On the Dynamics of the General Bianchi IX Spacetime Near the Singularity

Eur. Phys. J. C (2018) 78:691 https://doi.org/10.1140/epjc/s10052-018-6155-8

Regular Article - Theoretical Physics

On the dynamics of the general Bianchi IX spacetime near the singularity

Claus Kiefer1,a, Nick Kwidzinski1,b, Włodzimierz Piechocki2,c 1 Institute for Theoretical Physics, University of Cologne, Zülpicher Strasse 77, 50937 Cologne, Germany 2 Department of Fundamental Research, National Centre for Nuclear Research, Ho˙za 69, 00-681 Warsaw, Poland

Received: 24 July 2018 / Accepted: 11 August 2018 © The Author(s) 2018

Abstract We show that the complex dynamics of the gen- retained in the evolution towards singularities [10] (see [11] eral Bianchi IX universe in the vicinity of the spacelike sin- for an extended physical interpretation). This motivated the gularity can be approximated by a simplified system of equa- activity of the Landau Institute in Moscow to examining the tions. Our analysis is mainly based on numerical simulations. dynamics of homogeneous spacetimes [12]. A group of rel- The properties of the solution space can be studied by using ativists inspired by Lev Landau, including Belinski, Khalat- this simplified dynamics. Our results will be useful for the nikov and Lifshitz (BKL), started to investigating the dynam- quantization of the general Bianchi IX model. ics of the Bianchi VIII and IX models near the initial spacelike cosmological singularity [13]. After several years, they found that the dynamical behaviour can be generalized to 1 Introduction a generic solution of general relativity [14]. They did not present a mathematically rigorous proof, but rather a con- The problem of spacetime singularities is a central one in jecture based on deep analytical insight. It is called the BKL classical and quantum theories of gravity. Given some gen- conjecture (or the BKL scenario if specialized to the Bianchi- eral conditions, it was proven that general relativity leads to type IX model). The BKL conjecture is a locality conjecture singularities, among which special significance is attributed stating that terms with temporal derivatives dominate over to big bang and black hole singularities [1]. terms with spatial derivatives when approaching the singu- The occurrence of a singularity in a physical theory usu- larity (with the exception of possible ‘spikes’ [8,9]). Conse- ally signals the breakdown of that theory. In the case of gen- quently, points in space decouple and the dynamics then turn eral relativity, the expectation is that its singularities will out to be effectively the same as those of the (non-diagonal) disappear after quantization. Although a theory of quantum Bianchi IX universe. (In canonical gravity, this is referred to gravity is not yet available in finite form, various approaches as the strong coupling limit, see e.g. [2, p. 127].) exist within which the question of singularity avoidance can The dynamics of the Bianchi IX towards the singularity be addressed [2]. Quantum cosmological examples for such are characterized by an infinite number of oscillations, which an avoidance can be found, for example, in [3–6] and the give rise to a chaotic character of the solutions (see e.g [15]). references therein. Progress towards improving the mathematical rigour of the Independent of the quantum fate of singularities, the ques- BKL conjecture has been made by several authors (see e.g. tion of their exact nature in the classical theory, and in par- [16]), while numerical studies giving support to the conjec- ticular for cosmology, is of considerable interest and has a ture have been performed (see e.g. [17]). long history; see, for example [7,8], for recent reviews. This The dynamics of the diagonal Bianchi IX model, in the is also the topic of the present paper. Hamiltonian formulation, were studied independently from Already in the 1940s, Evgeny Lifshitz investigated the BKL by Misner [18,19]. Misner’s intention was to search for gravitational stability of non-stationary isotropic models of a possible solution to the horizon problem by a process that universes. He found that the isotropy of space cannot be he called “mixing”.1 Ryan generalized Misner’s formalism to the non-diagonal case in [20,21]. A qualitative treatment a e-mail: [email protected] b e-mail: [email protected] 1 This is how the diagonal Bianchi IX model received the name mix- c e-mail: [email protected] master universe.

