Holographic entropy bound and a special class of spherical systems in cosmology

TL;DR

The paper investigates how to preserve the holographic entropy bound in cosmology by defining special spherical systems whose entropy stays within the bound through cosmic evolution. It develops two approaches: (i) a bound based on a finite entropy per unit co-moving volume, which yields a radii scaling $R_c\propto a^3$ in certain cosmologies such as a universe with black holes and a cosmological constant; and (ii) a general framework using the maximal BH entropy as an auxiliary bound, leading to a universal $R_c$ expressed in terms of the Hubble rate and scale factor. The results show that while these definitions can enforce the bound under specific conditions, they encounter counterexamples (e.g., photon-dominated or singular early-universe scenarios) and do not yet provide a completely universal, component-independent construction. The work highlights the subtle role of the maximal entropy state and points to the need for a clearer identification of this state to formulate a universally valid holographic cosmological bound with practical significance.

Abstract

The holographic entropy bound is discussed in cosmology. Inspired by the work of Fischler and Susskind [hep-th/9806039], we aim to define a special class of spherical systems in cosmology, within which the entropy of matter remains compliant with the holographic entropy bound throughout the evolution of the universe, irrespective of the universe's components. It is found that if the entropy of matter per unit co-moving volume is bounded from above, such a special class of spherical systems indeed exists. Moreover, the matter contained within a unit co-moving volume can be replaced by a black hole of the same mass-energy. Provided that the entropy of the black hole consistently exceeds that of the matter it replaces, there is also a unified definition for these special spherical systems.