Oscillating Bianchi IX Universe in Horava-Lifshitz Gravity
aa r X i v : . [ h e p - t h ] A p r Oscillating Bianchi IX Universe in Hoˇrava-Lifshitz Gravity
Yosuke
Misonoh , ∗ Kei-ichi
Maeda ,
1, 2, † and Tsutomu Kobayashi
3, 4, ‡ Department of Physics, Waseda University, Okubo 3-4-1, Shinjuku, Tokyo 169-8555, Japan Waseda Research Institute for Science and Engineering, Okubo 3-4-1, Shinjuku, Tokyo 169-8555, Japan Hakubi Center, Kyoto University, Kyoto 606-8302, Japan Department of Physics, Kyoto University, Kyoto 606-8502, Japan (Dated: October 31, 2018)We study a vacuum Bianchi IX universe in the context of Hoˇrava-Lifshitz (HL) gravity. Inparticular, we focus on the classical dynamics of the universe and analyze how anisotropy changesthe history of the universe. For small anisotropy, we find an oscillating universe as well as abounce universe just as the case of the Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime.However, if the initial anisotropy is large, we find the universe which ends up with a big crunchafter oscillations if a cosmological constant Λ is zero or negative. For Λ >
0, we find a variety ofhistories of the universe, that is de Sitter expanding universe after oscillations in addition to theoscillating solution and the previous big crunch solution. This fate of the universe shows sensitivedependence of initial conditions, which is one of the typical properties of a chaotic system. If theinitial anisotropy is near the upper bound, we find the universe starting from a big bang and endingup with a big crunch for Λ ≤
0, while de Sitter expanding universe starting from a big bang forΛ > PACS numbers: 04.60.-m, 98.80.Cq, 98.80.-k
I. INTRODUCTION
Since the advent of the big bang theory, the initial sin-gularity problem is of prime importance in the field ofcosmology. As shown by Hawking and Penrose[1], gen-eral relativity (GR) predicts a spacetime singularity if acertain condition is satisfied. Their singularity theoremconcludes that our universe must have an initial singu-larity. However, once a singularity is formed, general rel-ativity becomes no longer valid. It must be replaced bymore fundamental gravity theory. Even in the frameworkof an inflationary scenario which resolves many difficul-ties in the early universe based on the big bang theory,the initial singularity cannot be avoided. New gravita-tional theory may be required to describe the beginningof the universe.Many researchers attempt to resolve this singularityproblem in the context of generalization or extensionof general relativity[2]. However, no success has beenachieved yet. Superstring theory, which is one of themost promising candidates for unified theory of funda-mental interactions, may solve it, but so far it has notbeen completed yet and is not so far able to describe anyrealistic strong gravitational phenomena. Loop quantumgravity theory may resolve the problem of a big bangsingularity via loop quantum cosmology[3]. However itis still unclear how to describe time evolution of quantumspacetime in loop quantum gravity because of the lack of“time” variable. ∗ Electronic address: y”underscore”misonou”at”moegi.waseda.jp † Electronic address: maeda”at”waseda.jp ‡ Electronic address: tsutomu”at”tap.scphys.kyoto-u.ac.jp
Among attempts to construct a complete quantumgravitational theory, Hoˇrava-Lifshitz (HL) gravity hasbeen attracted much interest as a candidate for sucha theory over the past years. HL gravity is charac-terized by its power-counting renormalizablity, which isbrought about by a Lifshitz-like anisotropic scaling as t → ℓ z t, ~x → ℓ~x , with the dynamical critical exponent z = 3 in the ultra-violet (UV) limit [4]. In order torecover general relativity (or the Lorentz invariance) inour world, one expects that the constant λ converges tounity in the infrared (IR) limit in the renormalizationflow. Although it has been argued that there exist somefundamental problems in HL gravity [5–16], some exten-sions are proposed to remedy these difficulties[10, 17–19].It is intriguing issue whether or not HL gravity can be acomplete theory of quantum gravity.There are a number of works on cosmology in HL grav-ity [7, 20–53]. As pointed out by earlier works, a bigbang initial singularity may be avoided in the frame-work of HL cosmology due to the higher order termsin the spatial curvature R ij in the action [38]. In thiscontext, many researchers have studied the dynamicsof the Friedmann-Lemaitre-Robertson-Walker (FLRW)universe in HL gravity [35–51]. In isotropic and ho-mogeneous spacetime, higher curvature terms with ar-bitrary coupling constants mimic various types of matterwith arbitrary sign of energy densities. The z = 2 and z = 3 scaling terms give “dark radiation” and “dark stiff-matter”, respectively. Although “dark radiation” termsin the models with the detailed balance condition canavoid the initial singularity, such terms may become irrel-evant to the dynamics when we include relativistic mat-ter fields, which may scale as z = 3 in the UV limit andbehave as a stiff matter[21].In our previous paper [47], we have studied the dynam-ics of vacuum FLRW spacetime in generalized HL gravitymodel without the detailed balance condition and shownthat “dark stiff-matter” can avoid the initial singular-ity of the universe. Even if we include relativistic mat-ter fields, when the contribution of “dark stiff-matter”is dominant, the singularity is avoided and an oscillatingspacetime or a bounce universe is obtained.Although we have shown a singularity avoidance in HLcosmology, the following question may arise: Is this sin-gularity avoidance generic? Is such a non-singular space-time stable against anisotropic and/or inhomogeneousperturbations? In order to answer for these questions,we have to study more generic spacetime than the FLRWuniverse.The initial state of the universe could be anisotropicand/or inhomogeneous. Before the singularity theorem,some people believed that the big bang singularity ap-pears because of its high symmetry and it may be re-solved if one studies anisotropic and/or inhomogeneousspacetime. Then they analyzed anisotropic Bianchi-type universes and their generalization. Although theyfound some interesting behaviours near the singularitysuch as chaos in Bianchi IX spacetime [54–57], theycould not succeed the singularity avoidance. It is sim-ply because the singularity theorem does not allow asingularity avoidance in GR. The situation becomesworse if we consider anisotropy and/or inhomogeneity.Even in the effective gravity model derived from super-string, which shows a singularity avoidance, with such aproperty[58, 59], once we include anisotropy and/or inho-mogeneity, the property of such a singularity avoidancemay