. Two three-dimensional Lie algebras are isomorpic if and only if they have the same Bianchi type, and in case of Bianchi type VI or VII additionally the same quantity η.
Proof. Given a Lie algebra of either class, the matrix n and the vector a defined in equations (45) admit the form n = diag (ν 1 , ν 2 , ν 3 ) and a = (a 1 , 0, 0) after applying a suitable change of basis. As a consequence, the Lie algebra falls in one of the types defined in Tables 4 and 5. From the transformation behaviour of n and a under a change of basis, see [Rin13, eq. (E.4)], we see that the number of non-vanishing diagonal elements of ν as well as the number of diagonal elements of ν having the same sign is fixed. As a conclusion, the given Lie algebra cannot fall in two different of the types from Tables 4 and 5. For Lie algebras of class A, scaling the basis such that ν i ∈ {−1, 0, 1}, i = 1, 2, 3, shows that each of the types in Table 4 has one unique representative, which concludes the proof for this class. For a Lie algebra of type V or IV, one can apply scaling to achieve a 1 = 1, ν 3 ∈ {0, 1}, and thereby uniqueness. In case of a Lie algebra of type VI or VII, the quantities a 1 and ν 2 , ν 3 cannot be scaled independently of one another but have to satisfy equation (47). Requesting ν 2 = 1 and ν 3 ∈ {±1} fixes a 1 up to sign, which by a change of direction in e 1 can be set to be positive. This concludes the proof.

The expansion-normalised variables from a four-dimensional spacetime point of view.
For this subsection and this subsection alone, we assume the existence of a fourdimensional spacetime from the start. That is, we assume that we already have a four-dimensional solution to Einstein's equations at our disposal, not only geometric initial data consisting of information on a three-dimensional manifold. With the four-dimensional information given, we recall the deduction of the expansion-normalised variables (Σ + ,Σ, ∆,Ã, N + ) from given structure constants γ δ αβ , as it was developed in [HW93]. In the following subsections, we then connect this to the initial data perspective we started with in the beginning of this paper.
We restrict ourselves to spacetimes with a Bianchi symmetry, i. e. with a three-dimensional Lie group G acting on the spacetime. We assume a stress-energy tensor which is either that of vacuum or that of a perfect fluid, and additionally assume that the fluid velocity u is orthogonal to the group orbits of G.
We further assume that we are given an orthonormal frame (e 0 , e 1 , e 2 , e 3 ) such that the only non-zero structure constants are , with A, B = 2, 3, which is the setting considered in [HW93].
Remark 11.7. In [HW93], the property that the only non-vanishing structure constants are those in (49) is justified by building upon the study of orthogonally transitive G 2 cosmologies, which are a generalisation of orthogonal Bianchi cosmologies. In [Wai79], a suitably adapted orthonormal frame (e 0 , e 1 , e 2 , e 3 ) of a G 2 spacetime is introduced, such that the vector fields e 2 , e 3 are tangential to the group orbits of the two-dimensional symmetry group G 2 , the vector field e 1 is spacelike, and the vector field e 0 is aligned with the fluid velocity. The study of orthogonally transitive G 2 cosmologies is continued in [HW90] where the authors, following a construction proposed by [Mac73], decompose the structure constants and deduce the corresponding evolution equations equivalent to the Einstein field equations of an orthogonal perfect fluid. This decomposition is explicitly adapted to the setting of Bianchi B orthogonal perfect fluid models by [HW93].
We do not make use of these arguments but instead treat the non-vanishing of all structure constants other than those in (49) as an assumption. In the following subsection, we discuss in more detail initial data sets where the three-dimensional manifold is a Lie group. We notice that by excluding one specific Bianchi type, we can ensure that we end up in a setting where the structure constants have the requested form, see Lemma 11.13 and the beginning of Subsection 11.4.
With structure constants as in (49) at hand, we set .
This gives a decomposition of the structure constants as proposed for general cosmological models by [Mac73] and deduced in the present setting by [HW93]: [e 1 , e A ] = ǫ AB n BC + a 1 δ C A e C , [e A , e B ] = 0, with A, B ∈ {2, 3}, ǫ AB the two-dimensional permutation symbol, and δ AB the Kronecker delta which is also used for lifting and lowering indices. The non-vanishing structure constants γ δ αβ , see (49), of the frame elements on the one hand and the set of variables (θ, σ AB , Ω 1 , a 1 , n AB ) as given in (51) on the other hand encode the same information.
The choice of orthonormal frame (e 0 , e 1 , e 2 , e 3 ) is not unique but allows for a rotation in the e 2 e 3 -plane. In the classification of Lie groups in Subsection 11.1 we have chosen a specific frame which additionally diagonalises n AB , but for the remainder of this section, we leave this rotation as a gauge freedom. Having this freedom of rotation in choosing the frame however becomes an issue when constructing the maximal globally hyperbolic development in Subsections 11.4-11.6.
Remark 11.8. The quantities appearing in the left column of (50) can be interpreted geometrically. In order to do so, assume that the spacetime is I × G, with I an open interval described by a time parameter t, and the frame is such that e 0 = ∂ t , and e i , i = 1, 2, 3, are tangential to every {t} × G and invariant under the action of G.
The decomposition of structure constants in a spacetime was originally proposed by [Mac73]. Upon comparison, one notices that the quantity θ ij given there (but in our index convention) equals the second fundamental form of {t} × G in the spacetime. Consequently, the trace θ of θ ij equals the mean curvature of {t} × G, and is the trace-free part of the second second fundamental form, where h is the spacetime metric restricted to {t} × G. We compute where ∇ denotes the four-dimensional Levi-Civita connection corresponding to the metric on I × G, which we here denote by ·, · . As σ ij is trace free, we find which should be compared to the first equation in (51). The quantity Ω 1 can be expressed in terms of the four-dimensional frame as follows: where ∇ denotes the four-dimensional Levi-Civita connection. The quantity Ω 1 thus describes a certain timelike derivative, but not one which is encoded in the second fundamental form θ ij .
Following the deduction of the expansion-normalised variables (Σ + ,Σ, ∆,Ã, N + ) as given in [HW93], one in a first step replaces n AB and σ AB , A, B ∈ {2, 3}, by their trace and trace-free part (with respect to the two-dimensional trace in the 23-components), i. e. one defines The information encoded in these new variables is equivalent to that encoded in n AB , σ AB . Additionally, one sets Assuming that the only non-vanishing structure constants are those given in (49) which are decomposed as above, and further assuming a stress-energy tensor of an orthogonal perfect fluid (2) with linear equation of state (3), the evolution equations for the variables (θ, σ + ,σ AB , a 1 , n + ,ñ AB ) have been given explicitly in [HW93, App. A]: with constraint equation and auxiliary equation Remark 11.9. Consider a four-dimensional spacetime with an orthonormal frame (e 0 , e 1 , e 2 , e 3 ).
Under the assumption that the only non-vanishing structure constants γ δ αβ are those given in (49) and they additionally satisfy e i γ δ αβ = 0, the system of evolution equations with constraints (54)-(57) holds if and only if the Ricci curvature of the spacetime is invariant under a three-dimensional Lie group action in the e 1 e 2 e 3 -space such that in the chosen basis the only non-vanishing terms are Ric(e 0 , e 0 ) = 1 2 (3γ − 2)µ, Ric(e i , e i ) = 1 2 (2 − γ)µ, i = 1, 2, 3, and the Jacobi identities hold in the e 1 e 2 e 3 -space. This follows from direct computation, expressing the Ricci curvature in terms of the structure constants γ δ αβ and then replacing them by the variables (θ, σ + ,σ AB , a 1 , n + ,ñ AB ). A Ricci curvature of this form is equivalent to Einstein's equations of a perfect fluid (2) with linear equation of state (3).
In detail, one finds that the evolution equations for the variables θ, σ + andσ AB correspond to the Ricci curvature terms Ric(e i , e j ) with i = 0 = j, while the evolution of a 1 , n + andñ AB follows from the Jacobi identities on the Lie group. The constraint equation (55) is equivalent to the momentum constraint, and equation (56) defining µ is equivalent to the Hamiltonian constraint. The evolution of µ is equivalent to the matter equation of a perfect fluid with linear equation of state.
