MultigravityMultigravityandand

Spacetime FoamSpacetime Foam

Remo GarattiniRemo Garattini

Università di BergamoUniversità di Bergamo

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

IRGAC 2006IRGAC 2006Barcelona, 15-7-Barcelona, 15-7-20062006

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).

can be seen as an eigenvalue can be seen as an eigenvalue

can be considered as an eigenfunctionijg

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.

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, (e.g. Schwarzschild)around a fixed background, (e.g. Schwarzschild) ggijij= g= gSS

ijij+ h+ hijij

66

0 1 23

1d x

V

77

Canonical DecompositionCanonical Decomposition

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

hhij ij represents the transverse-traceless 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

44

55

ij ij ijh x h x h

ijij

ij hxh

ix

*1 2 1 2 ij ij klD h h h

0

xhij

,ij kl

ijkl

h x h yK x y

))))))))))))) )))))))))))))) )

99

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

1010

Regularization Regularization

i

rm

ii

ii d

rmi

2

2

122

2

216,

• Zeta function regularization Equivalent to the Zero Point Energy subtraction procedure of the Casimir effect

2

12ln2ln

1

256,

2

2

2

4

rm

rm

i

ii

1111

Isolating the divergence

finitediv

finitedivergent

G

21218

rmrmGdiv 4

24132

1212

RenormalizationRenormalization

Bare cosmological constant changed intoBare cosmological constant changed into

div 0

The finite part becomes

rG

TTeff ,

8 210

1313

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

1414

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 32

G

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

Not satisfying

1515

Motivating MultigravityMotivating Multigravity

1)1) In a foamy spacetime, general relativity can be renormalized when a In a foamy spacetime, general relativity can be renormalized when a density of virtual black holes is taken under consideration coupled to N density of virtual black holes is taken under consideration coupled to N fermion fields in a 1/N expansionfermion fields in a 1/N expansion

[L. Crane and L. Smolin, Nucl. Phys. B (1986) 714.]. [L. Crane and L. Smolin, Nucl. Phys. B (1986) 714.].

2)2) When gravity is coupled to N conformally invariant scalar fields the When gravity is coupled to N conformally invariant scalar fields the evidence that the ground-state expectation value of the metric is flat evidence that the ground-state expectation value of the metric is flat space is falsespace is false

[J.B. Hartle and G.T. Horowitz, Phys. Rev. D 24, (1981) 257.].[J.B. Hartle and G.T. Horowitz, Phys. Rev. D 24, (1981) 257.].

Merging of point 1) and 2) with N gravitational fields (instead of scalars and fermions) leads to

multigravity

Hope for a betterCosmological constant

computation

1616

First Steps in MultigravityFirst Steps in Multigravity

Pioneering works in 1970s known under the name

strong gravitystrong gravity or

f-g theory (bigravity)[C.J. Isham, A. Salam, and J. Strathdee, Phys Rev. D 3, 867 (1971), A.

D. Linde, Phys. Lett. B 200, 272 (1988).]

1717

Structure of MultigravityStructure of Multigravity T.Damour and I. L. Kogan, Phys. Rev.T.Damour and I. L. Kogan, Phys. Rev.D 66D 66, ,

104024 (2002).104024 (2002).A.D. Linde, hep-th/0211048A.D. Linde, hep-th/0211048

N masslessN massless

gravitonsgravitons 0

1

signature wN

ii

S S g

41 8

2i i i i i ii

S g d x g R g G

0 int 1 2, , ,wTot i NS g S S g g g

1818

Multigravity gasMultigravity gas

: | 22

k ij klijkl ij ij

gD G R g g g

,k

iN N 0kiN For each action,

introduce the lapse and shift functions

Choose the gauge

Define the followingdomain

1 wk N

0Let No interaction

Depending on the structure You are looking, You could have a

‘ideal’gas of geometries.Our specific case:

Schwarzschild wormholes

1919

i j1

with when wN

i i j

Wave functionals do not overlapWave functionals do not overlap

Additional assumption

3

1ij ij ij

ij ij ij

D g g d x g

V D g g g

3

1

8k

kk k kij ij ijk k

k

k k kk kij ij ijk k

D g g d x g

V GD g g g

The single eigenvalueThe single eigenvalue problem turns intoproblem turns into

2020

And the total waveAnd the total wave functional becomes functional becomes

23

1ij Foam ij Foam ij

ij Foam ij Foam ij

D h h d x h

V D h h h

1 2 wTot N Foam

1

wN

i

The initial problem changes into

1 1,G

2 2,G

,w wN NG

2121

Further trivial assumptionFurther trivial assumptionR. Garattini - R. Garattini - Int. J. Mod. Phys. D 4 (2002) 635; gr-qc/0003090.Int. J. Mod. Phys. D 4 (2002) 635; gr-qc/0003090.

1 2

1 2

2 2 23 3 3

1 2

1 1 1Nw

wNw

N

d x d x d xV V V

1 2 wNG G G Nw copies of

the same gravity

Take the maximum

2222

231 1

w

Max d xN V

There are argumentsleading to

Nevertheless, there is noProof of this

2323

ConclusionsConclusions Wheeler-De Witt Equation Wheeler-De Witt Equation Sturm-Liouville Sturm-Liouville

Problem.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 Riemann zeta Eigenvalue Regularization with the Riemann zeta

function function Casimir energy graviton contribution Casimir energy graviton contribution to the cosmological constant.to the cosmological constant.

Renormalization and renormalization group Renormalization and renormalization group equation.equation.

Generalization to multigravity.Generalization to multigravity. Specific example: gas of Schwarzschild Specific example: gas of Schwarzschild

wormholes.wormholes.

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. Specific examples of interaction like the Linde bi-Specific examples of interaction like the Linde bi-

gravity model or Damour et al.gravity model or Damour et al. Possible generalization con N ‘different Possible generalization con N ‘different

gravities’?!?!gravities’?!?! Use a distribution of gravities!!Use a distribution of gravities!!