A quick review on Loop Qunatum Cosmology

1. A quick review on Loop Qunatum Cosmology Yi Ling Center for Gravity and Relativistic Astrophysics Nanchang University, China Nov. 5, 2007 KITPC & CCAST Workshop, Beijing

2. Outlines • The framework of loop quantum cosmology 1. The classical framework 2. Quantum theory • The resolution of cosmological singularity • Effective formalism and inflation

3. gr-qc/0702030, Ashtekar • gr-qc/0304074, Ashtekar, Bojowald, Lewandowski • gr-qc/0601085, Bojowald

4. The WDW theory • Good semi-classical limit. • No improvement on the classical short distance disasters like cosmological singularity. • The key differences from WDW theory in LQC • The classical framework is constructed based on the holonomy of SU(2) connection . • In quantum theory, Bohr compactification of the configuration space is employed in order to construct the representation of the holonomy algebras • The differential equation is replaced by the difference equation.

5. x The WDW theory LQC

6. The Classical framework • A quick view on standard FRW cosmology

7. The Classical framework Conjugate momenta Where In general constraints become

8. The Classical framework Ashtekar-Sen variables: SU(2) connection Barbero-Immirzi parameter A triplet vector field with density weight one

9. The Classical framework In the present isotropic and homogeneous setting Fiducial metric: Physical metric:

10. The Classical framework

11. The Classical framework The Hamiltonian constraint in full theory: In cosmological setting, it reduces to Thus, the total Hamiltonian constraint reads as Where

12. The Quantum theory • The phase space of gravity part • The Hilbert space The almost periodic functions constitute an ortho-normal basis in

13. The Quantum theory • Almost periodic functions

14. The Quantum theory • The action of the conjugate momentum Another well-defined operator: • The eigenbras and eigenvalues of volume operator:

15. The Quantum theory • The operator is well defined unitary operator, • but fails to be continuous with respect to • There is no operator corresponding to c on the Hilbert space • The well defined fundamental operators Related to the holonomy of connection.

16. The Quantum theory • The holonomy along the segment of length • in the i-th direction

17. The Quantum theory Classical constraints in full theory : (a) (b) After regularization

18. The Quantum theory The constraint in terms of well-defined fundamental variables:

19. The resolution of cosmological singularity The physical state Big bang corresponds to the state Given initial states One may determine all

20. The resolution of cosmological singularity • Cosmological singularity Closed universe：k=1, Scale factor Originated from a big-bang

21. The resolution of cosmological singularity Only valid at classical level

22. Effective formalism and inflation The effective or “semi-classical” Friedmann equations from LQC receive corrections from the following two facts : 1. The replacement of the inverse of scale factor: 2. The holonomy corrections.

23. Effective formalism and inflation The operator corresponding to the inverse of scale factor In standard quantum mechianics:

24. Effective formalism and inflation Ambiguities at semi-classical limit: 1. The representations of SU(2) for holonomy. 2. The operator ordering. j l

25. Effective formalism and inflation In general case

26. Effective formalism and inflation Effective Friedmann euqations:

27. Effective formalism and inflation 2. The holonomy corrections

28. Effective formalism and inflation From these effective equations, the following relevant phenomena have been investigated: 1. Super-inflation and inflations due to quantum geometry. 2. The big bounce universe. 3. The cosmological perturbation theory and scale invariance . 4. The resolution of the big rip in phantom cosmology.