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Casimir Energy and the Cosmological Constant in a de Sitter space

Convegno Informale di Fisica Teorica Sestri Levante 2008. Casimir Energy and the Cosmological Constant in a de Sitter space. Remo Garattini Università di Bergamo I.N.F.N. - Sezione di Milano. The Cosmological Constant Problem.

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Casimir Energy and the Cosmological Constant in a de Sitter space

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  1. Convegno Informale di Fisica Teorica Sestri Levante 2008 Casimir Energy and the Cosmological Constant in a de Sitter space 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 10123

  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. • Lcan be seen as an eigenvalue • Y[gij] can be considered as an eigenfunction

  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, gij= gSij+ hij

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

  7. 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)

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

  9. Isolating the divergence

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

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

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

  13. 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

  14. 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

  15. Explicit choice for f(R)

  16. De Sitter Case for a f(R) Theory

  17. AdS Case for a f(R) Theory

  18. 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. • 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

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