Rainbow Gravity and the Very Early Universe
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Rainbow Gravity and the Very Early Universe
Yi Ling Center for Gravity and Relativistic AstrophysicsNanchang University, ChinaNov. 4, 2006
2006 Workshop on Dark Universe, Suzhou
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Contents
• Introduction: the semi-classical effect of quantum gravity and cosmology
• From deformed special relativity to gravity’s rainbow
• Modified FRW universe from rainbow gravity
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• Two fundamental issues in quantum gravity
1. Dynamical problem
Gravity is a constrained system,
Diffeomorphism invariance ---observables
Hamiltonian constraint----problem of time
2. Classical limit of quantum gravity
The reconstruction of classical spacetime
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Some problems related to loop quantum gravity
• Special relativity: Lorentz symmetry
• Testable signature of quantum gravity : Gamma Ray Burst (2007 )
• Particle physics : Ultra high energy cosmic rays
• Quantum cosmology : Big bang , Inflation
• Classical limit : Coherent states , Gravitons
• Black Hole Physics: Black hole remnant and information Paradox
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The fate of Lorentz symmetry at Planck scale
• Threshold anomaly of ultra high energy cosmic rays :
GZK cutoff
S.Coleman and S.Glashow, Phys. Rev.D 59, 116008 (1999)
J. Magueijo and L. Smolin, Phys. Rev. Lett(88) 190403,2002.
Modified dispersion relations
11 200 10 10thE Gev ev
2 2 20 ( , )E m p E p
CMBp p
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CMBp p
2pth
p GZK
m mE E
E
2 2 20E m p 2 2 2
0 ( , )E m p E p
2 4
pp p
pthp
m
m m mE
E E
410E ev 17 2 2 20 210 ( ) (10 )ev E ev
2110 10thp GZKE ev E
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Modified dispersion relations as semi-classical effect of quantum gravity
Quantum mechanics General relativity
Planck Length:
Planck Mass:
3 34: / 10pl G c m
5 19/ 1/ 10p pm c G l Gev
G
2 2 4 2 2 2 2( )npE m c p c l p p c
41, | | 10n
1, 8 1/ 1/p p plc l G m E
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Three ways to obtain modified dispersion relations in semi-classcial approach to LQG
• The action of the Hamiltonian on the weave states in loop repr
esentation R.Gambini and J.Pullin, Phys. Rev. D 59, 124021 (1999)
• Kodama state in loop quantum gravity with a positive cosmological constant
L.Smolin, arXiv:hep-th/0209079.
• Coherent states for quantum gravity
H.Sahlmann and T.Thiemann, Class. Quant. Grav. 23, 909 (2006). arXiv:gr-qc/0207031.
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The fate of Lorentz symmetry at Planck scale
• Modified dispersion relations from semiclassical LQG:
2 (1 )p i ids dt dt l e e
2 2 2 3 ...pE p m l E
0( , ) ( ) ( , )A A A
ia iaE he 0
0
ˆ ˆ( , ) ( ) ( , , )
(1 ) ( ) ( , , )
ia ia
iap
E A E A T a
E l A T a
22 2( )
(1 )i
p
km g k k
l
.0( ) ( ) ( , ) 0grav matterH H A A
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Doubly Special Relativity(DSR)
• The relativity of inertial frames, two universal constant:
1) In the limit , the speed of a photon goes to a
universal constant, .
2) in the above condition is also a universal constant.
/ 0plE E
plE
c
2 2 2 2 21 2 0/ , / ,pl plE f E E p f E E m
As a result, the invariant of energy and momentum is modified to
1pl pE l
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Non-linear Lorentz transformation in DSR
• In momentum space
00
0
0
0
0
0
( )'
1 ( 1)
( )'
1 ( 1)
'1 ( 1)
'1 ( 1)
z
p p z
zz
p p z
xx
p p z
yy
p p z
p vpp
l p l vp
p vpp
l p l vp
pp
l p l vp
pp
l p l vp
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Two key issues in DSR
• The definition of position space
• Soccer problem
F
nonL
'p
px
1F'x
?
2 2 2 20
0 0
2 2 2 2 2 2 20 0
( )
,
( ) '( )
np
n n np p
E M P El E
M Nm E N P Np
m p N l m p l
?px
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• Gravity’s rainbow
Non-linear map on momentum space:
To keep the contraction between position and momentum linear
The dual space is endowed with an energy dependent quadratic invariant.
