the cosmological constantthe cosmological constant in

GR 100 years in Lisbon18-19 December 2015

The Cosmological ConstantThe Cosmological Constant ininThe Cosmological ConstantThe Cosmological Constant in in Distorted GravityDistorted Gravity

Remo GarattiniRemo GarattiniU i ità di BU i ità di BUniversità di BergamoUniversità di Bergamo

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

PlanPlan ofof the Talkthe Talk

••Building theBuilding the WheelerWheeler DeWittDeWitt EquationEquation••Building the Building the WheelerWheeler--DeWittDeWitt EquationEquation••The The WheelerWheeler--DeWittDeWitt EquationEquation asas a a SturmSturm--LiouvilleLiouville problemproblem••RelaxingRelaxing thethe LorentzLorentz SymmetrySymmetry in a MSSin a MSS approachapproach forfor a FLRWa FLRW••RelaxingRelaxing the the LorentzLorentz SymmetrySymmetry in a MSS in a MSS approachapproach forfor a FLRW a FLRW modelmodel..••TheThe CosmologicalCosmological ConstantConstant asas a Zero Point Energya Zero Point Energy••The The CosmologicalCosmological ConstantConstant asas a Zero Point Energya Zero Point EnergyComputationComputation in the in the GravityGravity’s ’s RainbowRainbow contextcontext

••ConclusionsConclusions andand OutlooksOutlooksConclusionsConclusions and and OutlooksOutlooks

34 4 31 2 2 82 matterS d x g R d x g K S G

M M

Relevant Action for Quantum Cosmology

Newton's Constant Cosmological Constant

G

GR 100 Years GR 100 Years 33LisbonLisbon

34 4 31 12S d x g R d x g K S

Relevant Action for Quantum Cosmology

22 matterS d x g R d x g K S

M M

ADM Decomposition

2 2 2

1 j

kk j

NN N N N N N

ADM Decomposition

33

2 2

k j

i i jiji ij

N Ng gN g N N Ng

N N

2 2 2ds i i j jijg dx dx N dt g N dt dx N dt dx

is the lapse function is the shift functioniN N

1 ijK N N K K 2

ijij ij i j j i ijK g N N K K g

N


2 2 2

1 j

kN

N N N N

2 22

33

2 2

dsk j

i i jiji ij

N N N N N Ng g g dx dxN g N N Ng

N N

33 2 31 2

2ij

ij matterIS dtd xN g K K K R S S

3

2

Legendre Transformation

jI

iiH d x N N H

H H Legendre Transformation iH d x N N H

H H

32 2 0 Classical Constraint Invariance by time2

ij klijkl

gG R

H

reparametrization

|2 0 Classical Constraint Gauss Lawi ijj H


WheelerWheeler--De De WittWitt EquationEquationqqB. S. B. S. DeWittDeWitt, , PhysPhys. Rev.. Rev.160160, 1113 (1967)., 1113 (1967).

2 2 0

2ij kl

ijkl ijg

G R g

2ijkl ijg

GG isis the superthe super metricmetric•• GGijklijkl isis the superthe super--metricmetric,,•• R R isis the scalar curvature in 3the scalar curvature in 3--dim.dim.Example:WDW for Tunneling

2 2 2 2 23ds N dt a t d

2 2

Example:WDW for Tunneling

2 2

2 42 2

9 04 3

qH a a a aa a a G

FormalFormal SchrSchröödingerdinger EquationEquation withwith zero zero eigenvalueeigenvalue whosewhose solutionsolution isis a a linearlinear combinationcombination ofofAiryAiry’s ’s functionsfunctions (q=(q=--1 1 VilenkinVilenkin PhysPhys. Rev. D 37, 888 (1988).) . Rev. D 37, 888 (1988).) containingcontaining expandingexpanding solutionssolutions



21 9

E=0E=0 isis highlyhighly degeneratedegenerate

2 42

1 9 04 3

qqH a a a a a E a E

a a a G

E 0 E 0 isis highlyhighly degeneratedegenerate

SturmSturm--LiouvilleLiouville EigenvalueEigenvalue ProblemProblemSturmSturm--LiouvilleLiouville Eigenvalue Eigenvalue ProblemProblem

0d dp x q x w x y xdx dx

b

dx dx

*

* Normalization with weight a

w x y x y x dx w x

2 2

2 43 3q q q

4 *

0

qa a a da

2 43 3 2 2 3

q q qp x a t q x a t w x a t y x aG G



2

2 42

1 9 04 3

qqH a a a a a

a a a G

SturmSturm LiouvilleLiouville EigenvalueEigenvalue ProblemProblem VariationalVariational procedureprocedure