be spoiled[60].Therefore it is important to study whether or not non-singular universes in the present HL cosmology still existwith anisotropy and/or inhomogeneity. In the presentpaper, we shall investigate the possibility of the singular-ity avoidance in homogeneous but anisotropic Bianchi IXuniverse. Since we are interested in a singularity avoid-ance, we focus on an oscillating universe and analyze howanisotropy changes the history of the universe. We willnot study the chaotic behaviour in detail, which may ap-pear near the big bang singularity, although it is one ofthe most popular and important properties in the BianchiIX spacetime and was discussed analytically in [52, 53].As we will show later, however, some property of non-integrable system, i.e. , sensitive dependence on initialconditions may be found in the fate of the universe inthe present analysis as well.The paper is organized as follows. After a shortoverview of HL gravity, we present the basic equationsfor the vacuum Bianchi IX universe in HL gravity inSec.II. In Sec.III, we study the stability of the closedFLRW universe against small anisotropic perturbations.In Sec.IV we analyze Bianchi IX universe numericallyand show a variety of histories of the universe, depend-ing on initial anisotropy. Summary and remarks followin Sec.V. In Appendix, we also analyze a bounce uni-verse with anisotropy as an another type of non-singular solution. II. BIANCHI IX UNIVERSE INHO ˇRAVA-LIFSHITZ GRAVITY
First we introduce our Lagrangian of HL gravity, bywhich we will discuss the Bianchi IX universe. The basicvariables in HL gravity are the lapse function, N , theshift vector, N i , and the spatial metric, g ij . These vari-ables are subject to the action [4, 19] S HL = 12 κ Z dtd x √ gN ( L K − V HL [ g ij ]) , (2.1)where κ = 1 /M ( M PL : the Planck mass) and thekinetic term is given by L K = K ij K ij − λ K (2.2)with K ij := 12 N ( ∂ t g ij − ∇ i N j − ∇ j N i ) (2.3) K := g ij K ij (2.4)being the extrinsic curvature and its trace. The poten-tial term V HL will be defined shortly. In GR we have λ = 1, only for which the kinetic term is invariant un-der general coordinate transformations. In HL gravity,however, Lorentz symmetry is broken in exchange forrenormalizability and the theory is invariant under thefoliation-preserving diffeomorphism transformations, t → ¯ t ( t ) , x i → ¯ x i ( t, x j ) . (2.5)As implied by the symmetry (2.5), it is most natu-ral to consider the projectable version of HL gravity, forwhich the lapse function depends only on t : N = N ( t )[4]. Since the Hamiltonian constraint is derived from thevariation with respect to the lapse function, in the pro-jectable version of the theory, the resultant constraintequation is not imposed locally at each point in space,but rather is an integration over the whole space. In thecosmological setting, the projectability condition resultsin an additional dust-like component in the Friedmannequation [7].The most generic form of the potential V HL is givenby [19] V HL = 2Λ + g R + κ (cid:16) g R + g R ij R ji (cid:17) + κ g ǫ ijk R iℓ ∇ j R ℓk + κ (cid:16) g R + g R R ij R ji + g R ij R jk R ki + g R ∆ R + g ∇ i R jk ∇ i R jk (cid:17) , (2.6)where Λ is a cosmological constant, R ij and R are theRicci and scalar curvatures of the 3-metric g ij , respec-tively, and g i ’s ( i = 1 , ...,
9) are the dimensionless cou-pling constants. By a suitable rescaling of time we set g = −
1. We also adopt the unit of κ = 1 ( M PL = 1)throughout the paper.Let us consider a Bianchi IX spacetime, which metricis written as ds = − dt + a e β ij ω i ω j , (2.7)where the invariant basis ω i is given by ω = − sin x dx + sin x cos x dx ,ω = cos x dx + sin x sin x dx ,ω = cos x dx + dx . (2.8)A typical scale of length of the universe is given by a ,which reduces to the usual scale factor in the case ofthe FLRW universe. We shall call it a scale factor inBianchi IX model as well. The traceless tensor β ij mea-sures the anisotropy of the universe. The spacelike sec-tions of the Bianchi IX is isomorphic to a three-sphere S , and a closed FLRW model is a special case of theabove metric in the isotropic limit ( β ij → β ij is diagonalized as β ij = diag (cid:16) β + + √ β − , β + − √ β − , − β + (cid:17) . (2.9) The basic equations describing the dynamics of BianchiIX spacetime in HL gravity are now given by the follow-ings: H = 23(3 λ − (cid:20)
3( ˙ β + ˙ β − ) + 64 a V ( a, β ± ) + 8 Ca (cid:21) , (2.10)˙ H + 3 H − λ − (cid:20) a ∂V∂a + 3 Ca (cid:21) = 0 , (2.11)¨ β ± + 3 H ˙ β ± + 323 a ∂V∂β ± = 0 , (2.12)where H = ˙ a/a is the Hubble expansion parameter, i.e. ,the volume expansion rate is given by K = 3 H . Theconstant C arises from the projectability condition andit could be “dark matter” [7], but here we assume C = 0just for simplicity.The potential V , which depends on a as well as β ± , isdefined by V ( a, β ± ) := a V HL = V ( β ± ) + V ( β ± ) a + V ( β ± ) a + V ( β ± ) a + Λ64 a , (2.13)where V ( β ± ) = e β + (cid:20) −
18 ( g + 3 g + g − g ) cosh(12 √ β − ) + 14 (3 g + 5 g + 3 g − g ) cosh(8 √ β − ) −
18 (15 g + 13 g + 15 g − g ) cosh(4 √ β − ) + 14 (5 g + 3 g + 5 g − g ) (cid:21) + e β + (cid:20)
14 (3 g + 5 g + 3 g − g ) cosh(10 √ β − ) −
14 (9 g + 7 g − g + 20 g ) cosh(6 √ β − )+ 12 (3 g + g − g + 12 g ) cosh(2 √ β − ) (cid:21) −
18 (15 g + 13 g + 15 g − g ) cosh(8 √ β − )+ 12 (3 g + g − g + 12 g ) cosh(4 √ β − ) + 18 (3 g + 9 g + 27 g − g )+ e − β + (cid:20)
12 (5 g + 3 g + 5 g − g ) cosh(6 √ β − ) + 12 (3 g + g − g + 12 g ) cosh(2 √ β − ) (cid:21) + e − β + (cid:20) −
18 (15 g + 13 g + 15 g − g ) cosh(4 √ β − ) −
18 (9 g + 7 g − g + 20 g ) (cid:21) + 14 (3 g + 5 g + 3 g − g ) e − β + cosh(2 √ β − ) −
116 ( g + 3 g + g − g ) e − β + , (2.14) V ( β ± ) = g h e β + (cid:16) cosh(10 √ β − ) − cosh(6 √ β − ) (cid:17) + e β + (cid:16) − cosh(8 √ β − ) + cosh(4 √ β − ) (cid:17) + 12 e − β + − e − β + cosh(2 √ β − ) + 12 e − β + (cid:21) , (2.15) V ( β ± ) = 116 e β + h ( g + 3 g ) cosh(8 √ β − ) − g + g ) cosh(4 √ β − ) + 3 g + g i + 14 ( g + g ) e β + h − cosh(6 √ β − ) + cosh(2 √ β − ) i + 18 e − β + h (3 g + g ) cosh(4 √ β − ) + g + g i −
14 ( g + g ) e − β + cosh(2 √ β − ) + 132 ( g + 3 g ) e − β + , (2.16) V ( β ± ) = − (cid:20) e β + (cid:16) − cosh(4 √ β − ) + 1 (cid:17) + e − β + cosh(2 √ β − ) − e − β + (cid:21) . (2.17)Although the similar potential was found in [53], we ex-tend it to the case without the detailed balance condition. III. LINEAR PERTURBATIONS OF THE FLRWUNIVERSE
In this section, we shall analyze the present systemwith small anisotropies by linear perturbations of theFLRW universe. We discuss the stability of the oscil-lating FLRW universe and present a new type of non-singular solutions.
