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Infinite Temperature is Not So Infinite: The Many Temperatures of de Sitter Space

Adel A. Rahman, Leonard Susskind

TL;DR

The paper clarifies that de Sitter holography entails multiple temperatures, analyzed within the concrete DSSYK$_{\infty}$ framework. It identifies Boltzmann, Tomperature, and cord temperatures and links them to bulk quantities via a bulk/dictionary including a blue-shift explanation that reconciles discrepancies between cord temperatures and Hawking temperatures. It shows that $T_B$ controls $1/N$ entropy corrections, while Tomperature aligns with the Hawking temperature seen at the pode and cord temperature describes the stretched-horizon local environment, with a large blue-shift between these scales. The work also discusses singlet cords that can probe bulk physics, and the flat-space limit in which cords resemble strings, proposing plausible directions for a flat-space cord theory. Overall, the results support the DSSYK$_{\infty}$–de Sitter duality across scales and temperatures, while outlining deep open questions about cord dynamics and cosmological implications.

Abstract

Several distinct concepts of temperature appear in the holographic description of de Sitter space. Conflating these has led to confusion and inconsistent claims. The double-scaled limit of SYK is a concrete model in which we can examine and explain these different concepts of temperature. This note began as an addendum to our paper ``Comments on a Paper by Narovlansky and Verlinde" but in the process of writing it we learned new things -- interesting in their own right -- that we wish to report here.

Infinite Temperature is Not So Infinite: The Many Temperatures of de Sitter Space

TL;DR

The paper clarifies that de Sitter holography entails multiple temperatures, analyzed within the concrete DSSYK framework. It identifies Boltzmann, Tomperature, and cord temperatures and links them to bulk quantities via a bulk/dictionary including a blue-shift explanation that reconciles discrepancies between cord temperatures and Hawking temperatures. It shows that controls entropy corrections, while Tomperature aligns with the Hawking temperature seen at the pode and cord temperature describes the stretched-horizon local environment, with a large blue-shift between these scales. The work also discusses singlet cords that can probe bulk physics, and the flat-space limit in which cords resemble strings, proposing plausible directions for a flat-space cord theory. Overall, the results support the DSSYK–de Sitter duality across scales and temperatures, while outlining deep open questions about cord dynamics and cosmological implications.

Abstract

Several distinct concepts of temperature appear in the holographic description of de Sitter space. Conflating these has led to confusion and inconsistent claims. The double-scaled limit of SYK is a concrete model in which we can examine and explain these different concepts of temperature. This note began as an addendum to our paper ``Comments on a Paper by Narovlansky and Verlinde" but in the process of writing it we learned new things -- interesting in their own right -- that we wish to report here.
Paper Structure (16 sections, 67 equations, 5 figures)

This paper contains 16 sections, 67 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The Boltzmann Temperature and Corrections to Entropy
  3. Tomperature and Hawking Temperature
  4. Chords, Cords, and Strings
  5. Almost Everything is Confined
  6. Singlets and the Flat Space Limit: A Challenge
  7. Back to Generic Cords
  8. Real and Fake Disks
  9. Correlation Functions and Hawking Temperature
  10. The Blue Shift Factor
  11. Summary
  12. Input Parameters
  13. The Temperatures
  14. The Transition Region
  15. Conclusions
  16. ...and 1 more sections

Figures (5)

  • Figure 1: The various types of temperature which appear in the analysis of DSSYK$_{\infty}$ along with their values in the two major units systems (cosmic and string).
  • Figure 2: The relationship between $\uptau_{\mathrm{cord}}$ and $\uptau_B$ measured in multiples of $\mathcal{J}_0$.
  • Figure 3: Top: A spatial slice of dS$_3$; Bottom: A spatial slice of NAdS$_2$ equipped with constant dilaton. The dashed grey line is the horizon.
  • Figure 4: The transition of a spatial slice from low to high temperatures. The dashed grey line is the horizon and the dashed orange lines are the Schwarzian boundaries/domain walls.
  • Figure 5: Spacetime (i.e. Penrose diagram) picture of the transition from low (left) to high (right) temperatures.