de Sitter space and Holography

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One can use AdS/CFT correspondence to study someproperties of this duality.dS slicing of dS:ds 2 dS d=R2cosh 2 (z) (ds2 dS d−1+ dz 2 )dS slicing of AdS:ds 2 AdS d=R2sinh 2 (z) (ds2 dS d−1+ dz 2 )Sods 2 1AdS d=tanh 2 (z) ds2 dS dWe can use this to map the physics of dS to dynamics in AdS,albeit with unusual actions. Namely this leads to• Scalars with position dependent masses.• Gravity with a position dependent Newton constant.By applying the AdS/CFT dictionary to the resulting system,this allows us to make a direct comparison of UV behavior ofthe d − 1 dual of dS d to the UV behavior of a strongly coupledCFT.50
g mn → f 2 g mnX → f −d−2 2 X√ −g → fd √ −g− √ −g(∂X) 2 → − √ −g(∂X) 2 − √ −g (d−2)2X 2 (∇ 2 ω)− √ −g (d−2)24X 2 (∇ω) 2√ −gR → fd−2 √ −g(R − 2(d − 1)(∇ 2 ω)−(d − 2)(d − 1)(∇ω) 2 )− √ −gξRX 2 → − √ −gξRX 2 + 2 √ −gξ(d − 1)(∇ 2 ω)+ √ −gξ(d − 2)(d − 1)(∇ω) 2−2 √ −gΛ → −2f d√ −gΛTable 1: Transformations under conformal rescaling;f = tanhz, ω = log f51
Page 1 and 2: de Sitter space and HolographyThird
Page 3: 1. Historical review and motivation
Page 7 and 8: 5. Technologies:There are a number
Page 9 and 10: Let us know study different coordin
Page 11 and 12: Conformal coordinates: T, θ i , i
Page 13 and 14: Planer (Inflationary) coordinates:
Page 15 and 16: The corresponding metric isdS 2 =
Page 17 and 18: It is easy to see thatX 0 + X d =
Page 20 and 21: 1.2 Causal structure (Penrose diagr
Page 22 and 23: IINorth PoleOSouth PoleNorth PoleOS
Page 24 and 25: The entire region of the dS space i
Page 26 and 27: For example• In static coordinate
Page 28 and 29: An observer moving along a timelike
Page 30 and 31: 3. dS/CFT correspondenceFrom what w
Page 32 and 33: In d dimensions the Einstein equati
Page 34 and 35: One can generalize it for a general
Page 36 and 37: • One can always decompose the me
Page 38 and 39: Consider the Brown-York stress tens
Page 40 and 41: Using the definition of Brown-York
Page 42 and 43: Using this diffeomorphism one may s
Page 44 and 45: lim dtdtr→∞∫I ′ dφdφ ′
Page 46 and 47: dS static patch (spatial)dg =0 g =1
Page 48 and 49: • Lower dimensional Plank mass is
Page 52 and 53: Under this conformal map one has:
Page 54 and 55: Conformal AnomalyAdS/CFT instructs
Page 56 and 57: • Naively one would think that gr
Page 58: This talk is based on the following

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