Loop Quantum Cosmology: Effective Dynamics Edison Montoya Escuela de Física, Universidad Industrial de Santander, Bucaramanga, Colombia. Julio Garavito Armero Meeting, ICRA-UIS 23-25 Noviembre Edison Montoya (UIS) LQC 1 / 47 Download-> http://math.ucr.edu/home/baez/week280.html Edison Montoya (UIS) LQC 2 / 47 The implications of the Big Bounce The qualitative picture is that non-perturbative quantum geometry corrections create a ’repulsive’ force. The universe follows a classical trajectory till the matter density ρ reaches about ∼ 1% of the Planck density ρPl. When the density is near to the Planck density, the universe bounces. Spacetime fluctuations are severely constrained on both sides of the bounce. At very early times, before the bounce, the universe was as classical as ours. The ’horizon problem’ disappears. LQC calculations provide an a priori justification for using classical general relativity during inflation. Edison Montoya (UIS) LQC 3 / 47 Introduction Loop Quantum Gravity Is a non-perturbative and background independent quantization of General Relativity. Some important results of the theory are the statistical description of the black holes entropy and the discrete spectrum of the volume and area operators, which gives a picture of a discrete space-time. Loop Quantum Cosmology is a reduce model that uses the Loop Quantum Gravity techniques in order to quantize cosmological models. A significant result from this theory is the resolution of the Big Bang singularity, which is replace with a Bounce that take place near to the Planck density. Loop quantum cosmology has provided a complete description of the quantum dynamics in the case of isotropic cosmological models and singularity resolution has been shown to be generic. The theory has been generalized to anisotropic universes. The idea is to solve the quantum dynamics for these models. Edison Montoya (UIS) LQC 4 / 47 Introduction Effective Dynamics Effective Dynamics This effective theory comes from the construction of the full quantum theory and we expect that it gives some insights about the quantum dynamics of semiclassical states. The accuracy of the effective equations has been established in the isotropic cases and thus we expect that they should give an excellent approximation of the full quantum evolution for semiclassical states. We analyzed the solutions to the effective equations that come from the improved LQC dynamics of the Bianchi I, II and IX models. These models are of interest to the issue of singularity resolution in the context of the Belinskii, Khalatnikov and Lifshitz (BKL) conjecture. Edison Montoya (UIS) LQC 5 / 47 Introduction BKL and Mixmaster Belinskii, Khalatnikov and Lifshitz (BKL) Conjecture Classical universes that have the BKL behavior are universes in which the dynamics near to the big bang singularity is dominated by the time derivatives, which turn been more important than the spatial derivatives. Therefore locally approach the Bianchi IX universe. Mixmaster In Bianchi IX when the universe approaches the big bang singularity, there is an oscillatory behavior between Bianchi I solutions with Bianchi II transitions. This is known as the mixmaster behavior, studied first by Misner. Edison Montoya (UIS) LQC 6 / 47 Introduction Bianchi Models We want to study homogeneous and anisotropic universes, these kind of universes are classified as Bianchi models. We are interested in Bianchi I, which the spacetime is like M = Σ × R where Σ is a spatial 3-manifold which have a 3-dimensional group of symmetries generated by three translations. 2 2 2 2 2 2 2 2 2 ds = −N dt + a1(t) d x + a2(t) dy + a3(t) dz (1) Bianchi II, which the spacetime is like M = Σ × R where Σ is a spatial 3-manifold which have a 3-dimensional group of symmetries generated by two translations and a rotation on a null 2-plane. 