Holographic Cosmology at Finite Time

Abstract

We investigate Cauchy Slice Holography in de Sitter spacetime. By performing a $T^2$ deformation of a (bottom-up) dS/CFT model, we obtain a holographic theory living on flat Cauchy slices of de Sitter, for which time is an emergent dimension, associated with an RG flow. In this $T^2$-deformed field theory, the dS/CFT is an IR fixed point rather than a UV fixed point, potentially affecting discussions of naturalness. As in the case of AdS/CFT, the terms in the $T^2$ deformation depend on the dimension and the bulk matter sector; in this article we consider gravity, plus optionally a scalar field of arbitrary mass. We compute scalar and graviton two-point correlation functions in the deformed boundary theory, and demonstrate precise agreement with finite-time wavefunction coefficients, which we calculate independently on the bulk side. The results are analytic in the scalar field dimension $\Delta$, and may therefore be continued to arbitrary generic values, including the principal series. Although many aspects of the calculations are similar to the AdS/CFT case, some new features arise due to the complex phases which appear in cosmology. Our calculations confirm previous expectations that the holographic counterterms are purely imaginary, when expressed in terms of wavefunction coefficients. But cosmological correlators, calculated by the Born rule, are shifted in a more complicated and nonlinear way.