Quantum Cosmologywith
Distorted Gravity
Remo GarattiniUniversità di Bergamo
I.N.F.N. - Sezione di Milano
Cagliari 18 Marzo 2016
Plan of the Talk
•Building the Wheeler-DeWitt Equation•The Wheeler-DeWitt Equation as a Sturm-Liouville problem•Relaxing the Lorentz Symmetry in a MSS approach for a FLRW model.•The Cosmological Constant as a Zero Point EnergyComputation in the Gravity’s Rainbow context
•Conclusions and Outlooks
34 4 31 2 2 82 Newton's Constant Cosmological Constant
matterS d x g R d x g K S G
G
M M
Relevant Action for Quantum Cosmology
34 4 31 122 matterS d x g R d x g K S
M M
Relevant Action for Quantum Cosmology
2 2 2
33
2 2
1
j
kk j
i i jiji ij
NN N N N N Ng g
N g N N NgN N
ADM Decomposition
2 2 2ds
is the lapse function is the shift function
i i j jij
i
g dx dx N dt g N dt dx N dt dx
N N
1 2
ijij ij i j j i ijK g N N K K g
N
33 2 3
3
3
1 22
Legendre Transformation
2 2 0 Classical Constraint Invariance by time2
ijij matterI
I
ii
ij klijkl
S dtd xN g K K K R S S
H d x N N H
gG R
H H
H
|
reparametrization2 0 Classical Constraint Gauss Lawi ij
j H
2 2 22
33
2 2
1
ds
j
kk j
i i jiji ij
NN N N N N Ng g g dx dx
N g N N NgN N
Wheeler-De Witt EquationB. S. DeWitt, Phys. Rev.160, 1113 (1967).
2 2 02
ij klijkl ij
gG R g
• Gijkl is the super-metric,• R is the scalar curvature in 3-dim.
2 2 2 2 23ds N dt a t d
2 2
2 42 2
9 04 3
qH a a a aa a a G
Formal Schrödinger Equation with zero eigenvalue whose solution is a linear combination ofAiry’s functions (q=-1 Vilenkin Phys. Rev. D 37, 888 (1988).) containing expanding solutions
Example:WDW for Tunneling
Wheeler-De Witt EquationB. S. DeWitt, Phys. Rev.160, 1113 (1967).
E=0 is highly degenerate
Sturm-Liouville Eigenvalue Problem
2
2 42
1 9 04 3
qqH a a a a a E a E
a a a G
0
* Normalization with weight b
a
d dp x q x w x y xdx dx
w x y x y x dx w x
2 2
2 43 3 2 2 3
q q qp x a t q x a t w x a t y x aG G
4 *
0
qa a a da
Wheeler-De Witt EquationB. S. DeWitt, Phys. Rev.160, 1113 (1967).
Sturm-Liouville Eigenvalue Problem Variational procedure
2
2 42
1 9 04 3
qqH a a a a a
a a a G
*min
*
b
aby x
a
d dy x p x q x y x dxdx dx
w x y x y x dx
0y a y b
Rayleigh-RitzVariational Procedure
Wheeler-De Witt EquationB. S. DeWitt, Phys. Rev.160, 1113 (1967).
Sturm-Liouville Eigenvalue Problem Variational procedure
2
2 42
1 9 04 3
qqH a a a a a
a a a G
22
20
4
0
3*23 min
3 2*
q q
aq
d da a a a dada da G
Ga a a da
0 0 0 De Witt Condition
Rayleigh-RitzVariational Procedure
2
0 0 for example for q=0
exp No Solutiona a
Relaxing Lorentz symmetryHořava-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)
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 necessary
At low energy all these models describe GR
Gravity’s Rainbow
2 2 2 2 21 2
1 2/ 0 / 0
/ /
lim / lim / 1P P
P P
P PE E E E
E g E E p g E E m
g E E g E E
Doubly 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.
Curved Space Proposal Gravity’s Rainbow[J. Magueijo and L. Smolin, Class. Quant. Grav. 21, 1725 (2004) arXiv:gr-qc/0305055].
