1. Quantum Cosmology From Three Different Perspectives Giampiero EspositoGiampiero Esposito, INFN, Naples; MG11 Conference, Berlin, 23-29 July 2006, COT5 Session

2. I: Familiar formulation via functional integrals (Misner 57, Hawking 79)

3. Hartle-Hawking quantum state (Phys. Rev. D28, 2960 (1983)). Quantum state of the Universe: an Euclidean functional integral over compact four-geometries matching the boundary data on the final surface, while the initial three-surface shrinks to a point (hence called “no boundary’’ proposal).

4. II: Renormalization-group approach If the scale-dependent effective action Gamma equals the classical action at the UV cut-off scale K, one uses the RG equation to evaluate Gamma(k) for all k less than K, and then sends k to 0 and K to infinity. The continuum limit as K tends to infinity should exist after ren. finitely many param. in the action, and is taken at a non-Gaussian fixed point of the RG-flow.

5. New paths: a new ultraviolet fixed point

6. Figure caption (from Lauscher- Reuter in HEP-TH/0511260) Part of theory space of the Einstein-Hilbert truncation with its Renormalization Group flow. The arrows point in the direction of decreasing values of k. The flow is dominated by a non-Gaussian fixed point in the first quadrant and a trivial one at the origin.

7. Cosmological applications Lagrangian and Hamiltonian form of pure gravity with variable G and Lambda, in A. Bonanno, G. Esposito, C. Rubano, Class. Quantum Grav. 21, 5005 (2004). Cf. earlier analysis of RG-improved equations for self-interacting scalar fields coupled to gravity, by A. Bonanno, G. Esposito, C. Rubano, Gen. Rel. Grav. 35, 1899 (2003).

8. Power-law inflation for pure gravity A. Bonanno, G. Esposito, C. Rubano, Class. Quantum Grav. 21, 5005 (2004); Int. J. Mod. Phys. A 20, 2358 (2005).

9. An accelerating Universe without dark energy Main assumption: existence of an infrared fixed point. By linearization of the RG-flow we evaluate the critical exponents and find how the fixed point is approached. We obtain a smooth transition between FLRW cosmology and the observed accelerated expansion. A. Bonanno, G. Esposito, C. Rubano, P. Scudellaro, Class. Quantum Grav. 23, 3103 (2006).

10. III: Perturbative quantum cosmology

11. Singularity avoidance at one loop? For pure gravity, one-loop quantum cosmology in the limit of small three- geometry describes a vanishing probability of reaching the singularity at the origin (of the Euclidean 4-ball) only with diff- invariant boundary conditions, which are a particular case of the previous scheme. All other sets of boundary conditions lead instead to a divergent one-loop wave function!

12. Peculiar property of the 4-ball? We stress we do not require a vanishing one-loop wave function. We rather find it, on the Euclidean 4-ball, as a consequence of diff-invariant boundary conditions. Peculiar cancellations occur on the Euclidean 4-ball, and the spectral (also called generalized) zeta function remains regular at the origin, despite the lack of strong ellipticity.

13. One-loop recent bibliography G. Esposito, G. Fucci, A.Yu. Kamenshchik, K. Kirsten, Class. Quantum Grav. 22, 957 (2005); JHEP 0509:063 (2005); J. Phys. A 39, 6317 (2006). G. Esposito, A.Yu. Kamenshchik, G. Pollifrone, “Euclidean Quantum Gravity on Manifolds with Boundary’’, Kluwer, Fundam. Theor. Phys. 85 (1997).

14. IV: Towards brane-world quantum cosmology

15. Braneworld effective action

16. Key open problem Does braneworld quantum cosmology preserve singularity avoidance at one-loop level? A.O. Barvinsky, HEP-TH/0504205; A.O. Barvinsky, D.V. Nesterov, Phys. Rev. D73, 066012 (2006).