[Paper Review] Spacetime Dimensionality from de Sitter Entropy

Arshad Momen, Rakibur Rahman|arXiv (Cornell University)|Jun 22, 2011

Cosmology and Gravitation Theories20 references3 citations

TL;DR

This paper argues that de Sitter universes with a small cosmological constant are entropically favored to have three spatial dimensions, based on the causal-patch description of de Sitter space, the holographic principle, and assumptions about Hawking/Unruh radiation. Using entropy-to-energy ratios derived from relativistic gas models in (d+1)-dimensional spacetime, it finds that entropy per unit energy peaks at d ≈ 2.97, strongly favoring three spatial dimensions.

ABSTRACT

We argue that de Sitter universes with a small cosmological constant are entropically favored to have three spatial dimensions. The conclusion relies on the causal-patch description of de Sitter space, where fiducial observers experience local thermal equilibrium up to a stretched horizon, on the holographic principle, and on some assumptions about the nature of gravity and the constituents of Hawking/Unruh radiation.

Motivation & Objective

• To investigate whether spacetime dimensionality can be explained by entropy maximization in the late-time de Sitter phase of the universe.
• To determine whether three spatial dimensions are entropically favored over other dimensions in asymptotically de Sitter universes.
• To apply the causal-patch approach and holographic principle to compute entropy in higher-dimensional de Sitter spaces.
• To model Hawking/Unruh radiation as a gas of photons and gravitons to estimate degrees of freedom and entropy in different dimensions.
• To derive a dimension-dependent entropy-to-energy ratio and identify its maximum in the range d ∈ [2,10].

Proposed method

• Adopt the causal-patch description of de Sitter space, where observers are confined to a finite region bounded by a stretched horizon.
• Use the Gibbons-Hawking temperature and Tolman redshift factor to model local thermal equilibrium at the stretched horizon.
• Assume the Hawking/Unruh radiation consists of massless photons and gravitons, with d−1 and ½(d+1)(d−2) degrees of freedom respectively.
• Derive the entropy-to-energy ratio S/E in (d+1)-dimensional spacetime using relativistic gas statistics and the holographic principle.
• Approximate the entropy expression for small Hubble parameter and finite cutoff at the Planck scale, yielding a dimension-dependent S/E function.
• Maximize S/E as a function of d over the range 2 ≤ d ≤ 10 to identify the dimension with maximal entropy per unit energy.

Experimental results

Research questions

• RQ1 Is there a preferred spacetime dimensionality that maximizes entropy in a de Sitter universe with a small cosmological constant?
• RQ2 How does the entropy-to-energy ratio depend on the number of spatial dimensions in a causal-patch description of de Sitter space?
• RQ3 Can the observed three spatial dimensions be explained by entropic selection in the late-time de Sitter phase?
• RQ4 What role do the degrees of freedom of photons and gravitons play in determining the dimensionality of the entropy-maximizing spacetime?
• RQ5 Does the entropy functional derived from relativistic gas models in higher dimensions exhibit a maximum near d=3?

Key findings

• The entropy-to-energy ratio S/E reaches a maximum at d ≈ 2.97 when d is treated as a continuous variable in the range 2 ≤ d ≤ 10.
• For discrete values of d from 2 to 10, the S/E ratio is highest at d=3, with a value of approximately 1.495, compared to 1.458 at d=4.
• The S/E ratio decreases slightly from d=3 to d=4 and then increases slowly for d > 4, indicating a clear preference for d=3.
• The entropy functional is derived using relativistic gas statistics and the holographic principle, with a finite cutoff at the Planck scale to regularize the horizon temperature.
• The analysis assumes that Hawking/Unruh radiation is dominated by photons and gravitons, with their degrees of freedom fixed by gauge invariance and spin constraints.
• The result implies that a universe evolving into a late-time de Sitter phase with a small cosmological constant is entropically favored to have three spatial dimensions.