1. Multigravity and Spacetime Foam Remo Garattini Università di Bergamo I.N.F.N. - Sezione di Milano IRGAC 2006  Barcelona, 15-7-2006

2. 2 The Cosmological Constant Problem  At the Planck era For a pioneering review on this problem see S. Weinberg, Rev. Mod. Phys. 61, 1 (1989). For more recent and detailed reviews see V. Sahni and A. Starobinsky, Int. J. Mod. Phys. D 9, 373 (2000), astro-ph/9904398; N. Straumann, The history of the cosmological constant problem gr-qc/0208027; T.Padmanabhan, Phys.Rept. 380, 235 (2003), hep-th/0212290. Recent measuresRecent measures A factor of 10 118

3. 3 Wheeler-De Witt Equation B. S. DeWitt, Phys. Rev.160, 1113 (1967).  can be seen as an eigenvalue  can be seen as an eigenvalue  G ijkl is the super-metric, 8G and  is the cosmological constant  R is the scalar curvature in 3-dim.

4. 4 Re-writing the WDW equation Where

5. 5 Eigenvalue problem Quadratic Approximation Let us consider the 3-dim. metric g ij and perturb around a fixed background, (e.g. Schwarzschild) g ij = g S ij + h ij

6. 6

7. 7 Canonical Decomposition  h is the trace  (L ij is the longitudinal operator  h  ij represents the transverse-traceless component of the perturbation  graviton M. Berger and D. Ebin, J. Diff. Geom.3, 379 (1969). J. W. York Jr., J. Math. Phys., 14, 4 (1973); Ann. Inst. Henri Poincaré A 21, 319 (1974).

8. 8 Integration rules on Gaussian wave functionals 12345

9. 9 Graviton Contribution W.K.B. method and graviton contribution to the cosmological constant

10. 10 Regularization Zeta function regularization  Equivalent to the Zero Point Energy subtraction procedure of the Casimir effect

11. 11 Isolating the divergence

12. 12 Renormalization  Bare cosmological constant changed into The finite part becomes

13. 13 Renormalization Group Equation  Eliminate the dependance on  and impose   must be treated as running

14. 14 Energy Minimization (  Maximization)  At the scale     has a maximum for with Not satisfying

15. 15 Motivating Multigravity 1) In a foamy spacetime, general relativity can be renormalized when a density of virtual black holes is taken under consideration coupled to N fermion fields in a 1/N expansion [L. Crane and L. Smolin, Nucl. Phys. B (1986) 714.]. [L. Crane and L. Smolin, Nucl. Phys. B (1986) 714.]. 2) When gravity is coupled to N conformally invariant scalar fields the evidence that the ground-state expectation value of the metric is flat space is false [J.B. Hartle and G.T. Horowitz, Phys. Rev. D 24, (1981) 257.]. [J.B. Hartle and G.T. Horowitz, Phys. Rev. D 24, (1981) 257.]. Merging of point 1) and 2) with N gravitational fields (instead of scalars and fermions) leads to multigravity Hope for a better Cosmological constant computation

16. 16 First Steps in Multigravity Pioneering works in 1970s known under the name strong gravity strong gravity or f-g theory (bigravity) [C.J. Isham, A. Salam, and J. Strathdee, Phys Rev. D 3, 867 (1971), A. D. Linde, Phys. Lett. B 200, 272 (1988).]

17. 17 Structure of Multigravity T.Damour and I. L. Kogan, Phys. Rev.D 66, 104024 (2002). A.D. Linde, hep-th/0211048 N massless gravitons gravitons

18. 18 Multigravity gas For each action, introduce the lapse and shift functions Choose the gauge Define the following domain No interaction Depending on the structure You are looking, You could have a ‘ideal’gas of geometries. Our specific case: Schwarzschild wormholes

19. 19  Wave functionals do not overlap Additional assumption The single eigenvalue problem turns into problem turns into

20. 20 And the total wave functional becomes The initial problem changes into 

21. 21 Further trivial assumption R. Garattini - Int. J. Mod. Phys. D 4 (2002) 635; gr-qc/0003090. N w copies of the same gravity Take the maximum

22. 22 There are arguments leading to Nevertheless, there is no Proof of this

23. 23 Conclusions  Wheeler-De Witt Equation  Sturm-Liouville Problem.  The cosmological constant is the eigenvalue.  Variational Approach to the eigenvalue equation (infinites).  Eigenvalue Regularization with the Riemann zeta function  Casimir energy graviton contribution to the cosmological constant.  Renormalization and renormalization group equation.  Generalization to multigravity.  Specific example: gas of Schwarzschild wormholes.

24. 24 Problems  Analysis to be completed.  Beyond the W.K.B. approximation of the Lichnerowicz spectrum.  Discrete Lichnerowicz spectrum.  Specific examples of interaction like the Linde bi- gravity model or Damour et al.  Possible generalization con N ‘different gravities’?!?!  Use a distribution of gravities!!