A. Oscillating Closed FLRW Universe
First we summarize the result of the closed FLRWspacetime, which metric is given by ds = − dt + a (cid:18) dr − r + r d Ω (cid:19) . (3.1)We find the Friedmann equation as12 ˙ a + U ( a ) = 0 , (3.2)where U ( a ) = 13 λ − (cid:20) − Λ3 a − g r a − g s a (cid:21) . (3.3)The coefficients g r and g s are defined by g r := 6( g + 3 g ) ,g s := 12(9 g + 3 g + g ) . (3.4)The conditions for an oscillating FLRW universe toexist were already given in [47], which are summarizedas follows:(a) Λ = 0 g r > , − g ≤ g s < . (3.5)(b) Λ = 3 /ℓ > g r > , ˜ g [1 , − )s (˜ g r ) ≤ ˜ g s ( < ≤ ˜ g [1 , (˜ g r ) , (3.6) (a) U ( a ) for Λ = 0(b) U ( a ) for Λ = 3 / U ( a ) for Λ = − / λ = 1, g r = 6, and(a) g s = − /
25, (b) g s = − /
25, and (c) g s = − / (c) Λ = − /ℓ < g r > , ˜ g [ − , − )s (˜ g r ) ≤ ˜ g s < , (3.7)where ˜ g r = g r /ℓ , ˜ g s = g s /ℓ , and ˜ g [1 , ± )s (˜ g r ) is definedby ˜ g [ ǫ, ± )s (˜ g r ) := 19 ǫ h − ǫ ˜ g r ± − ǫ ˜ g r ) / i (3.8)with ǫ = ± U ( a ) inFig. 1 for the coupling parameters which we use inour numerical analysis. For an oscillating universe, a is bounded in a finite range as a min ≤ a ≤ a max . B. Linear Stability of the FLRW Universe
As we mentioned in Introduction, a non-singularFLRW universe such as an oscillating universe should bestable against anisotropic perturbations. Otherwise sucha spacetime may not be realized in the history of the uni-verse. Hence, in this subsection we study stability of theFLRW universe against linear anisotropic perturbations.When the anisotropic part of metric vanishes, i.e. , β ± = 0, Eq. (2.10) reduces to the usual Friedmann equa-tion (3.2) for a closed universe. For the stability analysis,we expand the potential V ( a, β ± ) around the FLRW uni-verse with β ± = 0 to second order of β ± , so that V ≃
316 (9 g + 3 g + g ) −
92 (9 g − g − g − g ) (cid:0) β + β − (cid:1) , (3.9) V ≃ g (cid:0) β + β − (cid:1) , (3.10) V ≃
332 ( g + 3 g ) + 32 ( g − g ) (cid:0) β + β − (cid:1) , (3.11) V ≃ − (cid:2) − (cid:0) β + β − (cid:1)(cid:3) . (3.12)The total potential V ( a, β ± ) is thus approximated by V ( a, β ± ) ≃ U ( a ) + U ( a ) (cid:0) β + β − (cid:1) , (3.13)where U ( a ) = 316 (9 g + 3 g + g ) + 332 ( g + 3 g ) a − a + Λ64 a , (3.14) U ( a ) = −
92 (9 g − g − g − g ) + 9 g a + 32 ( g − g ) a + 38 a . (3.15)In order to ensure that the FLRW universe is stableagainst small anisotropic perturbations, we impose thecondition U ( a ) >
0. The sufficient condition for sta-bility is obtained if all coefficients in U ( a ) are positivebecause a is positive, i.e. , − g + g + 3 g + 4 g ≥ ,g ≥ ,g − g = g r − g ≥ , (3.16) ⇐⇒ g ≤ g r ,g ≥ ,g + g + g ≥ g s . (3.17) The necessary and sufficient conditions for stability areobtained by taking into account the dynamics of thebackground FLRW universe, i.e. , the time evolution ofa scale factor a . We will cover a wider range of the cou-pling parameters than the above. To show the explicit necessary and sufficient conditions, just for simplicity,we restrict our analysis to the case with g = 0, i.e. , theparity-conserved theory. In this case, U can be recast in U ( a ) = 38 (cid:20)(cid:16) a − g + g r (cid:17) + 48( g + g + g ) − g s − (cid:16) g − g r (cid:17) (cid:21) . (3.18)The stability condition is such that U ( a ) > ∀ a ∈ [ a min , a max ] (the range in which the oscillation occurs).When Λ = 0, a min and a max are given explicitly by a ≡ h g r − p g + 12 g s i a ≡ h g r + p g + 12 g s i . (3.19)We thus find that either of the following three conditionsmust be satisfied to ensure the stability: g ≤ (cid:16) g r − p g + 12 g s (cid:17) ,g + g + g ≥ g s + 1288 (24 g − g r ) × (cid:16) g r − p g + 12 g s (cid:17) , (3.20)or g ≥ (cid:16) g r + p g + 12 g s (cid:17) ,g + g + g ≥ g s + 1288 (24 g − g r ) × (cid:16) g r + p g + 12 g s (cid:17) , (3.21)or (cid:12)(cid:12)(cid:12) g − g r (cid:12)(cid:12)(cid:12) < p g + 12 g s ,g + g + g ≥ g s + 1432 (36 g − g r ) . (3.22)The above inequalities give the necessary and sufficientconditions for a stable oscillating FLRW universe in thecase of Λ = 0 and g = 0. Since in more general caseswith Λ = 0 and/or g = 0, the necessary and sufficientconditions could be obtained straightforwardly but wouldbe much more involved, we dare not list the full condi-tions in the present paper. It may be sufficient to demon-strate that the stability range of coupling parameters ex-ists without any fine tuning. C. Perturbation around a static universe
Next we provide a simple and illustrative example inwhich even small anisotropies β ± can bring in a possiblyinteresting cosmological dynamics.Let us consider the case with g + 3 g >
0, ( g +3 g ) + 4(9 g + 3 g + g ) = 0, and Λ = 0, so that wefind a static FLRW universe with the constant scale fac-tor a = a S := √ g + 3 g . We then add small anisotropicperturbations β ± to this static background. The basicequations governing the system are given by H = 23 λ − (cid:20) ( ˙ β + ˙ β − ) + 643 a V ( a, β ± ) (cid:21) , (3.23)¨ β ± + 323 a ∂V∂β ± = 0 , (3.24)where V ( a, β ± ) ≃ U ( a ) + U ( a ) (cid:0) β + β − (cid:1) . (3.25)Integrating Eqs. (3.23) and (3.24), we find conservedanisotropic energies E β ± defined by E β ± := 12 (cid:20) ˙ β ± + 643 a S U ( a S ) β ± (cid:21) . (3.26)Using the constants E β ± we obtain the equation for thescale factor a as12 ˙ a + 4(3 λ − a S ( a − a S ) = 2 a S λ − (cid:0) E β + + E β − (cid:1) . (3.27)This equation gives an oscillating solution for a with thefrequency ω a := 8(3 λ − a S . (3.28)Similarly, β ± also oscillate with the frequencies ω β ± := 64 U ( a S )3 a S . (3.29)The ratio of two frequencies ω a and ω β ± is given by ω β ± ω a = s λ − U ( a S )3 a S . Those frequencies give the typical values of the presentoscillating system. They and their ratio are fixed onlyby the coupling parameters ( g i ’s) because a S and U ( a S )are given by them.The above perturbative analysis around a static uni-verse shows that an oscillating solution newly appears inthe presence of the small anisotropies β ± . As the argu-ment here is based on perturbations | β ± | ≪
1, one maywonder whether or not there exist similar oscillating so-lutions with large anisotropies. We are going to performnumerical calculations to explore the anisotropic cosmo-logical dynamics arising from more general setups beyondperturbations.
