de Sitter space and Holography

In d dimensions the Einstein equations with positivecosmological constant can be derived from the actionI= I bulk + I GH∫1− d d x √ −g(R + 2Λ) + 116πG8πGM∫∂Md d−1 x √ hKhere• I GH is Gibbons-Hawking surface term which is needed toget a well-defined Euler-Lagrange variation.• M is d-dimensional Manifold with Newton’s constant Gwith spatial Euclidean boundary ∂M.• g µν is the bulk metric.• h µν and K are induced metric and the trace of the extrinsiccurvature of the boundary. The extrinsic curvature is definedby K µν = −∇ (µ n ν) where n ν is outward pointing unit vector.• A useful length scale in the model is given byl =√(d − 1)(d − 2)2ΛFor example in the vacuum dS solution, l is the radius of dSspace.32