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Multigravity and Spacetime Foam

IRGAC 2006 Barcelona, 15-7-2006. Multigravity and Spacetime Foam. Remo Garattini Università di Bergamo I.N.F.N. - Sezione di Milano. The Cosmological Constant Problem. For a pioneering review on this problem see S. Weinberg, Rev. Mod. Phys. 61 , 1 (1989).

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Multigravity and Spacetime Foam

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

  2. The Cosmological Constant Problem 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 measures • At the Planck era A factor of 10118

  3. Wheeler-De Witt Equation B. S. DeWitt, Phys. Rev.160, 1113 (1967). • Gijkl is the super-metric, k =8pG and L is the cosmological constant • R is the scalar curvature in 3-dim. L can be seen as an eigenvalue

  4. Re-writing the WDW equation Where

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

  6. Canonical Decomposition 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). • h is the trace • (Lx)ij is the longitudinal operator • h^ij represents the transverse-traceless component of the perturbation  graviton

  7. Integration rules on Gaussian wave functionals

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

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

  10. Isolating the divergence

  11. Renormalization • Bare cosmological constant changed into The finite part becomes

  12. Renormalization Group Equation • Eliminate the dependance on m and impose L0 must be treated as running

  13. Energy Minimization (L Maximization) • At the scale m0 L0 has a maximum for with Not satisfying

  14. Motivating Multigravity • 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.]. • 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.]. 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

  15. First Steps in Multigravity Pioneering works in 1970s known under the name 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).]

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

  17. 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

  18. Additional assumption The single eigenvalue problem turns into • Wave functionals do not overlap

  19. And the total wave functional becomes The initial problem changes into S

  20. Further trivial assumptionR. Garattini - Int. J. Mod. Phys. D 4 (2002) 635; gr-qc/0003090. Nw copies of the same gravity Take the maximum

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

  22. 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.

  23. 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!!

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