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De Sitter Space, Double-Scaled SYK, and the Separation of Scales in the Semiclassical Limit

Leonard Susskind

TL;DR

The paper analyzes a two-scale separation in the semiclassical limit of de Sitter space and its proposed holographic dual, the infinite-temperature double-scaled SYK model (DSSYK_infty). It develops a concrete dictionary linking cosmic scales (L_c, S_ds) with microscopic and emergent string scales (L_m, L_s) and shows how quantities such as H_c and H_s, along with the parameter λ = (L_m/L_s)^2, map across the dS/dSSYK_infty duality. Through perturbation theory, emergent string-scale dynamics, tomperature, scrambling, and correlation functions, the work argues for a parallel decoupling of cosmic and micro sectors in the semiclassical limit, with 1/N corrections providing the bridge between sectors. It also discusses the algebraic structure of observables, proposing a two-sector (cosmic and string) algebra that captures the distinct scales and their interplay, and highlights both the supporting evidence and the need for further rigorous proofs of the duality. The findings have potential implications for understanding holographic descriptions of de Sitter space and the role of emergent scales in quantum gravity models.

Abstract

In the semiclassical limit of de Sitter gravity a separation of scales takes place that divides the theory into a "cosmic" sector and a "microscopic" sector. A similar separation takes place in the double-scaled limit of SYK theory. We examine the scaling behaviors that accompany these limits and find parallels that support the previously conjectured duality between Jackiw-Teitelboim gravity (with positive cosmological constant), and double-scaled SYK. This paper is a companion to "dS JT Gravity and Double-Scaled SYK" by Adel Rahman, to appear simultaneously with this paper.

De Sitter Space, Double-Scaled SYK, and the Separation of Scales in the Semiclassical Limit

TL;DR

The paper analyzes a two-scale separation in the semiclassical limit of de Sitter space and its proposed holographic dual, the infinite-temperature double-scaled SYK model (DSSYK_infty). It develops a concrete dictionary linking cosmic scales (L_c, S_ds) with microscopic and emergent string scales (L_m, L_s) and shows how quantities such as H_c and H_s, along with the parameter λ = (L_m/L_s)^2, map across the dS/dSSYK_infty duality. Through perturbation theory, emergent string-scale dynamics, tomperature, scrambling, and correlation functions, the work argues for a parallel decoupling of cosmic and micro sectors in the semiclassical limit, with 1/N corrections providing the bridge between sectors. It also discusses the algebraic structure of observables, proposing a two-sector (cosmic and string) algebra that captures the distinct scales and their interplay, and highlights both the supporting evidence and the need for further rigorous proofs of the duality. The findings have potential implications for understanding holographic descriptions of de Sitter space and the role of emergent scales in quantum gravity models.

Abstract

In the semiclassical limit of de Sitter gravity a separation of scales takes place that divides the theory into a "cosmic" sector and a "microscopic" sector. A similar separation takes place in the double-scaled limit of SYK theory. We examine the scaling behaviors that accompany these limits and find parallels that support the previously conjectured duality between Jackiw-Teitelboim gravity (with positive cosmological constant), and double-scaled SYK. This paper is a companion to "dS JT Gravity and Double-Scaled SYK" by Adel Rahman, to appear simultaneously with this paper.
Paper Structure (24 sections, 79 equations, 13 figures)

This paper contains 24 sections, 79 equations, 13 figures.

Table of Contents

  1. Introduction
  2. J-T from 3-D De Sitter
  3. Mass and Length Scales in dS
  4. 4-D
  5. 3-D
  6. String Scale
  7. DSSYK$_{\bf {\infty}}$
  8. Perturbation Theory
  9. The Emergent String Scale
  10. Tomperature and the Cosmic Scale
  11. DSSYK- De Sitter Correspondences
  12. Gauge and DSSYK$_{\infty}$ Correpondences
  13. Gauge Theory and DSSYK$_{\infty}$ Scaling
  14. Note on L$_m$
  15. DSSYK Is Like the Flat Space Limit of AdS/CFT
  16. ...and 9 more sections

Figures (13)

  • Figure 1: Penrose diagram for de Sitter space showing two static patches. The green surface represents the stretched horizon of the right-side static patch.
  • Figure 2: Mass scales in 4 dimensional dS. The horizontal axis represents $\log{M}.$ Mass scales increase to the right and length scales decrease.
  • Figure 3: Mass scales in 3 dimensional dS
  • Figure 4: Mass scales in 3 dimensional including the string scale. Typically the string mass scale will lie near but somewhat below the geometric mean of $M_{min}$ and the Planck mass.
  • Figure 5: The simplest vacuum melon diagram.
  • ...and 8 more figures