In the next step of the construction, one introduces the variables which are invariant under the freedom of rotation and satisfyσñ = δ 2 + * δ 2 . The constraint equation (55) implies * δ = 3a 1 σ + , which can be used to eliminate * δ in the following. One further finds that there is a constant κ such that In the case of Bianchi type VI and VII, this relation immediately follows from the definition of the Bianchi group parameter η in equation (47), and the constant satisfies κ = 1/η. In all other Bianchi cases, one computes directly from the form of n AB and a 1 that equation (59) holds for κ = 0. One further sets (60)ã = 9a 2 1 for simplification. The resulting variables (θ, σ + ,σ, δ,ã, n + ) are called the basic variables and evolve according to They satisfy (62)σ ≥ 0,ã ≥ 0,ñ ≥ 0, σ 2 + +σ +ã +ñ ≤ θ 2 , and the constraint The matter µ is given by In the final step, the basic variables are normalised with appropriate powers of the rate of expansion scalar θ. By excluding the trivial solution in basic variables, we ensure that θ(t) = 0 at all times t, see Lemma 11.11. Setting then (67) 11.3. Lie groups as initial data subsets. In this subsection, we consider initial data sets where the three-dimensional manifold is a Lie group, and the initial metric and second fundamental form are invariant under the group action. In addition to the purely three-dimensional Lie group properties which we have discussed in Subsection 11.1, the Hamilton and momentum constraint equations pose restrictions on the two-tensor k which we now investigate in detail. In the following subsections, these restrictions are used to connect geometric initial data to expansion-normalised variables and the construction of the maximal globally hyperbolic development.
We start with initial data (G, h, k, µ 0 ) as in Def. 1.5: A Lie group G of class B or of type I or II, a left-invariant Riemannian metric h on G, a left-invariant symmetric covariant two-tensor k on G, and a constant µ 0 ≥ 0, satisfying the constraint equations In a first step, we fix a two-dimensional Abelian subalgebra g 2 of the Lie algebra g corresponding to the Lie group G. In case of a group of Bianchi class B, this is the uniquely defined subalgebra which is the kernel of the one-form H, see Lemma 11.2. In case of a group of Bianchi type I or II, the existence of such a subalgebra is ensured by Lemma 11.1. We then choose an orthonormal basisẽ 2 ,ẽ 3 of this subalgebra. Having fixed g 2 , these two vectors can be chosen uniquely up to rotation and reflection. Using the given initial metric h to fix a third unit vectorẽ 1 in the Lie algebra orthogonal to the span of those two yields a basis of g. This choice of basis gives structure constants γ k ij , i, j, k ∈ {1, 2, 3}, or equivalently n ij , a k , see (45) and (46) in Subsection 11.1. For a fixed g 2 , the chosen frame of the Lie algebra g is unique up to rotation and reflection in g 2 in case ofẽ 2 ,ẽ 3 , and a choice of sign in case ofẽ 1 . We can therefore use the frame from the classification of Lie groups in Subsection 11.1, see also Remark 11.4. Note that in the case of a Bianchi type II Lie group, we rename the basis elements such thatẽ 2 andẽ 3 commute. For this special frame, the structure constants are such that a 2 = a 3 = 0, the matrix n is diagonal with n 11 = 0, and a 1 = 0 for G of Bianchi class B. The reflection inẽ 1 and as well as the rotation and reflection in g 2 do not affect a i or n 1i = n i1 , i = 2, 3.
Let us now have a closer look at the constraint equations, which the metric h and two-tensor k have to satisfy by Def. 1.5. As both tensors are left-invariant, the momentum constraint reduces to In terms of the chosen orthonormal frame, we find where Γ k ij denote the Christoffel symbols Γ k ijẽ k = ∇ẽ iẽj corresponding to the Levi-Civita connection ∇ on the Riemannian manifold (G, h). We conclude that the momentum constraint is equivalent to As a consequence of Koszul's formula, the Christoffel symbols can be expressed in terms of the structure constants via Using the definitions for n ij and a k , equations (45), as well as the symmetry of k we then find We conclude that there is a non-vanishing solution (k 12 , k 13 ) of the system of equations (74) if and only if 9a 2 1 + n 22 n 33 = 0. In Bianchi class B, this is only possible for Bianchi type VI due to a 2 1 > 0 and the signs of n ii , compare Table 5. Furthermore, the invariance of the parameter η, equation (47), implies that only the case η = −1/9 allows for a non-vanishing solution. In total, we have shown the following: Lemma 11.13. Let G be a Lie group of class B, h a left-invariant Riemannian metric on G, and k a left-invariant symmetric covariant two-tensor on G satisfying the momentum constraint Let g 2 be the kernel of the one-form H from Lemma 11.2 which is an Abelian subalgebra of the Lie algebra g corresponding to G. Let (ẽ 1 ,ẽ 2 ,ẽ 3 ) an orthonormal basis of g such thatẽ 2 andẽ 3 span g 2 . If k 12 or k 13 is non-vanishing, then the Lie group G is of Bianchi type VI −1/9 .
Note that in the previous Lemma, the subalgebra g 2 is defined geometrically, as the kernel of a well-defined one-form. Consequently, whether (75) k 12 = k 21 = 0 = k 13 = k 31 or not is also a well-defined geometric property, and independent of the exact choice of orthonormal basis, as long asẽ 2 andẽ 3 span g 2 .
For a Lie group of Bianchi class A, a basis can be chosen which satisfies the properties of i) in the classification and diagonalises both the metric and the second fundamental form, see [Rin09,Cor. 9.14]. Therefore in this basis the relations (75) hold as well.
Lemma 11.13 is the reason why we have explicitly excluded Bianchi type VI −1/9 Lie groups in the definition of geometric initial data, Def. 1.5: By doing so, we ensure that equations (75) hold. In fact, we do not have to exclude the special Bianchi type VI −1/9 altogether. It can be included in the discussion as long as we adopt equation (75) as an additional assumption.
Remark 11.14. It has been shown in [EM69] that condition (75) is equivalent to 'non-exceptionality' of a Bianchi spacetime. More precisely, this reference considers the trace-free part of k, where θ = tr h k is the mean curvature, see also Remark 11.8, and makes use of the equivalent formulation This notion of 'non-exceptionality' is a property of the four-dimensional spacetime, namely not being one of the 'exceptional' spacetimes which [EM69] denote by Bbii. These 'exceptional' spacetimes are those which admit an orthonormal frame e i , i = 0, 1, 2, 3, such that e 1 , e 2 , e 3 are tangential to the spacelike hypersurfaces and σ 12 σ 13 = 0, η = −1/9, n 22 = 3a 1 σ 13 σ 12 , n 33 = − 3a 1 σ 12 σ 13 holds.
The property of being 'non-exceptional' is further related to the notion of orthogonally transitive G 2 cosmologies, see [Wai79, Thm 3.1(i)] for a characterisation of orthogonal transitivity. In combination with several results from [EM69], in particular Lemma 4.1 and Thm 5.1 in that reference, we conclude the following: A Bianchi class B spacetime admits an Abelian subgroup acting orthogonally transitively if and only if it is 'non-exceptional'. For this reason, we use the terms 'orthogonal' and 'non-exceptional' initial data interchangeably.
In terms of the Bianchi classification, only certain initial data sets for one specific parameter in one Bianchi type of class B satisfy the 'exceptional' properties. However, due to the additional degree of freedom, the phase space effectively has a higher dimension than that of all remaining Bianchi class B types. For this reason, the notion 'exceptional' should be interpreted as 'having exceptional behaviour', not 'being special enough to discard'. In fact, towards the initial singularity such spacetimes are expected to show chaotic oscillatory behaviour, see [HHW03]. 11.4. Construction part I: From geometric initial data to the expansion-normalised evolution equations. In this subsection and the two following ones, we carry out the construction of the maximal globally hyperbolic development for given geometric initial data, by which we mean initial data to Einstein's field equations.