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1 ¯ ¯ ¯ ¯ ¯ of the dynamics for all the Bianchi models may be found in σ =−sin(ψ)dθ + cos(ψ)sin(θ)dφ, the review article by Jantzen [22], and we make reference to σ 2 = cos(ψ)¯ dθ¯ + sin(ψ)¯ sin(θ)¯ dφ,¯ (2) it whenever we get similar results. σ 3 = (θ)¯ φ¯ + ψ.¯ Part of the BKL conjecture is that non-gravitational (‘mat- cos d d ter’) terms can be neglected when approaching the singular- The σ i , i = 1, 2, 3, are dual to the vector fields ity. An important exception is the case of a massless scalar (ψ)¯ field, which has analogies with a stiff fluid (equation of state =− (ψ)∂¯ + cos ∂ + (θ)∂¯ , X1 sin θ¯ ¯ φ¯ cos ψ¯ p = ρ) and, in Friedmann models, has the same dependence sin(θ) ρ ∝ −6 (ψ)¯ of the density on the scale factor as anisotropies ( a ). ¯ sin ¯ (3) X2 = cos(ψ)∂θ¯ + ∂φ¯ + cos(ψ)∂ψ¯ , As was rigorously shown in [23], such a scalar field will sup- sin(θ)¯ press the BKL oscillations and thus is relevant during the X3 = ∂ψ¯ , evolution towards the singularity. Arguments for the impor- tance of stiff matter in the early universe were already given which together with X0 = ∂t form an invariant basis of by Barrow [24]. the Bianchi IX spacetime. The Xi are constructed from the In our present work, we shall mainly address the general Killing vectors that generate the isometry group SO(3, R) (non-diagonal) Bianchi IX model near its singularity. Our (see [27] for more details). The basis one-forms satisfy the main motivation is to provide support for a rather simple relation asymptotic form of the dynamics that can suitably model its 1 dσ i =− Ci σ j ∧ σ k , (4) exact complex dynamics. We expect this to be of relevance in 2 jk the quantization of the general Bianchi IX model, which we with Ci = ε being the structure constants of the Lie alge- plan to investigate in later papers; see, for example [25]. Apart jk ijk so( , R) [ , ]=− k from a few particular solutions that form a set of measure bra 3 .TheXi obey the algebra Xi X j CijXk. zero in the solution space, no general analytic solutions to We parametrize the metric coefficients in this frame as fol- the classical equations of motion are known. Therefore, we lows will restrict ourselves to qualitative considerations which will k l ¯ hij = Oi O j hkl, (5) be supported by numerical simulations. The examination of the non-diagonal dynamics presented in [26], though it is where √ √ mathematically satisfactory, is based on the qualitative theory α β + β β − β − β h¯ ≡ h¯ = e2 diag e2 + 2 3 − , e2 + 2 3 − , e 4 + of differential equations, which is of little use for our purpose. ij Our paper is organized as follows. Section 2 contains ≡ diag ( 1, 2, 3) . (6) the formalism and presents our main results for a gen- The variables α, β+, and β− are known as the Misner vari- eral Bianchi IX model. We first specify the kinematics and ables. The scale factor exp(α) is related to the volume, while dynamics. We then consider a matter field in the form of the anisotropy factors β+ and β− describe the shape of this (tilted) dust. This is followed by investigating the asymptotic model universe. The variables 1, 2 and 3 were used by regime of the dynamics near the singularity. Our conclusions BKL in their original analysis [31]. are presented in Sect. 3. The numerical methods used in our j We introduced here a matrix O ≡ Oi ≡ Oθ Oφ Oψ numerical simulations are described in the Appendix. (i corresponding to rows and j corresponding to columns), which is an SO(3, R) matrix that can be parametrized by another set of Euler angles, {θ,φ,ψ} ∈[0,π]×[0, 2π]× 2 The general Bianchi IX spacetime [0,π]. Explicitly, ⎛ ⎞ 2.1 Kinematics cos(ψ) sin(ψ) 0 ⎝ ⎠ Oψ = − sin(ψ) cos(ψ) 0 , The general non-diagonal case describes a universe with 001 ⎛ ⎞ rotating principal axes. The metric in a synchronous frame 10 0 ⎝ ⎠ can be given as follows (see for the following e.g. [27,28]): Oθ = 0 cos(θ) sin(θ) , (7) 2 =− 2 2 + σ i ⊗ σ j , 0 − sin(θ) cos(θ) ds N dt hij (1) ⎛ ⎞ cos(φ) sin(φ) 0 where N is the lapse function. Spatial hypersurfaces in the ⎝ ⎠ Oφ = − sin(φ) cos(φ) 0 . spacetime are regarded topologically as S3 (describing closed 001 universes), which can be parametrized by using three angles θ,¯ φ,¯ ψ¯ ∈[0,π]×[0, 2π]×[0,π]. The basis one-forms The Euler angles θ, φ, and ψ are now dynamical quantities read and describe nutation, precession, and pure rotation of the 123 Eur. Phys. J. C (2018) 78:691 Page 3 of 10 691 principal axes, respectively. In the case of Bianchi IX space- The Lagrangian in the gauge N i = 0 then takes the form ( , R) time, the group SO 3 is the canonical choice for the 2 ˙2 ˙2 2 2 3 2 1 2 diagonalization of the metric coefficients. For a treatment of α −˙α + β+ + β− + I1 ω 3 + I2 ω 1 + I3 ω 2 L = Ne3 other Bianchi models, see [22]. 2N 2 (3) R + , (15) 2.2 Dynamics 12

