The Cosmological constant as an eigenvalue of a Sturm-Liouville problem in modified gravity theories

1
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

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.

• At the Planck era

• Recent measures

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.
• L can be seen as an eigenvalue
• Ygij 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)
• hij 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
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.
14
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

15
(No Transcript)
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