Asymptotically de Sitter dilatonic space - time , holographic RG flow and conformal anomaly from ( dilatonic ) dS / CFT correspondence

The number of asymptotically de Sitter (non-singular) solutions of 5d dilatonic gravity with positive cosmological constant is found. These solutions are similar to the previously known asymptotically AdS spaces where dilaton may generate the singularity. Using these solutions the consistent c-function is proposed in the same way as in AdS/CFT. The consistency of RG flow gives further support for dS/CFT correspondence. From holographic RG flow equations we calculate the holographic 4d conformal anomaly with dilatonic contributions. This conformal anomaly turns out to be the same as in AdS/CFT correspondence. PACS numbers: 04.50.+h, 04.70.Dy, 11.25.Db [email protected] 2 [email protected], [email protected] 1 The universality of holographic principle manifests itself in the form of AdS/CFT correspondence [1] as well as recently proposed dS/CFT correspondence [2, 3]. Despite the numerous attempts (for not complete list of refs. on dS/CFT correspondence, see [4, 5, 6]) dS/CFT set-up is mainly studied from the gravitational side and it is not well understood. One of the fundamental problems in the development of dS/CFT is the lack of the suitable dual CFT which is presumably Euclidean and not unitary. Probably it is related with the fact that de Sitter space unlike to AdS is very difficult to realize as vacuum space of string theory. Nevertheless, it is became clear that most of relations discussed in AdS/CFT have their analogs also in dS/CFT [5]. In the present letter we extend the study of holographic RG flows developed in AdS/CFT to asymptotically de Sitter dilatonic spaces. In particulary, the asymptotically de Sitter (non-singular) solution of 5d dilatonic gravity is found. Using this solution (with non-trivial dilaton) the definition of c-function in dS/CFT correspondence is proposed. The proposed c-function which is similar (up to the sign of the potential) to the AdS cfunction gives further support for dS/CFT set-up. The holographic RG flow equations are written. Using them the dilatonic conformal anomaly which turns out to be the same as in AdS/CFT is found. We start from the following action of dilatonic gravity in d+1 dimensions: S = − 1 16πG ∫ dx √ −G (R− Λ− αG∂μφ∂νφ) . (1) We now assume λ ≡ Λ to be positive. From the variation of the action (1) with respect to the metric G , one obtains 0 = Rμν − 1 2 GμνR + Λ 2 Gμν − α ( ∂μφ∂νφ− 1 2 GμνG ∂ρφ∂σφ ) (2) and from that of dilaton φ 0 = ∂μ (√ −GG∂νφ ) . (3) 3 The conventions of curvatures are given by R = GRμν Rμν = −Γμλ,κ + Γλμκ,λ − ΓημλΓκη + ΓημκΓλλη Γημλ = 1 2 G ην (Gμν,λ + Gλν,μ − Gμλ,ν) .