IV. ANISOTROPIC OSCILLATING UNIVERSEIN HO ˇRAVA-LIFSHITZ GRAVITY
In the previous section, we have shown that an os-cillating FLRW solution in HL gravity is stable againstsmall anisotropies, | β ± | ≪
1, for a wide range of thecoupling parameters. We now proceed to investigate therich variety of the dynamics of oscillating universes in thecontext of Bianchi IX spacetime, extending the analysisto the case with large anisotropies.If β ± are not small, the previous perturbative approachis no longer valid. To extend the analysis to include thecase with large anisotropies, we employ a numerical ap-proach and solve the governing equations without anyperturbative expansion. With this, we intend to uncoverthe rich variety of anisotropic cosmology and clarify theresultant fate of the universe.We have already given the basic equations for theBianchi IX universe in HL gravity. It will be convenientto rewrite the equations as H = 23(3 λ − (cid:20) σ + 64 a V ( a, β ± ) + 8 Ca (cid:21) , (4.1)˙ H + 3 H = 83(3 λ − (cid:20) a ∂V∂a + 3 Ca (cid:21) , (4.2)˙ β ± = σ ± (4.3)˙ σ ± + 3 Hσ ± + 323 a ∂V∂β ± = 0 , (4.4)where σ := 12 σ αβ σ αβ = 3 (cid:0) σ + σ − (cid:1) . (4.5)( σ αβ ) = diag( σ + + √ σ − , σ + − √ σ − , − σ + ) is the sheartensor of a timelike normal vector perpendicular to thehomogeneous three-space, and σ is its magnitude. It maybe convenient to introduce the dimensionless shear byΣ ± = σ ± H and Σ = σH , (4.6)which measure the relative anisotropies to the expansionrate H . We also introduce the phase variable ϕ definedby ϕ := arctan (cid:18) σ − σ + (cid:19) , (4.7)which parameterizes the direction of the anisotropic ex-pansion.We have the five first-order evolution equations for H , β ± and σ ± , i.e. , Eqs. (4.2), (4.3)and (4.4), supplementedwith one constraint (4.1). We have to set up the initialvalues for five of the six variables, a, H, β ± , Σ and ϕ .The other one is fixed by the constraint equation. Sincewe are interested in how the cosmological dynamics isaltered by the introduction of anisotropies, we start withthe isotropic oscillating universe by setting β ± , to vanishand a to be a local minimum a M of U ( a ), with arbitraryshear (Σ and ϕ ) at the initial moment. So we shallgive the initial data for a , β ± , , Σ and ϕ , and thendetermine H (or ˙ a ) by the constraint (4.1).Without any loss of generality, we can analyze only therange of 0 ≤ ϕ ≤ π/ π/ V ( a, β ± ). Since U ( a M ) isnegative (therefore V ( a M , β ± = 0) is positive) for theoscillating FLRW universe, the possible range of initialshear Σ is limited from Eq.(4.2) as0 ≤ Σ < Σ := 3(3 λ − . (4.8) As we discussed in the previous section, the FLRWuniverse is stable against small anisotropic perturbationsif U ( a ) defined by Eq. (3.18) is positive. However onemay suspect that it becomes unstable when anisotropyis large. For stability against large anisotropy, we haveone natural indicator, which is the potential V ( a, β ± ). Ifthe potential is unbounded from below for large | β ± | , weexpect that if initially large | β ± | will diverge in time andthe universe evolves into a singularity.From Eq. (2.14), we find that the potential is boundedfrom below, if and only if one of the following conditionsis satisfied:(i) g >
18 ( g + 3 g + g ) , (4.9)(ii) g ≥ , g + 7 g + 5 g ≥ , and g = 18 ( g + 3 g + g ) with g + (5 g + 7 g + 5 g ) = 0 , (4.10)(iii) 3 g + g ≥ , g ≥ , g = 0 , g = −
15 (7 g − g ) , and g = 15 g with (3 g + g ) + g = 0 , (4.11)(iv) g ≥ , g ≤ , g = − g , g = − g , and g = g = g = 0 . (4.12)We then classify the potential V into four types: SS,US, SU, and UU, where the first S (stable) or U (un-stable) denotes the stability against small perturbationsaround FLRW spacetime, while the second S (stable) orU (unstable) corresponds to the stability against largeanisotropies. Since a cosmological constant will alsochange the fate of the universe, we shall discuss eighttypes of cosmological models; Models I-SS, I-SU, I-US,I-UU, II-SS, II-SU, II-US, and II-UU, depending on thesign of Λ (I for Λ ≤ >
0) and the potentialtypes.Note that a singularity, of course, may appear evenfor small anisotropies because the Bianchi IX spacetimeincludes a closed FLRW model.We have performed numerical calculations for all pos-sible models and various initial data. Now we show our
Model Λ
V g g g g FiguresI-SS 0 SS 1 0 −
225 3100
Figs. 2-9I-SU 0 SU 1 − Figs. 10-12II-SS
SS 1 − Figs. 13-17TABLE I: The values of nontrivial coupling parameters g i ’sand a cosmological constant Λ, which are used in our numer-ical analysis. We also choose λ = 1, g = g = g = 0. Thetypes (SS and SU) of the potential V are described in thetext. numerical results for each model. We find several typesof fates of the universe depending on the magnitude ofanisotropy, which we shall describe one by one.In Table I, we list up the values of parameters for whichwe present the figures in this paper. A. Model I-SS ( Λ ≤ and Type-SS potential) First we discuss Model I-SS, in which Λ ≤ V is the SS type. Since the universe is closed,there are two fates: an eternal oscillation or a big crunch.Depending on the strength of initial anisotropy, we findthe following three types of histories of the universe.(A) Anisotropic oscillation : (small anisotropy)Since we have an oscillating universe for the FLRWspacetime, we find an eternally oscillating non-singular solution if Σ is sufficiently small (Σ < Σ / a is regularly oscillating with timejust as the FLRW solution with the same initialscale factor a , which is shown by the dotted redcurves as a reference.This oscillating solution shows only small deviationfrom the isotropic FLRW universe. The scale factor a (and then the volume) oscillates very regularly.Its orbit in the phase space shows an ellipse (a crosssection of a torus) (see Fig. 2(b)). The radiusis slightly larger than that of the FLRW universebecause of the existence of shear (see Eq. (4.1)) (a) the time evolution of scale factor a (b) the phase space of ( a, ˙ a )FIG. 2: The time evolution of non-singular oscillating uni-verse. The solid blue line and dashed red line representBianchi IX universe and FLRW universe with the same initial a , respectively. We show the scale factor a in (a) and theorbit of a in the phase space in (b), respectively. We set theinitial values as a = 2 √ /