In the following, we shall discuss the Hamiltonian formula- where the ‘moments of inertia’ are given by tion of this model. In order to keep track of the diffeomor- √ phism (momentum) constraints, we replace the metric (1)by 2 3I1 ≡ sinh 3β+ − 3β− , the ansatz √ 2 3I ≡ sinh 3β+ + 3β− , 2 =− 2 2 + i + σ i ⊗ j + σ j , 2 ds N dt hij N dt N dt (8) √ 2 3I3 ≡ sinh 2 3β− . (16) where N i are the shift functions. The Hamiltonian formulation was ﬁrst derived in a series of papers by Ryan: the 1 2 2 3 2 Note, in particular, that the term I1 ω 3 + I2 ω 1 + symmetric (non-tumbling) case obtained by constraining ψ, 2 1 2 φ to be constant and keeping θ dynamical is discussed in I3 ω 2 would formally correspond to the rotational [20], and the general case can be found in [21]. We write the energy of a rigid body if the moments of inertia were con- Einstein–Hilbert action in the well known ADM form, stant. The canonical momenta conjugate to the Euler angles are given by 1 1 2 3 SEH = σ ∧ σ ∧ σ 16πG 3α e 2 √ pθ = I sin(ψ) sin(φ)ω ik jl ij kl (3) 1 3 × dtN h h h − h h KijKkl + R , N −I (ψ) (φ)ω3 + I (θ)ω1 , (9) 2 cos sin 1 3 cos 2 3α (17) e 2 3 where pφ = I cos(ψ)ω + I sin(ψ)ω , N 1 3 2 1 1 ˙ 3α K = h − 2D( N ) e 1 ij ij i j pψ = I ω . 2N N 3 2 is the extrinsic curvature, and Di is the spatial covari- { } It is convenient to introduce the following (non-canonical) ant derivative in the non-coordinate basis Xi . We will 3 σ 1 ∧ σ 2 ∧ σ 3 = angular momentum like variables: set 4πG 1 for simplicity. The three- ( ) dimensional curvature 3 R on spatial hypersurfaces of con- e3α e3α e3α l ≡ I ω2 , l ≡ I ω1 and l ≡ I ω2 . stant coordinate time is given by 1 N 1 3 2 N 2 3 3 N 3 1 −2α √ (18) (3) e −8β+ −2β+ R =− e − 4e cosh 2 3β− 2 √ The relation to the canonical momenta can now explicitly be 4β+ + 2e cosh 4 3β− − 1 . (10) given by

