Casimir Energy, the Casimir Energy, the Cosmological Constant andCosmological Constant and

massivemassive gravitonsgravitons

Remo GarattiniRemo Garattini

Università di BergamoUniversità di Bergamo

I.N.F.N. - Sezione di MilanoI.N.F.N. - Sezione di Milano

Cala Gonone, 12-9-2005

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.

47110 GeVC •Recent measuresRecent measures

44710 GeVC

A factor of 10118

33

Wheeler-De Witt Equation Wheeler-De Witt Equation B. S. DeWitt, Phys. Rev.B. S. DeWitt, Phys. Rev.160160, 1113 (1967)., 1113 (1967).

GGijklijkl is the super-metric, is the super-metric, 88G and G and cc is the is the cosmological constantcosmological constant

R is the scalar curvature in 3-dim.R is the scalar curvature in 3-dim. cc can be seen as an eigenvalue can be seen as an eigenvalue

ioneigenfunctan as considered becan ijg

44

Re-writing the WDW equationRe-writing the WDW equation

Where Where Rg

G klijijkl

2

2ˆ

C

gx

55

Eigenvalue problemEigenvalue problem

66

Quadratic ApproximationQuadratic Approximation

Let us consider the 3-dim. metric Let us consider the 3-dim. metric ggijij and and perturb around a fixed background, (e.g. perturb around a fixed background, (e.g. Schwarzschild)Schwarzschild)

ggij= ij= ggSSij+ ij+ hhijij

77

Canonical DecompositionCanonical Decomposition

h is the traceh is the trace (L(Lijij is the longitudinal operator is the longitudinal operator hh

ij represents the transverse-traceless ij represents the transverse-traceless component of the perturbation component of the perturbation graviton graviton

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).

ijijijij hLhgh 3

1

88

Integration rules on Gaussian wave functionals

11

22

33

ijijij hxhxh

ij

ij

ij hxh

ix

klijij hhh 2*121 D

99

44

55

0

xhij

yxKyhxh

ijklklij ,

1010

Graviton ContributionGraviton Contribution

operator czLichnerowi modified theis 2

r)(Propagato

2:,

klij

yhxhyxK ijkl

iakl

a

jijkl xxKxxKGgxdV

ijkl ,2

1,2

4

1ˆ2

,13

W.K.B. method and graviton contribution to the cosmological constant

1111

Regularization Regularization

i

rm

ii

ii d

rmi

2

2

122

2

216,

Mm curvature

Mmr

MGrm

Mmr

MGrm

20

203

0

22

203

0

21

mass 3

3

• Riemann zeta function Equivalent to the Zero Point Energy subtraction procedure of the Casimir effect

2

12ln2ln

1

256,

2

2

2

4

rm

rm

i

ii

1212

Isolating the divergence

finitediv

finitedivergent

G

21218

rmrmGdiv 4

24132

1313

RenormalizationRenormalization

Bare cosmological constant changed intoBare cosmological constant changed into

div 0

The finite part becomes

rG

TTeff ,

8 210

1414

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

1515

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

32,

40

000

Gr

e

Mm 1

4 20

20

1616

If gravitons are massive….If gravitons are massive….

Pauli e Fierz (M. Fierz and W. Pauli, Proc. Roy. Soc. Pauli e Fierz (M. Fierz and W. Pauli, Proc. Roy. Soc. Lond. A 173, 211 (1939)) introduce a mass term Lond. A 173, 211 (1939)) introduce a mass term in the gravitational action.in the gravitational action.

The term does not introduce neither ghosts nor The term does not introduce neither ghosts nor instabilities, but….instabilities, but….

We need to introduce the Pauli-Fierz term

1717

It breaks the gauge invariance It breaks the gauge invariance of the formof the form

2hh

Gauge invariance can be recovered with the Stückelberg methodE.C.G. Stückelberg (Helv. Phys. Acta 30, 209 (1957).

1818

No mass in General RelativityNo mass in General Relativity

D.G. Boulware and S. Deser, Phys. D.G. Boulware and S. Deser, Phys. Rev. Rev. D 12D 12 3368 (1972). 3368 (1972).

1919

Rubakov ProposalRubakov Proposal

If we chooseIf we choose

We recover the Pauli-Fierz term

V.A. Rubakov, Lorentz-Violating Graviton Masses: getting around ghosts, low strong coupling scale and VDVZ discontinuity. hep-th/0407104.

0 0 224

23

22

21

20 mmmmmm

iijjii

ijij

ii

m hhmhhmhhmhhmhhmgxdS 0024

23

220

02100

0020

4 228

1

2020

A very particular choice

ijij

gijij

m hhmgxdhhmgxdS 2422

4

8

1

8

1

In terms of the linearized Hamiltonian

ijijg

m hhgxNdm

H 32

8

2121

The effective mass becomesThe effective mass becomes

Mmmr

MGmrm

Mmmr

MGmrm

gg

gg

20

23

0

222

20

23

0

221

3

3

3 Case3 Case

Mmmg20

2 a) Case

Mmmg20

2 b) Case

Mmmg20

2 c) Case

2222

Repeating the procedure as in the Repeating the procedure as in the mass-less casemass-less case

Case a)Case a)

• Case b)

• Case c)

2323

ConclusionsConclusions 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 (infinites).Variational Approach to the eigenvalue equation (infinites). Eigenvalue Regularization with the Riemann zeta function Eigenvalue Regularization with the Riemann 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. In the mass-less graviton case, gravity contributes with a In the mass-less graviton case, gravity contributes with a

‘’‘’curvature masscurvature mass’’’’ A massive graviton (in this specific case), behaves as a A massive graviton (in this specific case), behaves as a

cosmological constant.cosmological constant. 3 different cases.3 different cases. Case mCase mgg=m=m00(M) gives the highest contribution to (M) gives the highest contribution to C.C.

2424

ProblemsProblems

Analysis to be completed.Analysis to be completed. Beyond the W.K.B. approximation of the Beyond the W.K.B. approximation of the

Lichnerowicz spectrum.Lichnerowicz spectrum. Discrete Lichnerowicz spectrum.Discrete Lichnerowicz spectrum. Trace Contribution to be included in the Trace Contribution to be included in the

massive graviton case.massive graviton case. A CURIOUS Thing A CURIOUS Thing

Factor