5, ˙ a = 0 . = 0 . ϕ = 2 π/ The orbit of the anisotropy ( β + , β − ) is depicted inFig. 3(a). The anisotropic variables β ± are trappedaround the origin of ( β + , β − )-space by the poten-tial wall. It looks complicated but definitely pe-riodic. The shear is also regularly oscillating asshown in Fig. 3(b), but the oscillation period ismuch shorter than that of the scale factor a . Theoscillation amplitude of the shear σ is then modu- (a) the orbit of ( β + , β − )(b) the evolution of the shear σ FIG. 3: The orbit of anisotropy ( β + , β − ) and the time evolu-tion of the shear σ of the oscillating universe given in Fig.2. lated by a -oscillation. We can estimate those oscil-lation frequencies from the result in § III C. Usingthe coupling parameters of the present model (seeTable I), we find ω a = 2 and ω β ± = 2 √
10 fromEqs. (3.28) and (3.29), which are almost the sameas the frequencies in Figs 2(a) and 3(b).The resultant universe is regarded as an isotropicspacetime with small anisotropic perturbations.The anisotropic oscillation may continue eternally.(B)
Big crunch after oscillations : (large anisotropy)If Σ is large as Σ / < Σ < Σ , an ini-tially oscillating universe eventually collapses into abig crunch ( a = 0) after many oscillations becauseof increase of the anisotropy. A typical example ofsuch a singular universe is shown in Fig. 4. (a) scale factor a (b) phase space ( a, ˙ a )FIG. 4: The time evolution of the unstable oscillating BianchiIX universe. The dashed red line shows the FLRW spacetime.We set the initial values as a = 2 √ /
5, ˙ a = 0 . =2 . ϕ = 2 π/ The oscillation period is almost the same as that inFig. 2(a). As shown in Fig. 4 (b), the orbit of thescale factor a in the phase space is initially almostan ellipse (a cross section of a torus), but its “ra-dius” gradually increases because of the increasinganisotropy, and a finally evolves into a big crunchsingularity (( a, ˙ a ) = (0 , −∞ )).The behaviour of anisotropy is shown in Fig. 5(a),which shows that the orbit of ( β + , β − ) is trappedand reflected many times by the potential wall. Theshear σ is initially oscillating and eventually di-verges at a big crunch as shown in Fig. 5(b). Before (a) the orbit of ( β + , β − )(b) the evolution of shear σ FIG. 5: The time evolution of the shear of the unstable oscil-lating universe given in Fig. 4. this divergence, we can see the increase of σ , whichleads the leave from the oscillating phase. Then theuniverse eventually evolves into a singularity withfinite values of β ± . Note that the relative shear Σis finite at the end, which means that the shear isnot responsible for the singularity.In Fig. 6, we show the curvature invariant K ij K ij = 3 H + 2 σ , (4.13)which really diverges at a big crunch singularity.The universe evolves into a big crunch after manyoscillations. FIG. 6: The time evolution of the extrinsic curvature square K ij K ij of the unstable oscillating universe given in Fig. 4.For the Bianchi IX universe (solid blue line), K ij K ij oscil-lates in the beginning, but it eventually diverges at the endof the evolution, while it just oscillates periodically for theFLRW universe. (C) From big bang to big crunch :(near maximally large anisotropy)Another type of singular solution is found for theextremely large initial anisotropy (Σ ∼ Σ ).In the case of the closed FLRW universe in GR, the spacetime starts from a big bang and ends up with abig crunch. Bianchi IX universe in GR also evolvesfrom zero volume (a big bang) to zero volume (a bigcrunch) through a finite maximum volume. Henceeven for the case with the oscillating FLRW uni-verse, if we add a sufficiently large anisotropy, wemay expect such a non-oscillating simple evolution.We show one example. As shown in Fig. 7, thescale factor evolves from an initial finite value a toa singularity ( a = 0), which is called a big crunch.If we calculate the time reversal one from the sameinitial data, we will find a singularity ( a = 0), whichis called a big bang. As a result, this universe startsfrom a big bang and ends up with a big crunch.There is no oscillation in the evolution of a just asthe closed FLRW universe in GR. FIG. 7: The time evolution of the scale factor a of the BianchiIX universe with the potential given in Fig. 2.6 is shownby the solid blue line. The dashed red line represents theoscillating FLRW universe. We set set the initial values as a = 2 √ /
5, ˙ a = 2 . = 0 . (= 2 . ϕ = π/ As for the anisotropy, as shown in Fig. 8, the orbitof β ± is oscillating around the origin and reflectingat the potential wall until formation of a singularity. β ± is finite even at a big crunch.The shear σ is also oscillating, but the frequencyis not so regular compared with the previous twocases (A) and (B).The shear diverges at a big crunch ( a = 0), whichis really singular because the extrinsic curvaturesquare K ij K ij also diverges there as shown in Fig.9. The behaviour of K ij K ij is very similar to thatof the shear square. However the relative shearΣ does not diverge at a big crunch, which meansthat the singularity is similar to that of the FLRWuniverse. The shear does not dominate in the dy-namics (Compare it with next example (C) ′ ).In the case of a negative cosmological constant(Λ < (a) anisotropy ( β + , β − )(b) shear square σ FIG. 8: (a) The orbit of anisotropy ( β + , β − ) and (b) the timeevolution of the shear square σ of the universe given in Fig.7.FIG. 9: The time evolution of the extrinsic curvature square K ij K ij for the solution shown in Fig. 7. The solid blue lineand dashed red line represent Bianchi IX universe and FLRWuniverse, respectively. It diverges at the end of evolution,which is a singularity. B. Model I-SU ( Λ ≤ and Type-SU potential) When the potential V is unbounded from be-low, the universe may be unstable against largeanisotropic perturbations (see Fig. 10). We alsofind three types of histories of the universe as (A),(B), and (C) in Model I-SS. The difference fromModel I-SS appears when anisotropy gets large.That is, the anisotropy β ± diverges when a sin-gularity appears. We show one example with nearmaximally large initial anisotropy.(C) ′ From big bang to big crunch :(near maximally large anisotropy)In this case, just as the history (C) of Model I-SS,the Bianchi IX universe expands from zero volumeto zero volume without oscillation (Fig. 11).