We now turn to the calculation of the kinetic term and the dif- pθ = sin(ψ) sin(φ)l1 − cos(ψ) sin(φ)l2 + cos(φ)l3 , feomorphism constraints. For this purpose, we deﬁne an anti- pφ = cos(ψ)l1 + sin(ψ)l2 , (19) i symmetric angular velocity tensor ω j by the matrix equation pψ = l3 . ⎛ ⎞ ω1 −ω3 0 2 1 It is readily shown that the variables l obey the Poisson ω = ωi = ⎝ −ω1 ω2 ⎠ ≡ T ˙ . i j 2 0 3 O O (11) bracket algebra l , l =−Ck l . After the usual Legendre ω3 −ω2 i j ij k 1 3 0 transform, we obtain the Hamiltonian constraint,

An explicit calculation of the right-hand side gives, using (7), −3α H = e ω2 = (ψ)φ˙ + (ψ) (φ)θ,˙ 2 3 cos sin sin (12) 2 2 2 6α l l l e ( ) 3 ˙ ˙ × − 2 + 2 + 2 + 1 + 2 + 3 − 3 . ω 1 = sin(ψ)φ − cos(ψ) sin(φ)θ, (13) pα p+ p− R I1 I2 I3 6 ω1 =ψ˙ + (φ)θ.˙ 2 cos (14) (20) 123 691 Page 4 of 10 Eur. Phys. J. C (2018) 78:691

From (9), we find that the diffeomorphism constraints equation for the spatial components of the four velocity can (∂ L/∂ N i = 0) can be written as then be written as j kl 0(∂ ) − k j = . Hi = 2C ilh jk p , (21) u t ui Cijuku 0 (26) where The geodesic equation implies the existence of a constant √ of motion. To see this explicitly, we compute the expression h ij ik jl ij kl = , , u˙ u and convince ourselves that it vanishes iden- p = h h − h h Kkl i 1 2 3 i i 24N tically. Thus the Euclidean sum is the ADM momentum. From this expression we can finally 2 2 2 2 C ≡ (u1) + (u2) + (u3) (27) compute the diffeomorphism constraints in terms of the angu- T lar momentum-like variables and obtain is a constant of motion. Defining u ≡ (u1, u2, u3) ,the geodesic equation (26) can be rewritten in vector notation, j Hi = Oi l j , (22) N u × (Oh¯−1 OT u ) ∂ = , that is, we can identify the diffeomorphism constraints with t u (28) 1 + uT Oh¯−1 OT u a basis of the generators of SO(3, R). The full gravitational Hamiltonian then reads where “×” denotes the usual cross product in the three dimensional Euclidean space. Defining for convenience v ≡ = H + i H . T 2 2 2 H N N i (23) O u /C,wehave(v1) +(v2) +(v3) = 1, and the geodesic equation simplifies to From the diffeomorphism constraints (22) we conclude that ¯−1 in the vacuum case li = 0 and that therefore no rotation is NC v × (h v) (∂t + ω) v = . (29) possible, that is, we recover the diagonal case. If we want to 1 + C2v T h¯−1v obtain a Bianchi IX universe with rotating principal axes, we ωv = v × ω are thus forced to add matter to the system. A formalism for Note that we can also write , where obtaining equations of motion for general Bianchi class A T ω ≡{ωi }= N l1 , l2 , l3 . models filled with fluid matter was developed by Ryan [28]. 3α e I1 I2 I3 For simplicity, we will only consider the case of dust as dis- ω cussed by KuchaˇrandBrownin[29]. If we were, for example, It will thus be possible to eliminate from the geodesic interested in the study of the quantum version of this model, equation by using the diffeomorphism constraints. it would be desirable to introduce a fundamental matter field We now add homogeneous dust to the system. The for- instead of an ideal fluid. Standard scalar fields alone cannot malism developed in [29] leads to the following form of the lead to a rotation for Bianchi IX models. The easiest way to Hamiltonian and diffeomorphism constraints for dust in a Bianchi IX universe, achieve this is, to our knowledge, the introduction of a Dirac field [30]. −3α 2 2 2 (m) e 2 2 2 l1 l2 l3 H + H = −pα + p+ + p− + + + 2 I1 I2 I3 2.3 Adding dust to the system 6α (30) e (3) 3α ij − R + 2e pT 1 + h ui u j , The energy momentum tensor for dust reads Tμν = ρuμuν. 6 μν The local energy conservation ∇μT = 0 leads to a ( ) H + H m = O j l − Cp v , geodesic equation for the positions of the dust particles. Let i i i j T j us start therefore by considering the geodesic equation for a where pT denotes the momentum canonically conjugate to single dust particle, whose four-velocity we can express in T , where T is the global ‘dust time’. Since the Hamiltonian i the non-coordinate frame σ used above by the Pfaffian form does not explicitly depend on T , the momentum pT is a constant of motion. The fact that l2 + l2 + l2 commutes i 1 2 3 u = u0dt + ui σ with u, u =−1 . (24) H 2 + 2 + 2 = ( )2 with implies that l1 l2 l3 CpT is a conserved We partially fix the gauge by setting N i = 0. The normal- quantity. This is consistent with (27). We note that a similar ization condition implies form of the constraints (30) was already presented in [20, 21]. The formalism is not entirely canonical and must be ij complemented by the geodesic Eq. (26). u0 =−N 1 + h ui u j . (25) For our numerical purposes, it will be convenient to rewrite We have chosen here the minus sign because this guarantees the equations of motion in the variables i introduced in (6). that the proper time in the frame of the dust particle has We find that the choice of the variables log i allow for a the same orientation as the coordinate time t. The geodesic better control over the error in the Hamiltonian constraint. 123 Eur. Phys. J. C (2018) 78:691 Page 5 of 10 691

