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

99

Isolating the divergence

finitediv

finitedivergent

G

21218

rmrmGdiv 4

24132

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

1515

1616

Explicit choice for f(R)

expf R A R

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.