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Entropy Bounds and Holographic Dark Energy: Conflicts and Consensus

Manosh T. Manoharan

TL;DR

The work interrogates the UV–IR entropic bound proposed by Cohen–Kaplan–Nelson, clarifying how different entropy definitions and causal considerations yield distinct cosmological implications. It introduces a new von Neumann entropy bound based on entanglement across a boundary, and compares it with the CKN and Bekenstein bounds within the causal entropy bound (CEB) framework, showing how each saturates to matter-, radiation-, or black-hole-like scaling. The analysis critically examines holographic dark energy (HDE), highlighting circularities in using $L_{IR} \sim H^{-1}$ or the future event horizon, and shows that IR cut-offs involving derivatives of the Hubble function (GO, Ricci, Gauss–Bonnet) can reproduce late-time acceleration but rely on integration constants that are not predicted by quantum field theory. The overarching message is that the CKN UV–IR relation does not fix a cosmological constant, but rather explains why its observed value is small through UV–IR mixing and horizon thermodynamics, while HDE offers a phenomenological track that often introduces unresolved constants or model dependencies. Collectively, the paper clarifies how entropy bounds interface with cosmology, horizon thermodynamics, and the search for a holographic description of dark energy, pointing toward a consistent framework where the bounds inform but do not unilaterally determine the late-time dynamics of the universe.

Abstract

Cohen, Kaplan, and Nelson's influential paper established that the UV-IR cut-offs cannot be arbitrarily chosen but are constrained by the relation $Λ^2 L \lesssim M_p$. Here, we revisit the formulation of the CKN entropy bound and compare it with other bounds. The specific characteristics of each bound are shown to depend on the underlying scaling of entropy. Notably, employing a non-extensive scaling with the von Neumann entropy definition leads to a more stringent constraint, $S_{\text{max}} \approx \sqrt{S_{\text{BH}}}$. We also clarify distinctions between the IR cut-offs used in these frameworks. Moving to the causal entropy bound, we demonstrate that it categorises the CKN bound as matter-like, the von Neumann bound as radiation-like, and the Bekenstein bound as black hole-like systems when saturated. Emphasising cosmological implications, we confirm the consistency between the bounds and the first laws of horizon thermodynamics. We then analyse the shortcomings in standard Holographic Dark Energy (HDE) models, highlighting the challenges in constructing HDE using $Λ^2 L \lesssim M_p$. Specifically, using the Hubble function in HDE definitions introduces circular logic, causing dark energy to mimic the second dominant component rather than behaving as matter. We further illustrate that the potential for other IR cut-offs, like the future event horizon in an FLRW background or those involving derivatives of the Hubble function, to explain late-time acceleration stems from an integration constant that cannot be trivially set to zero. In brief, the CKN relation doesn't assign an arbitrary cosmological constant; it explains why its value is small.

Entropy Bounds and Holographic Dark Energy: Conflicts and Consensus

TL;DR

The work interrogates the UV–IR entropic bound proposed by Cohen–Kaplan–Nelson, clarifying how different entropy definitions and causal considerations yield distinct cosmological implications. It introduces a new von Neumann entropy bound based on entanglement across a boundary, and compares it with the CKN and Bekenstein bounds within the causal entropy bound (CEB) framework, showing how each saturates to matter-, radiation-, or black-hole-like scaling. The analysis critically examines holographic dark energy (HDE), highlighting circularities in using or the future event horizon, and shows that IR cut-offs involving derivatives of the Hubble function (GO, Ricci, Gauss–Bonnet) can reproduce late-time acceleration but rely on integration constants that are not predicted by quantum field theory. The overarching message is that the CKN UV–IR relation does not fix a cosmological constant, but rather explains why its observed value is small through UV–IR mixing and horizon thermodynamics, while HDE offers a phenomenological track that often introduces unresolved constants or model dependencies. Collectively, the paper clarifies how entropy bounds interface with cosmology, horizon thermodynamics, and the search for a holographic description of dark energy, pointing toward a consistent framework where the bounds inform but do not unilaterally determine the late-time dynamics of the universe.

Abstract

Cohen, Kaplan, and Nelson's influential paper established that the UV-IR cut-offs cannot be arbitrarily chosen but are constrained by the relation . Here, we revisit the formulation of the CKN entropy bound and compare it with other bounds. The specific characteristics of each bound are shown to depend on the underlying scaling of entropy. Notably, employing a non-extensive scaling with the von Neumann entropy definition leads to a more stringent constraint, . We also clarify distinctions between the IR cut-offs used in these frameworks. Moving to the causal entropy bound, we demonstrate that it categorises the CKN bound as matter-like, the von Neumann bound as radiation-like, and the Bekenstein bound as black hole-like systems when saturated. Emphasising cosmological implications, we confirm the consistency between the bounds and the first laws of horizon thermodynamics. We then analyse the shortcomings in standard Holographic Dark Energy (HDE) models, highlighting the challenges in constructing HDE using . Specifically, using the Hubble function in HDE definitions introduces circular logic, causing dark energy to mimic the second dominant component rather than behaving as matter. We further illustrate that the potential for other IR cut-offs, like the future event horizon in an FLRW background or those involving derivatives of the Hubble function, to explain late-time acceleration stems from an integration constant that cannot be trivially set to zero. In brief, the CKN relation doesn't assign an arbitrary cosmological constant; it explains why its value is small.

Paper Structure

This paper contains 13 sections, 28 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Entropy Bounds -- Origin and Assumptions
  3. Cosmologists versus CKN
  4. IR cut-off vs Boundary
  5. Von Neumann Entropy Bound
  6. Dynamic Boundary and Causality
  7. Causal Entropy Bound & Cosmology
  8. Implications of Causal Entropy Bounds
  9. Holographic Dark Energy -- Connections with Thermodynamics
  10. What was wrong with the Hubble cut-off?
  11. What about particle and future event horizons?
  12. Why is only Dark Energy considered holographic in HDE?
  13. Infrared Cut-off with Derivatives of the Hubble Function

Figures (2)

  • Figure 1: A lattice of IR cut-off $L_{IR}$ and UV cut-off $\epsilon\sim1/\Lambda$ along with an arbitrary boundary at $r_{A}$.
  • Figure 2: Scaling of entropy bounds against the black hole entropy. Grey: violation of all entropy bounds. White: The CKN and von Neumann bounds are violated. Blue: The von Neumann bound violation. Red: All bounds are satisfied.