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Review on the Generic Cosmological Solution Near the Singularity The Oscillatory Regime Cosmological Chaos Non Classical Properties Conclusions and Bibliography Review on the Generic Cosmological Solution Near the Singularity Riccardo Benini and Giovanni Montani Sapienza Universit`adi Roma ICRA & ICRANet III Stueckelberg Workshop Pescara, 17th July 2008 Riccardo Benini and Giovanni Montani Review on the Generic Cosmological Solution Near the Singularity The Oscillatory Regime Cosmological Chaos Non Classical Properties Conclusions and Bibliography Introduction The relevance of the Belinski-Khalatnikov-Lifshitz (BKL) work relies on two points 1 It provided a piecewise analytical solution of the Einstein equations 2 The corresponding generic cosmological solution exhibits a chaotic behaviour. Why believe that Universe behaviour is related to a chaotic cosmology? 1 The Standard Cosmological Model is based on the highly symmetric FRW model, and observations agree with these assumptions. 2 A correspondence between theory and data exists even for an inflationary scenario. The validity of the BKL regime must be settled down in pre-inflationary evolution, although unaccessible to present observations. Riccardo Benini and Giovanni Montani Review on the Generic Cosmological Solution Near the Singularity The Oscillatory Regime Cosmological Chaos Non Classical Properties Conclusions and Bibliography Introduction The relevance of the Mixmaster dynamics can be identified as follows 1 The oscillatory regime can describe the mechanism of transition to a classical cosmology. 2 The FRW model is backward unstable with respect to tensor perturbations. 3 The inflationary scenario offers an efficient isotropization mechanism, able to reconcile the primordial inhomogeneous Mixmaster with the local high isotropy of the sky sphere at the recombination age. 4 At Planckian scale, quantum fluctuations can be correlated at most on the causal scale, thus we should regard global symmetries as approximated toy models. Riccardo Benini and Giovanni Montani Review on the Generic Cosmological Solution Near the Singularity The Oscillatory Regime Cosmological Chaos Non Classical Properties Conclusions and Bibliography Outline 1 The Oscillatory Regime BKL analysis of the Mixmaster The Generic Cosmological solution Minisuperspace Description 2 Cosmological Chaos Chaotic properties of the Mixmaster: the Gauss Map and the Invariant Measure Lyapunov Exponent and the long discussion on Chaos Covariance Cosmological Chaos as a dimensional and matter dependent phenomenon 3 Non Classical Properties WDW Equation in Misner variables Semi-classical and quantum properties in Misner-Chitr´evariables 4 Conclusions and Bibliography Riccardo Benini and Giovanni Montani Review on the Generic Cosmological Solution Near the Singularity The Oscillatory Regime BKL analysis of the Mixmaster Cosmological Chaos The Generic Cosmological solution Non Classical Properties Minisuperspace Description Conclusions and Bibliography The Oscillatory Regime Riccardo Benini and Giovanni Montani Review on the Generic Cosmological Solution Near the Singularity The Oscillatory Regime BKL analysis of the Mixmaster Cosmological Chaos The Generic Cosmological solution Non Classical Properties Minisuperspace Description Conclusions and Bibliography Einstein Equations for a homogeneous model Form of the 3-metric 2 2 α β Synchronous frame of reference, the metric is taken diagonal ds = dt − hαβ dx dx 2 2 2 hαβ = a (t)lαlβ + b (t)mαmβ + c (t)nαnβ α, β; γ = 1; 2; 3 Einstein field equation for the Bianchi models : (_abc) 1 » “ ”2– −Rl = + λ2a4 − µb2 − νc2 = 0 'l abc 2a2b2c2 $ : (abc_ ) 1 » “ ”2– −Rm = + µ2b4 − λa2 − νc2 = 0 m abc 2a2b2c2 : (abc_) 1 » “ ”2– −Rn = + ν2c4 − λa2 − µb2 = 0 n abc 2a2b2c2 ¨a b¨ c¨ −R0 = + + = 0 0 a b c for the Bianchi VIII model: (1; 1; −1) (λ, µ, ν) specifies& the homogeneous model % for the Bianchi IX model: (1; 1; 1) Riccardo Benini and Giovanni Montani Review on the Generic Cosmological Solution Near the Singularity The Oscillatory Regime BKL analysis of the Mixmaster Cosmological Chaos The Generic Cosmological solution Non Classical Properties Minisuperspace Description Conclusions and Bibliography BKL analysis of the Bianchi type IX model This system of equations is not analytically integrable... ...but a piece-wise solution can be constructed! BKL construction of the solution 1 a ∼ t2p1 ; b ∼ t2p2 ; c ∼ t2p3 ; Assume the potential term to be negligible at a certain t∗ 2 2 2 p1 + p2 + p3 = p + p + p = 1 then the solution is Kasner like. 1 2 3 1 p3 one index is always lower than 0 (say p ) 0.8 2 1 0.6 3 p2 so that a(t) ∼ t4p1 ! 1 as t ! 