2 2 2 22 2 2 2
2 2 21 2 22
2
0 0 0
sin/ / /
1 /
exp 2 is the redshift function
is the shape function Condition ,
P P PP
N r dt dr r rds d db rg E E g E E g E E
g E Er
N r r r
b r b r r r r
Gravity’s Rainbow Application to Inflation[R. Garattini and M. Sakellariadou, Phys. Rev. D 90 (2014) 4, 043521; arXiv:1212.4987 [gr-qc]]
22 2 2
2 22 2
1 2
2 22 22
2
3 /1 0
2 / 3 /
3 41 0 12 3
P
P P
effeff
g E Eq aa aa a a Gg E E g E E
aq Ga a Va a a G
2 22 2 2
32 21 2
Distorted / /
N t a tds dt d FLRW metric
g E Ep g E Ep
22 2 22 2
2 21 2
3 /1 0
2 / 3 /P
P P
g E Eq aa aa a a Gg E E g E E
But we can go beyond this…indeed if 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
Gravity’s Rainbow Application to Hořava-Lifshitz theory[R. Garattini and E.N.Saridakis, Eur.Phys.J. C75 (2015) 7, 343; arXiv:1411.7257 [gr-qc]]
2
2
/1/
P
P P
dg E a t E dEA tdE dag E a t E 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
21
2 422 1 22 4
If we fix / , =1
/ 1
P
PP P
g E E f a t a
E a t E a tg E E c c
E E
2 2 2 2
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=3
Gravity’s Rainbow Application to Hořava-Lifshitz theory[R. Garattini and E.N.Saridakis, Eur.Phys.J. C75 (2015) 7, 343; arXiv:1411.7257 [gr-qc]]
It is possible to build a mapAlso for SSM
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. G., P.R.D 86 123507 (2012) 7, 343; arXiv:0912.0136 [gr-qc]]
Gravity’s Rainbow Application to Hořava-Lifshitz theory[R. Garattini and E.N.Saridakis, Eur.Phys.J. C75 (2015) 7, 343; arXiv:1411.7257 [gr-qc]]
The WDW equation becomes
Albrecht, Barrow, Harko, Maguejio, Moffat..
MSS in a VSL CosmologyR.G. and M.De Laurentis, arXiv:1503.03677
MSS in a VSL CosmologyR.G. and M.De Laurentis, arXiv:1503.03677
0 PSetting a kl
A Brief Mention to GUP
[R. Garattini and Mir Faizal; N.P. B 905 (2016) 313 arXiv:1510.04423 [gr-qc]]
Flat space
Deformed Momentum
Deformed U.P.
Trial Wave Function
Higher Order Derivative
From Mini-SuperSpaceto
Field Theory in 3+1 Dimensions
The Cosmological Constantas a Zero Point Energy
Calculation
Generalization
Solve this infinite dimensional PDE with a VariationalApproach without matter fields contribution
is a trial wave functional of the gaussian typeSchrödinger Picture
Spectrum of depending on the metricEnergy (Density) Levels
Wheeler-De Witt EquationB. S. DeWitt, Phys. Rev.160, 1113 (1967).
3
1
Tij j
D h h d x h
V D h h h
D h D h D D h J
InducedCosmological‘‘Constant’’
Eliminating Divergences usingGravity’s Rainbow
[R.G. and G.Mandanici, Phys. Rev. D 83, 084021 (2011), arXiv:1102.3803 [gr-qc]]
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
42
, : (Propagator)2
ij kl
ijkl
h x h yK x y
g E
2
Modified Lichnerowicz operator
Standard Lichnerowicz operator
4
2
aia j ijij
jl a aij ijkl ia j ja iij
h R h Rh
h h R h R h R h
2
2 22
E= ijijh h
g E
One loop Graviton Contribution
We can define an r-dependent radial wave number
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
22 2 2 3
3 ' 36 12 2
' 36 12 2
b r b r b rm r
r r r r
b r b r b rm r
r r r r
3222
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 2
18 16 i
iim r
i i
dG m r
Standard Regularization
Eliminating Divergences usingGravity’s Rainbow
[R.G. and G.Mandanici, Phys. Rev. D 83, 084021 (2011), arXiv:1102.3803 [gr-qc]]
Gravity’s Rainbow and the Cosmological ConstantR.G. and G.Mandanici, Phys. Rev. D 83, 084021 (2011), arXiv:1102.3803 [gr-qc]
1
2
Popular Choice...... Not Promising
/ 1
/ 1
n
PP
P
Eg E EE
g E E
Failure of Convergence
2
1 2
2
/ exp 1
/ 1
PP P
P
E Eg E EE E
g E E
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
Conclusions and Outlooks• The Wheeler De Witt equation can be considered as a
Sturm-Liouville Problem Rayleigh-Ritz Variationalprocedure.
• In ordinary GR, we need a cut-off or a regularization/renormalization scheme.
• Application of Gravity’s Rainbow can be considered tocompute divergent quantum observables.
• Neither Standard Regularization nor Renormalization are required. This also happens in NonCommutativegeometries. A tool for ZPE Computation
• A connection between Horava-Lifshits theory withoutdetailed balanced condition and with projectability and Gravity’s Rainbow seems possible, at least in a FLRW metric. This is expected also for a VSL
• Repeating the above procedure for a SSM• Technical Problems with Kerr and other complicated
metrics. Comparison with Observation.