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Localized Gravity, de Sitter, and the Horizon Criterion

Bjoern Hassfeld, Arthur Hebecker, Daniel Schiller

TL;DR

This work refines the Horizon Criterion for quantum gravity by focusing on spacetimes with end-of-the-world branes and introducing the notion of brane-brane horizons, which occur when two observers on the same ETW brane cannot remain in causal contact. It derives explicit thresholds on brane-localized and bulk scalar potentials—$V(\phi) \sim \frac{c^2}{\phi^2}$ with $c>c_{\rm crit}$ on the brane and $V(\phi)=V_0 e^{-\lambda\phi}$ with $\lambda<\lambda_{\rm crit}$ in the bulk—that guarantee brane-brane horizons, using autonomous-system analyses and Friedmann-like equations on the brane. The paper analyzes spherical and flat ETW brane embeddings, showing how backreaction and boundary conditions modify the horizon criteria and produce distinct critical exponents, including $c_{\rm crit}=\sqrt{2(4d-9)}$ and $\lambda_{\rm crit}=1/\sqrt{d-2}$. It also discusses how RS-type horizons can exist without ruling out models in string theory, suggesting that horizons alone may not fully single out the swampland and highlighting the need for a deeper, possibly causal-isolation-based refinement. Overall, the results sharpen our understanding of when cosmological horizons on branes signal fundamental inconsistencies and point to concrete potential realizations and obstructions within string theory settings.

Abstract

Realizing de Sitter-like solutions in string theory remains challenging, prompting speculation about which specific feature might be responsible for their inconsistency in quantum gravity. In this work, we focus on the `Horizon Criterion', which identifies spacetimes as problematic if they exhibit cosmological horizons. In particular, we study the implications for spacetimes with dynamical boundaries. We argue that requiring inertial observers localized on an end-of-the-world (ETW) brane to be in causal contact with every other observer is too restrictive as there exist string-theoretic solutions without this property. Hence, if one does not want to abandon the idea of cosmological horizons being the fundamental issue with de Sitter, a refined condition is needed. The requirement that inertial, boundary-localized observers should be in causal contact with all other observers on the same ETW brane is such an appropriate refinement. We explore the consequences of this criterion for ETW branes whose energy density is governed by a scalar field, considering two cases: First, with a scalar field confined to the ETW brane, and second, with a bulk modulus subject to a brane-localized potential.

Localized Gravity, de Sitter, and the Horizon Criterion

TL;DR

This work refines the Horizon Criterion for quantum gravity by focusing on spacetimes with end-of-the-world branes and introducing the notion of brane-brane horizons, which occur when two observers on the same ETW brane cannot remain in causal contact. It derives explicit thresholds on brane-localized and bulk scalar potentials— with on the brane and with in the bulk—that guarantee brane-brane horizons, using autonomous-system analyses and Friedmann-like equations on the brane. The paper analyzes spherical and flat ETW brane embeddings, showing how backreaction and boundary conditions modify the horizon criteria and produce distinct critical exponents, including and . It also discusses how RS-type horizons can exist without ruling out models in string theory, suggesting that horizons alone may not fully single out the swampland and highlighting the need for a deeper, possibly causal-isolation-based refinement. Overall, the results sharpen our understanding of when cosmological horizons on branes signal fundamental inconsistencies and point to concrete potential realizations and obstructions within string theory settings.

Abstract

Realizing de Sitter-like solutions in string theory remains challenging, prompting speculation about which specific feature might be responsible for their inconsistency in quantum gravity. In this work, we focus on the `Horizon Criterion', which identifies spacetimes as problematic if they exhibit cosmological horizons. In particular, we study the implications for spacetimes with dynamical boundaries. We argue that requiring inertial observers localized on an end-of-the-world (ETW) brane to be in causal contact with every other observer is too restrictive as there exist string-theoretic solutions without this property. Hence, if one does not want to abandon the idea of cosmological horizons being the fundamental issue with de Sitter, a refined condition is needed. The requirement that inertial, boundary-localized observers should be in causal contact with all other observers on the same ETW brane is such an appropriate refinement. We explore the consequences of this criterion for ETW branes whose energy density is governed by a scalar field, considering two cases: First, with a scalar field confined to the ETW brane, and second, with a bulk modulus subject to a brane-localized potential.
Paper Structure (30 sections, 157 equations, 1 figure)

This paper contains 30 sections, 157 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Cosmological horizons, ETW branes, and the swampland:
  3. The Refined Horizon Criterion:
  4. Preliminary Reflections on Horizon Criteria
  5. On different types of observers
  6. Brane Observer vs. Bulk Observer:
  7. Brane Observer vs. Rindler Observer:
  8. Potential counterexamples
  9. Alternative formulations of a criterion against de-Sitter-like spacetimes
  10. On a conceivable 'Eternal Compact Universe Criterion':
  11. On a conceivable 'Local Asymptotic Acceleration Criterion':
  12. ETW Brane with brane-localized Scalar
  13. Spherical ETW Brane
  14. ETW Branes with flat Slicing
  15. Implications for the Potential
  16. ...and 15 more sections

Figures (1)

  • Figure 1: The Penrose diagram of $d$-dimensional AdS with a flat Randall-Sundrum UV brane embedded (red-green dashed line). The gray region corresponds to a portion of a Poincaré patch of AdS. The worldline $\sigma$ of a brane-bound observer is illustrated in green. It coincides with the location of the RS brane in this illustration, which is why the spacetime boundary is illustrated as a red-green dashed line. One observes that there exists a point p whose future domain of influence does not intersect $\sigma$.