We start with initial data (G, h, k, µ 0 ) as in Def. 1.5. We fix a two-dimensional Abelian subalgebra g 2 of the Lie algebra g corresponding to the Lie group G. As we detailed in the previous subsection, this is the kernel of the one-form H from Lemma 11.2 in case of a Lie group of class B, otherwise such a subalgebra exists due to Lemma 11.1. We then choose an orthonormal basisẽ 1 ,ẽ 2 ,ẽ 3 of g such thatẽ 2 ,ẽ 3 span g 2 . Such a choice of basis gives initial structure constants γ k ij (0), i, j, k ∈ {1, 2, 3}, and this information is equivalent to n ij (0), a k (0) due to (45) and (46) in Subsection 11.1. We have explained in the previous subsection that in this basis one has a 2 = a 3 = 0, the matrix n is diagonal with n 11 = 0, and a 1 = 0, in case of a Lie group G of Bianchi class B. For groups of type I or II, we can assume that a 1 = a 2 = a 3 = 0 and the matrix n is diagonal with n = diag (0, n 22 , n 33 ). In particular, the initial structure constants are such that the third and fourth equation in (51) hold. We further set initial values for θ and σ ij in accordance with Remark 11.8. For the initial value Ω 1 (0), we choose an arbitrary value. This is not determined by the geometric initial data. We will see further down that one can without loss of generality assume Ω 1 ≡ 0, but at this stage this is a non-determined variable. From Lemma 11.13, we conclude that the 12-and 13-component of σ ij (or equivalently of k ij ) vanish.
In particular, we are now in a position to connect geometric initial data to the construction carried out in Subsection 11.2, as the only non-vanishing initial quantities are those appearing on the left-hand side of equations (50).
With the definitions made so far, we can introduce quantities We stress that these are merely numbers at this point. Only after we have constructed the spacetime can they be interpreted as the structure constants on the hypersurface {0} × G of a suitable four-dimensional frame. In combination with the three-dimensional structure constants γ k ij (0) we found above, we have now constructed, at time t = 0, a set of numbers as appear on the right-hand side of equations (51).
In the next step, we apply the algebraic operations (52) and (53) to the set of numbers to obtain a set of numbers (θ, σ + ,σ AB , a 1 , n + ,ñ AB )(0).

Direct computation shows that the Hamiltonian constraint
which the geometric initial data has to satisfy is equivalent to equation (56), using for example [Rin13, eq. [E.12]] to compute the three-dimensional scalar curvature R. The i = 1-component of the momentum constraint implies equation (55), as can be concluded from equation (73). We therefore interpret (θ, σ + ,σ AB , a 1 , n + ,ñ AB )(0) as initial data for the evolution equations (54)-(57). Note that from this point on, the arbitrary value Ω 1 (0) does no longer appear. Applying now also the transformations (58) and (60), one obtains a set of numbers which can be interpreted as initial data in basic variables. Note that equation (64) holds due to the underlying Lie group structure of the geometric initial data we started with. We assume that we have started out with initial data (G, h, k, µ 0 ) such that after the steps carried out so far, the initial data in basic variables is different from the trivial data, i. e.
Further down in Lemma 11.16, we show that this restriction is equivalent to geometric initial data whose maximal globally hyperbolic development is not isometric to a quotient of Minkowski space. Going through the construction so far, looking for a solution to (54)-(57) for the initial values we constructed, we can assume that θ = 0 at all times due to Lemma 11.11. We therefore assume that θ > 0.
Once we have obtained a spacetime which admits the geometric initial data induced on a hypersurface, this choice of sign corresponds to the choice of normal vector e 0 , or in other words fixes the orientation of time t.
Essentially, the transformation of geometric initial data into initial values in basic variables is a one-to-one correspondence up to sign. We can recover the signs from the geometric initial data, see the third part of the construction.
In the next step, we normalise the initial values via equations (67) to obtain initial values These are interpreted as initial data to the evolution equations in expansion-normalised variables (5)-(11), where we also change to an expansion-normalised time τ which we choose to satisfy τ (t = 0) = 0.

Construction part II: From the expansion-normalised evolution equations to a solution in basic variables.
At the end of the first part of the construction, Subsection 11.4, we have obtained dynamical initial data (Σ + ,Σ, ∆,Ã, N + )(0). One can apply the evolution equations (5)-(11) for the expansion-normalised variables, which form a system of polynomial differential equations on a compact subset of R 5 , see Remark 3.3. Consequently, for given dynamical initial data, there exists a unique solution which is defined at all times τ ∈ (−∞, ∞). In this part of the construction, this solution is now translated back to a solution in basic variables, and then in the following subsection even further, to a spacetime solving the correct Einstein's equations and with correct initial data.
In order to retrieve basic variables at all times t, one considers the evolution equation for the basic variable θ, equation (70). The initial value θ(0) is given from the geometric initial data, see equation (76). Consequently this evolution equation has a unique maximal solution θ(τ ). As q ∈ [−1, 2] due to the relations in (16), there are two cases to consider: i) q = −1 at some time. This corresponds to γ = 0 and Ω = 1, which implies This point is an equilibrium point of the evolution equations (5), from which we conclude that q = −1 at all times and θ = θ(0) constant. ii) q > −1 at all times. In case of inflationary matter Ω > 0, γ ∈ [0, 2/3), we have shown in Prop. 4.2 that the only solution whose α-limit set contains the point Σ + =Σ = ∆ = A = N + = 0 is the constant solution, which is excluded in this case. Consequently, 1 + q is bounded from below by a positive number for τ sufficiently negative. In vacuum and in the remaining matter cases Ω > 0, γ ∈ [2/3, 2], equation (14) and equation (15) even yield 1 + q > 1. With this, equation (70) implies that θ is monotone and, for sufficiently negative times, |θ| can be estimated from above and below by exponential functions of τ with a negative exponent. Due to choice of sign we have made in the first part of the construction, θ > 0 at all times and therefore θ monotone decaying. Thus, θ converges to some non-negative number θ ∞ as τ → ∞, and diverges to ∞ as τ → −∞.
To replace the expansion-normalised time τ , we define a different time scale t via equation (69) and the initial condition τ (t = 0) = 0. Due to θ > 0, this definition immediately yields that t and τ have the same time orientation. In case q = −1, the time t is simply a rescaling of τ , and the basic variables are defined at all times t ∈ (−∞, ∞). In case q > −1, our discussion in ii) implies that time τ → −∞ corresponds to diverging θ and a finite time t, while τ → ∞ corresponds to θ → θ ∞ ≥ 0 and t → ∞.
In total, we obtain a curve in basic variables (θ, σ + ,σ, δ,ã, n + )(t). By our construction, they are related to the solution in expansion-normalised variables (Σ + ,Σ, ∆,Ã, N + )(τ ) exactly as in Subsection 11.2. We have noted there, in Remark 11.12, that the evolution equations for the expansion-normalised variables and the evolution equations for the basic variables are equivalent. Therefore, the construction carried out here yields the maximal solution to equations (61)-(66). The resulting maximal interval of existence is (t − , ∞), with t − > −∞ apart from when γ = 0 and Ω = 1, in which case t − = −∞.
11.6. Construction part III: From a solution in basic variables to a spacetime. Having found a solution in basic variables (θ, σ + ,σ, δ,ã, n + )(t) via the previous two parts of the construction, Subsections 11.4 and 11.5, we now translate this into a spacetime with suitable Lorentzian metric which is consistent with the geometric initial data (G, h, k, µ 0 ) we started with.
