1. 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
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

2. 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/ ; N. Straumann, The history of the cosmological
constant problem gr-qc/ ; T.Padmanabhan, Phys.Rept. 380, 235 (2003),
hep-th/
At the Planck era
Recent measures
A factor of 10123

3. 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

4. Re-writing the WDW equation
Where

5. Quadratic Approximation
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. The finite part becomes
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
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: [gr-qc].

14. 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

16. Explicit choice for f(R)

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

18. AdS Case for a f(R) Theory

19. 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