Cosmological Analyticity and Bootstrap

Cosmological bootstrap methods use symmetry, analytic singularities, and unitarity to reconstruct late-time quantum-field data. Their strength depends on identifying the object being reconstructed and the ambiguities those constraints leave.

Enter this chapter

Begin with Wavefunction and Correlator Object Dictionary to distinguish wavefunction coefficients, in/in correlators, and scattering amplitudes.

Helpful background. Momentum-Space Correlators and Conformal Ward Identities and Celestial and Cosmological Correlators: Comparing the Axioms develop the definitions used by the chapter’s opening pages.

Topics

Wavefunction and Correlator Object Dictionary — How are cosmological wavefunction coefficients, in-in correlators, boundary objects, amplitudes, and probability weights related without identifying inequivalent objects?
Cosmological Symmetry and Ward Identities — Which de Sitter, conformal, dilation, and rotation Ward identities constrain cosmological boundary objects, and how are those identities modified by slow-roll breaking?
Contact, Exchange, and Seed Solutions — How are contact and exchange seed solutions constructed for scalar and spinning cosmological interactions while retaining state, boundary, and normalization data?
Total-Energy and Factorization Singularities — Which total-energy, partial-energy, and factorization singularities of cosmological boundary objects encode bulk locality and flat-space scattering data?
Cosmological Cutting Identities and Unitarity Constraints — Under which state, contour, reality, and boundary assumptions do cosmological discontinuity and cutting identities express perturbative unitarity?
Locality, Contact Ambiguities, and Field-Redefinition Equivalence — Which analytic and large-momentum tests diagnose bulk locality once contact ambiguities, boundary terms, and local field redefinitions are quotiented correctly?
Dispersion Relations, Subtractions, and Qualified Positivity — When do analyticity, growth, state, and subtraction hypotheses permit cosmological dispersion relations or qualified positivity constraints?
Reconstruction from Symmetry, Singularities, and Cuts — How far can symmetry, singularities, factorization, cuts, and locality reconstruct a cosmological boundary object, and what irreducible data remain?
Cosmological Bootstrap: Loops, Initial States, and Validity — Which cosmological bootstrap statements remain valid with loops, nonstandard initial states, infrared effects, slow-roll breaking, and observational projection?

How the ideas fit together

Factorization singularities encode how lower-order processes combine, while symmetry constrains their momentum dependence. Reconstructing a wavefunction coefficient from such data can still leave local contact terms that obey the same homogeneous constraints Arkani-Hamed et al. 2020, §§ 2–5.

Converting the coefficient to an in/in correlator requires its proper wavefunction combination and state prescription. Likewise, a flat-space amplitude appears only through a specified analytic limit. Cutting relations, dispersion subtractions, and field redefinitions must therefore be tracked before calling a reconstructed expression unique.

The comparison below separates the assumptions and tests needed for these different conclusions.

Domain and failure conditions

Object or constraint	Normalization, continuation, and state	Symmetry or analytic datum	Singular or cut information	Local/contact and field-basis freedom	Licensed result	Decisive check	Failure or downgrade
Wavefunction coefficient $\psi_n$	Field basis, late cutoff, ket branch, initial state, counterterms	Analytic coefficient of $\log\Psi$	May contain total- and partial-energy singularities	Imaginary local phases and boundary counterterms move analytic pieces	A branch-dependent boundary kernel	Reconstruct the Gaussian probability covariance	Calling $\psi_n$ an observable correlator
Equal-time in-in correlator	Density matrix or $\Psi\Psi^\ast$ , both branches, operator normalization	Real probability moments and connected cumulants	Inherits combinations of coefficient singularities	Composite-operator contacts and field redefinitions remain	Observable for the prepared state and slicing	Direct in-in calculation and positivity of covariance	Using one branch or omitting the state measure
Ward-identity solution	Exact de Sitter or declared slow-roll source, weights, helicities	Dilation, rotation, and special-conformal equations	Constrains momentum dependence	Homogeneous and anomalous local solutions remain	Symmetry-compatible solution space	Apply every Ward operator including contact sources	Claiming uniqueness from symmetry alone
Contact seed	Vertex, derivative count, boundary prescription, state	Polynomial or rational energy dependence	Total-energy behavior without exchange channel pole	Basis changes and boundary terms are substantial	Normalized local interaction representative	Direct time integral and soft/large-momentum limits	Treating one seed basis as invariant physics
Exchange seed	Internal mass/spin, propagator, branch and boundary conditions	Casimir or differential equation plus source	Partial-energy poles or branch points and factorized residues	Added contact solutions do not change nonlocal exchange data	Nonlocal channel data modulo contacts	Compare direct integral, differential equation, and residue	Ignoring other sheets or induced contacts
Total-energy singularity	Bunch–Davies-type analytic continuation, local tree interaction, external normalization	$k_T=\sum_a k_a\to0$ on a specified complex sheet	Leading residue can encode a flat-space amplitude	Analytic terms and some subleading poles are convention dependent	Flat-space residue under stated locality and state hypotheses	Match coupling, polarization, powers, and continuation phase	Calling every $k_T$ feature an S-matrix element
Cosmological cut	Unitary time evolution, Hermitian action, specified density matrix and branch conjugation	Discontinuity or branch relation	Products of lower-point data with state-dependent weights	Boundary and initial-state terms may add cuts or interference	Perturbative unitarity identity for that object	Explicit two-branch exchange calculation	Importing a Cutkosky rule unchanged
Locality quotient	Derivative expansion and declared boundary conditions	Large-energy and spurious-pole cancellation tests	Nonlocal residues preserved	Integration by parts, equations of motion, local redefinitions, contacts	Equivalence class of bulk interactions	Compare on-shell nonlocal data and observables	Using a contact coefficient as a field-basis invariant
Dispersion relation	Analytic domain, boundedness, contour at infinity, infrared regulator, $N$ subtractions	Discontinuity along declared cuts	Reconstructs nonlocal part from spectral data	Polynomial of degree at most $N-1$ remains	Subtracted representation, sometimes sign information	Vary subtraction point and include massless exchange	Automatic flat-space positivity without hypotheses
Reconstruction	All rows above plus a bounded local operator basis	Ward data, residues, cuts, locality tests	Fixes nonlocal singular part channel by channel	Finite contact or subtraction basis remains	Solution class or unique result only with extra input	Construct two solutions with identical nonlocal data	Silently selecting one contact completion
Loop or excited-state extension	Renormalization scheme, contour, density matrix, infrared prescription	Logs, anomalous scaling, folded singularities, secular terms	Multiparticle cuts and state-dependent interference	New counterterms and boundary operators enter	Order-by-order controlled extension	Second formalism, regulator, and standard-state limit	Promoting a tree-level Bunch–Davies identity to a universal theorem

Where to go next

Continue to Validity, Breakdown, and the Need for Quantum Gravity to set a target observable, resolution, duration, and accuracy.

References

Arkani-Hamed, N., D. Baumann, H. Lee, and G. L. Pimentel. “The Cosmological Bootstrap: Inflationary Correlators from Symmetries and Singularities.” Journal of High Energy Physics 2020, no. 04 (2020): 105. DOI. Open PDF.