[Paper Review] Holography and Cosmology

Willy Fischler, Leonard Susskind|ArXiv.org|Jun 4, 1998

Black Holes and Theoretical PhysicsPhysics and Astronomy3 references220 citations

TL;DR

This paper proposes a cosmological version of the holographic principle, asserting that entropy in any spatial region must not exceed the area of its boundary in Planck units. It derives a bound on the equation of state (γ < 1) to prevent entropy from exceeding area, ensuring consistency with special relativity and the holographic principle across cosmic evolution.

ABSTRACT

A cosmological version of the holographic principle is proposed. Various consequences are discussed including bounds on equation of state and the requirement that the universe be infinite.

Motivation & Objective

• To extend the holographic principle—originally formulated for black holes—to the context of cosmology.
• To investigate whether the holographic principle imposes constraints on the equation of state of the universe.
• To examine the consistency of the holographic principle with flat, open, and closed Friedmann-Robertson-Walker universes.
• To determine whether the universe must be infinite to satisfy the holographic bound over all time.
• To analyze the entropy-to-area ratio in different cosmological eras and spacetime geometries.

Proposed method

• Formulates the holographic principle in cosmology by requiring that the entropy passing through a past light cone from a spatial boundary never exceeds the area of that boundary.
• Applies the condition S/A < 1 in Planck units to a spherical region of coordinate size R, leading to the inequality σR_H^d < [aR_H]^{d-1} for comoving entropy density σ.
• Uses the scale factor a(t) ~ t^p to derive a lower bound on the expansion rate, p > 1/d, which translates to a bound on the equation of state γ < 1.
• Analyzes flat anisotropic (Kasner) universes, showing that S/A remains constant when ∑p_i = 1 and ∑p_i² = 1, allowing saturation of the holographic bound.
• Extends the analysis to closed universes using the S³ metric, deriving S/A as a function of the horizon coordinate χ_H and showing inconsistency when χ_H approaches π/(K-1).
• Considers open universes and finds that late-time expansion allows the holographic bound to be satisfied without strong constraints on the equation of state.

Experimental results

Research questions

• RQ1 Can the holographic principle be consistently applied to cosmological spacetimes, particularly in the context of expanding universes?
• RQ2 What constraints does the holographic principle impose on the equation of state parameter γ in a homogeneous and isotropic universe?
• RQ3 Is it possible for the entropy-to-area ratio to remain below one throughout cosmic evolution, and under what conditions is it saturated?
• RQ4 Are closed universes with positive curvature compatible with the holographic principle, or do they inevitably violate it?
• RQ5 How does the holographic bound behave in open and anisotropic universes, and what are the implications for cosmic evolution?

Key findings

• The holographic principle implies a bound on the equation of state: γ < 1, which corresponds to a lower bound on the expansion rate p > 1/d, ensuring entropy does not exceed area in Planck units.
• In the flat, radiation-dominated era, the entropy-to-area ratio ρ = S/A ∼ t^{-1/2} remains below 1 for all times after the Planck time, indicating the early universe was holographically consistent.
• For flat anisotropic (Kasner) universes, the entropy-to-area ratio S/A is constant in time when the Kasner exponents satisfy ∑p_i = 1 and ∑p_i² = 1, allowing the holographic bound to be saturated.
• Closed universes with K > 1 (e.g., matter- or radiation-dominated) inevitably violate the holographic bound as χ_H approaches π/(K-1), suggesting such models may be inconsistent with the principle.
• Open universes do not face strong constraints from the holographic principle at late times because volume and area grow in fixed proportion, allowing arbitrarily slow expansion.
• The agreement between the derived bound γ < 1 and the relativistic causality bound (sound speed < c) provides strong evidence that the cosmological holographic principle is physically viable.