3α Moreover,√ we pick the quasi-Gaussian gauge N = e = be written as N i = 1 2 3, 0. Recall that the singularity is reached in a v ˙ = Cv × (Mv ) where ﬁnite amount of comoving time (corresponding to the gauge 3α diag ( 2 3, 1 3, 1 2) N = 1). The choice N = e allows to resolve the oscil- M = 2 2 2 2 lations in the approach towards the singularity. With these 1 2 3 + C 2 3v + 1 3v + 1 2v 1 2 3 choices the Hamiltonian constraint becomes + 2 3 , 1 3 , 1 2 . · · · · · · pT diag 2 2 2 − (log 1) (log 2) − (log 2) (log 3) − (log 1) (log 3) [ 2 − 3] [ 3 − 1] [ 1 − 2] + 2 + 2 + 2 − 2( + + ) (34) 1 2 3 1 2 3 1 2 3 2 2 2 l2 l2 l2 Together with the constraint v + v + v = 1thisisallwe + 1 + 2 + 3 1 2 3 24 need for numerical integration. Note that all dependence on I1 I2 I3 (31) the Euler angles and their momenta has dropped out from the + | | + 2 v2 + v2 + v2 equations of motion (33, 34). The numerical method we use 2 pT 1 2 3 C 2 3 1 1 3 2 1 2 3 is described in Appendix. = 0, 2.4 The tilted dust case where the moments of inertia are 2 2 ( 3 − 2) ( 1 − 3) A qualitative picture for the dynamics of the universe can be I1 = , I2 = , 12 3 2 12 1 3 obtained by considering the Hamiltonian (30) in the quasi- ( − )2 Gaussian gauge N = e3α, N = 0. The Hamiltonian can = 1 2 . i I3 (32) then be interpreted as the “relativistic energy” of a point par- 12 1 2 ticle (called the universe point) with “spacetime coordinates” The diffeomorphism constraints read l = p Cv and can be i T i (α, β+,β−). The universe point is subject to the forces gen- used to eliminate the angular momentum variables from the erated by the dynamical potential equations of motion. These equations can then be written as l2 l2 l2 e6α ( )·· = ( − )2 − 2 + 2 2 1 + 2 + 3 − (3) + 3α + ij , log 1 2 3 1 2pT C R 2e pT 1 h ui u j (35) I1 I2 I3 6 ( + )v2 ( + )v2 × 1 3 1 3 2 + 1 2 1 2 3 which is depicted in Fig. 1. The contour lines of the curva- 3 3 6α ( 1 − 3) ( 1 − 2) − e (3) ture potential 12 R are represented by solid black lines. p ( + 2C2v2 ) The curvature potential is exponentially steep and takes its + T 1 2 3 1 2 3 , minimum at the origin β± = 0. When the universe evolves + 2 v2 + v2 + v2 1 2 3 C 2 3 1 1 3 2 1 2 3 towards the singularity (α →−∞), the curvature poten- ·· (log ) = ( − )2 − 2 tial walls move away from the origin while becoming effec- 2 3 1 2 2 2 tively hard walls in the vicinity of the singularity. The term 1 2( 1 + 2)v 2 3( 2 + 3)v + p 2C2 3 + 1 2 T 3 3 ( 2 − 1) ( 2 − 3) 2 2 p ( 1 2 3 + 2C v 1 3) + T 2 , + 2 v2 + v2 + v2 1 2 3 C 2 3 1 1 3 2 1 2 3 ·· (log ) = ( − )2 − 2 3 1 2 3 2 2 1 3( 1 + 3)v 3 2( 3 + 2)v + p 2C2 2 + 1 2 T 3 3 ( 3 − 1) ( 3 − 2) 2 2 p ( 1 2 3 + 2C v 1 2) + T 3 , + 2 v2 + v2 + v2 1 2 3 C 2 3 1 1 3 2 1 2 3 (33) ≡ where we have set pT 12pT for convenience. Note that these equations are exact. (In [13], the matter terms were neglected.) We use the diffeomorphism constraints to elim- Fig. 1 Potential picture for the tumbling case. The black contour lines inate ω from the geodesic equation (29). If expressed in the represent the curvature potential while the dotted (blue) and dashed 3α gauge N = e and using the i , the geodesic Eq. (29) can (red) lines represent the rotation and centrifugal walls, respectively 123 691 Page 6 of 10 Eur. Phys. J. C (2018) 78:691

