1. Casimir Energy, the Cosmological Constant and massive gravitons Remo Garattini Università di Bergamo I.N.F.N. - Sezione di Milano Cala Gonone, 12-9-2005

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).  G ijkl is the super-metric, 8G and  c is the cosmological constant  R is the scalar curvature in 3-dim.   c can be seen as an eigenvalue

4. 4 Re-writing the WDW equation Where

5. 5 Eigenvalue problem

6. 6 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

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 123

9. 9 45

10. 10 Graviton Contribution W.K.B. method and graviton contribution to the cosmological constant

11. 11 Regularization Riemann zeta function  Equivalent to the Zero Point Energy subtraction procedure of the Casimir effect

12. 12 Isolating the divergence

13. 13 Renormalization  Bare cosmological constant changed into The finite part becomes

14. 14 Renormalization Group Equation  Eliminate the dependance on  and impose   must be treated as running

15. 15 Energy Minimization (  Maximization)  At the scale     has a maximum for

16. 16 If gravitons are massive….  Pauli e Fierz (M. Fierz and W. Pauli, Proc. Roy. Soc. Lond. A 173, 211 (1939)) introduce a mass term in the gravitational action.  The term does not introduce neither ghosts nor instabilities, but…. We need to introduce the Pauli-Fierz term

17. 17 It breaks the gauge invariance of the form Gauge invariance can be recovered with the Stückelberg method E.C.G. Stückelberg (Helv. Phys. Acta 30, 209 (1957).

18. 18 No mass in General Relativity  D.G.  D.G. Boulware and S. Deser, Phys. Rev. D 12 12 3368 (1972).

19. 19 Rubakov Proposal  If we choose We recover the Pauli-Fierz term V.A. Rubakov, Lorentz-Violating Graviton Masses: getting around ghosts, low strong coupling scale and VDVZ discontinuity. hep-th/0407104.

20. 20 A very particular choice In terms of the linearized Hamiltonian

21. 21 The effective mass becomes 3333 Case

22. 22 Repeating the procedure as in the mass-less case  Case a) Case b) Case c)

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.  In the mass-less graviton case, gravity contributes with a ‘’curvature mass’’  A massive graviton (in this specific case), behaves as a cosmological constant.  3 different cases.  Case m g =m 0 (M) gives the highest contribution to  C.

24. 24 Problems  Analysis to be completed.  Beyond the W.K.B. approximation of the Lichnerowicz spectrum.  Discrete Lichnerowicz spectrum.  Trace Contribution to be included in the massive graviton case.  A CURIOUS Thing Factor