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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 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

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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, 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:

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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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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.

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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. 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

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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!

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