Recall how the construction started: In order to obtain initial data in basic variables from the geometric initial data, one introduced a frameẽ 1 ,ẽ 2 ,ẽ 3 and then had to choose an arbitrary initial value Ω 1 (0), whose information was lost in the first algebraic manipulation. This corresponded to introducing variables which were explicitly invariant under the choice of frame, in the sense that they did no longer depend on the freedom of rotation in theẽ 2ẽ3 -plane. In order to now recover from a maximal solution (θ, σ + ,σ, δ,ã, n + )(t), t ∈ (t − , ∞), a spacetime frame with fourdimensional structure constants and then a spacetime metric, one needs to break this gauge invariance and reverse this process by choosing a suitable frame. This is done in several steps: i) Given an arbitrary, sufficiently smooth Ω 1 defined on the same t-time interval (t − , ∞) as the solution in basic variables, one retrieves the variables (σ AB , θ, n AB , a 1 )(t). ii) With this, one defines a manifold and constructs a frame e 0 , e 1 , e 2 , e 3 . iii) Then, one checks that this construction holds through time, independently of the choice of Ω 1 . iv) Once the frame is obtained, one defines this frame as orthonormal, which uniquely defines the corresponding spacetime metric on the manifold (t − , ∞) × G. v) Finally, one checks that in the resulting spacetime Einstein's equations are satisfied and the correct initial data induced.
We begin with the first step, which is achieved via constructing a solution in the variables (θ, σ + ,σ AB , a 1 , n + ,ñ AB ) satisfying equations (54)-(57). With this solution at our disposal, we can immediately deduce the variables (σ AB , θ, n AB , a 1 )(t). It is the first part which is more intricate, as we have to be careful about notation. We start with a solution in basic variables, which for the sake of precision, we denote by (θ basic , σ +,basic ,σ basic , δ basic ,ã basic , n +,basic )(t) , as the solution satisfies the evolution equations for basic variables, equations (61)-(66).
We want to find a solution (θ, σ + ,σ AB , a 1 , n + ,ñ AB ) to equations (54)-(57). Even though we do not know that the functions θ basic , σ +,basic and n +,basic evolve correctly, we carry them over, but keep the subscript as a reminder. The variable a 1 is supposed to satisfyã = 9a 2 1 , see equation (60). Asã = 0 is an invariant set due to the evolution equations (61), the sign of a 1 cannot change and is determined by the initial data. From the knowledge ofã basic and the initial data, we can uniquely define a function a 1,basic . Upon comparison with the evolution equation forã in (61), we find that a 1,basic satisfies the evolution equation for a 1 in (54), if one replaces σ + and θ by σ +,basic and θ basic . Now consider the evolution equations ofσ AB andñ AB in (54). Together, they form a system of linear differential equations whose coefficients are the function Ω 1 and-after adding the subscript basic at all necessary places-quantities whose time development is known from the solution in basic variables. For every sufficiently smooth function Ω 1 , there is a maximal solution (ñ AB,basic ,σ AB,basic ) to this system. Due to linearity of the differential equations, the maximal solution is defined on the whole time interval on which the coefficients are defined. By construction, this is the same time interval as for the solution in basic variables.
Next, we consider the expressions withñ basic defined by equation (64). These expressions should be compared to the constraint equation (55) and equations (58). The derivative of (Fσ, Fñ, F δ , F constraint ) is a homogeneous system of equations, and as all four functions vanish initially due to their components having been constructed from the same geometric initial data, we see that all four functions vanish identically. In particular, the constructed functionsñ basic,AB andσ basic,AB are related to the basic variablesσ basic ,ñ basic and δ basic as in equations (58), and we can replace these expressions at every occurence. This in particular implies that the function µ basic , defined via equation (65) in basic variables, also satisfies equation (56). Further, we conclude from the argumentation above that the constraint equation (55) holds for the functions indexed basic . It remains to check that the variables θ basic , σ +,basic and n +,basic satisfy the evolution equations (54). To this end, consider the initial data to the evolution equations (54)-(57) which we, in the construction in Subsection 11.4, obtained from geometric initial data and then subsequently used to obtain initial data to the evolution in basic variables. There is a unique solution with this initial data, and in order to distinguish its individual variables from the ones constructed above, we denote this solution by The initial values of these variables coincide with the initial values of the ones constructed, and uniqueness of systems of ordinary differential equations yields that θ i.d. = θ basic at all times, and equivalently for all other variables. We conclude that via this construction, we have obtained the unique solution to equations (54)-(57) to the initial data coming from the geometric initial data.
In this transformation, the maximal interval of the solution (θ, σ + ,σ AB , a 1 , n + ,ñ AB ) cannot exceed that of the solution in basic variables, as otherwise this would yield an extension of the solution in basic variables, a contradiction. Consequently, the maximal interval of existence of the solution after transformation coincides with the interval (t − , ∞). With this solution at hand, we use equations (52) and uniquely retrieve the variables σ AB and n AB , defined on the same interval.
In the next step, we construct a four-dimensional manifold with a frame whose structure constants have the correct form. To this end, we define scalar functions γ δ αβ (t) on the time interval (t − , ∞) via setting , and setting all other γ δ αβ to vanish identically. The form of these scalar functions coincides with the form of the structure constants in equation (51). By construction, these objects are consistent with the initial data: At time t = 0, the spacelike ones γ k ij (0) coincide with the Lie group structure constants we chose in Subsection 11.4, and the remaining ones γ j 0i (0) coincide with the structure constant-like objectγ j 0i (0) defined there. We construct now a four-dimensional frame whose structure constants coincide with these γ δ αβ (t) at all times. Note that it is at this point that the structure constant-like objectγ j 0i (0) can finally be interpreted geometrically. From the initial data, we have obtained an initial frameẽ 1 ,ẽ 2 ,ẽ 3 on the Lie group G. We consider the manifold (t − , ∞) × G and in this manifold extend this frame globally, i. e. timeindependently to every {t} × G. Further, we choose e 0 = ∂ t globally. This gives a fourdimensional frame e 0 ,ẽ 1 ,ẽ 2 ,ẽ 3 , but the relation between its structure constants and the solution (θ, σ AB , Ω 1 , a 1 , n AB )(t) does not necessarily fulfill equations (51) at times other than t = 0. The final frame e 0 , e 1 , e 2 , e 3 which does satisfy these relations can be constructed as follows.
We wish the frame vectors e 2 , e 3 to be tangent to the subalgebra g 2 which we have chosen in the beginning of Subsection 11.4, which implies that we need to construct time-dependent Assuming the existence of such functions gives the following commutator relation: Consequently, solving the system yields the final frame vectors e 2 and e 3 . Note that in order to do so, we have to fix a function Ω 1 (t).
For the missing spatial frame component e 1 , we make the ansatz After a computation similar to the above, the resulting commutator reads gives the following systems: In particular, f A 1 (t) = 0 at all times. The solution to this system of equations is independent of Ω 1 , and yields the final frame vector e 1 .
Having found functions f B A , f A 1 , f 1 1 , we have constructed a frame which satisfies all but the third commutator relation in (51). These remaining equations can be considered as constraint equations. From the construction of the final frame vectors, we obtain 1A as defined in (77), in order for our construction to by consistent. Thus, we have to check whether holds. Initially, the left-hand side and right-hand side coincide, and in order to prove that the relation holds at all times, it is therefore enough to prove an evolution equation of the form for some time-dependent functions M C B . Using the derivatives of the functions f 1 1 , f A 1 and f B A which we determined above, we find that the derivative of the left-hand side of (78) reads For the right-hand side, we find The time-derivative of the structure constants can be obtained by expressing γ B 1A in terms of the variables (θ, σ + ,σ AB , a 1 , n + ,ñ AB ), applying their evolution equations (54)-(57), and converting the results back to structure constants. This yieldṡ which is the same structure we also found for the evolution of the left-hand side, equation (79). We conclude that the constraint is preserved independently of the choice of Ω 1 . For all our purposes, we can assume Ω 1 ≡ 0 at all times. The above construction results in a time-dependent left-invariant frame e 1 (t), e 2 (t), e 3 (t) on the Lie group, which in combination with e 0 = ∂ t yields a spacetime frame admitting the correct structure constants (51). In checking this consistency, we have completed the third step of our construction of the spacetime.
