The d-Dimensional Cosmological Constant and the Holographic Horizons

TL;DR

This paper introduces a novel approach to the cosmological constant as an eigenvalue in a Sturm--Liouville problem with holographic boundary conditions, deriving spectra and exploring implications for horizons and entropy in various dimensions.

Contribution

It formulates a new boundary condition-based framework for the cosmological constant problem, deriving spectra in d-dimensions and analyzing horizon entropy modifications.

Findings

Derived the spectrum of vacuum energy density in d-dimensions.

Solved the 2D case analytically using hypergeometric functions.

Discussed implications of fractal horizons and alternative entropy formulas.

Abstract

This article is dedicated to establishing a novel approach to the cosmological constant, in which it is treated as an eigenvalue of a certain Sturm--Liouville problem. The key to this approach lies in the proper formulation of physically relevant boundary conditions. Our suggestion in this regard is to utilize the ``holographic boundary condition'', under which the cosmological horizon can only bear a natural (i.e., non-fractional) number of bits of information. Under this framework, we study the general d-dimensional problem and derive the general formula for the discrete spectrum of a positive energy density of vacuum. For the particular case of two dimensions, the resultant problem can be analytically solved in the degenerate hypergeometric functions, so it is possible to define explicitly a self-action potential, which determines the fields of matter in the model. We conclude the article by taking a look at the d-dimensional model of a fractal horizon, where the Bekenstein's formula for the entropy gets replaced by the Barrow entropy. This gives us a chance to discuss a recently realized problem of possible existence of naked singularities in the $D\neq 3$ models.