0 0.4 0.2 0.2 0.4 0.6 0.8 1 this solution is unstable backward in time!! 1/u -0.2 1 1 − p 3 If a4 term is taken into account 8a2 = 2jp1jΛ <> cosh(2jp1jΛτ) # 2 2 b = b0 exp [2Λ (p2 − jp1j) τ] cosh (2jp1jΛτ) :> 2 2 Bianchi type II dynamics c = c0 exp [2Λ (p3 − jp1j) τ] cosh (2jp1jΛτ) Riccardo Benini and Giovanni Montani Review on the Generic Cosmological Solution Near the Singularity The Oscillatory Regime BKL analysis of the Mixmaster Cosmological Chaos The Generic Cosmological solution Non Classical Properties Minisuperspace Description Conclusions and Bibliography BKL map [ BKL Adv. Phys 19, 525 (1970).] The perturbation induces a transition from a Kasner epoch to another 8 >a ∼ exp [−Λp1τ] p0 p0 p0 > a ∼ t l ; b ∼ t m ; c ∼ t n <b ∼ exp [Λ (p2 + 2p1) τ] τ ! 1 : ) >c ∼ exp [Λ (p3 + 2p1) τ] 0 > abc = Λ t :t ∼ exp [Λ (1 + 2p1) τ] ANOTHER KASNER EPOCH! BKL Map 0 jp1j 0 2jp1j − p2 0 p3 − 2jp1j 0 pl = pm = − pn = Λ = (1 − 2jp1j)Λ 1 − 2jp1j 1 − 2jp1j 1 − 2jp1j The second Kasner epoch starts with a different negative Kasner index (say p2). The a4 term starts decreasing, while b4 starts increasing.... ...increasing up to induce a new \transition" to another Kasner epoch Riccardo Benini and Giovanni Montani Review on the Generic Cosmological Solution Near the Singularity The Oscillatory Regime BKL analysis of the Mixmaster Cosmological Chaos The Generic Cosmological solution Non Classical Properties Minisuperspace Description Conclusions and Bibliography Piecewise solution Kasner Epochs and Eras −u p1 (u) = 1 + u + u2 9 8 0 pl = p1(u) pl = p2 (u − 1) 1 + u = < 0 u parametrization: p2 (u) = pm = p2(u) ) pm = p1 (u − 1) 1 + u + u2 0 pn = p3(u) ; : pn = p3 (u − 1) u (1 + u) p3 (u) = 1 + u + u2 The full u-map: u0 = u − 1 for u > 2 1 u0 = for u ≤ 2 u − 1 Riccardo Benini and Giovanni Montani Review on the Generic Cosmological Solution Near the Singularity The Oscillatory Regime BKL analysis of the Mixmaster Cosmological Chaos The Generic Cosmological solution Non Classical Properties Minisuperspace Description Conclusions and Bibliography Formulation of the Generic cosmological problem BKL in the 70's derived this solution and showed how its dynamics resembles the one of Bianchi types VIII and IX. The construction can be achieved in two steps 1 firstly considering the generic solution for the individual Kasner epoch, 2 then providing a general description of the alternation of two successive epochs. Results 1 Generalized Kasner Solution (GKS) 2 A Replacement rule for the homogeneous indices Riccardo Benini and Giovanni Montani Review on the Generic Cosmological Solution Near the Singularity The Oscillatory Regime BKL analysis of the Mixmaster Cosmological Chaos The Generic Cosmological solution Non Classical Properties Minisuperspace Description Conclusions and Bibliography Generalized Kasner solution [I.M. Khalatnikov and E.M. Lifshitz.Adv. Phys. 12, 185 (1963)] ( 2 α β KL in 1963 showed that the Kasner solution dl = hαβ dx dx can be generalized to the inhomogeneous 2 2 2 hαβ = a lαlβ + b mαmβ + c nαnβ case and near the singularity a ∼ tpl ; b ∼ tpm ; c ∼ tpn γ γ γ 2 γ 2 γ 2 γ pl (x ) + pm(x ) + pn(x ) = pl (x ) + pm(x ) + pn(x ) = 1 ¨ and the frame vectors l; m; n are now arbitrary functions of the coordinates© but this solution requires an additional condition to be stable l · r ^ l = 0 and this make GKS not general (not enough arbitrary functions) Total functions = 12 (9 vector components + 3 Kasner indices) (2 Kasner relations + 3 0α eqns + 3 conditions from the Constraints = 9 invariance under three-dimensional coordinate transforma- tions + l · r ^ l = 0.) Arbitrary functions = 3 A general solution possesses 4 functions Riccardo Benini and Giovanni Montani Review on the Generic Cosmological Solution Near the Singularity The Oscillatory Regime BKL analysis of the Mixmaster Cosmological Chaos The Generic Cosmological solution Non Classical Properties Minisuperspace Description Conclusions and Bibliography The Generic Cosmological solution [BKL Adv. Phys 31, 639 (1982)] 1 1 −R0 = χ_ α + χαχβ = 0 Near the singularity, the matter energy- 0 2 α 4 β α momentum tensor in the 00- and αβ- β 1 “p β ” β components may be neglected −Rα = p @t hχα + Pα = 0 2 h β The GKS is obtained neglecting the three-dimensional Ricci tensor Pα l m n −2 l m n Constraints: Pl ; Pm ; Pn t ; Pl Pm ; Pn q q q Necessary and sufficient conditions: a k=Λ 1 ; b k=Λ 1 ; c k=Λ 1 1=k ∼ spatial distances over which the metric significantly changes. As t decreases, these conditions may be violated r k atr = ∼ 1 Λ Riccardo Benini and Giovanni Montani Review on the Generic Cosmological Solution Near the Singularity The Oscillatory Regime BKL analysis of the Mixmaster Cosmological Chaos The Generic Cosmological solution Non Classical Properties Minisuperspace Description Conclusions and Bibliography The Generic Cosmological solution [BKL Adv. Phys 31, 639 (1982).] Einstein Equations (_abc): a2 (abc_): a2 −Rl = + λ2 = 0 −Rn = − λ2 = 0 l abc 2b2c2 n abc 2b2c2 (abc_ ): a2 ¨a b¨ c¨ −Rm = − λ2 = 0 −R0 = + + = 0 m abc 2b2c2 0 a b c which differ from those of the IX type only for the quantity l · r ^ l λ(x) = l · [m × n] no longer being a constant, but a function of the space coordinates.
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