There is for every t a unique metric t g on {t} × G such that the frame e 1 (t), e 2 (t), e 3 (t) is orthonormal. With this, we define the spacetime where I is the existence interval of the basic variables and contains 0, i. e. I = (t − , ∞), with −∞ < t − < 0 apart from the case γ = 0, Ω = 1, where t − = −∞. By our construction above, the only non-vanishing structure constants of the four-dimensional frame (e 0 , e 1 , e 2 , e 3 ) are those in (49), and as the variables composing the structure constants satisfy the evolution equations (54)-(57), we can deduce from Remark 11.9 that the spacetime constructed in this way solves Einstein's field equations for an orthogonal perfect fluid with linear equation of state.
On the t = 0 timeslice, the frame vectors e i andẽ i coincide, i = 1, 2, 3. The metric induced on this timeslice by the spacetime metric (80) therefore coincides with the initial metric h. Due to our choice of metric, the vector field e 0 is the timelike unit normal to every {t} × G timeslice. It follows from the construction that the second fundamental form of the timeslice {0} × G in (I × G, g) coincides with the initial two-tensor k. We conclude that the constructed spacetime induces the correct initial data. This completes the fifth step, and as a consequence the following definition is reasonable.
Definition 11.15. Given initial data as in Def. 1.5, we call a spacetime (I × G, g, µ) constructed as in the three parts of the construction, Subsections 11.4-11.6, a Bianchi B development of the data.
Due to the choice of frameẽ 2 ,ẽ 3 in the beginning of the construction, the spacetime constructed this way is not necessarily unique.
In order to transform to expansion-normalised variables in Subsection 11.4, we had to exclude initial data with zero mean curvature θ. We now prove that this corresponds to excluding Bianchi B developments which are part of Minkowski spacetime.
Lemma 11.16. Consider initial data as in Def. 1.5. Then either its universal covering space is initial data for the four-dimensional Minkowski spacetime, or the mean curvature θ = tr h k is non-vanishing at all times.
The proof proceeds similar to that of [Rin09, Lemma 20.6] for Bianchi class A developments.
Proof. As in the first part of the construction, Subsection 11.4, we choose a suitable frame and carry out the transformations into initial data in basic variables, i. e. initial data for the evolution equations (61)-(66). From the construction, we see that the mean curvature coincides with the initial value for θ. Using Lemma 11.11, we find that if the initial value θ(0) vanishes, then the solution in basic variables is the trivial solution.
With this trivial solution in basic variables at hand, we retrace the steps of the third part of the construction, Subsection 11.6, and recover a spacetime with an orthonormal frame (e 0 , e 1 , e 2 , e 3 ) such that the initial data is correctly induced. We have been able to choose Ω 1 ≡ 0 in this construction, and from this we conclude that all structure constants γ δ αβ vanish identically. Consider now the Lie group G with structure constants γ k ij = 0. The three-dimensional Christoffel symbols Γ k ij vanish due to equation (72), and we conclude that the three-dimensional Ricci curvature We have explained in Remark 11.8 that the second fundamental form of the timeslice {t} × G can be computed from θ and the trace-free variable σ ij , which satisfies σ 12 = σ 13 = 0. We can therefore conclude that the second fundamental form vanishes. This concludes the proof. 11.7. Properties of Bianchi B developments of initial data. We now show that a development as in Def. 11.15 of given geometric initial data, i. e. one which we obtain through our construction in the previous three subsections, is in fact the maximal globally hyperbolic development of the geometric initial data. The proof of global hyberbolicity works identically as in the case of Bianchi class A and has been carried out in [Rin09,p. 217]. We only state the proposition: Proposition 11.17. Let (G, h, k, µ 0 ) be initial data as in Def. 1.5, and (I × G, g, µ) a Bianchi B development of the data. Then {t} × G is a Cauchy hypersurface for every t ∈ I.
For the proof of maximality, we wish to apply the following proposition, which is part of [Rin09,Prop. 18.16]: Proposition 11.18. Let (M, g) be a connected and time oriented Lorentzian manifold and assume that the following holds: • Then either θ ≡ 0 and f 0 is a constant, or d ds Proof. By construction of the Bianchi B development in Subsections 11.4-11.6, we know that the trace-free part σ ij of the second fundamental form θ ij of the submanifold {t} × M satisfies For a fixed time s, we apply a rotation in the e 2 e 3 -plane to diagonalise σ ij (and simultaneously θ ij ), then define for i = 1, 2, 3.
As f 0 (s) = c ′ (s), e 0 | c(s) and c is a geodesic, we find The only case not covered by this last statement is when µ > 0, γ = 0, in which case one can interpret the stress-energy tensor T αβ as that of vacuum with a positive cosmological constant Λ > 0.
We have to distinguish between the two situations which we encountered in Subsection 11.5: q ≡ −1 and q > −1. In the first case, θ = θ 0 is constant and all expansion-normalised variables vanish at all times. One concludes from the construction of Bianchi B development that the commutators have the form From the second relation, we can conclude that a left-invariant metric on the universal covering group of the Lie group G is isometric to Euclidean three-space (R 3 , δ), while the first relation implies with t g and t k the metric and second fundamental form of {t}×G with respect to the development. Combined, the second fundamental form of {t} × G equals One compares this with the spacetime with H = Λ/3, which can be isometrically embedded in the well-known de Sitter spacetime, see [Sch56,eq. (52)]. Upon comparison, one realises that the t = const. slices in this spacetime have the same induced metric and second fundamental form as we computed for our case γ = 0, q ≡ −1. We conclude that a Bianchi B development, which by Prop. 11.17 is a globally hyperbolic development of the given initial data, corresponds to a development of a hypersurface in a quotient of the de Sitter spacetime. This concludes the case γ = 0, q = −1.
In case q > −1, we can show that the development we constructed cannot be isometrically embedded into a globally hyperbolic spacetime which extends the development to the past. By construction, the curve also maximises the length between q and every earlier timeslice {t}× G, t ∈ (t − , 0) and therefore the causal vector c ′ is orthogonal to every one of these hypersurfaces. We conclude that in M , the curve c has the form c : s → (t(s), q).
Assume that s 0 ∈ (s − , s + ) is the curve parameter such that c(s 0 ) ∈ ∂M ⊂M and choose a sequence of times s n ∈ (s 0 , s + ) such that s n ց s 0 . Set t n to be the time satisfying c(s n ) ∈ {t n } × M , for every n ∈ N.
Let now E i , i = 1, 2, 3, be vector fields which are parallelly propagated along c and such that they form an orthonormal basis of T c(0) ({0} × G). Consequently, they are orthonormal along the whole curve and span the orthogonal complement to c ′ (s), for every s ∈ Note that the subscript v denotes derivation with respect to the second component.
We see that x is a variation of c with fixed endpoint x(s − , ·) = c(s − ) = q. By assumption, the curve c maximises the length between q and the hypersurface {t n } × G. We can therefore apply [Rin09,Lemma 18.7] to find We are interested in estimate (86) in the limit n → ∞. Due to the assumption on extendibility, the Ricci curvature is bounded along c, consequently However, the mean curvature θ(t = t n ) diverges to ∞ as n → ∞ by construction of the development, which is a contradiction.
We can conclude from the previous proof that in a Bianchi B development with µ > 0, γ = 0 and θ covering (θ ∞ , ∞), a causal curve connecting the hypersurfaces {t 1 } × G and {t 2 } × G has length at most |t 1 − t 2 |. In particular, causal curves have finite length towards the past. This stands in contrast to the de Sitter spacetime, where causal curves have infinite length towards the past.
With Prop. 11.22 at hand, we can now prove that in the remaining case µ > 0 with γ = 0 but q > −1, the Bianchi B development is isometric to the maximal globally hyperbolic development.
Corollary 11.23. Consider a Bianchi B development of initial data as in Def. 1.5. If µ > 0, γ = 0, and the development does not correspond to a development of a hypersurface in a quotient of the de Sitter spacetime, it is isometric to the maximal globally hyperbolic development of the initial data.
