de Sitter and Cosmological Holography

Cosmological holography lacks the single asymptotic framework that makes AdS dictionaries comparatively sharp. The dS/CFT proposal of Strominger 2001, late-time wavefunctions, Euclidean boundary data, static-patch algebras, horizon entropy, higher-spin models, and lower-dimensional matrix constructions encode different objects. This chapter develops their calculations and evidence without merging them, and closes with a dated completeness test through 10 August 2026.

Helpful background. de Sitter Infrared Regimes: States, Observables, Gauges, and Limits supplies state and infrared control. In-In Cosmological Correlators supplies normalized observables. Celestial and Cosmological Correlators: Axiom and Handoff Audit separates related boundary kinematics. Holographic Duality: Claims, Dictionaries, and Regimes fixes claim types; Euclidean Preparation and Lorentzian State Dictionaries fixes the contour/state distinction.

Enter with an object, state, and observer

Before using the word “boundary,” identify whether it means future conformal infinity, a horizon screen, a stretched timelike surface, an observer algebra, or the boundary of a lower-dimensional path integral. Then state the quantum state and contour, the patch and observer, the desired observable, and the inner product. The same $SO(d+1,1)$ symmetry can constrain several inequivalent objects and cannot choose among them.

The basic semiclassical regime is

$$G_NH^{d-1}\ll1.$$

This suppresses gravitational loops but does not control late-time infrared logs or create a top-down de Sitter vacuum. A string claim must separately bound $g_s$, $\alpha'H^2$, curvature, Kaluza–Klein modes, moduli, and decay rates. A higher-spin model has no higher-spin gap; a two-dimensional dilaton model has no local graviton. Those limitations cannot be averaged together.

Route through the chapter

Order	Page	Central task
1	de Sitter and Cosmological Holography	Classify proposals by encoded object, observer, state, inner product, and evidence.
2	de Sitter Observables, States, and Horizon Patches	Compare global, late-time, relational, and static-detector observables without conflation.
3	Late-Time Wavefunctions and Boundary Data	Derive a Gaussian wavefunction and separate local phase from nonlocal data.
4	dS/CFT Dictionaries and Analytic Continuation	Track radii, contours, phases, weights, reality, and positivity through AdS-to-dS continuation.
5	Wavefunction Coefficients versus In-In Correlators	Convert coefficient vertices into normalized expectation values.
6	Cosmological Bootstrap as Holographic Input	Import symmetry, singularity, factorization, and cutting data without claiming a duality.
7	Static-Patch Algebras and Observer Dependence	Construct the accessible algebra and state with horizon, clock, and dressing qualifications.
8	de Sitter Entropy and Finite-Hilbert-Space Proposals	Separate thermodynamic and generalized entropy from conjectural exact state counts.
9	Higher-Spin de Sitter Examples	Evaluate an explicit continued wavefunction model while retaining nonunitarity and locality limits.
10	Matrix and Lower-Dimensional de Sitter Models	Identify amplitudes, contours, genus expansions, and completion dependence in solvable models.
11	de Sitter Holography: Dictionary Completeness, Obstructions, and Status	Test every proposal against one fixed checklist and current primary evidence.

Pages 1–2 fix proposal and observable domains. Pages 3–6 move from a defined wavefunction through continuation, Born-rule conversion, and bootstrap constraints. Pages 7–10 develop observer, entropy, higher-spin, and lower-dimensional alternatives. Page 11 asks which structures are actually complete.

A common dictionary test

For every proposal, record:

1. Boundary object: wavefunction, generating functional, algebra, screen theory, amplitude, partition function, or ensemble quantity.
2. State and contour: Bunch–Davies/Euclidean, alpha-state, mixed state, observer KMS state, or model-specific integration cycle.
3. Observable: late-time coefficient, in-in correlator, relational quantity, detector response, algebraic entropy, or lower-dimensional boundary amplitude.
4. Access and reconstruction: global, expanding-patch, static-patch, or asymptotic; include a map rather than assuming one.
5. Reality and probability: conjugation, inner product, reflection positivity or its failure, Born-rule normalization, and contact phases.
6. Approximation: perturbative order, late-time and infrared limit, $G_N$, $g_s$, $\alpha'$, Kaluza–Klein and curvature hierarchy, and metastable lifetime.
7. Completion: fixed theory versus ensemble, topology weights, integration contour, factorization, and nonperturbative ambiguity.
8. Falsifier: a state, patch, observer, gauge, contour, positivity, or finite-time request that the claimed dictionary must answer.

The protocol makes absences informative. A proposal can successfully compute late-time coefficients while lacking a positive inner product; another can define an observer algebra while lacking global reconstruction.

Synthesis problem

For a massive scalar in de Sitter, compute the Bunch–Davies Gaussian wavefunction at a late cutoff and identify its local phase and nonlocal kernel. Continue the corresponding EAdS kernel, tracking every phase and counterterm. Use $|\Psi|^2$ to derive the equal-time two-point function and add a cubic coefficient to derive the tree-level three-point function.

