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The Cosmological constant as an eigenvalue of a

Sturm-Liouville problem in modified gravity

theories

SMFT 2008THE XIV WORKSHOP ON STATISTICAL

MECHANICS AND NON PERTURBATIVE FIELD THEORY Bari

3-9-2008

- 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). 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).

- 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
- Ygij can be considered as an eigenfunction

Re-writing the WDW equation

- Where

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) - hij 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

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) TheoriesS. Capozziello and

R.G., Class. Quant. Grav., 24, 1627 (2007)

- A straightforward generalization is a f(R) theory

substituting the classical Lagrangian with

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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