The Effective Equations for Bianchi IX Loop Quantum Cosmology

Transcription

1 The Effective Equations for Bianchi IX Loop Quantum Cosmology Asieh Karami 1,2 with Alejandro Corichi 2,3 1 IFM-UMICH, 2 CCM-UNAM, 3 IGC-PSU Mexi Lazos 2012 Asieh Karami Bianchi IX LQC Mexi Lazos / 12

2 Introduction Introduction Bianchi models are spatially homogeneous models such that the symmetry group S acts simply and transitively on a space manifold Σ = S. The symmetry group for Bianchi IX model is generated by three spatial rotations on a 3-sphere. We identify this group with SU(2). The fiducial cell is a 3-sphere with radius a o (=2) and its volume is V o = 2π 2 a 3 o. We define l o = V 1/3 o and ϑ = l o /a o = (2π 2 ) 1/3. The fiducial frame and co-frame satisfy [ o e i, o e j ] = 2 a o o ɛ k ij o e k, d o ω k = 1 a o o ɛ k ij o ω i o ω j (1) The physical metric is given by q ab = a 2 i o ω i a o ω bi (2) Asieh Karami Bianchi IX LQC Mexi Lazos / 12

3 Introduction For diagonal Bianchi IX model A i a = c io ω i a/l o and E a i = p i o q o e a i /l 2 o where p i s in terms of scale factors a i are p i = l 2 oa j a k, and the volume is V = p 1 p 2 p 3. The nonzero Poisson brackets are given by {c i, p j } = 8πGγδ ij. The classical hamiltonian constraint in terms of the phase space variables used in loop quantum gravity, a connection A i a and a densitized triad E a i, is C H = V N [ ɛij k Ea i Eb j 16πGγ 2 q ] ( ) Fab k (1 + γ2 )Ω k ab + H matter d 3 x (3) Asieh Karami Bianchi IX LQC Mexi Lazos / 12

4 Introduction Therefore the classical Hamiltonian constraint for Bianchi IX is ( 1 p1 p 2 p1 p 3 p2 p 3 C H = 8πGγ 2 λ 2 c 1 c 2 + c 1 c 3 + c 2 c 3 p 3 p 2 p 1 [ ] p1 p 2 p2 p 3 p1 p 3 + 2ϑ c 3 + c 1 + c 2 p 3 p 1 p 2 ϑ 2 (1 + γ 2 ) [2 p3/ p3/ p3/2 3 (p 1p 2 ) 3/2 p2 p 3 p1 p 3 p1 p 2 (p 1p 3 ) 3/2 p 5/2 2 (p 2p 3 ) 3/2 ]) p 5/2 + ρv 0 1 p 5/2 3 (4) Asieh Karami Bianchi IX LQC Mexi Lazos / 12

5 Quantization Quantization To construct the quantum kinematics, we have to select a set of elementary observables such that their associated operators are unambiguous. In loop quantum gravity they are the holonomies h e defined by the connection A i a along edges e and the fluxes of the densitized triad Ei a across surfaces. For this model we choose holonomies and p i. The first term in the classical Hamiltonian constraint, ɛ ij k Ea i Eb j / q, as in loop quantum gravity, can be treated by using Thiemann s strategy. ɛ ijk E ai E bj q = i 1 2πγGµ o ɛ abc o ωctr(h i (µ) i {h (µ) 1 i, V }τ k ) (5) Asieh Karami Bianchi IX LQC Mexi Lazos / 12

6 Quantization To find an operator related to the curvature Fab k, for isotropic models and Bianchi I, one can consider a square ij in the i j plane which is spanned by two of the fiducial triads (for the closed isotropic model since triads do not commute, to define this plane we use a triad and a right invariant vector o ξi a), with each of its sides having length µ i. Therefore, Fab k is given by F k ab = 2 lim Area 0 ( h µ o ɛij k Tr ij I )o µ τ k ω i µ a i o ω j b. (6) j Since in loop quantum gravity, the area operator does not have a zero eigenvalue, one can take the limit of above equation to the point where the area is equal to the smallest eigenvalue of the area operator, λ 2 = 4 3πγlp, 2 instead of zero. Then, µ i a i = λ. We take µ i = µ il o where µ i is a dimensionless parameter and, by previous considerations, is equal to µ i = λ p i / p j p k, (i j k). Asieh Karami Bianchi IX LQC Mexi Lazos / 12

