Remo Garattini Università di Bergamo I.N.F.N. - Sezione di Milano SM&FT 2008THE XIV WORKSHOP ON STATISTICAL MECHANICS AND NON PERTURBATIVE FIELD THEORY Bari 3-9-2008 The Cosmological constant as an eigenvalue of a Sturm-Liouville problem in modified gravity theories Remo Garattini Università di Bergamo I.N.F.N. - Sezione di Milano

The Cosmological Constant Problem R. Garattini Low Energy Quantum Gravity 20 July 2007 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. At the Planck era Recent measures A factor of 10123

Wheeler-De Witt Equation B. S. DeWitt, Phys. Rev.160, 1113 (1967). R. Garattini Low Energy Quantum Gravity 20 July 2007 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 Y[gij] can be considered as an eigenfunction

Re-writing the WDW equation Where

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

Form of the background N(r)  Lapse function b(r)  shape function for example, the Ricci tensor in 3 dim. is

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 (spin 0) (Lx)ij is the gauge part [spin 1 (transverse) + spin 0 (longitudinal)] h^ij represents the transverse-traceless component of the perturbation  graviton (spin 2)

Graviton Contribution: Regularization Zeta function regularization  Equivalent to the Zero Point Energy subtraction procedure of the Casimir effect

Isolating the divergence

The finite part becomes Renormalization Bare cosmological constant changed into The finite part becomes

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

Energy Minimization (L Maximization) At the scale m0 L0 has a maximum for with

De Sitter Case Adopting the same procedure of the Schwarzschild case with a running G instead of a running L Remark  The AdS background leads to an infinite set of solutions Not only L  the same method can be applied to the Maxwell charge, i.e. the electric (magnetic) charge [R.G. P.L.B 666 (2008), 189. arXiv: 0807.0082 [gr-qc].

Extension to f(R) Theories [S. Capozziello and R. G. , Class. Quant Extension to f(R) Theories [S. Capozziello and R.G., Class. Quant. Grav., 24, 1627 (2007)] A straightforward generalization is a f(R) theory substituting the classical Lagrangian with

Explicit choice for f(R)

De Sitter Case for a f(R) Theory

AdS Case for a f(R) Theory

Conclusions, Problems and Outlook Wheeler-De Witt Equation  Sturm-Liouville Problem. The cosmological constant is the eigenvalue. Variational Approach to the eigenvalue equation (infinites). Eigenvalue Regularization with the zeta function  Casimir energy graviton contribution to the cosmological constant. Renormalization and renormalization group equation.  Application to the Maxwell charge. Analysis to be completed. Beyond the W.K.B. approximation of the Lichnerowicz spectrum. Discrete Lichnerowicz spectrum. Introducing massive graviton. In progress, spectrum of spherically symmetric metrics