De Sitter Space and Some Related Matters RongGen

De Sitter Space and Some Related Matters Rong-Gen Cai (蔡荣根) Institute of Theoretical Physics Chinese Academy of Sciences

Contents: A. What is de Sitter Space? B. Why de Sitter Space? C. Some Related Matters (Puzzles)

A. What is de Sitter Space? (W. de Sitter, 1917) 1) Λ<0 anti-de Sitter 2) Λ=0 Minkowski 3) Λ>0 de Sitter (Willem de Sitter, 1872 -1934)

Definitions: 1): Conformal Flat

2): Topology A four dimensional de Sitter space is a hyperboloid embedded in a five dimensional Minkowski space!

The coordinates often used i) The globe coordinates:

ii) The planar coordinates: O-gauge!

The Penrose Diagram of de Sitter Space in the Planar Coordinates

iii) The static coordinates:

B. Why the de Sitter Space? 1) Maximally symmetric curved space 2) Inflation model for the early universe Inflation Model⊕Dark Matter ⊕ Dark Energy 22% ⊕ 73% 3) Current accelerated expanding universe

astro-phys/9812133

astro-phys/9812133

astro-phys/9812133

astro-phys/9812133

C. Some Related Matters (Puzzles) (1) The CC problem -----the Cosmological Constant problem QFT

(2) What is the statistical degrees of freedom of the de Sitter space? Area of cosmological horizon (G. W. Gibbons and S. Hawking, 1977)

(3) The cosmological constant has any relation to SUSY? In general: Fitting data: (T. Banks, 2000) why not is a critical limit of M theory! ?

(4) The vacuum for QFT in de Sitter Space? α vacuum What is the vacuum in the inflation model? (Bunch-Davies Vacuum, Trans-Planck Physics)

(5) Is there the d. S/CFT correspondence? (A. Strominger, 2001) However, hep-th/0202163 by L. Dyson, J. Lindesay & L. Susskind d. S complementarity precludes the existence of the appropriate limits. We find that the limits exist only in approximations in which the entropy of the de Sitter Space is infinite. The reason that the correlators exist in quantum field theory in the de Sitter Space background is traced to the fact that horizon entropy is infinite in QFT.

(6) The cosmological constant has any relation to inflation model? (T. Banks and W. Fischler, 2003) Cosmological Entropy Bound: (Cai, JCAP 0402(2004)007)

(7) How to define conserved quantities for asymptotically de Sitter space? a) AD mass (L. Abbott and S. Deser, 1982) b) Surface counterm method (V. Balasubramanian et al, 2001) (8) Are there corresponding descriptions for thermodynamics of black hole horizon and cosmological horizon in terms of CFTs?

(9) The de Sitter space can be realized in string theory? (KKLT Model, hep-th/0301240) “de Sitter Vacua in String Theory” (10) Entropy of black hole-de-Sitter spacetime? (This can be derived only for the lukewarm black hole) Cai, Ji and Soh, CQG 15, 2783 (1998), Cai and Guo, PRD 69, 104025 (2004).

D. Defining conserved charges in asymptotically d. S spaces As an example, consider an (n+2)-dimensional Sd. S spacetime Narirai Black Holes

Path integral method to quantum gravity For (asymptotically) d. S space:

The action: A finite action could be obtained as: Counterterms

The counterterms:

Beyond the cosmological horizon:

The Brown-York “Tensor”: For a Killing vector, there is a conserved charge!

The conserved mass for the Killing vector

For the Sd. S spacetime: e. g.

A Conjecture for Mass Bound in d. S Spaces? (V. Balasubramanian et al, 2001) For an asymptotically d. S spaqce if its mass is beyond the mass of a pure d. S space, there must be a singularity.

Topological d. S spaces: (Cai, Myung and Zhang, PRD 65, 2002)

(Cai, Myung and Zhang, PRD 65, 2002)

E. Thermodynamics of black hole horizon and cosmological horizon in d. S space (1) Black Hole Horizon: r_+ (2) Cosmological Horizon: r_c

Cardy-Verlinde Formula ------An Entropy Formula for a CFT (J. Cardy, 1986, E. Verlinde, 2000) in (n+1) dimensions (Cai, PRD 63, 2001; Cai, Myung & Ohta, CQG 18, 2001, Cai & Zhang, PRD 64, 2001)

(1) Cosmological horizon in Sd. S spacetime: (Cai, PLB 525, 2002)

(2) Black Hole horizon in Sd. S spacetime (Cai, NPB 628, 2002)

F. Dyanamics of a Brane in Sd. S Spacetime For a closed FRW universe with a positive CC: If , then (E. Verlinde, 2000)

If , we introduce Then (Cai & Mung, PRD 67, 2003)

The dynamics of the brane is governed by The equation of motion:

Consider a radial timelike geodesic satisfying then the reduced metric on the brane:

Define Case 1: The Penrose diagram for the Sd. S spacetime

Case 2:

Case 3:

Holography on the brane: Suppose Then and

On the brane, one has Entropy density Energy density Temperature

In particular, one has It coincides with the Friedmann equation when the brane crosses the black hole horizon!

Thanks !