Title: The Cosmological constant as an eigenvalue of a Sturm-Liouville problem in modified gravity theories
1The 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
2The 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.
A factor of 10123
3Wheeler-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
4Re-writing the WDW equation
5Eigenvalue problem
Quadratic Approximation Let us consider the
3-dim. metric gij and perturb around a fixed
background, gij gSij hij
6Form of the background
N(r) ?? Lapse function b(r) ?? shape function
for example, the Ricci tensor in 3 dim. is
7Canonical 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)
8Graviton Contribution Regularization
- Zeta function regularization ?? Equivalent to
the Zero Point Energy subtraction procedure of
the Casimir effect
9Isolating the divergence
10Renormalization
- Bare cosmological constant changed into
The finite part becomes
11Renormalization Group Equation
- Eliminate the dependance on m and impose
L0 must be treated as running
12Energy Minimization (L Maximization)
L0 has a maximum for
with
13De 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.
14Extension 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
15(No Transcript)
16Explicit choice for f(R)
17De Sitter Case for a f(R) Theory
18AdS Case for a f(R) Theory
19Conclusions, 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