SturmSturm--LiouvilleLiouville Eigenvalue Eigenvalue ProblemProblem VariationalVariational procedureprocedure

b d d

*min

*

aby x

d dy x p x q x y x dxdx dx

w x y x y x dx

Rayleigh-RitzVariational Procedure

*a

w x y x y x dx

0y a y b 0y a y b GR 100 Years GR 100 Years 88LisbonLisbon


2

2 42

1 9 04 3

qqH a a a a a

a a a G

SturmSturm LiouvilleLiouville EigenvalueEigenvalue ProblemProblem VariationalVariational procedureprocedure

SturmSturm--LiouvilleLiouville Eigenvalue Eigenvalue ProblemProblem VariationalVariational procedureprocedure

2

23* q qd d d

22

0

4

3*23 min

3 2*

q q

aq

d da a a a dada da G

Ga a a da

Rayleigh-RitzVariational Procedure

0

*qa a a da

0 0 0 De Witt Condition

0 0 for example for q=0

0 0 0 De Witt Condition


2exp No Solutiona a

RelaxingRelaxing LorentzLorentz symmetrysymmetryRelaxingRelaxing LorentzLorentz symmetrysymmetryHořava Lifshitz theory UV Completion problems with scalar graviton in IRHořava-Lifshitz theory UV Completion, problems with scalar graviton in IR

Varying Speed of Light Cosmology Solve problems in the Inflationary phase(horizon flatness particle production)(horizon,flatness, particle production)

Gravity’s Rainbow Like VSL. Moreover it allows finite calculation to one loop.The set of the Rainbow’s functions is too large A selection procedure is necessaryThe set of the Rainbow s functions is too large. A selection procedure is necessary

At low energy all these models describe GR

Gravity’s RainbowGravity s RainbowDoubly Special Relativity

G. Amelino-Camelia, Int.J.Mod.Phys. D 11, 35 (2002); gr-qc/001205.G. Amelino-Camelia, Phys.Lett. B 510, 255 (2001); hep-th/0012238.

2 2 2 2 21 2/ /P PE g E E p g E E m

1 2/ 0 / 0lim / lim / 1

P PP PE E E E

g E E g E E

Curved Space Proposal Gravity’s Rainbowp p y[J. Magueijo and L. Smolin, Class. Quant. Grav. 21, 1725 (2004) arXiv:gr-qc/0305055].

2 2 2 22 2 2 2

2 2 2 sinN r dt dr r rds d d

2 2 21 2 22

2/ / /

1 /

exp 2 is the redshift function

P P PP

b rg E E g E E g E Eg E E

r

N r r r

0 0 0 is the shape function Condition ,b r b r r r r


Gravity’sGravity’s Rainbow Application to Rainbow Application to InflationInflation[[R. Garattini and M. Sakellariadou, R. Garattini and M. Sakellariadou, PhysPhys. Rev. D 90 (2014) 4, 043521; arXiv:1212.4987 [gr. Rev. D 90 (2014) 4, 043521; arXiv:1212.4987 [gr--qcqc]]]]

2 22 2 2 Di d

N t a td d d FLRW i

2 2 232 2

1 2

Distorted / /

ds dt d FLRW metricg E Ep g E Ep

22 2 22 2

2 2

3 /1 0Pg E Eq aa a

2 21 2

02 / 3 /P P

a aa a a Gg E E g E E

2 22 2

22

3 41 0 12 3

effeff

aq Ga a Va a a G


GravityGravity’s ’s RainbowRainbow ApplicationApplication toto HořavaHořava--LifshitzLifshitz theorytheory[R. Garattini and [R. Garattini and E.N.SaridakisE.N.Saridakis, , Eur.Phys.J.Eur.Phys.J. C75 (2015) 7, 343; C75 (2015) 7, 343; arXivarXiv:1411.7257 [:1411.7257 [grgr--qcqc]]]]

22 2 22 2

2 2

3 /1 0

2 / 3 /Pg E Eq aa a

G E E E E 2 2

1 22 / 3 /P Pa a a Gg E E g E E

But we can go beyond this…indeed if thenE E a tg y thenE E a t

2

2 2 2 21 2

1 33 / , where , 1 2ijij P

aK K K g E E f a t a f a t a a t A t A t a tN t a

2 /1/

Pdg E a t E dEA tdE dag E a t E E


2 / P P dE dag E a t E E

GravityGravity’s ’s RainbowRainbow ApplicationApplication toto HořavaHořava--LifshitzLifshitz theorytheory