800 0

600 -2000

400 -4000 200 -6000 0 -8000 -200

-400 -10000

-600 -12000

-800 -14000 -500 0 500 00.511.522.5 10 6

Fig. 2 Numerical simulation of the tumbling case. In the region where 0 < t < 5 × 105 (arbitrary units) one can see a typical Kasner era. The solution bounces in the valley formed between the curvature and one of the centrifugal potential walls. Increasing t corresponds to evolution towards the singularity

α β 3α ij = 2 2 + = e pT 1 + h ui u j can be interpreted as three rotational For the i variables it means that 1 e e 2 and = 2α −4β+ = potential walls. These potentials are rather unimportant in 3 e e . With this choice we obtain I3 0 and = = 2 ( β ) the examination of the asymptotic dynamics, since they move 3I1 3I2 sinh 3 + . Most importantly, the geodesic l2 l2 l2 Eq. (34) is trivially satisfied, that is, v = v = 1/2 and away from the origin with unit speed. The term 1 + 2 + 3 1 2 I1 I2 I3 v = 0 for all times. When setting C = 0 we obtain the can be interpreted as three singular centrifugal potential 3 diagonal case which contains the isotropic case of a closed walls. They are represented by the dashed red lines. Asymp- Friedmann universe. The simulation plotted in Fig. 2 was totically close to the singularity, these walls are expected to performed for the tumbling case, that is, the v are chosen to become static. In general, however, the centrifugal potential i be non-zero. walls are dynamical and change in a complicated manner dictated by the geodesic Eq. (34). The centrifugal walls will prevent the universe point from penetrating certain regions 2.4.2 The asymptotic regime close to the singularity of the configuration space. Misner [19] and Ryan [20,21] employed these facts to obtain approximate solutions in a In order to simplify the dynamics of the general case, BKL diagrammatic form. The other Bianchi models can be treated made two assumptions based on qualitative considerations in a similar way [22]. of the equations of motion. The first assumption states that anisotropy of space grows without bound. This means that the solution enters the regime 2.4.1 Special classes of solutions 1 2 3. (37) Before doing the numerics, we comment on particular classes The ordering of indices is irrelevant. In fact, there are six of solutions: one class of solutions is obtained if we choose, possible orderings of indices which each correspond to the for example, the initial conditions v = 0 = v and v = 1. 1 2 3 universe point being constrained to one of the six regions The geodesic Eq. (34) implies now that the velocities stay bounded by the rotation and centrifugal walls sketched in constant in time. This implies that at all times l1 = 0 = Fig. 1. The region 1 > 2 > 3 corresponds to the right l2 and l3 = pψ = pT C. This class of solutions is known region above the line β− = 0inFig.1. More precisely, the as the non-tumbling case. Furthermore, there are classes of inequality (37) means that solutions which are rotating versions of the Taub solution. These solutions should be divided into two subclasses: one 2/ 1 → 0 and 3/ 2 → 0. (38) class that oscillates between the centrifugal walls and the curvature potential and one class that runs through the valley Our numerical simulations support the validity of this straight into the singularity. We set assumption (see the plot of the ratios 2/ 1 and 3/ 2 in Fig. 3). We provide plots of the two ratios 2/ 1, 3/ 2 v 1 and the velocities in order to provide a sanity check of the v = v = ,v= 0 and β− = 0. (36) 1 2 2 3 approximation we will perform later on. 123 Eur. Phys. J. C (2018) 78:691 Page 7 of 10 691

