1. Spherically symmetric gravity with variable G G. Esposito, INFN, Naples (GRG18 Conference, Sydney, July 2007), with A. Bonanno, C. Rubano, P. Scudellaro

2. Main ideas and results The asymptotic safety suggested by Weinberg in 1979 seems to work: the running couplings have a finite limit at large k, and a new non-Gaussian fixed point exists. We find that the treatment of cosmological term and Newton parameter as dynamical variables, jointly with a fixed-point condition, gives consistent chances of emulating dark matter in long-range gravitational interactions, at least on galactic scales.

3. Renormalization-group approach A scale-dependent effective action is built. If this 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 parameters in the action, and is taken at a non-Gaussian fixed point of the RG-flow.

4. A new ultraviolet fixed point 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 Lauscher-Reuter in hep-th/0511260

5. Action functional

6. Spherical symmetry: 3-metric of the leaves

7. Action from Legendre transform

8. Constraints

9. Effective Hamiltonian Hamilton equations

10. First-order operator in the Hamilton equations: General form of Hamilton equations:

11. The U functions are

12. The V functions read as

13. Fixed-point condition

14. Hamiltonian constraint

15. All V functions vanish weakly

16. All V functions …

17. Nonlinear equation for G(r)

18. Novel form of the Newton parameter It holds in the singular limit of vanishing Work is in progress on the generic case

19. Approximate linear growth of G(r)

20. Large-r behaviour of radial velocities of galaxies

21. Convenient parameters

22. Equation for G with these parameters

23. Vanishing cosmological constant This is possibly more relevant than the previous case

24. Open problems (among the many) 1. Can the solution with vanishing cosmological constant, or the previous solution, reproduce the flat rotation curves of galaxies? 2. Horizon and gravitational collapse