FIG. 10: The unstable potential V ( a, β ± ) against largeanisotropic perturbations is shown for a = 1.FIG. 11: The time evolution of the scale factor a of theBianchi IX universe is shown by the solid blue line. Thedashed red line represents the oscillating FLRW universe.We set set the initial values as a = 2 √ /
5, ˙ a = 6 . = 2 . ϕ = π/ The difference appears in the behaviour of β ± ,which diverges at a big crunch. The orbit of β ± initially oscillates around the origin and reflects onthe potential wall, but it eventually evolves to in-finity over the potential hill because the potentialis not bounded from below (see Fig. 12(a)). Wealso show the time evolution of the shear σ , whichdiverges at the end of the universe. The extrin-sic curvature square K ij K ij also diverges there,which means it is really a singularity.This singularity is different from one appeared inthe history (B) or (C). To show it, we depict thetime evolution of the relative shear Σ, which di-verges at a big crunch. It means that the shearbecomes dominant at the end. The increase of theanisotropic shear is responsible for the formationof a singularity. Note that Σ diverges also in themiddle of the evolution but its divergence appearsbecause of H = 0.At a big bang, which appears in the time reversalone, we suspect that the shear diverges but β ± isfinite just as the beginning of Bianchi IX universein GR.1 (a) anisotropy ( β + , β − )(b) shear square σ FIG. 12: (a) The orbit of anisotropy ( β + , β − ) and (b) thetime evolution of the shear σ of the universe given in Fig.11. .FIG. 13: The time evolution of the relative shear square Σ of the universe shown in Fig. 11 C. Model II-SS ( Λ > and Type-SS potential) Next we discuss the case of Λ >
0. In this case, we findanother fate of the universe, which is an exponentiallyexpanding universe by a positive cosmological constant.If the effect of anisotropy is smaller than the contri-bution of a cosmological constant, the universe will beisotropized. The asymptotic equation for a is given by¨ a ≈ − a a + 2Λ3 λ − a , (4.14)if we neglect the anisotropic terms in Eq. (2.11), find-ing an exponentially expanding FLRW spacetime, i.e. ,de Sitter spacetime; a ≈ e H t (4.15)with H = s λ − . (4.16) Along with the cosmic expansion the potential is flat-tened to be negligible in the evolution equation for theanisotropy because a increases rapidly, so that we find˙ σ ± ≈ − H σ ± , (4.17)from Eq. (3.24). This implies that σ ± → i.e. , locally de Sitter spacetime. Note that this does notmean that it is globally de Sitter spacetime because theasymptotic values of β ± do not vanish. However thespacetime is exponentially expanding, and the observ-able region such as a horizon scale becomes effectivelyisotropic. Hence we shall still call this asymptotic space-time de Sitter universe.As a result, we have three fates of the universe in thecase with Λ >
0: oscillating universe, a big crunch, andde Sitter expanding universe. We then find five types ofhistories of the universe: two new types with asymptoti-cally de Sitter universe in addition to the previous threetypes (A), (B) and (C) discussed in IV A.Two new types are the similar to the histories (B) and(C), but different from those in their final states, i.e. , (D)de Sitter expansion after oscillation, and (E) de Sitterexpansion from a big bang.For small anisotropy (Σ < ∼ Σ / / < Σ < Σ ), the universe is either Type(B) or Type (D). When the initial anisotropy is extremelylarge enough, i.e. , Σ < ∼ Σ , Type (C) or Type (E)is obtained. We shall describe new types (D) and (E)below.(D) de Sitter expansion after oscillation :(large anisotropy)We show one example in Fig. 14. FIG. 14: The time evolution of the scale factor a of the uni-verse with Λ >
0. The solid blue line and dashed red line rep-resent Bianchi IX universe and FLRW universe, respectively.We choose the coupling constants as g = 1, g = − / g = 1 / g = g = g = g = 0, Λ = 3 /
10, and λ = 1, andset the initial values as a = 1 . a = 0 . = 2 . ϕ = π/ If Σ is as large as Σ / < Σ < Σ ,an initially oscillating universe eventually evolves2 (a) σ (b) β + (c) β − FIG. 15: The time evolution of the shear square σ andanisotropies β ± of the universe shown in Fig. 14. The shearoscillates initially, but the universe suddenly leaves the oscil-lating phase to de Sitter expanding phase, which drops theshear to zero rapidly. The initially oscillating β ± finally set-tles to finite values after small bump. into an exponentially expanding de Sitter universebecause of a cosmological constant.The initially oscillating universe leaves the oscilla-tion phase when the anisotropy increases beyondsome critical value. We also show the evolution ofthe shear σ in Fig.15. We find that it is oscillat-ing regularly for two-third of the whole period, buteventually increases. Then the universe leaves theoscillating phase and evolves into de Sitter phase.Because of rapid expansion of the universe, theshear vanishes soon [61].We also show the time evolution of the anisotropy( β ± ) in Fig. 15(b),(c). The initially oscillatinganisotropy increases as a burst and then decreasesto a small finite constant.How the universe choose its fate ((B) or (D)) isas follows: If the spacetime is expanding when itleaves from the oscillating phase, it evolves into deSitter phase (D), while if it is contracting, it col- lapses to a big crunch (B).(E) de Sitter expansion from big bang :(near maximally large anisotropy)If Σ is close to the maximum value (Σ = 3)and the universe is initially expanding ( ˙ a > FIG. 16: The time evolution of the scale factor a of the uni-verse with Λ > On the other hand, if the universe is initially con-tracting ( ˙ a < i.e. , from a big bang, which appears inthe time reversal one, to a big crunch without os-cillation. FIG. 17: The time evolution of the shear σ of the universeshown in Fig. 16. Initially oscillating shears drops to zeroafter de Sitter expansion starts. D. Model II-SU ( Λ > and Type-SU potential) In this case, we also find the similar histories ofthe universe to Types (A), (B), (C), (D) and (E),depending on initial anisotropies. The differencesbetween Models II-SS and II-SU are qualitativelythe same as those between Models I-SS and I-SU.3Only one difference from Model I is that there existsde Sitter phase as the fate of the universe becauseof a positive cosmological constant.
E. Models with the unstable potential againstsmall perturbations around FLRW spacetime
In the previous four subsections, we discuss thecosmological models with the stable potentialagainst small perturbations around FLRW space-time. When the potential is unstable against smallperturbations around FLRW spacetime, we also find qualitatively similar results. The main differ-ence is that oscillations around the FLRW space-time never happen. Even if the universe starts fromnear FLRW spacetime, it evolves into spacetimewith large anisotropy because the FLRW space-time is unstable. As a result, in the case of Type-UU potential, the universe collapses to a singularityfor Model I-UU. No oscillating phase is found. IfΛ > i.e. , for Model II-UU, some universe col-lapses to a singularity without oscillations, and theother one evolves into de Sitter expanding universe,depending on initial conditions. F. Dependence of anisotropy on the date of the universe
In Table II, we summarize the fate of the universe. We assume the coupling parameters by which there existsan oscillating FLRW universe. For Models I-SS and II-SS, we find an oscillating FLRW universe with anisotropy inthe case of small initial anisotropy. When we increase the strength of anisotropy, the spacetime leaves the initiallyoscillating phase and eventually evolves into a singularity or de Sitter spacetime. If the initial anisotropy is near themaximum value, the oscillating phase disappears and a simply expanding and contracting universe is found just as aclosed universe in GR for Λ ≤
0. When Λ >
0, an initially expanding universe evolves into de Sitter spacetime, whilean initially contracting universe evolves into a big crunch.