10 -3 1.5 1

0.5 1

0.5 -0.5

-1 0 01234 5 012345 10 5 10 5

Fig. 3 Plots of the ratios 2/ 1, 3/ 2 and the velocity v . The plots correspond to the numerical solution presented in Fig. 2. The peaks in the ratios appear during bounces of the universe point with the centrifugal walls. As we can see, the height of these peaks decreases in the evolution towards the singularity

2 The second assumption made by BKL states that the Euler ≡ , ≡ 2 2 v(0) , a 1 b 2pT C 3 2 angles assume constant values: 2 ≡ 4 4 v(0)v(0) , c 4pT C 1 3 3 (42) (θ,φ,ψ)→ (θ0,φ0,ψ0), (39) we arrive at a simplified Hamiltonian constraint and equa- that is, the rotation of the principal axes stops for all prac- tions of motion, tical purposes and the metric becomes effectively diagonal. · · · · · · (log a) (log b) + (log a) (log c) + (log b) (log c) The analysis of BKL [31] supports the consistency of making both assumptions at the same time. Similar heuristic consid- = a2 + b/a + c/b, ·· ·· (43) erations can possibly be applied to other Bianchi models as (log a) = b/a − a2,(log b) well [22]. In the dust model under consideration, this assump- ·· = a2 − b/a + c/b,(log c) = a2 − c/b, tion is equivalent to the statement that the dust velocities v (0) assume constant values v → v . Our numerical results indi- which coincides with the asymptotic form of equations cate that this is in fact true (see Fig. 3). BKL then arrive at obtained in [31]. Equation (43) can now be treated by the the simplified effective set of equations. numerical methods which we have used in the previous sec- Let us now carry out the approximation and apply it to tions. One must ensure that initial conditions are chosen such our equations of motion. The kinetic term stays untouched that the simulation starts close to the asymptotic regime (37). during the approximation. The first step in the approximation is to ignore the rotational potential. In view of the strong inequality (37), we approximate the curvature potential via 3 Conclusions 2 + 2 + 2 − ( + + ) ≈ 2 . 1 2 3 2 1 2 3 1 2 3 1 (40) The numerical simulations indicate that the non-diagonal Bianchi IX solutions, with tilted dust, evolve into the regime Furthermore, we approximate the centrifugal potential by where 1 2 3 and vi ≈const. The results motivate us to formulate the conjecture: 2 2 2 l l l ( ) 2 ( ) 2 1 + 2 + 3 ≈ C2 p2 3 v 0 + 2 v 0 . Given a tumbling solution to the general Bianchi IX model 12 T 1 3 I1 I2 I3 2 1 filled with pressureless tilted matter, there exists t0 ∈ R such (41) that the solution is well approximated by a solution to the asymptotic equations of motion for all times t > t0 describ- Note that one centrifugal wall was ignored completely. Hav- ing the vicinity of the singularity. ing Fig. 1 in mind, this approximation is well motivated since To make the notion of “approximation” mathematically only two of the centrifugal walls are expected to have a signif- more precise, a suitable measure of the “distance” on the set icant influence on the dynamics of the universe point. After of solutions is needed. For this purpose, we propose to use defining the new variables the following simple measure: 123 691 Page 8 of 10 Eur. Phys. J. C (2018) 78:691