Proof. Recall the construction we carried out in Subsections 11.4-11.6. We argued in the second part of the construction that there are two different cases which can occur: q ≡ −1 and q > −1. We showed above that the first case results into a development which is part of the de Sitter spacetime, and is hence excluded by assumption. As q > −1, the mean curvature θ(t) covers the interval (θ ∞ , ∞), see the end of Subsection 11.5. As a result, we can apply Prop. 11.22 to see that the Bianchi B development cannot be extended to the past as a globally hyperbolic development. We further know from Lemma 11.20 that every timelike geodesic is future complete. This concludes the proof.
We collect the statements we made about Bianchi B developments in the different cases in the following proposition. Consider a causal geodesic inM . The behaviour of any geometric quantity in the incomplete direction of this geodesic is the same as the behaviour of this geometric quantity on {t} × G as t → t − , while the complete direction corresponds to t → t + .
Proof. If we exclude spacetimes isometric to (quotients of) the Minkowski spacetime, we can carry out the construction from Subsections 11.4-11.6, see also Lemma 11.16. We have seen in Corollaries 11.21 and 11.23 that the resulting Bianchi B development is isometric to the maximal globally hyperbolic development of the data, if in case γ = 0 we exclude de Sitter spacetime.
The properties of t − , t + and the convergence behaviour of θ are a consequence of the construction, see also at the end of Subsection 11.5. The one case with t − = −∞ mentioned there corresponds to γ = 0 and Ω = 1, or in other words q ≡ −1, which implies de Sitter spacetime and is therefore excluded by assumption.
Consider an inextendible causal geodesic inM . Using the isometry betweenM and the Bianchi B development M , we obtain an inextendible causal geodesic in M . All geometric quantities are invariant under isometries, and in addition constant on every timeslice {t} × G due to the metric being invariant under the action of the Lie group G.
As in the proof of Corollary 11.23, the case q = −1 is excluded by assumption, and θ covers the interval (θ ∞ , ∞). Consequently, Prop. 11.22 applies and in combination with Lemma 11.20 implies that the geodesic in M is future complete and past incomplete. The incomplete direction of an inextendible geodesic in bothM and M therefore corresponds to t → t − , and the complete direction to t → t + . This concludes the proof. 11.8. Solutions with additional symmetry. In the results of this paper, several types of geometric initial data sets as well as invariant subsets are of interest which have additional properties. In particular, those with local rotational symmetry and those describing plane wave equilibrium solutions appear frequently. We have defined both notions twice, first in the setting of geometric initial data and then in the setting of expansion-normalised variables. In this subsection, we show that these definitions are consistent provided we transform between geometric initial data sets and expansion-normalised variables the way we described in Subsections 11.4-11.6.
The notion of locally rotationally symmetric geometric initial data was given in Def. 1.6, where such initial data is defined via the properties of a specific basis of the associated Lie algebra. Given more generally some orthonormal basis e 1 , e 2 , e 3 of the Lie algebra g such that [e 1 , e 3 ] e 2 , k = diag(k 11 , k 22 , k 33 = k 11 ), but not necessarily with e 2 , e 3 spanning a particular subalgebra, we find that these properties are preserved under a rotation in the e 1 e 3 -plane. The term local rotational symmetry is thereful meaningful and should be compared to the definition of local rotational symmetry in Bianchi class A spacetimes in [Rin09,Def. 19.16] and in more general spacetimes in [SE68].
Let us now discuss Definition 1.6 in connection with the different Bianchi types. For a Lie group of type V or IV, the form of the structure constants from Table 5 reveals for a basis chosen as in the classification of Lie groups in Subsection 11.1. Note that the basis elements e 2 , e 3 span g 2 , see Remark 11.4. There is no rotation in the e 2 e 3 -plane such that the commutators of the rotated basis have the form requested for local rotational symmetry. Consequently, no initial data set of Bianchi type V or IV can have local rotational symmetry. Let us therefore assume that the Lie group is of type VI or VII. When comparing the properties of local rotational symmetry from Def. 1.6 with the decomposition of the structure constants in (51), we see that the vanishing of [e 2 , e 1 ] and [e 2 , e 3 ] is equivalent to a 1 = −n 32 and n 33 = 0, while [e 1 , e 3 ] e 2 holds if and only if a 1 = n 23 . In particular, we can consider the linear map A 2 = ad v1 | g2 , v 1 ∈ g \ g 2 , from Lemma 11.5 and conclude that local rotational symmetry implies det A 2 = 0. The image of either ad v1 or A 2 , which coincides with [g, g], is one-dimensional and spanned by e 2 . After a rotation in the e 2 e 3 plane, we can assume that n is diagonal, but lose the properties a 1 = n 23 = −n 23 and n 33 = 0, see Remark 11.4. A vanishing determinant of A 2 implies that η = −1, hence the Lie group can only be of type VI −1 .
Alternatively, let e 1 ,ẽ 2 ,ẽ 3 a basis of the Lie algebra g withẽ 2 ,ẽ 3 spanning g 2 and such that a = (a 1 , 0, 0), n = diag (0, ν 2 , ν 3 ), i. e. a basis as in the classification of Lie groups in Subsection 11.1. Let us further assume that η = −1, i. e. the Lie group is of type VI −1 . In this case, we find from the matrix representation of A 2 that where the sign depends on the sign of a 1 . We further find that if we set e 2 and e 3 to be unit vectors in g 2 , one spanning [g, g] and one orthogonal to it, and by a change of sign in e 1 ensure that a 1 is negative, then we recover that the commutator of this new basis satisfy the properties from the definition of local rotational symmetry, Def. 1.6. Whether geometric initial with an associated Lie algebra of type VI −1 is locally rotationally symmetric then depends solely on the two-tensor k.
Given geometric initial data as in Def. 1.6, we can deduce from the commutators the corresponding basic variables via expressions (50), (52), (53) and (58) and find that local rotational symmetry implies that 3σ 2 + =σ and σ + n + = δ. Further, the Lie group has to be of class A or of type VI −1 due to the above discussion. The expansion-normalised variables result from applying (67) and (68), and one recovers the definition of locally rotationally solutions in expansion-normalised variables, Def. 3.4, as Bianchi class A corresponds toÃ = 0.
For the notion of plane wave equilibrium initial data, we turn to Def. 1.8. In terms of structure constants for the four-dimensional spacetime, the property γ B 1A + γ A 1B = −2k AB is equivalent to (87) γ 2 12 = γ 2 02 , γ 3 13 = γ 3 03 , γ 3 12 + γ 2 13 = γ 3 02 + γ 2 03 . This should be compared with the definition given in [HKL04], where the special spacelike direction is set to be e 1 instead of e n . Applying an appropriate permutation of the basis elements to fit with our choice of frame, the definition given there reads In terms of geometric initial data, the right-hand side is not defined, but its symmetrisation can be replaced by the expression −k AB , see equations (51) and Remark 11.8. This tensor is symmetric by definition, and also replacing the left-hand side by its symmetrisation one obtains Def. 1.8.
Remark 11.25. In terms of initial data, the Kasner parabola K corresponds to vacuum Bianchi type I data. This can easily be deduced from Table 2 and using the condition for vacuum, Ω = 0, in equation (10). The two special Taub points have the following characterisation: • The point Taub 1 corresponds to initial data of Bianchi type I which is of plane wave equilibrium type. • The point Taub 2 corresponds to initial data of Bianchi type I which is locally rotationally symmetric and additionally the unique eigenvalue of k in the rotation plane is smaller than the unique eigenvalue along the rotation axis. In a basis satisfying the conditions for local rotation symmetry from Definition 1.6, this means that k 11 < k 22 holds. This follows from the consistency check we carried out before. Geometric initial data being locally rotationally symmetric is equivalent to the corresponding expansion-normalised variables satisfying Def. 3.4, and geometric initial data of plane wave equilibrium type corresponds to Def. 1.17, including the possibility that Σ + = −1. Intersection with the Kasner parabola K, which consist of all Bianchi type I vacuum spacetimes, yields the characterisation: Taub 1 is the unique point on the Kasner parabola which satisfies the plane wave equilibrium point relations. Taub 2 is one of two intersection points between the Kasner parabola K and the set of LRS solutions, the one with Σ + > 0. This last property translates into 2k 11 − k 22 − k 33 < 0, and as k 11 = k 33 for LRS geometric initial data, the statement follows.

