Casimir Energy and the Casimir Energy and the Cosmological ConstantCosmological Constant
in a de Sitter space in a de Sitter space
Remo GarattiniRemo Garattini
Università di BergamoUniversità di Bergamo
I.N.F.N. - Sezione di MilanoI.N.F.N. - Sezione di Milano
Convegno Informale di Convegno Informale di Fisica TeoricaFisica Teorica
Sestri Levante 2008 Sestri Levante 2008
22
The Cosmological Constant The Cosmological Constant ProblemProblem
At the Planck eraAt the Planck era
For a pioneering review on this problem see S. Weinberg, Rev. Mod. Phys. For a pioneering review on this problem see S. Weinberg, Rev. Mod. Phys. 6161, 1 (1989)., 1 (1989).For more recent and detailed reviews see V. Sahni and A. Starobinsky, Int. J. Mod. Phys.For more recent and detailed reviews see V. Sahni and A. Starobinsky, Int. J. Mod. Phys.D 9D 9, 373 (2000), astro-ph/9904398; N. Straumann, , 373 (2000), astro-ph/9904398; N. Straumann, The history of the cosmologicalThe history of the cosmologicalconstant problemconstant problem gr-qc/0208027; T.Padmanabhan, Phys.Rept. gr-qc/0208027; T.Padmanabhan, Phys.Rept. 380380, 235 (2003),, 235 (2003),hep-th/0212290.hep-th/0212290.
76 410C GeV •Recent measuresRecent measures
44710 GeVC
A factor of 10123
33
Wheeler-De Witt Equation Wheeler-De Witt Equation B. S. DeWitt, Phys. Rev.B. S. DeWitt, Phys. Rev.160160, 1113 (1967)., 1113 (1967).
2 2 02
ij klijkl ij
gG R g
GGijklijkl is the super-metric, is the super-metric, 88G and G and is the cosmological constant is the cosmological constant R is the scalar curvature in 3-dim.R is the scalar curvature in 3-dim.
can be seen as an eigenvaluecan be seen as an eigenvalue [g[gijij] can be considered as an eigenfunction] can be considered as an eigenfunction
44
Re-writing the WDW equationRe-writing the WDW equation
Where Where Rg
G klijijkl
2
2ˆ
C
gx
ij ij ij ij ij ijxD g g g D g g g
55
Eigenvalue problemEigenvalue problem
3
1ij ij ij
ij ij ij
D g g d x g
V D g g g
Quadratic ApproximationQuadratic Approximation
Let us consider the 3-dim. metric Let us consider the 3-dim. metric ggijij and and perturb perturb
around a fixed background, around a fixed background, ggijij= g= gSSijij+ h+ hijij
66
Form of the background
0 1 23
1d x
V
2
2 2 2 2 2 2 2sin
1
drds N r dt r d d
b r
r
N(r) Lapse function
b(r) shape function
for example, the Ricci tensor in 3 dim. is
77
Canonical DecompositionCanonical Decomposition
ijijijij hLhgh 3
1
h is the trace (spin 0)h is the trace (spin 0) (L(Lijij is the gauge part [spin 1 (transverse) + spin 0 is the gauge part [spin 1 (transverse) + spin 0
(longitudinal)](longitudinal)] hh
ij ij represents the transverse-traceless component of represents the transverse-traceless component of the perturbation the perturbation graviton (spin 2) graviton (spin 2)
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).
The canonical decomposition is equivalent
to a gauge fixing procedure with the gauge
10
3i ij
ij ij ijh g h g h
88
Graviton Contribution: RegularizationGraviton Contribution: Regularization
2
12ln2ln
1
256,
2
2
2
4
rm
rm
i
ii
irm
ii
ii d
rmi
2
2
122
2
216,
• Zeta function regularization Equivalent to the Zero Point Energy subtraction procedure of the Casimir effect
21 2 2 3
22 2 2 3
3 ' 361
2 2
' 361
2 2
b r b r b rm r
rr r r
b r b r b rm r
rr r r
1010
RenormalizationRenormalization
Bare cosmological constant changed intoBare cosmological constant changed into
div 0
The finite part becomes
rG
TTeff ,
8 210
1111
Renormalization Group EquationRenormalization Group Equation
Eliminate the dependance on Eliminate the dependance on and impose and impose
d
rd
G
TTeff ,
8
1 0
must be treated as running
0
42
41000 ln
16,,
rmrm
Grr
1212
Energy Minimization Energy Minimization (( Maximization) Maximization)
At the scale At the scale
2
1
4ln
16,
20
204
0000 Mm
MmG
r
has
a maximum for
40
0 0 2
2G
e
e
Mm 1
4 20
20
with
2 21 03
0
2 22 03
0
3
effective mass
due to the curvature3
MGm r m M
r
MGm r m M
r
1313
De Sitter CaseDe Sitter Case
3 23
b r r R
Adopting the same procedure of the Schwarzschildcase with a running G instead of a running
220
20 0
4 2ln
8 32
Max
G Ge
Remark The AdS background leads to an infinite set of solutions
1414
Extension to f(R) TheoriesExtension to f(R) Theories[[S. Capozziello and R.G., Class. Quant. Grav., 24, 1627 (2007)]S. Capozziello and R.G., Class. Quant. Grav., 24, 1627 (2007)]
A straightforward generalization is a f(R) theory substituting A straightforward generalization is a f(R) theory substituting the classical Lagrangian withthe classical Lagrangian with
2 and defining 'cg f R V P g f R R f R L
1717
De Sitter Case for a f(R) TheoryDe Sitter Case for a f(R) Theory
0 0
3 exp 2 2 1
4 exp 2 4
A
A G
0 0
0 0
exp 2Case a)
2
3 2 1Case b)
4 4
A G
A
A G
Two Approximations
a) 1; b) 1
2 20 02
2 20 0
2
0 0
8In Case b) 0 if with
1212 1
GA
G
G
1818
AdS Case for a f(R) TheoryAdS Case for a f(R) Theory
0 0
3 exp 2 2 1
4 exp 2 4
A
A G
0 0
Case a) No solution
3 2 1Case b) -
4 4
A
A G
Two Approximations
a) 1; b) 1
2 20 02
2 20 0
2
0 0
8In Case b) 0 if with
1212 1
GA
G
G
1919
Conclusions, Problems and OutlookConclusions, Problems and Outlook
Analysis to be completed.Analysis to be completed. Beyond the W.K.B. approximation of the Lichnerowicz Beyond the W.K.B. approximation of the Lichnerowicz
spectrum.spectrum. Discrete Lichnerowicz spectrum.Discrete Lichnerowicz spectrum. Introducing massive graviton.Introducing massive graviton. In progress, spectrum of spherically symmetric metricsIn progress, spectrum of spherically symmetric metrics
Wheeler-De Witt Equation Wheeler-De Witt Equation Sturm-Liouville Problem. Sturm-Liouville Problem. The cosmological constant is the eigenvalue.The cosmological constant is the eigenvalue. Variational Approach to the eigenvalue equation Variational Approach to the eigenvalue equation
(infinites).(infinites). Eigenvalue Regularization with the zeta function Eigenvalue Regularization with the zeta function
Casimir energy graviton contribution to the Casimir energy graviton contribution to the cosmological constant.cosmological constant.
Renormalization and renormalization group equation.Renormalization and renormalization group equation.