Qing-Guo Huang based on arXiv: (to appear in PLB) done with F.L.Lin Institute of High Energy Physics, CAS State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, CAS 05/22/2012 1

2 Supernova cosmology project collaboration, S. Perlmutter et al, APJ 517(1999)565 Supernova search team collaboration, A.G. Riess et al, APJ 116(1998)1009

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 Why is the cosmological constant so small? 5

 Why is it comparable to nowadays matter energy density? (cosmic coincidence) (Anthropic???) scale factor a(t) cosmological constant matter radiation 6

 Why is there a positive cosmological constant? 7  Why it is exponentially small compared to known fundamental energy scale?  How does it fit in a self-consistent quantum theory?

 Why is there a positive cosmological constant? 8  Why it is exponentially small compared to known fundamental energy scale?  How does it fit in a self-consistent quantum theory?

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10 ‘t Hooft, gr-qc/ Susskind, hep-th/

Black Hole M, Q, J Black Hole M, Q, J ??? Classical picture 11

Black Hole Hawking Radiation Quantum picture 12 e-e- e+e+

Complementarity principle for black hole: The process of formation and evaporation of a black hole, as viewed by a distant observer, can be described entirely within the context of standard quantum theory. In particular, there exists a unitary S-matrix which describes the evolution from infalling matter to outgoing Hawking radiation. No information loss. ‘t Hooft, Nucl.Phys.B 335(1990)138 Susskind, Thorlacius, Uglum, Phys.Rev.D 48(1993)

Black Hole complementarity principle 14 e-e- e+e+

de Sitter space is the maximally symmetric vacuum solution to Einstein equations with a positive cosmological constant Λ. FRW coordinates Static coordinates 15

observer-dependent horizon Gibbons, Hawking, Phys.Rev.D 15(1977) R observer

Similar to black hole, Complementarity principle for de Sitter space: To an observer who never crosses the horizon, the horizon can absorb, thermalize and re-emit all information that falls on it. No information loss. Banks, Fischler, hep-th/ Banks, Fischler, Paban, JHEP 0212(2002)062 Dyson, Lindesay, Susskind, JHEP 0208(2002)045 Susskind, hep-th/ Dyson, Kleban, Susskind, JHEP 0210(2002)011 17

Question: How fast we can re-construct one qubit from Hawking radiation? 18

 Suppose the degrees of freedom are arranged in a D dimensional system.  The total number of d.o.f scales with N.  Awaring of thermalization is process of diffusion in which the initial perturbation spreads in space to a distance of order t 1/2. size ~ N 1/D power law 19

20 scrambling time for de Sitter space Susskind, arXiv:

21 inflation  Flatness, horizon, structure formation, ……  It must end in the early universe.

22 During inflation  Alice crosses Bob’s event horizon at a moment t c.  Based on the complementarity principle, Bob can reconstruct it after the moment of t c +t *. Bob Alice

23  Inflation must end in the early universe (t end ). At the end of inflation, the proper distance between Alice and Bob is  The minimum distance for the case in which Bob can re-construct the qubit from Hawking radiation during inflation is given by

24 Question: Whether can we clone a q-bit if inflation lasted long enough?

25  How far a photon can travel in an expanding universe after inflation?  Since L r,m -> ∞ in a decelerating universe, Bob can get the qubit carried by Alice after inflation sooner or later, and therefore the qubit can be cloned.

26 Quantum superposition + Unitarity Wootters and Zurek, Nature 299(1982)802 two arbitrary states: If we can clone an unknown quantum state, if U is unitary This is not the case for two arbitrary states!

27 Naively,

28 A better estimation, Assuming that the vacuum energy driving inflation instantaneously decays into radiation at the end of inflation, we roughly have

29  A positive cosmological constant is postulated to preserve the unitarity in quantum mechanics if a long-lasting inflation happened in the early universe.  The scale of cosmological constant can be exponentially small compared to inflation scale.  In fact, a similar argument for more general dark energy is also applicable. Our arguments cannot be used to select dark energy models.  The fate of our universe should be in a state with accelerating expansion.

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