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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
GR 100 Years GR 100 Years 44LisbonLisbon
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
GR 100 Years GR 100 Years 55LisbonLisbon
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
GR 100 Years GR 100 Years 66LisbonLisbon
WheelerWheeler--De De WittWitt EquationEquationqqB. S. B. S. DeWittDeWitt, , PhysPhys. Rev.. Rev.160160, 1113 (1967)., 1113 (1967).
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
GR 100 Years GR 100 Years 77LisbonLisbon
WheelerWheeler--De De WittWitt EquationEquationqqB. S. B. S. DeWittDeWitt, , PhysPhys. Rev.. Rev.160160, 1113 (1967)., 1113 (1967).
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
WheelerWheeler--De De WittWitt EquationEquationqqB. S. B. S. DeWittDeWitt, , PhysPhys. Rev.. Rev.160160, 1113 (1967)., 1113 (1967).
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
GR 100 Years GR 100 Years 99LisbonLisbon
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
GR 100 Years GR 100 Years 1111LisbonLisbon
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
GR 100 Years GR 100 Years 1212LisbonLisbon
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
GR 100 Years GR 100 Years 1313LisbonLisbon
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
GR 100 Years GR 100 Years 1414LisbonLisbon
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]]]]
GR 100 Years GR 100 Years 1515LisbonLisbon
[[ ,, ( )( ) , ;, ; [[gg qq ]]]]
GR 100 Years GR 100 Years 1616LisbonLisbon
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
GR 100 Years GR 100 Years 1717LisbonLisbon
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
GR 100 Years GR 100 Years 1818LisbonLisbon
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
GR 100 Years GR 100 Years 1919LisbonLisbon
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
GR 100 Years GR 100 Years 2121LisbonLisbon
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
GR 100 Years GR 100 Years 2222LisbonLisbon
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
GR 100 Years GR 100 Years 2323LisbonLisbon
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
GR 100 Years GR 100 Years 2424LisbonLisbon
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
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