2If fi / 1E E f

[R. Garattini and [R. Garattini and E.N.SaridakisE.N.Saridakis, , Eur.Phys.J.Eur.Phys.J. C75 (2015) 7, 343; C75 (2015) 7, 343; arXivarXiv:1411.7257 [:1411.7257 [grgr--qcqc]]]]

21

2 42

If we fix / , =1

/ 1

Pg E E f a t a

E a t E a tE E

2 2 2 2

22 1 22 4/ 1P

P P

g E E c cE E

2 2 2 2

2 22 2 2then using the "normal" dispersion relation and P

P P

k k k kE Ea t a l G

2 2 2 2

2 16 256 16 256b G G b R R P t ti l t f th 22 2 4 2

0 0

16 256 16 256/ 1 1 Pb G G b R Rg E E

a t a t R R

Potential part of the Projectable Hořava-Lifshitz theorywithout detailed balancedCondition z=3Condition z=3

It is possible to build a mapAlso for SSM


Also for SSM

GravityGravity’s ’s RainbowRainbow ApplicationApplication toto HořavaHořava--LifshitzLifshitz theorytheory[R. Garattini and [R. Garattini and E.N.SaridakisE.N.Saridakis, , Eur.Phys.J.Eur.Phys.J. C75 (2015) 7, 343; C75 (2015) 7, 343; arXivarXiv:1411.7257 [:1411.7257 [grgr--qcqc]]]]

10 12 1 g g 0b c GR

Applying the Rayleigh-Ritz procedure we can find candidate eigenvaluesdepending on the combination of the coupling constants[R. [R. G., G., P.R.DP.R.D 86 123507 (2012) 86 123507 (2012) 7, 343; 7, 343; arXivarXiv:0912.0136 :0912.0136 [[grgr--qcqc]]]]


[[ ,, ( )( ) , ;, ; [[gg qq ]]]]

MSS in a VSL MSS in a VSL CosmologyCosmologygygyR.G. and R.G. and M.DeM.De Laurentis, Laurentis, arXiv:1503.03677

Albrecht, Barrow, Harko, Maguejio, Moffat..

The WDW equation becomes


MSS in a VSL MSS in a VSL CosmologyCosmologygygyR.G. and R.G. and M.DeM.De Laurentis, Laurentis, arXiv:1503.03677

0 PSetting a kl


A A BriefBrief MentionMention toto GUPGUP

[[R. Garattini and R. Garattini and Mir Faizal;Mir Faizal; arXivarXiv:1510.04423:1510.04423 [[grgr--qcqc]] ]]

Deformed Momentum

Deformed U.P.

Trial Wave Function

Flat space Higher Order Derivative


Higher Order Derivative

G li tiGeneralizationFrom Mini-SuperSpace

totoField Theory in 3+1

DimensionsDimensions

The Cosmological Constantas a Zero Point Energyas a Zero Point Energy

Calculation

GR 100 Years GR 100 Years LisbonLisbon 2020

WheelerWheeler--DeDe WittWitt EquationEquationWheelerWheeler--De De WittWitt EquationEquationB. S. B. S. DeWittDeWitt, , PhysPhys. Rev.. Rev.160160, 1113 (1967)., 1113 (1967).

3

1D h h d x h

InducedCosmological‘‘Constant’’

T

V D h h h

Solve Solve thisthis infinite infinite dimensionaldimensional PDE with a PDE with a VariationalVariational

Tij jD h D h D D h J

ApproachApproach withoutwithout mattermatter fieldsfields contributioncontribution isis a trial a trial wavewave functionalfunctional ofof the the gaussiangaussian typetype

SchrSchröödingerdinger PicturePictureSpectrumSpectrum ofof dependingdepending on the on the metricmetric

Energy (Density)Energy (Density) LevelsLevelsEnergy (Density) Energy (Density) LevelsLevels


EliminatingEliminating DivergencesDivergences usingusingG itG it ’’ R i bR i bGravityGravity’s ’s RainbowRainbow

[[R.G. and R.G. and G.MandaniciG.Mandanici, , PhysPhys. Rev. D 83, 084021 (2011), . Rev. D 83, 084021 (2011), arXivarXiv:1102.3803 [:1102.3803 [grgr--qcqc]]]]

