Strongly coupled CFT in FRW universe from AdS/CFT correspondence

We develop a formalism to calculate the effective action of the strongly co upled conformal field theory (CFT) in curved spacetime. The effective actio n of the CFT is obtained from AdS/CFT correspondence. The anti de-Sitter (A dS) spacetime has various slicing which give various curved spacetime on it s boundary. We show the de Sitter spacetime and the Friedmann-Robertson-Wal ker (FRW) universe can be embedded in the AdS spacetime and derive the scal ar two-point function of the conformal fields in those spacetime. In curved spacetime, the two-point function depends on the vacuum state of the CFT. A method to specify the vacuum state in AdS/CFT calculations is shown. Beca use the classical action in AdS spacetime diverges near the boundary, we ne ed the counter terms to regulate the result. The simple derivation of the c ounter terms using the Hamilton-Jacobi equation is also presented in the ap pendix.