2 2 2 2 2 2 2 2 2 ds = −N dt + a1(t) (dx − αz dy) + a2(t) dy + a3(t) dz (2) where α is a switch (Bianchi I α = 0, Bianchi II α = 1). Edison Montoya (UIS) LQC 7 / 47 Introduction Bianchi Models Bianchi IX, which the spacetime is like M = Σ × R where Σ is a spatial 3-manifold which have a 3-dimensional group of symmetries generated by three rotations that can be identified by the symmetry group SU(2). The Bianchi IX metric can be construct from the fiducial co-triads o 1 !a = sin β sin γ(dα)a + cos γ(dβ)a; o 2 !a = − sin β cos γ(dα)a + sin γ(dβ)a; (3) o 3 !a = cos β(dα)a + (dγ)a; i i o i i with physical co-triads !a = a (t) !a and 3-metric qab := !a!bi . The spacetime metric is gµν = −nµnν + qµν (4) Edison Montoya (UIS) LQC 8 / 47 Introduction Phase space variables We use the fiducial triads and co-triads to introduce a convenient a i parametrization of the phase space variables, Ei ; Aa given by a −1p o o a i i −1 o i Ei = pi Li V0 j qj ei and Aa = c Li !a: (5) Thus, a point in the phase space is now coordinatized by six real numbers i (pi ; c ) with Poisson brackets given by i i fc ; pj g = 8πGγ δj : (6) Additionally to the matter degrees of freedom, e.g., the scalar field φ and its momenta pφ, with Poisson bracket fφ, pφg = 1 : The relation between the new and the old variables is pi = aj ak Lj Lk ; with i 6= j 6= k 6= i, Li fiducial lengths, V0 = L1L2L3 . Edison Montoya (UIS) LQC 9 / 47 Introduction Constraints The choice of the physical triads and connections has fixed the Gauss and vector constraint and only remain the Hamiltonian constraint Z " NE aE b # i j ij k 2 i j 3 CH = p k Fab − 2(1 + γ )K[aKb] + NHmatt d x (7) V 16πG jqj k k k i j with Fab = 2@[aAb] + ij AaAb and Hmatt Hamiltonian of matter. At classical level the Hamiltonian constraint for Bianchi I and II is given by " 1 C = − p c p c + p c p c + p c p c + αp p c HBII 8πGγ2 1 1 2 2 1 1 3 3 2 2 3 3 2 3 1 # αp p 2 p2 −(1 + γ2) 2 3 + φ ≈ 0 (8) 2p1 2 2 p pφ with lapse N = V = p1p2p3 and Hmatt = 2V . Edison Montoya (UIS) LQC 10 / 47 Bianchi Models −1q pj pk Directional scale factors, ai = L . i pi 0 p0 0 0 ai 1 j pk pi Hubble parameters, Hi = = + − . ai 2 pj pk pi V 0 Expansion, θ = V = H1 + H2 + H3 . 2 2 pφ pφ Matter density, ρ = 2 = . 2V 2p1p2p3 24πGρ Density parameter, Ω = θ2 . 2 1 2 2 2 Shear, σ = 3 [(H1 − H2) + (H1 − H3) + (H2 − H3) ] . 2 3σ2 Shear parameter, Σ = 2θ2 3 (3)R 2 Curvature parameter, K = − 2θ2 , where Ω + Σ + K = 1 for the classical Bianchi. Ricci scalar R. Edison Montoya (UIS) LQC 11 / 47 Bianchi Models Intrinsic curvature, one feature of Bianchi II and IX models is that the spatial curvature is different from zero. The intrinsic spatial curvature is given by 1 (3)R = − x 2 + x 2 + x 2 − 2(x x + x x + x x ) ; (9) 2 1 2 3 1 2 1 3 2 3 where s s s p2p3 p1p3 p1p2 x1 = α1 3 ; x2 = α2 3 ; x3 = α3 3 : (10) p1 p2 p3 The values of αi are: Bianchi I, α1 = α2 = α3 = 0. Bianchi II, α1 = 1; α2 = 0; α3 = 0 or permutations. 2 1=3 Bianchi IX, α1 = α2 = α3 = l0 , with l0 = V0 . The isotropic FLRW model with k = 1, is obtained from the Bianchi IX model by setting x1 = x2 = x3. Edison Montoya (UIS) LQC 12 / 47 Introduction Quantum Dynamics The elementary functions on the classical phase space are the momenta i pi and the holonomies of the gravitational connection Aa . grav The elementary functions are promoted to operators on Hkin . The Hamiltonian constraint is written in terms of the elementary variables and promoted to operator. In the last step there are some subtleties, namely The curvature is calculated from the connection and not from the k k k i j holonomies, Fab = 2@[aAb] + ij AaAb . The connection is calculated from the holonomies X 1 (lk ) (lk ) −1 k sin(¯µk ck ) o k Aa = lim hk − (hk ) ) Aa = !a lk !2µ¯k 2lk Lk µ¯k Lk k with r r r p1 p2 p3 µ¯1 = λ ; µ¯2 = λ ; µ¯3 = λ : p2p3 p1p3 p1p2 p 2 2 The value of λ is chosen such that λ = 4 3πγ`Pl. Edison Montoya (UIS) LQC 13 / 47 Introduction Effective constraint p p p h i H = 1 2 3 sin µ¯ c sin µ¯ c + sin µ¯ c sin µ¯ c + sin µ¯ c sin µ¯ c BII 8πGγ2λ2 1 1 2 2 2 2 3 3 3 3 1 1 " # 1 α(p p )3=2 αp p 2 p2 + 2 3 µ¯ − ( + γ2) 2 3 − φ ≈ 2 p sin 1c1 1 0 8πGγ λ p1 2p1 2 p p p H = − 1 2 3 sin µ¯ c sin µ¯ c + sin µ¯ c sin µ¯ c + sin µ¯ c sin µ¯ c BIX 8πGγ2λ2 1 1 2 2 2 2 3 3 3 3 1 1 # (p p )3=2 (p p )3=2 (p p )3=2 − 1 2 µ¯ + 2 3 µ¯ + 3 1 µ¯ 2 p sin 3c3 p sin 1c1 p sin 2c2 8πGγ λ p3 p1 p2 #2(1 + γ2) p p 2 p p 2 p p 2 − ( 2 + 2 + 2) − 1 2 − 2 3 − 3 1 2 2 p1 p2 p3 32πGγ p3 p1 p2 p2 + φ ≈ 0 2 p with lapse N = V = p1p2p3 .