7 Quantization For Bianchi IX, we cannot use this method because the resulting operator is not almost periodic, therefore we express the connection A i a in terms of holonomies and then use the standard definition of curvature F k ab. A i 1 a = lim (h (li) h (li) 1 ) li 0 2l i To be consistent with other models, we choose l i = 2µ i The operators corresponding to the connection are given by sin ĉ i = µ i c i. (7) µ i Asieh Karami Bianchi IX LQC Mexi Lazos / 12

8 Quantization The terms related to the curvatures, Fab k and Ωk ab, contain some negative powers of p i which are not well defined operators. To solve this problem we use the same idea as Thiemann s strategy. p i (l 1)/2 pi l o = 16πGγ µ i l Tr(τ ih ( µ i) i {h ( µ i) 1 i, p i l/2 }), (8) For these three different operators we have three different curve lengths (µ, µ, µ) where µ and µ can be some arbitrary functions of p i. For simplicity we choose all of them to be equal to µ. With this choice, the eigenvalues for the operator p i 1/4 are given by J i (p 1, p 2, p 3 ) = h(v ) V c j i p 1/4 j, (9) with h(v ) = V + V c V V c, and V c = 2πγλl 2 p. (10) Asieh Karami Bianchi IX LQC Mexi Lazos / 12

9 Quantization By using these results and choosing some factor ordering, we can construct the total constraint operator. By solving the constraint equation ĈH Ψ = 0, we can obtain the physical states and the physical Hilbert space H phys. Working with full quantum theories of the models is difficul. It is shown that for isotropic models, the behavior of the effective or semiclassical equations, which are classical equations with some quantum corrections, are good approximations to the numerical quantum evolutions even near the Planck scale. Asieh Karami Bianchi IX LQC Mexi Lazos / 12

10 Effective Equation Effective Equation To obtain the effective equations, one way is calculating the expectation value of the Hamiltonian operator respect to a semiclassical state and keeping leading terms. V H eff = 8πGγ 2 λ 2 (sin µ 1c 1 sin µ 2 c 2 + sin µ 1 c 1 sin µ 3 c 3 + sin µ 2 c 2 sin µ 3 c 3 ) ( ϑ p1 p 2 + 8πGγ 2 sin µ 3 c 3 + p 2p 3 sin µ 1 c 1 + p ) 1p 3 sin µ 2 c 2 λ p 3 p 1 p 2 ϑ2 (1 + γ 2 ) 32πGγ 2 (p 1p 3 ) 3/2 p 5/2 2 (2 p3/ p3/ p3/2 3 (p 1p 2 ) 3/2 p2 p 3 p1 p 3 p1 p 2 (p 2p 3 ) 3/2 ) p 5/2 + ρv 0 1 p 5/2 3 (11) Asieh Karami Bianchi IX LQC Mexi Lazos / 12

11 H eff = V 4 A(V )h 6 (V ) 8πGVc 6 γ 2 λ 2 ) + sin µ 2 c 2 sin µ 3 c 3 Effective Equation To gain qualitative insights into the quantum effects for small volumes we add some corrections which come from the inverse triad terms. ( sin µ 1 c 1 sin µ 2 c 2 + sin µ 1 c 1 sin µ 3 c 3 + p 2 1p 2 3 sin µ 2 c 2 ) [ + ϑa(v ( )h4 (V ) 4πGVc 4 γ 2 p 2 λ 1p 2 2 sin µ 3 c 3 + p 2 2p 2 3 sin µ 1 c 1 ϑ2 (1 + γ 2 )A(V )h 4 ( [ ] (V ) 2V p p p 2 3 8πGV 4 c γ 2 (p 1 p 2 ) 4 + (p 1 p 3 ) 4 + (p 2 p 3 ) 4 ] h 6 (V ) V 6 c ) + ρv 0 where A(V ) = 1 { V/Vc V < V (V + V c V V c ) = c 2V c 1 V V c is a correction term which comes from ɛ ij k Ea i Eb j / q. Asieh Karami Bianchi IX LQC Mexi Lazos / 12

12 Effective Equation Thank You! Asieh Karami Bianchi IX LQC Mexi Lazos / 12