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The Cosmological constant as an eigenvalue of a SturmLiouville problem in modified gravity theories

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Zeta function regularization Equivalent to the Zero Point Energy subtraction ... Eigenvalue Regularization with the zeta function Casimir energy graviton ... – PowerPoint PPT presentation

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Title: The Cosmological constant as an eigenvalue of a SturmLiouville 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
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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
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