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De Sitter Space and Some Related Matters. Rong-Gen Cai ( 蔡荣根 ) Institute of Theoretical Physics Chinese Academy of Sciences. Contents:. What is de Sitter Space? Why de Sitter Space? C. Some Related Matters (Puzzles). What is de Sitter Space? (W. de Sitter, 1917).

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De Sitter Space and Some Related Matters

Rong-Gen Cai (蔡荣根)

Institute of Theoretical Physics

Chinese Academy of Sciences


  • What is de Sitter Space?

  • Why de Sitter Space?

  • C. Some Related Matters (Puzzles)

  • What is de Sitter Space?

  • (W. de Sitter, 1917)

1) Λ<0

anti-de Sitter

2) Λ=0


3) Λ>0

de Sitter

(Willem de Sitter,1872-1934)



Conformal Flat



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:


The Penrose Diagram of de Sitter Space in the Planar Coordinates

iii) The static coordinates:

B. Why the de Sitter Space?

  • Maximally symmetric curved space

  • 2) Inflation model for the early universe

  • Inflation Model⊕Dark Matter⊕ Dark Energy

  • 22% ⊕ 73%

  • 3) Current accelerated expanding universe





C. Some Related Matters (Puzzles)

  • The CC problem

  • ----------the Cosmological Constant problem


(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:

why not ?

(T. Banks, 2000)

is a critical limit of M theory!

(4) The vacuum for QFT in de Sitter Space?


What is the vacuum in the inflation model?

(Bunch-Davies Vacuum, Trans-Planck Physics)

(5) Is there the dS/CFT correspondence?

(A. Strominger, 2001)

However, hep-th/0202163

by L. Dyson, J. Lindesay & L.Susskind

dS 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?

  • AD mass

  • (L. Abbott and S. Deser, 1982)

  • 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, CQG15,2783 (1998),

Cai and Guo, PRD69, 104025 (2004).

D. Defining conserved charges

in asymptotically dS spaces

As an example, consider an (n+2)-dimensional SdS spacetime

Narirai Black Holes

Path integral method to quantum gravity

For (asymptotically) dS space:

The action:

A finite action could be obtained as:


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 SdS spacetime:


A Conjecture for Mass Bound in dS Spaces?

(V. Balasubramanian et al, 2001)

For an asymptotically dS spaqce if its mass is beyond the mass of a pure dS space, there must be a singularity.

Topological dS spaces:

(Cai,Myung and Zhang, PRD65, 2002)

(Cai,Myung and Zhang, PRD65, 2002)

E. Thermodynamics of black hole horizon

and cosmological horizon in dS space

  • 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, CQG18, 2001, Cai & Zhang, PRD64, 2001)

(1) Cosmological horizon in SdS spacetime:

(Cai, PLB525,2002)

(2) Black Hole horizon in SdS spacetime

(Cai, NPB628, 2002)

F. Dyanamics of a Brane in SdS Spacetime

For a closed FRW universe with a positive CC:

If , then

(E. Verlinde, 2000)

If , we introduce


(Cai & Mung, PRD67,2003)

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:


Case 1:

The Penrose diagram for the SdS spacetime

Case 2:

Case 3:

Holography on the brane:




On the brane, one has

Entropy density

Energy density


In particular, one has

It coincides with the Friedmann equation when

the brane crosses the black hole horizon!

Thanks !

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