2 2 2 2 21 2 0, ,E f E p f E m
iidx dp dtdE dx dp
2 2 22 2
1 2
1 1
( , ) ( , )ds dt dx
f E f E
0, , ( / ) , ( / )i i pl pl iU E p U U f E E E g E E p
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From doubly special relativity to gravity’s rainbow or deformed general relativity
• Rainbow gravity as an extension of DSR into a general relativity:
1) Correspondence principle
2) Modified equivalence principle
Freely falling observers who take measurement with energy E, will observe the laws of physics to be the same as modified special relativity.
0
lim ( )
pl
ab abE
E
g E g
0 0
1 1;
/ /i i
pl pl
e e e ef E E g E E
1/ plR E E
J. Magueijo and L. Smolin, Class.Quant.Grav.21, 1725 (2004). arXiv:gr-qc/0305055
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Rainbow metric and rainbow Einstein equations
• Rainbow metric
• Rainbow Einstein equation
0 0
1 1;
/ /i i
pl pl
e e e ef E E g E E
( ) aba bg E e e
( ) 8 ( ) ( ) ( )G E G E T E g E
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Modified FRW universe
• The modified metric
22 2
2 21 2
1
( ) ( )i j
ij
a tds dt dx dx
f E f E
22
22 2
1 1
21
8 ( )( )
3 3
4 3 ( )( )
3 3
3 0
fa K EG E
a f a f
a p EG E
a f
ap p
a
( ) 8 ( ) ( ) ( )G E G E T E g E
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Modified FRW universe
• Generalization:
22 ( ) 1f E
The probe is identified with the radiation particle
( )E E t
( ) ; 0; 0; ij ijG E G K
Ansatz:
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Modified FRW universe • The connection components:
• Non-trivial components of Riemann tensor:
• Ricci tensor components:
0 0 200 0, , i i
ij ij j j
f af aa
f a
00
0 20
2
i i ij j j
i j ij ij
i i ijkm k jm m jk
a afR
a af
R f aa ffaa
R f a
00
2 2
22
2
3 3
2
6
ij ij
a afR
a af
R f aa a ffaa
a af aR f
a af a
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Modified FRW universe
• Energy-momentum tensor:
• Unit vector:
• Conservation equation:
0
3 0
E T
H P
( )T u u P g u u
1( ,0,0,0) 1u f u u g
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Modified FRW universe
• Generalization of the modified FRW equations:
22
2
8
3
4
3 0
GH
f
P fH G H
f f
H P P
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Modified FRW universe
• Specify the function:
2 2 21 pf l E
2 2 4 2pE l E p
2 2 22
11 1 4
2 pp
E l pl
max
11/(2 ),
2p
p
p l El
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Modified FRW universe
• The averaged effect:
2
2 2
3 1
8
3 1 p
a
a
a G
a l E
2 2 21 pf l E
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Modified FRW universe
• The averaged effect:
• Define
2
2
1pl xx C
x
2 , 4a
Ea
2px l
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Modified FRW universe
• The solution:
2
1
t
2 1 11lnp
xl xt
C x x
0pl
0t 4
2 4
1p
p p
l
l Ct l
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Modified FRW universe
• The modified equations from LQG:
2 81
3
4 1 2
c
c
GH
H G P
12 4 41 pf l E
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Remark 1: Statistics of photons with modified dispersion relations
• The state density
2
3
4( ) 2
V p dpg d
h
2 22 2 2 2 2
20
1 ( )p
p cE E p f c
l E
E
22
3 3 3 3
4 8 (1 '/ )( ) 2
V p dp V f fg d d
h h c f
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Remark 1: Statistics of photons with modified dispersion relations
• Modified Stefan-Boltzmann law
2
24/0
1 4 100( ) 1
1 21 pkT
Ug d T l kT
V V e c
/ 21ln(1 ) ( ) (1 )
3kTPV
e g d P TkT
3 2/0
1( ) 1
1kTN g d VT T
e
2(1 )U
E kT TN
25.88
28.17
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Remark 2:
• Impact on black hole physics
the modified black holes will not evaporate totally, but have a remnant which can be viewed as a candidate for dark matter.
• Unruh effect
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Summary and Conjectures
• Rainbow gravity or deformed general relativity can be viewed as an effective theory at the semi-classical limit of quantum gravity. In this framework there is no single or fixed background spacetime, it depends on the energy of probes or test particles.
• We have considered some possible impacts on the FRW universe as well as Unruh effect and black holes in a heuristic way. More strict mathematical consideration is needed.
• The scheme presented here can be generalized to other sorts of modified black hole solutions.
• Conjectures 1)Tunneling effect 2) MOND
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