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 1.Good semi-classical limit. 2.No improvement on the classical short distance disasters like cosmological singularity. The key differences from WDW theory in LQC 1.The classical framework is constructed based on the holonomy of SU(2) connection. 2.In quantum theory, Bohr compactification of the configuration space is employed in order to construct the representation of the holonomy algebras 3.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 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 : After regularization (a) (b)

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

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

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.