Corianò and Frampton (2019) proposed combining holographic dark energy — in which the dark energy density is set by the future event horizon of the universe — with the Baum–Frampton cyclic universe, in which entropy is jettisoned at each turnaround under the 'Come Back Empty' (CBE) hypothesis so that the contracting universe is empty of matter and black holes and cycles can repeat indefinitely. The authors themselves flagged two unresolved foundational issues: the future event horizon as infrared cutoff is counterintuitive because it refers to the future evolution of the universe (a circularity/causality objection); and the CBE hypothesis is a speculation with no mathematical justification, its detailed dynamics — the number of causally disconnected patches at turnaround and the probability that a patch is empty — being undetermined. The observational consistency of the model, notably whether the contracting phase generates the nearly scale-invariant density perturbations observed in the cosmic microwave background and large-scale structure, remains an open question.

Background

The cosmological constant problem — why the vacuum energy density is some 120 orders of magnitude below naive quantum-field-theory estimates — motivated the holographic principle ('t Hooft 1993), which states that the number of independent degrees of freedom in a spatial region is bounded by its surface area rather than its volume. Cohen, Kaplan and Nelson (1999) turned this into a relation between the ultraviolet (UV, short-distance) cutoff and the infrared (IR, long-distance) cutoff of any effective quantum theory in a region of size LL: L3ρUV≲MP2LL^3 \rho_{UV} \lesssim M_P^2 L, with MPM_P the Planck mass. Hsu (2004) noted that saturating this relation with LL the Hubble radius gives dark energy with equation-of-state ratio w=p/ρ=0w = p/\rho = 0, which is observationally excluded. Li (2004) instead took the IR cutoff LIRL_{IR} to be the future event horizon — the distance beyond which events can never send us light signals, which exists whenever the expansion accelerates — giving the holographic dark energy (HDE) density ρDE=3c2MP2LIR−2\rho_{DE} = 3 c^2 M_P^2 L_{IR}^{-2} (with cc an order-one parameter) and the observed late-time acceleration with w≃−1w \simeq -1. This choice has an acknowledged conceptual cost: the present dark energy density then depends on the future expansion history of the universe, a circularity and causality problem that has been listed as a standing puzzle of HDE (Kim, Lee and Lee 2012; Lee, Kim and Lee 2019). Separately, phantom dark energy with w<−1w < -1 (Caldwell 2002) leads to a 'Big Rip' — a finite-time future singularity in which all bound structures are torn apart. Tolman (1931) showed that in any cyclic (periodically expanding and contracting) universe the entropy grows from cycle to cycle, so extrapolating to the past one still reaches an initial singularity: this is the Tolman entropy conundrum (TEC). Baum and Frampton (2007) proposed to evade the TEC with phantom-driven dark energy in a brane-corrected Friedmann cosmology: a turnaround occurs a fraction of a second before the would-be Big Rip, at which the universe fragments into a very large number NN of causally disconnected patches; almost all patches are empty of matter and black holes and contract adiabatically with nearly zero entropy, while the few containing matter bounce prematurely and are 'failed universes'. The assumption that our universe contracts empty — the 'Come Back Empty' (CBE) hypothesis — discards the accumulated entropy into the patches that are jettisoned and permits infinite cyclicity. Corianò and Frampton (2019) recast this program in holographic language: the holographic dark energy equation of state can be w=−1w = -1 exactly (a cosmological constant, with turnaround induced by a brane-type correction term ρDE−ρDE2/ρc\rho_{DE} - \rho_{DE}^2/\rho_c to the Friedmann equation) or slightly phantom, so the cyclic model no longer requires a Big Rip. In their discussion they flag as open issues the counterintuitive future-event-horizon cutoff and the mathematically unjustified CBE hypothesis with its undetermined patch dynamics, while presenting the scale-invariant density perturbations claimed to arise from quantum fluctuations during the contraction (Frampton 2016) as shown at lowest order. A neighboring-subfield reader needs one more piece of terminology: 'infrared cutoff' here means the large-distance scale entering the holographic energy density, not a regulator in the field-theory sense.

Problem Statement

The problem is to establish — or rigorously refute — the mathematical and observational viability of the holographic cyclic cosmology framework proposed by Corianò and Frampton (2019), by resolving the three jointly linked foundational issues that the framework's own construction leaves open. These are the viability conditions of one model, not independent questions: the dark energy that drives the cycles is holographic, the cycles survive only if the contracting universe comes back empty, and the model is tenable only if its perturbations match the observed sky. Specifically: (1) Provide a principled derivation of the future event horizon as the infrared cutoff of holographic dark energy — one that answers the circularity and causality objection that the present dark energy density must not depend on the future evolution of the universe, whether by expressing the present density without a future-directed integral or by showing that its dependence on the future event horizon is benign and self-consistent — or demonstrate that no such derivation exists within the original model. (2) Justify the Come Back Empty hypothesis mathematically: derive, within the model's Friedmann-plus-brane-correction cosmology near turnaround, the number NN of causally disconnected patches into which the universe fragments and the probability that a given contracting patch is empty of matter and black holes (thereby adjudicating the contested claims that the Hubble length divergence at turnaround prevents fragmentation, and that black holes survive phantom accretion), or prove that the required fragmentation cannot occur and the cyclic model fails. (3) Determine whether the density perturbations generated during the empty, radiation-dominated contraction phase — claimed at lowest order to be scale-invariant — are consistent with the observed cosmic microwave background and large-scale-structure data, including the measured spectral index and its deviations from exact scale invariance. Because these three issues are the viability conditions of a single model, a rigorous refutation of any one of them suffices to establish that the model must be abandoned; a positive verdict requires all three to be resolved affirmatively.

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