Proof of the main theorems
In this final section, we give the proofs of the main theorems stated in the introduction. Four of them, Thm 1.18, Thm 1.21, Thm 1.23, and Thm 1.25, are stated in the setting of expansion-normalised variables, and these are proven first. The remaining statements, Thm 1.10 and Thm 1.11, are translated versions of Thm 1.21 and Thm 1.23, giving the results in terms of geometric initial data and the corresponding maximal globally hyperbolic development.
In the previous sections, we have determined the asymptotic behaviour of solutions to the evolution equations (5)-(11). We have found possible α-limit sets and determined in detail the solutions which converge to the Kasner parabola K or the plane wave equilibrium points L κ as τ → −∞. Thm 1.21, Thm 1.23, and Thm 1.25 state that apart from a short list of exceptional solutions, either the Kretschmann scalar R αβγδ R αβγδ or the contraction of the Ricci tensor with itself R αβ R αβ or both become unbounded along solutions as τ → −∞. This implies Strong Cosmic Censorship in the C 2 -sense. We start with a discussion of these two geometric invariants and their form and asymptotic behaviour in terms of expansion-normalised variables, before we are then in a position to give the proof of these three theorems.
We then continue this section with a proof of Thm 1.18, where we show that apart from a 'small' subset, all solutions to the evolution equations (5)-(11) converge to a Kasner point to the right of Taub 2, as τ → −∞. For this proof as well, we heavily rely on the results on asymptotic behaviour of solutions which we obtained in the previous sections.
At the end of this section, we conclude with the proofs of the main statements in the initial data perspective, Thm 1.10 and Thm 1.11. The equivalent statements in expansion-normalised variables are given in Thm 1.21 and Thm 1.23. Using the transformation between this set of variables and the maximal globally hyperbolic development to given geometric initial data which we constructed in Subsections 11.4-11.6, we re-translate the statements back to the setting of geometric initial data. Applying the proof to µ > 0, γ = 0, i. e. the stress-energy tensor of a positive cosmological constant in vacuum, even justifies the statement given in Remark 1.14 as a re-translated version of Thm 1.25.
The central idea of proof has already been given in [Rin00b], where the Bianchi A case was discussed. It only relies upon equation (88), which follows immediately from the assumption on the stress-energy tensor and is independent of the Bianchi class of the Lie group. Note that only the evolution equations for θ and Ω come into play. In particular, this statement can be obtained without knowledge on the detailed asymptotic properties of the individual variables.
As Ω > 0 is an invariant set by equation (11), and θ = 0 by construction of the expansionnormalised coordinates in Subsections 11.4-11.6, the statement follows. Proof. Vacuum is equivalent to vanishing Ricci curvature, consequently R αβγδ R αβγδ = C αβγδ C αβγδ in this case, while for Ω > 0, γ = 0, the argument from the previous proof implies that R αβγδ R αβγδ − C αβγδ C αβγδ = 2R αβ R αβ − 1 3 S 2 = 1 3 4 + (3γ − 2) 2 µ 2 is a constant. Direct computation shows that on the Kasner parabola K one finds which is non-zero if and only if Σ + / ∈ {−1, 1/2}. The statement then follows from equation (89) and the fact that θ → ∞ as τ → −∞, due to equation (70). Proof. This follows from direct computation, which we in the case of the Taub points already carried out in the previous proof, and where the set L κ is defined in Def. 1.17.
Remark 12.4. On the Taub points 1 and 2, on the plane wave equilibrium points L κ , and on the point Σ + =Σ = ∆ =Ã = N + = 0, the contraction of the Weyl tensor with itself, expression (89), vanishes due to Lemma 12.3. Together with the result from Lemma 12.1 and equation (88), this implies that both the Kretschmann scalar R αβγδ R αβγδ and the contraction of the Ricci tensor with itself R αβ R αβ remain bounded for the constant solutions in the points Taub 1 and Taub 2, the plane wave equilibrium points and the point Σ + =Σ = ∆ =Ã = N + = 0.
We now give the proofs of the theorems stated in expansion-normalised variables.
Proof of Thm 1.21. This statement is an immediate consequence of Lemma 12.1.
Proof of Thm 1.23. Prop. 4.2 restricts the possible α-limit sets of solutions in vacuum Ω = 0 to the Kasner parabola K and the plane wave equilibrium points L κ . Non-constant solutions with α-limit set in L κ are immediately excluded by the same statement. One knows further from Prop. 6.1 that solutions with α-limit set in K have a unique α-limit point, i. e. they converge as τ → −∞. We now discuss the different possible locations of limit points for solutions to equations (5)-(11) and check in which cases the Kretschmann scalar possibly remains bounded. Lemma 12.2 states that the Kretschmann scalar becomes unbounded upon convergence to every Kasner points but the points Taub 1 and 2. The only solution converging to the point Taub 1 is the constant one, see Prop. 5.1. All solutions converging to Taub 2 are characterised in Thm 7.2. This concludes the proof.
Proof of Thm 1.25. The proof proceeds similarly to the previous one. Prop. 4.2 restricts the possible α-limit sets of solutions with Ω > 0, γ = 0, to the Kasner parabola K, the plane wave equilibrium points L κ , and the point Σ + =Σ = ∆ =Ã = N + = 0. Non-constant solutions converging to the point Σ + =Σ = ∆ =Ã = N + = 0 are immediately excluded by the same proposition, which also states that solutions whose α-limit set intersects both K \ T1 and L κ do not exist. One knows further from Prop. 6.1 and Prop. 4.4 that solutions with α-limit set in either K or L κ have a unique α-limit point, i. e. they converge as τ → −∞. We now discuss the different possible locations of limit points and check in which cases the Kretschmann scalar possibly remains bounded.
Let us now turn to solutions converging to a plane wave equilibrium point, i. e. solutions contained in C. We apply Thm 10.3 and find that the case −(3γ − 2)/4 < s < 0 is excluded by γ = 0. Every solution in C is contained in the union of submanifolds L ′ m or L ′′ m , and the limiting value s and the parameter κ have to satisfy relation (44). In Remark 10.4, we list the possible values depending on the Bianchi type. Setting the union of submanifolds {L m } to include all L ′ m and L ′′ m gives the statement. The arguments regarding solutions converging to a point on the Kasner parabola K are identical to the ones given in the previous proof. Note that only the non-constant solutions, i. e. locally rotationally symmetric solutions of Bianchi type I, II or VI −1 can satisfy Ω > 0. This concludes the proof.
For the case of vacuum, our statement Thm 1.23 makes precise and proves a claim by [WE97,p. 165] saying that all solutions except those contained in the unstable manifold of the point Taub 2 have an initial curvature singularity.
Proof of Thm 1.18. The three sets describing solutions of Bianchi type VI η , VII η and IV are open sets of R 5 . Restricted to the constraint equations (8) and (9), they form open subsets of the set defined by these equations. Consequently, these three Bianchi sets are of dimension four. Restricting to vacuum Ω = 0 yields sets of dimension three. The set describing solutions of Bianchi type V is a two-dimensional closed subset of R 5 and contained in the set defined by the constraint equations (8) and (9). Restricted to vacuum Ω = 0, this Bianchi type yields a set of dimension one. We prove the theorem by showing that all solutions with convergence behaviour different from the one in the statement are contained in countable unions of smooth submanifolds of positive codimension.
Independently of the Bianchi type, the α-limit set of a solution to equations (5)-(11) with either Ω = 0 or Ω > 0, γ ∈ [0, 2/3) is given in Prop. 4.2: We exclude constant solutions as they are contained in two smooth curve arcs together with one additional point, namely the Kasner parabola K, the plane wave equilibrium points L κ and the point Taub 1. In Bianchi type V, we additionally notice that the intersection between these two arcs and the subset defining this Bianchi type consists of two points.
