Modave Lecture Notes on de Sitter Space & Holography

TL;DR

The notes survey de Sitter space from its classical geometry to horizon thermodynamics and semi-classical probes, creating a framework for holographic interpretations. They juxtapose global dS/CFT concepts with observer-centered static-patch approaches, emphasizing timelike boundaries, horizon entropy, and quasi-local thermodynamics. The discussion highlights concrete holographic proposals in various dimensions: conformal boundary data for $d\ge4$, solvable $T\bar{T}+\Lambda_2$ deformations in $d=3$, and open quantum-mechanical duals in $d=2$, while noting ongoing challenges such as observer-dependence, finite entropy, and potential nonunitarity. Collectively, the notes map out promising directions to translate dS physics into a holographic language and connect them with lessons from AdS/CFT and holographic chaos.

Abstract

These lecture notes provide an overview of different aspects of de Sitter space and their plausible holographic interpretations. We start with a general description of the classical spacetime. We note the existence of a cosmological horizon and its associated thermodynamic quantities, such as the Gibbons-Hawking entropy. We discuss geodesics and shockwave solutions, that might play a role in a holographic description of de Sitter. Finally, we discuss different approaches to quantum theories of de Sitter space, with an emphasis on recent developments in static patch holography.

Figures (14)

• Figure 1: (a) Temperature fluctuations of the CMB, taken from refId0. Measurements are consistent with cosmic inflationary models. (b) Hubble diagram for the Union 2.1 compilation of type Ia supernovae. The solid line is the best-fit of the data, consistent with a $\Lambda$CDM flat cosmology with $\Omega_\Lambda \sim 0.7$. Figure taken from Suzuki_2012. In dashed blue, we added for comparison the prediction obtained for a flat Universe with no cosmological constant.
• Figure 2: $\Lambda$CDM model: $68.3\%, 95.4\%,$ and $99.7\%$ confidence regions of the $(\Omega_m,\Omega_\Lambda)$ plane from SNe Ia combined with the constraints from BAO and CMB. Figure from Suzuki_2012.
• Figure 3: De Sitter in 2d as the hyperboloid embedded in 3d Minkowski space. Constant $X_0$ slices are circles with varying radius, some of which are drawn in blue.
• Figure 4: Penrose diagram of de Sitter space.
• Figure 5: The intersection between the regions that can be affected by and affect an observer is called the static patch. In \ref{['fig:static_3']}, we show constant $t$ slices in dotted black and constant $r$ slices in dashed brown.