0.09 25 0.08

0.07 20

0.06 15 0.05

0.04 10 0.03

0.02 5 0.01

0 0 00.511.522.53 -800 -600 -400 -200 0 10 6 (a) (b)

Fig. 4 The difference between the exact and the asymptotic solutions: a evolution towards the singularity, b evolution away from the singularity

2 ¯ 2 2 (t) ≡ (log 1(t) − log a¯(t)) + log 2(t) − log b(t) + (log 3(t) − log c¯(t)) , (44) where { 1, 2, 3} denotes the numerical solution to the Employing the asymptotic form of the equations of motion exact equations of motion (33)–(34), and may enable one to study the chaotic behaviour and other 2 properties of the solution space for the general model. This is =¯, = 2 2 v(0) ¯, a a b 2pT C 3 b also important for quantizing the general Bianchi IX model, 2 where the quantization of the exact dynamics seems to be = 4 4 v(0)v(0) ¯ c 4pT C 1 3 c (45) quite difﬁcult, whereas the quantization of the asymptotic denote the numerical solution to the asymptotic equations of case seems to be feasible [25]. motion (43). = Acknowledgements We are grateful to Vladimir Belinski and Claes We have evolved the exact system of equations from t 0 Uggla for helpful discussions. This work was supported by the German- 6 forward in time until t = 3 × 10 . There we used the same Polish bilateral project DAAD and MNiSW, No. 57391638, “Model of initial conditions as the ones we used to obtain the solution stellar collapse towards a singularity and its quantization”. shown in Fig. 2. We then took the ﬁnal state at t = 3 × 106 Open Access This article is distributed under the terms of the Creative as an initial condition for the asymptotic system of equations Commons Attribution 4.0 International License (http://creativecomm and evolved it backwards in time towards the re-bounce until ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, t =−980. and reproduction in any medium, provided you give appropriate credit Figure 4 presents the measure (44) as a function of time. to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. We can see fast decrease of with increasing time (evo- Funded by SCOAP3. lution towards the singularity) and fast increase of with decreasing time (evolution away from the singularity). Our numerical simulations give strong support to the conjecture concerning the asymptotic dynamics of the general Appendix: Numerical analysis Bianchi IX spacetime put forward long ago by Belinski et al. [31]. We remark that approximating the diagonal or non- Numerical simulations of the diagonal Bianchi IX model tumbling case by the asymptotic dynamics (43)isinvalid(see were already carried out in the late 1980s and early 1990s [32] for more details). (see e.g. [33,34]; for a modern account, see [7]). Our main It is sometimes stated that “matter does not matter” in the interest here is in the nondiagonal case. With given initial H = asymptotic regime, but this does not mean that one is allowed conditions (respecting the constraints 0), the system to use the dynamics of the purely diagonal case. One only (33) and (34) can be integrated by using a suitable numeri- encounters an effectively diagonal case, which is expressed cal method. In this work we employ the MATLAB R2016b in terms of the directional scale factors {a, b, c}. So there solver ode113 [35]. This code is an implementation of an exist serious differences between the purely diagonal and Adams–Bashforth–Moulton method. It turns out to lead to effectively diagonal cases (see [32] for more details). the best results when compared to other MATLAB solvers. The relative error tolerance of the solver was chosen to be 123 Eur. Phys. J. C (2018) 78:691 Page 9 of 10 691

10 -16 1 1.5 0 -1 -2 1 -3 -4 -5 0.5 -6 -7 -8 -9 0 0 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 10 6 10 6 (a) (b)

Fig. 5 a Shows the error in the Hamiltonian constraint. b Shows the the Hamiltonian constraint becomes relevant and cannot be neglected speed of the universe point in the beta plane as measured in “α-time”. in the approach towards the singularity (see [7] for a more detailed dis- This plot can be viewed as another check of the numerics. Between cussion). Both plots correspond to the numerical solution shown in Fig. two successive bounces from the potential walls this quantity should be 2 close to one. If this ceased to be true it would indicate that the error in

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