4 ah R h Rh

One loop Graviton Contribution

2

Modified Lichnerowicz operator

4

2

ia j ijij

jl a a

h R h Rh

h h R h R h R h

2

2E= ijh h

Standard Lichnerowicz operator

2 jl a aij ijkl ia j ja iij

h h R h R h R h

2 22

ijijh h

g E

3 2

1,2 1323

2 2

1ˆ 2 , ,4 2

ijkl

ijkl ijkl

g E g Ed x gG K x x K x x

V g E g E

4, : (Propagator)ij kl

h x h yK x y

4

2

, : ( op g o )2ijkl

x yg E


EliminatingEliminating DivergencesDivergences usingusingG itG it ’’ R i bR i bGravityGravity’s ’s RainbowRainbow

[R.G. and [R.G. and G.MandaniciG.Mandanici, , PhysPhys. Rev. D 83, 084021 (2011), . Rev. D 83, 084021 (2011), arXivarXiv:1102.3803 [:1102.3803 [grgr--qcqc]] ]]

We can define an r-dependent radial wave number 3 ' 36 b b b p

22 2

2 22

1, ,

/nl

nl iP

l lEk r l E m r r r xg E E r

21 2 2 3

2

3 ' 36 12 2

' 36

b r b r b rm r

r r r r

b r b r b r

22 2 2 3

36 12 2

b r b r b rm r

r r r r

3221 Ed 222

1 22 21 2*

1 ( / ) ( / ) ( )3 ( / )8

ii P P i i

i i PE

EdE g E E g E E m r dEdE g EG E

2

2

122 2

18 16 i

iim r

dG

Standard Regularization

2 2 28 16 i

i iG m r


GravityGravity’s’s RainbowRainbow and theand the CosmologicalCosmological ConstantConstantGravityGravity s s RainbowRainbow and the and the CosmologicalCosmological ConstantConstantR.G. and R.G. and G.MandaniciG.Mandanici, , PhysPhys. Rev. D 83, 084021 (2011), . Rev. D 83, 084021 (2011), arXivarXiv:1102.3803 [:1102.3803 [grgr--qcqc]]

Popular Choice...... Not Promising

1

p g

/ 1n

PP

Eg E EE

Failure of Convergence

2 / 1Pg E E

2

1 2/ exp 1PP P

E Eg E EE E

2 / 1P P

Pg E E

Minkowski de Sitter Anti de Sitter

2 2 2 2 21 2 0 0

- - -

/ P

Minkowski de Sitter Anti de Sitter

m r m r m r x m r E


ConclusionsConclusions andand OutlooksOutlooksConclusionsConclusions and and OutlooksOutlooks•• The The WheelerWheeler De De WittWitt equationequation can can bebe consideredconsidered asas a a

SturmSturm--LiouvilleLiouville ProblemProblem RayleighRayleigh--RitzRitz VariationalVariationalprocedureprocedureprocedure.procedure.

•• In In ordinaryordinary GR, GR, wewe needneed a cuta cut--off or a off or a regularizationregularization//renormalizationrenormalization schemescheme..

•• ApplicationApplication ofof GravityGravity’s ’s RainbowRainbow can can bebe consideredconsidered totocomputecompute divergentdivergent quantum quantum observablesobservables..

•• NeitherNeither StandardStandard RegularizationRegularization nornor RenormalizationRenormalization areare•• NeitherNeither Standard Standard RegularizationRegularization nornor RenormalizationRenormalization are are requiredrequired. . ThisThis alsoalso happenshappens inin NonCommutativeNonCommutativegeometriesgeometries. . A A tooltool forfor ZPE ZPE ComputationComputationAA titi b tb t HH Lif hitLif hit thth ith tith t•• A A connection connection betweenbetween HoravaHorava--LifshitsLifshits theorytheory withoutwithoutdetaileddetailed balancedbalanced conditioncondition and and withwith projectabilityprojectability and and GravityGravity’s ’s RainbowRainbow seemsseems possiblepossible, at , at leastleast in a FLRW in a FLRW

t it i ThiThi ii t dt d ll ff VSLVSLmetricmetric. . ThisThis isis expectedexpected alsoalso forfor a VSLa VSL•• RepeatingRepeating the the aboveabove procedure procedure forfor a SSMa SSM•• TechnicalTechnical ProblemsProblems withwith KerrKerr andand otherother complicatedcomplicated•• TechnicalTechnical ProblemsProblems withwith KerrKerr and and otherother complicatedcomplicated

metricsmetrics. . ComparisonComparison withwith ObservationObservation..2525LisbonLisbonGR 100 Years GR 100 Years

the cosmological constantthe cosmological constant in

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