Franken, Partouche, Rondeau and Toumbas (JHEP 05 (2024) 219) conjectured that closed Friedmann–Lemaître–Robertson–Walker (FRW) cosmologies with ∣γ∣≥1|\gamma|\geq1 — where γ\gamma is the exponent of the scale factor in conformal time — admit a holographic encoding on two screens tracked by antipodal comoving observers, with entanglement entropies of screen subsystems computed by a bilayer, constrained generalized-entropy extremization, realizing the ER=EPR conjecture in a time-dependent setting. The prescription was postulated by analogy with de Sitter static-patch holography and the quantum extremal surface formalism, not derived; the authors themselves call for a derivation from the bulk gravitational replica path integral (Lorentzian, Schwinger–Keldysh type). This problem asks for that derivation — or a demonstration that the prescription fails — together with a first-principles settlement of the conjectured breakdown of two-screen encoding for Big Bang/Big Crunch cosmologies with 1/(n−1)≤γ<11/(n-1)\le\gamma<1, including all moduli-dominated cases. Both cores are open in the literature to date: the closest work (Irakleous, 2026) reproduces the generalized-entropy structure only conditionally, on the assumption of a screen-based dual description.

Background

In the AdS/CFT correspondence, the entanglement entropy of a spatial region of the boundary theory is computed holographically from bulk geometry: the Ryu–Takayanagi (RT) prescription and its covariant generalization by Hubeny, Rangamani and Takayanagi (HRT) state that the entropy equals one quarter of the area, in Planck units, of an extremal codimension-2 bulk surface homologous to the boundary region. The quantum extremal surface (QES) prescription of Engelhardt and Wall incorporates quantum corrections by extremizing instead the generalized entropy, Sgen=A/4GNℏ+SsemiclS_{\rm gen}=\mathcal{A}/4G_N\hbar+S_{\rm semicl} — the area term plus the von Neumann entropy of bulk quantum fields (including gravitons) evaluated semiclassically on the enclosed region. The classical area law was derived by Lewkowycz and Maldacena from a gravitational replica path integral: Rényi entropies are computed on mm-fold replicated bulk geometries and analytically continued to m→1m\to1, and this replica derivation is the foundation on which the QES and island rules rest (the island rule adds to the entropy of radiation outside an evaporating black hole a contribution from an interior region bounded by a quantum extremal surface).

Our universe is not asymptotically anti-de Sitter. For spacetimes without a spatial asymptotic boundary — de Sitter space (the maximally symmetric positively curved solution relevant to inflation and to the far future of our cosmos) and general cosmologies — no accepted holographic dual is known, and the replica machinery has no boundary on which to anchor the replicated geometry. A conservative proposal for de Sitter is static patch holography: the dual degrees of freedom live on the cosmological horizon bounding the causally accessible region (the static patch) of a single observer. Franken, Partouche, Rondeau and Toumbas developed this into a two-screen framework with two antipodal observers (a pode and an antipode), each with its horizon. Two entropy prescriptions were put forward in that setting, the monolayer and bilayer proposals; they argued that only the bilayer proposal — in which the generalized entropy receives contributions from the two horizon interiors and from the exterior region between them, with extremal surfaces constrained to lie in the relevant causal diamonds — is self-consistent, and that the entire de Sitter spacetime can then be reconstructed from the two horizon screens.

The source paper extends this framework to generic closed FRW cosmologies. A closed FRW universe in n+1n+1 spacetime dimensions has metric ds2=a(η)2(−dη2+dθ2+sin⁡2θ dΩn−12)ds^2=a(\eta)^2(-d\eta^2+d\theta^2+\sin^2\theta\,d\Omega_{n-1}^2), where η\eta is conformal time, θ\theta the polar angle on the spatial sphere SnS^n, and dΩn−12d\Omega_{n-1}^2 the metric of the unit sphere Sn−1S^{n-1}. Filled by a perfect fluid with constant equation-of-state index w=p/ρw=p/\rho, the scale factor near the Big Bang and Big Crunch singularities behaves as a(η)∝∣η∣γa(\eta)\propto|\eta|^\gamma with γ=2/(n(1+w)−2)\gamma=2/(n(1+w)-2): moduli (massless scalar fields, w=1w=1) give γ=1/(n−1)\gamma=1/(n-1), radiation (w=1/nw=1/n) gives γ=2/(n−1)\gamma=2/(n-1), pressureless matter (w=0w=0) gives γ=2/(n−2)\gamma=2/(n-2), and a cosmological constant (w=−1w=-1) is de Sitter with γ=−1\gamma=-1. Values γ≤−1\gamma\le-1 describe non-singular bouncing universes; γ≥1\gamma\ge1 gives singular Big Bang/Big Crunch evolutions whose Penrose diagrams (conformal causal diagrams) are taller than wide. In these spacetimes the apparent horizons — the boundaries of the regions of trapped surfaces (codimension-2 spacelike surfaces whose area decreases along both future light-cone directions) or anti-trapped surfaces (area increasing) — play the role of the de Sitter cosmological horizons. Using the Bousso covariant entropy bound (which limits the entropy flux through converging light sheets by the bounding area in Planck units), the authors argue that in the expanding phase the holographic screens must sit on the two apparent horizons; in the contracting phase a whole family of screen trajectories is possible, organized into equivalence classes whose effective theories are related by integrating out degrees of freedom of a parent two-screen system, in analogy with renormalization-group flow.