Prop. 5.1 states that only the constant solution has Taub 1 as an α-limit point, so in particular we exclude solutions with Taub 1 as an α-limit point. We conclude from Prop. 4.2 that the αlimit set is contained in either the Kasner parabola K without the point Taub 1 or in the plane wave equilibrium points L κ , and also see that latter set can only occur for inflationary matter, i. e. for Ω > 0, γ ∈ [0, 2/3).
In Prop. 6.1 and Prop. 4.4, we further showed that the α-limit sets consist of exactly one point meaning that solutions converge as τ → −∞. All that remains to show now is that the set of solutions converging to a plane wave equilibrium point L κ or a point on the Kasner parabola K with −1 < Σ + ≤ 1/2, resp. the complement of this set of solutions, has the necessary properties.
We start with solutions converging to a point on the Kasner parabola K. According to Thm 7.2, the only Bianchi class B solutions converging to the point Taub 2 are those of Bianchi type VI −1 which are locally rotationally symmetric. They are characterised by κ = −1,Ã > 0, 3Σ 2 + =Σ, Σ + N + = ∆, and are solutions of Bianchi type VI −1 . In particular, no Bianchi type VII η , type IV or type V solution converges to the point Taub 2. The set is a C 1 submanifold of dimension three, and its restriction to vacuum of dimension two. In particular, it forms a subset of positive codimension in the set of all Bianchi type VI −1 solutions, both in case of vacuum and inflationary matter. The solutions converging to a Kasner point to the left of Taub 2 are given in Thm 8.5. In case of Bianchi type VI η , solutions either satisfyÃ > 0, ∆ = 0 = N + , 3Σ 2 + + κΣ = 0 (Bianchi type VI η with n α α = 0, see Table 3), or have to be contained in a countable union of C 1 submanifolds satisfyingÃ > 0 and ∆, N + not both vanishing identically. These submanifolds are contained either in the set of all non-vacuum solutions or the set of all vacuum solutions, and in the respective set have codimension at least one. Solutions of Bianchi type VII η cannot converge to a Kasner point to the left of Taub 2 due to κ > 0. In the remaining two cases, Bianchi type IV and V, the restriction κ = 0 implies that solutions can only converge to the Kasner point with s = 0.
For non-constant solutions converging to a plane wave equilibrium point, Thm 10.3 states that they converge to the arc L κ ∩{Σ + ≤ −(3γ −2)/4}, which includes the point L κ ∩{Σ + = 0} due to the assumption on γ. All solutions are contained in the countable union of C 1 submanifolds L ′ m or L ′′ whose dimension is at most two. This is enough to identify them as subsets of positive codimension in the Bianchi type VI η , VII η and IV. For Bianchi type V solutions, we have argued in Remark 10.4 that the solution is contained in the subset Σ + =Σ = ∆ = N + = 0, which is a subset of positive codimension in non-vacuum.
Collecting the different partial results, we conclude the following: In the case of Bianchi type VI η or VII η , if a solution does not converge to a Kasner point to the right of Taub 2, then it is contained in a countable union of C 1 submanifolds of positive codimension, both in case of vacuum and inflationary matter. Therefore, this set of exceptions has the right properties: its complement is of full measure and a countable intersection of open and dense sets. We argue similarly in case of Bianchi type IV, with the additional possibility of convergence to the Kasner point with s = 0. For the case of Bianchi type V, convergence to Kasner point to the right of Taub 2 is not possible, as all such Bianchi solutions satisfy Σ + = 0 by definition. We have shown that all non-constant solutions converge to the point on the Kasner parabola K satisfying s = 0.
We conclude this section with the proof of the two remaining theorems, which are stated in the setting of geometric initial data. As large portions of the proofs are identical, we combine them into one.
Proofs of Thm 1.10 and Thm 1.11. We consider initial data which is of Bianchi class B. This implies in particular that the Minkowski spacetime or quotients of this space cannot occur as development of the data. This excludes the case θ = tr h k = 0, see Lemma 11.16, and we can therefore consider a development of the data (I × G, g, µ) as in Def. 11.15, i. e. one constructed as in Subsections 11.4-11.6. According to Corollary 11.21, this development is isometric to the maximal globally hyperbolic development of the given initial data. We further know from Prop. 11.24 that the behaviour of any geometric quantity in the incomplete direction of causal geodesics in M corresponds to the behaviour of this geometric quantity on {t} × G as t → t − .
We now switch to the point of view of expansion-normalised variables, as explained in Subsection 11.4. The behaviour of any geometric quantity in the incomplete direction of causal geodesics in M consequently corresponds to the behaviour of this geometric quantity as τ → −∞, see also the construction of the development, specifically the end of Subsection 11.5.
The result in the matter case µ 0 > 0, γ > 0, which is the setting of Thm 1.10 is an immediate consequence of Thm 1.21, as the definition of the density parameter in equation (68) yields Ω > 0 at τ = 0, and this property is conserved by the evolution equation (11).
Consider now the vacuum case µ 0 = 0, i. e. the setting of Thm 1.11. In the setting of expansionnormalised variables this corresponds to Ω = 0, and we have listed all exceptions to curvature blow-up as τ → −∞ in Thm 1.23. To prove the theorem for geometric initial data, we have to carry over the individual exceptions to the geometric initial data perspective. As we only consider initial data with a Lie group of Bianchi class B, a number of these exceptions cannot occur by assumption, namely those assuming Bianchi type I and II. Note also that the point Taub 1 is a constant Bianchi type I solution.
We have argued in Subsection 11.8 that the definitions of local rotational symmetry in terms of geometric initial data, Def. 1.6, and in terms of expansion-normalised variables, Def. 3.4, carry equivalent information. This proves the statement.
Remark 12.5. We can adapt the proof to justify the statement made in Remark 1.14. Assuming µ > 0 and γ = 0, we switch to expansion-normalised variables. The de Sitter spacetime or quotients of this space cannot occur as development of the data due to the assumption on the Bianchi class. It therefore follows from Prop. 11.24 that the spacetime constructed as in Subsections 11.4-11.6 is in fact the maximal globally hyperbolic development of the given geometric initial data, otherwise the arguments are identical to those given in the previous proof. The exceptions to curvature blow-up are LRS Bianchi VI −1 solutions and solutions converging to plane wave equilibrium points, see Thm 1.25. The following theorem states that these manifolds exist, at least locally, around the compact set K, and that they contain the different maximal invariant sets. Theorem B.3. Let M be a C r+1 manifold, 1 ≤ r < ∞, let Y be a C r vector field on M . Let E be a C r submanifold consisting entirely of equilibrium points of Y , and K ⊂ E a compact subset such that the number of non-vanishing eigenvalues of T Y | K situated in the left half-plane, on the imaginary axis, and in the right half-plane, respectively, is constant. Then there is a C r centre-stable manifold C s , a C r centre-unstable manifold C u , and a C r centre manifold C for Y near K. Furthermore, there is a neighborhood U of K such that In addition, the following uniqueness properties hold: • Let D s be any C r manifold which is locally invariant relative to U and tangent to C s at a point m ∈ A + (U ). Then C s and D s have contact of order r at m. • Let D u be any C r manifold which is locally invariant relative to U and tangent to C u at a point m ∈ A − (U ). Then C u and D u have contact of order r at m. • Let D be any C r manifold which is locally invariant relative to U and tangent to C at a point m ∈ I(U ). Then C and D have contact of order r at m.
The statement appears as Thm 9.1 (i) and (iv) in [Fen79], where one considers not only one vector field Y , but a family X ǫ , ǫ ∈ (−ǫ 0 , ǫ 0 ). The theorem stated here is a direct consequence of restricting to a vector field which remains unchanged in ǫ, i. e. X ǫ = X 0 = Y , and ignoring the ǫ-direction in the definition of centre-stable, centre-unstable, and centre manifolds.