Model cosmological potential Σ constant V small ✄ large ✄ near maximally largeI-SS SS (A) OSC ✄ (B) OSC → SING1 ✄ (C) SING1I-US Λ ≤ ✄ (B) OSC → SING1 ✄ (C) SING1I-SU SU (A) OSC ✄ (C) ′ SING2I-UU UU (C) ′ SING2II-SS SS (A) OSC ✄ (B) · (D) OSC → deS/SING1 ✄ (E) deS [(C) SING1]II-US Λ > ✄ (B) · (D) OSC → deS/SING1 ✄ (E) deS [(C) SING1]II-SU SU (A) OSC ✄ (E) ′ deS [(C) ′ SING2]II-UU UU (E) ′ deS [(C) ′ SING2]TABLE II: Classification of Bianchi IX cosmological models by a cosmological constant Λ and types of the potential. OSC,SING1, SING2 and de S represent the anisotropic oscillation, two types of big crunch singularities (one with finite anisotropyand the other with infinite anisotropy), and de Sitter spacetime, respectively. deS/SING1 means that the spacetime evolveseither de Sitter phase or a big crunch with finite anisotropy. deS [SING1 or SING2] denotes that the fate is either de Sitteruniverse if the universe is initially expanding or a big crunch if contracting.
For Models I-US and II-US, the histories of the uni-verses are similar to those in Models I-SS and II-US,respectively, although deviation from isotropy becomeslarge even for initially small anisotropy.In the cases of Models I-SU and II-SU, unless an initialanisotropy is small, an initially expanding universe turnsto contract and collapses into a big crunch for Λ ≤ >
0. Thissingularity at a big crunch is different from that in Mod-els I-SS,-US, and II-SS, US. The anisotropy β ± for thepresent model diverges, while that for the other cases is finite even at a singularity.Models I-UU and II-UU are not so interesting. Thereis no oscillating phase. If Λ ≤
0, the spacetime is simplya spacetime evolving from a big bang to a big crunch.For Λ >
0, the initially expanding universe evolves intode Sitter spacetime, while the contracting one collapsesinto a singularity.One may wonder whether the initial anisotropy classi-fies the fate of the universe. Are there any critical valuesof the initial anisotropies for their transitions in Table II? We understand naively that such a transition occurs4as the anisotropy increases.We find there exists a critical value for the transition(B) to (C) or (D) to (E), which is about (0 . − . × Σ . For the transition from (A) to (B) (or (D)), it isnot so clear whether there exists a critical value of Σ ornot. If Σ is sufficiently small, we find the history (A),while when it is large, we find the history (B) or (D).However, because we analyze the system numerically, weare not sure whether the model with small anisotropyoscillates forever or will turn to collapse long after. Ifthe latter case is true, the case with small anisotropy isclassified into the history (B) or (D). So there may notbe exactly the history (A) except for the exact FLRWspacetime.More interesting fact is found in the history (B) · (D) forΛ >
0. In order to study how the fate of the universe de-pends on the initial data, we solve the basic equations inModel II-SS assuming various initial values of anisotropy(Σ and ϕ ). We summarize the results in Fig. 18. Aswe see from Fig. 18, the spacetime with ˙ a > (a) ˙ a > a < and ϕ . We judge the fate of the uni-verse at t = 100 t PL ( t PL : the Planck time). The histories (A),(B), (C), (D) and (E) are represented by a filled green circle,yellow triangle, red cross, purple diamond and blue square,respectively. The empty yellow triangle and purple diamondare classified to the histories (B) and (D), respectively, butthose oscillating periods are longer than 100 t PL . We have set a = 1 . a is fixed by the constraint equation [(a) ˙ a > a < ing from near maximum anisotropy evolves into de Sitteruniverse (the history (E)), while it collapses into a sin-gularity if ˙ a < , wefind an oscillating universe (the history (A)). If Σ isbetween the above two cases, however, the fate of theuniverse is not so simple. The history of such a universeis classified either (B) OSC → SING1 or (D) OSC → deS.However such a history does not shift monotonically from(B) to (D) as the initial anisotropy increases. How theuniverse choose its fate ((B) or (D)) as follows: If thespacetime is expanding when it leaves from the oscillat-ing phase, it evolves into de Sitter phase (D), while if itis contracting, it collapses to a big crunch (B). As a re-sult, the fate of the universe is sensitively dependent oninitial conditions. If one changes the initial conditions,the fate changes drastically. It is because the presentsystem is non-integrable. Such a property is found in adynamical system with chaos. Since our model is BianchiIX, which shows chaotic behaviour near singularity inGR[55–57], we understand why we find such a compli-cated basin structure of the fate in Fig. 18, which can befractal[57]. V. SUMMARY AND REMARKS
We have explored a singularity avoidance in a vacuumBianchi IX universe in HL gravity. We have studied anoscillating cosmological solution with anisotropy. In thecase of small anisotropy ( | β ± | ≪ V . We find five types ofthe histories of the universe: (A) an oscillating universewith anisotropy, (B) a big crunch after oscillations, (C)from a big bang to a big crunch, (D) de Sitter expansionafter oscillations and (E) from a big bang to de Sitterexpansion, as summarized in Table II.The stable oscillating universe (A) is found if initialanisotropy is small in the case that the coupling param-eters ( g i ′ s) satisfy the stability condition. When initialanisotropy is large, the oscillating universe evolves into asingular big crunch (B) for Λ ≤
0. In the case of Λ > , but the present system shows sensitivedependence of initial conditions just as one of the typicalproperties of chaos. The anisotropic bounce universe isalso obtained for the model satisfied the stability condi-tion if initial anisotropy is small.Since we adopt the unit of M PL = 1, the oscillation pe-riod and oscillation amplitude are the Planck scale, unlessthe coupling constants are unnaturally large. Hence in5order to obtain a macroscopic universe, we need a posi-tive cosmological constant Λ >
0, which provides us a deSitter expanding phase.In a more realistic situation, this cosmological constantshould be replaced by a potential V φ of an inflaton scalarfield φ . Reheating after inflation may give an initial stateof a macroscopic big bang universe.When we include a scalar field, however, we have totake into account modification of a scalar field action inthe UV limit similar to HL gravity action. The action S φ may be given by S φ := Z dtd xN √ g ˙ φ N − φ O φ − V φ ( φ ) ! , (5.1)where O := C M ∆ + C M ∆ + C ∆ (5.2)with M being a typical mass scale and C i ′ s ( i = 1 − H = 23(3 λ − (cid:20)
3( ˙ β + ˙ β − ) + 64 a V ( a, β ± )+ 8 Ca + 12 ˙ φ + V φ ( φ ) (cid:21) , (5.3)˙ H + 3 H = 83(3 λ − (cid:20) a ∂V∂a + 3 Ca + 34 V φ ( φ ) (cid:21) , (5.4)¨ β ± + 3 H ˙ β ± + 323 a ∂V∂β ± = 0 , (5.5)¨ φ + 3 H ˙ φ + ∂V φ ∂φ = 0 . (5.6)Although the action (5.1) contains higher spatial deriva-tives, there exists no difference from the conventionalcanonical kinetic term of a scalar field for a homogeneousspacetime. We expect a usual inflationary scenario oncede Sitter exponential expansion starts. There exists re-heating after slow-roll inflation, finding a big bang uni-verse.If we have an oscillating phase before inflation, we mayexpect one interesting effect, which is modification of pri-mordial perturbations. We should stress that a classicaltransition from an oscillating phase to an inflationarystage never happens in the FLRW model (See our discus-sion in [47] for quantum transition). So anisotropy maybe important in the pre-inflationary oscillating phase. Asa result, we may find large non-Gaussian density per-turbations. To confirm our scenario, we should explorethe dynamics of a scalar field in the pre-inflationary erabecause V φ in an inflationary model may be fluctuatedby the oscillating scale factor in the pre-inflationary erawhile Λ is a constant in our present analysis.Another interesting possibility is anisotropic inflation,which is discussed in the model with higher curvatureterms[62] or with a vector field[63]. It may leave distin-guishable imprints on the primordial perturbations. Acknowledgments
TK would like to thank RESCEU, the University ofTokyo, where a large part of this work was completed.This work was supported in part by JSPS Grant-in-Aid for Research Activity Start-up No. 22840011 (TK),by JSPS Grant-in-Aids for Scientific Research FundNo.22540291 (KM) and by JSPS under the Japan-RussiaResearch Cooperative Program (KM).
