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 Astrophysics
• Nanchang University, China
• Nov. 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 gravitys
  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

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Modified dispersion relations as semi-classical
effect of quantum gravity

• Quantum mechanics
  General relativity
• Planck Length
• Planck Mass

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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 representation
• R.Gambini and J.Pullin, Phys. Rev. D 59, 124021
  (1999)
• Kodama state in loop quantum gravity with a
  positive cosmological constant
• L.Smolin, arXivhep-th/0209079.
• Coherent states for quantum gravity
• H.Sahlmann and T.Thiemann, Class.
  Quant. Grav. 23, 909 (2006). arXivgr-qc/0207031.

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The fate of Lorentz symmetry at Planck scale

• Modified dispersion relations from semiclassical
  LQG

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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.

As a result, the invariant of energy and momentum
is modified to
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Non-linear Lorentz transformation in DSR

• In momentum space

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Two key issues in DSR

• The definition of position space
• Soccer problem

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• Gravitys 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.

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From doubly special relativity to gravitys
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.

J. Magueijo and L. Smolin, Class.Quant.Grav.21,
1725 (2004). arXivgr-qc/0305055
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Rainbow metric and rainbow Einstein equations

• Rainbow metric
• Rainbow Einstein equation

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Modified FRW universe

• The modified metric

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Modified FRW universe

• Generalization

The probe is identified with the radiation
particle
Ansatz
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Modified FRW universe

• The connection components
• Non-trivial components of Riemann tensor
• Ricci tensor components

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Modified FRW universe

• Energy-momentum tensor
• Unit vector
• Conservation equation

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Modified FRW universe

• Generalization of the modified FRW equations

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Modified FRW universe

• Specify the function

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Modified FRW universe

• The averaged effect

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Modified FRW universe

• The averaged effect
• Define

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Modified FRW universe

• The solution

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Modified FRW universe

• The modified equations from LQG

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Remark 1 Statistics of photons with modified
dispersion relations

• The state density

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Remark 1 Statistics of photons with modified
dispersion relations

• Modified Stefan-Boltzmann law

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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
• 
  THANK YOU!