The central conjecture is that for ∣γ∣≥1|\gamma|\ge1 the entire spacetime is encoded on the two screens, and that the fine-grained entropy of any subsystem AA of the two-screen system is computed by a bilayer constrained extremization,

S(A)=min⁡χL,χE,χR ∑i∈{L,E,R}[A(χi)4GNℏ+Ssemicl(ΣAi)],S(A)=\min_{\chi_L,\chi_E,\chi_R}\ \sum_{i\in\{L,E,R\}}\left[\frac{\mathcal{A}(\chi_i)}{4G_N\hbar}+S_{\rm semicl}(\Sigma_{A_i})\right],

where, for each of the three regions — the left observer's interior LL, the exterior EE between the screens, and the right observer's interior RR — χi\chi_i is a codimension-2 surface homologous to the part AiA_i of AA lying on the screen of that region, restricted to the causal diamond of that region (a constraint implemented with Lagrange multipliers and auxiliary fields, allowing the surfaces to lie on the diamond boundary), and ΣAi\Sigma_{A_i} is the bulk portion bounded by AiA_i and χi\chi_i. The minimal trapped or anti-trapped sphere in the exterior causal diamond acts as an effective Einstein–Rosen bridge between the screens, realizing the ER=EPR conjecture of Maldacena and Susskind (entanglement equals geometrical connectivity) in a time-dependent setting: for Big Bang/Big Crunch universes the bridge pinches off on the singularity a finite time before the screens arrive there, at which moment the screens disentangle; for bouncing universes the entanglement entropy saturates a finite bound even as the screen areas diverge.

Two features of this construction are conjectural. First, the entropy prescription and its constrained extremization are postulated by analogy with de Sitter static-patch holography and the QES formalism — no replica-path-integral derivation exists, and the authors explicitly state that obtaining one 'would involve Lorentzian Schwinger–Keldysh path integrals' (real-time contour integrals suited to mixed states and time-dependent backgrounds), with the black-hole replica literature as the only existing model. Second, the proposal is claimed only for ∣γ∣≥1|\gamma|\ge1: for Big Bang/Big Crunch cosmologies with 1/(n−1)≤γ<11/(n-1)\le\gamma<1 — wider-than-tall Penrose diagrams in which the antipodal causal patch are disjoint for the entire evolution, including all moduli-dominated cases, matter-dominated cases for n>4n>4, and radiation-dominated cases for n>3n>3 — the paper argues that two screens cannot encode the full spacetime, but does not prove this or construct what replaces them. Both gaps are acknowledged by the authors as open problems.

Problem Statement

The problem is to give the two-screen holographic proposal of Franken–Partouche–Rondeau–Toumbas for closed FRW cosmologies a first-principles foundation, or to refute it.

(1) Derive the bilayer constrained-extremization prescription — the formula for S(A)S(A) stated in the background, together with its entanglement wedges (the bulk regions bounded by each screen subsystem and its associated extremal surface) — from the bulk gravitational replica path integral for closed FRW spacetimes with ∣γ∣≥1|\gamma|\ge1. This would involve constructing the replicated geometries appropriate to a time-dependent cosmology without spatial boundaries (Lorentzian, Schwinger–Keldysh-type replica contours), specifying the bulk gluing and boundary conditions at the holographic screens, identifying the dominant saddles (including any replica-wormhole saddles — replica geometries in which the copies are joined through the bulk, the mechanism that produces islands — and the one-loop quantum corrections that lift classical degeneracies of the extremal surfaces), and showing that the m→1m\to1 limit reproduces precisely the constrained generalized-entropy extremization over the three regions (the two interiors and the exterior), recovering the classical area terms, the O((GNℏ)0)\mathcal{O}((G_N\hbar)^0) corrections, and the de Sitter results as the γ=−1\gamma=-1 special case. Alternatively, demonstrate that no consistent replica derivation exists and identify precisely where the prescription fails: wrong saddle structure, wrong constraints, or screens with insufficient degrees of freedom.

(2) Within the same framework, settle the domain of validity of the two-screen encoding: prove or disprove the claim that for Big Bang/Big Crunch cosmologies with 1/(n−1)≤γ<11/(n-1)\le\gamma<1 — which includes all moduli-dominated cases (w=1w=1, γ=1/(n−1)\gamma=1/(n-1)), matter-dominated cases for n>4n>4, and radiation-dominated cases for n>3n>3, all with wider-than-tall Penrose diagrams and mutually causally disconnected antipodal patches — two screens associated with antipodal observers cannot encode the full spacetime.

These two deliverables are linked parts of one objective — establishing whether the two-screen/bilayer proposal is a consequence of semiclassical quantum gravity and fixing where it applies. A successful derivation validates the prescription for ∣γ∣≥1|\gamma|\ge1; a refutation of the prescription resolves deliverable (1) negatively, while deliverable (2) — the encoding capacity for 1/(n−1)≤γ<11/(n-1)\le\gamma<1 — remains a separate obligation that must be proved or disproved on its own. Explicitly out of scope, as separate open directions flagged by the source: the identification of concrete dual Hamiltonians or degrees of freedom, microscopic (for example double-scaled SYK, a solvable limit of the Sachdev–Ye–Kitaev quantum many-body model) realizations of the time-dependent ER=EPR bridge, controlled treatment of the Big Bang/Big Crunch singularities themselves, and extensions to time-dependent ww or multiple fluids.

References

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