Then compare that late-time data with a finite-time geodesic detector response in the static patch. Ask whether a dS/CFT, static-algebra, higher-spin/$Sp(N)$, and dS JT proposal supplies the boundary object, state, inner product, reconstruction map, and nonperturbative definition needed for the same task. Add one contact wavefunction phase, one alpha-state deformation, and one alternative matrix contour as adversarial controls.

Your conclusion must preserve the distinction among a calculational continuation, a normalized in-in observable, an observer algebra, an entropy interpretation, and a model-specific completion. It must date any frontier status claim.

Review the chapter

A satisfactory answer should:

• name the boundary object, state, contour, patch, and observer before invoking holography;
• distinguish a late-time coefficient, $|\Psi|^2$ probability vertex, in-in correlator, and detector response;
• derive the scalar Gaussian kernel and identify local phase versus nonlocal data;
• track the AdS-to-dS radius, mass, contour, source, and counterterm continuation;
• retain complex weights and failed reflection positivity as explicit limitations;
• include normalization and lower-point contractions in coefficient-to-correlator conversion;
• use bootstrap singularities as bulk constraints while retaining contact and initial-state ambiguity;
• distinguish type-III QFT and type-II$_1$ gravitational patch algebras from finite matrix algebras;
• explain why $A/(4G_N)$ is not by itself $\log\dim\mathcal H$;
• state why the calculable $Sp(N)$ model is nonunitary and why dS JT is dimension-specific;
• declare $G_N$, $g_s$, $\alpha'H^2$, curvature, Kaluza–Klein, infrared, contour, and completion limits;
• stop at the evidence ceiling rather than reporting a complete de Sitter dual.

Exits

Detailed in-in, wavefunction, bootstrap, infrared, stochastic, and detector calculations remain in Quantum Fields in Curved Spacetime and Cosmology. Euclidean conformal representation and positivity questions remain in Conformal Bootstrap. The AdS higher-spin starting point remains in the preceding higher-spin chapter. New proposals and changes in comparative status require a dated holography research dossier rather than an undated textbook consensus.

Chapter-scale structure and validity checks

The chapter-scale structure map locates this page’s result inside the full reasoning chain. Follow the solid arrows through the declared inputs and checks; the dashed final arrow marks the point where an additional inference would be required.

Late-time wavefunctions, in-in correlators, and static-patch observables are different objects; no single continuation supplies a complete de Sitter dual. The diagram is an original schematic, is not to scale, and uses the dashed final arrow to mark the claim boundary.

Accessible figure data (JSON)

The companion validity map turns three common overclaims into explicit failure tests. Read each row from its declared object to the diagnostic, then compare the licensed conclusion with the dashed “not” endpoint.

Late-time wavefunctions, in-in correlators, and static-patch observables are different objects; no single continuation supplies a complete de Sitter dual. Each row pairs a diagnostic with the strongest supported conclusion and an explicitly unsupported promotion. The diagram is an original schematic and is not to scale.

Accessible figure data (JSON)

Claim-domain comparison

The table below gives a screen-reader-friendly comparison of three representative claims. It keeps the required declaration, approximation status, evidence timing, counterevidence, falsifier, failure condition, and licensed conclusion in one reading order.

Representative claim domains and validity boundaries for de Sitter and Cosmological Holography

Claim object	State, ensemble, and conventions	Approximation, status, and evidence timing	Uncertainty and counterevidence	Falsifier	Failure condition	Licensed conclusion
late-time wavefunction	Declare state, boundary data, and phase convention; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: de Sitter state and observer patch → wavefunction or in-in observable → analytic continuation and dictionary → entropy and static-patch tests → proposal status. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “Ward and analytic-continuation check” check is counterevidence to the promoted claim.	Ward and analytic-continuation check	ordinary Euclidean CFT probabilities	wavefunction coefficients
in-in correlator	Declare closed-time contour and operator ordering; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: de Sitter state and observer patch → wavefunction or in-in observable → analytic continuation and dictionary → entropy and static-patch tests → proposal status. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “reality and cutting checks” check is counterevidence to the promoted claim.	reality and cutting checks	the same object as a wavefunction coefficient	cosmological expectation value
static-patch proposal	Declare observer algebra and horizon state; use the volume conventions unless the page states a local replacement.	Proposal or conditional construction. Control chain: de Sitter state and observer patch → wavefunction or in-in observable → analytic continuation and dictionary → entropy and static-patch tests → proposal status. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “entropy and finite-system consistency” check is counterevidence to the promoted claim.	entropy and finite-system consistency	complete global de Sitter holography	a model-dependent observer description

Download the structured table data (JSON).

References

• Chandrasekaran, V., Longo, R., Penington, G., and Witten, E. (2023). “An Algebra of Observables for de Sitter Space.” Journal of High Energy Physics 2023(2), 082. DOI.
• Cotler, J., and Jensen, K. (2024). “Non-Perturbative de Sitter Jackiw–Teitelboim Gravity.” Journal of High Energy Physics 2024(12), 016. DOI.
• Strominger, A. (2001). “The dS/CFT Correspondence.” Journal of High Energy Physics 2001(10), 034. DOI.