Appendix A: Anisotropic bounce universe inHoˇrava-Lifshitz gravity
In the text, we have considered oscillating universesas a possible way to avoid an initial singularity. In thisappendix, we study another way to singularity avoidance, i.e. , a bounce solution [47]. For a closed FLRW universe,a bounce solution exists only only in the case of Λ > > U ( a ). The first class issuch that U (0) = −∞ , which we call Type A, and theother is such that U (0) = + ∞ , which we call Type B.We show two typical potentials in Fig. 19. For Type- (a) Type A(b) Type BFIG. 19: Typical shapes of potential U ( a ). (a) Type A: Wechoose the coupling constants as g r = − g s = 27 /
25 andΛ = 3 /
10. The bounce point in the isotopic FLRW model isgiven by a T = 3 . g r = 6, g s = − /
25 and Λ = 3 /
10. The bouncepoint, the maximum and minimum oscillating radii in theisotopic FLRW model are given by a T = 2 . a max =1 . a min = 0 . A potential, in addition to a bounce solution, we find aFLRW spacetime starting from a big bang and collapsingto a big crunch, while for Type-B potential, we have an6oscillating FLRW solution as well as a bounce spacetime.This difference gives rise to different fates of anisotropicBianchi IX universe as we will show later.When initial anisotropy is small, it is easy to findanisotropic bounce solutions because anisotropy can betreated as perturbations around the isotropic closedFLRW universe. In order for this type of stable solu-tions to exist, the condition U ( a ) > ∀ a > a T ) mustbe satisfied. The sufficient condition for stability is thesame as Eq. (3.17).If the stability condition U ( a ) > < ∼ − ), a bounce solu-tion can be found even if the condition is not satisfied.It is because the the bounce occurs before the unstablemode grows enough.In Table III, we list up the values of parameters forwhich we present the figures and tables here. U Λ g g g g Figures & TablesA − −
12 1100 3100
Figs. 18(a), 19, Table IVB − Figs. 18(b), 20, Table VTABLE III: The values of nontrivial coupling parameters g i ’sand a cosmological constant Λ, which are used in our numer-ical analysis. We also choose λ = 1, g = g = g = 0. Thetypes (A and B) of the potential U are described in the text.
1. Type-A bounce universe
First we show the results for Type-A potential inFig. 20, where we find two typical evolutions of the uni-verse.
FIG. 20: The time evolutions of the scale factor a of theuniverse with Type-A potential. The solid blue, and dashedred lines denote de Sitter universe via a bounce, and a col-lapsing universe, respectively. We have set a = 3 . ϕ = 5 π/
18, and the shear square Σ = 0 . = 1 . a is determined bythe constraint equations as ˙ a = − . a = − . We assume the coupling parameters which guaranteethe existence of the FLRW bounce universe. We theninclude anisotropy and study how anisotropy changes thefate of the universe. We find the following results: • If the shear is small, we still have a regular bouncesolution (the solid blue curve). • When the shear becomes large, the initially con-tacting universe just collapses to a singularity (thedash-dotted red line). No singularity avoidance isobtained.Hence, there seems to exist a critical value of Σ ,beyond which the spacetime collapses to a singularity.The critical values Σ for Type-A potential are listedin Table IV, which is strongly dependent on the initialscale factor a . However, the corresponding values of σ are not so much different. Hence we may concludethat the critical value is determined by the absolute valueof the shear but not by the relative value to the Hubbleparameter. ϕ a shear 0 π π π π π π Σ .
539 0 .
963 0 .
861 0 .
786 0 .
738 0 .
708 0 . . a T σ .
426 0 .
285 0 .
264 0 .
246 0 .
237 0 .
231 0 . .
225 0 .
114 0 .
102 0 .
093 0 .
087 0 .
081 0 . . a T σ .
207 0 .
144 0 .
135 0 .
129 0 .
126 0 .
120 0 . andthe corresponding absolute shear square σ for Type-Bbounce universe. for Type-A bounce universe. If the initialshear exceeds this critical value, the universe evolves into sin-gularity.
2. Type-B bounce universe
In this case, we find the following three evolutionaryhistories of the universe: • A simple bounce solution just as Type AThis is possible if the deviation from the “back-ground” isotropic universe is sufficiently small. • A big crunch solution just as Type AThe universe collapses into a singularity if Σ islarger than some critical value Σ . • A bounce solution after some oscillationsWe also find an oscillating phase before a bounce,for which Σ is very close to Σ initially. This7 FIG. 21: The time evolutions of the scale factor a of theuniverse with Type-B potential. The solid blue, dash-dottedred, and dotted green lines denote de Sitter universe via abounce, a collapsing universe, and de Sitter universe afterseveral oscillations, respectively. We have set a = 3 . ϕ = π/
9, and the shear square Σ = 0 . = 1 . = 1 . a is determined by the constraint equations as ˙ a = − . a = − . a = − . type of solution requires a fine-tuning to some de-gree, and hence in this sense the solutions are notgeneric. The typical evolution of Type-B universe is presentedin Fig. 21. The critical values Σ for Type-B bounceuniverses are listed in Table V. As Type-A bounce solu-tion, the critical value of σ does not strongly dependon the initial scale factor a , but Σ does.Near the critical anisotropy, we suspect that which fateof the universe is realized may depend sensitively on ini-tial conditions. ϕ a shear 0 π π π π π π Σ .
254 1 .
188 1 .
143 1 .
086 1 .
065 1 .
062 1 . . a T σ .
288 0 .
276 0 .
267 0 .
255 0 .
252 0 .
252 0 . .
153 0 .
135 0 .
120 0 .
114 0 .
105 0 .
105 0 . . a T σ .
150 0 .
141 0 .
132 0 .
129 0 .
123 0 .
123 0 . andthe corresponding absolute shear square σ for Type-Bbounce universe. If the initial shear exceeds this critical value,the universe evolves into singularity.[1] R. Penrose, Phys. Rev. Lett. , 57 (1965); S.W. Hawk-ing, Proc. Roy. Soc. Lond., A300 , 187 (1967); S.W.Hawking and R. Penrose,Proc. Roy. Soc. Lond.,
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