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Entropy-area law and temperature of de Sitter horizons from modular theory

Edoardo D'Angelo, Markus B. Fröb, Stefano Galanda, Paolo Meda, Albert Much, Kyriakos Papadopoulos

Abstract

We derive an entropy-area law for the future horizon of an observer in diamonds inside the static patch of de Sitter spacetime, taking into account the backreaction of quantum matter fields. We prove positivity and convexity of the relative entropy for coherent states using Tomita--Takesaki modular theory, from which the QNEC for diamonds follows. Furthermore, we show that the generalized entropy conjecture holds. Finally, we reveal that the local temperature which is measured by an observer at rest exhibits subleading quantum corrections with respect to the well-known cosmological horizon temperature $H/(2π)$.

Entropy-area law and temperature of de Sitter horizons from modular theory

Abstract

We derive an entropy-area law for the future horizon of an observer in diamonds inside the static patch of de Sitter spacetime, taking into account the backreaction of quantum matter fields. We prove positivity and convexity of the relative entropy for coherent states using Tomita--Takesaki modular theory, from which the QNEC for diamonds follows. Furthermore, we show that the generalized entropy conjecture holds. Finally, we reveal that the local temperature which is measured by an observer at rest exhibits subleading quantum corrections with respect to the well-known cosmological horizon temperature .
Paper Structure (6 sections, 28 equations, 3 figures)

This paper contains 6 sections, 28 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Relative entropy
  3. Half-sided modular inclusions and QNEC
  4. Entropy-area law from the modular flow and backreaction
  5. Local temperature
  6. Conclusion and outlook

Figures (3)

  • Figure 1: Geometry of the patches of de Sitter space relevant for our universe. On the left, the expanding Poincaré patch in conformally flat coordinates $(\eta,{\mathbfi{x}})$ with $r = {\mathopen{}\mathclose{\left\lvert{{\mathbfi{x}}}\right\rvert}}$; on the right, the static patch in static coordinates $(T,{\mathbfi{X}})$ with $R = {\mathopen{}\mathclose{\left\lvert{{\mathbfi{X}}}\right\rvert}}$. The cosmological horizon is situated at $R = H^{-1}$.
  • Figure 2: The diamonds relevant for the half-sided modular inclusion. Note that their future boundary coincides, and that we take a test function $f$ whose support is inside the smaller diamond.
  • Figure 3: De Sitter diamond of size $2\ell$ whose future coincides with the cosmological horizon at $v = 0$. Since we are considering massless fields in even dimensions, the horizon is perturbed only for some null time $u \in [u_0,u_1] \subset [-2\ell,0]$.