Holographic Cosmology at Finite Time

TL;DR

This work develops a holographic framework for cosmology by applying a $T^2$-deformation to a seed dS/CFT model, yielding a boundary theory that lives on finite-time Cauchy slices with time emerging from RG flow. The authors compute scalar and graviton two-point functions both from bulk wavefunctions and via the boundary flow, finding precise agreement after holographic renormalization with imaginary counterterms and a nonlinear mapping to cosmological correlators. They analyse complex scaling dimensions (including the principal series) and construct deformation operators across several $ extDelta$-strips, demonstrating analytic continuation that connects complementary and principal series results. The results illuminate how CSH can reproduce finite-time cosmological observables and offer insights into naturalness, RG flow interpretation of time, and the emergence of spacetime from holographic data, with potential applications to more general FLRW cosmologies.

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 $Δ$, 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.

Figures (6)

• Figure 1: A schematic illustration of how the hypothetical boundary "dS/CFT’’ theory can be deformed by an appropriately defined irrelevant$T^2$ deformation operator (denoted schematically by $\hat{O}_{T^2}$, whose precise form depends on the bulk field content) to produce the holographically renormalised wavefunction $\Psi_{\text{ren}}$ at finite time. To reconstruct the unrenormalised bulk wavefunction $\Psi_{\text{bulk}}$, one must then reintroduce the power-law divergent terms through the addition of counterterms that are relevant from a boundary RG perspective.
• Figure 2: The patch covered by our planar coordinates. The bulk Hilbert space will live on these slices. Notice that as $\eta \to-\infty$ the slice approaches the null horizon, which agrees with the d-volume $a \to 0$. In Section \ref{['sec:Boundary']} we will show how to interpret this time flow as a $T^2$ deformation on the dual boundary side, as indicated by the blue arrow here.
• Figure 3: The bulk state $\Psi$ is defined on an abstract spatial slice $\Sigma$ with boundary $\partial\Sigma$. The extra dimension is emergent, via the fact that the state satisfies the Hamiltonian constraint. The fundamental equation of CSH is shown. On the RHS we have a QFT partition function living on the same $\Sigma$. On the boundary $\partial\Sigma$ one has to input a QFT state, which will be mapped to the bulk state $\Psi$ via the evaluation of the partition function $Z$.
• Figure 4: The seed CFT is deformed by an irrelevant operator quadratic in the stress-tensor. The flow effectively deforms the algebra of local (anomalous) scale invariance to the "algebra" of local diffeomorphism invariance. The Weyl anomaly constraint gets deformed to the Hamiltonian constraint. The value of the deformation parameter is in one-to-one correspondence with the conformal time labelling bulk slices in Figure \ref{['fig:dS']}.
• Figure 5: Phase of the tree-level contribution (i.e. leading in $1/N$) to the central charge $\mathbf{c}$ in the complex plane for different dimensions and signs of the cosmological constant. For $\Lambda>0$ the central charge $\mathbf{c}_{\text{dS}}$ rotates counterclockwise by $\pi/2$ with each increment in dimension.