Cosmological Analyticity and Bootstrap

Cosmological bootstrap methods reconstruct parts of late-time quantum-field-theory data from symmetry, singularities, factorization, and unitarity. Their force comes from using those constraints together. Their limit is equally important: a wavefunction coefficient is not an in-in correlator, Ward identities admit local solutions, cut data miss subtraction terms, and a flat-space residue emerges only under declared state, locality, and analytic-continuation hypotheses.

Helpful background. Momentum-space conformal Ward identities supply the symmetry operators. Cosmological correlators on the in-in contour fix the observable, Cutkosky rules supply the flat-space comparison, and celestial and cosmological handoffs distinguish boundary analytic objects from scattering amplitudes.

Analytic objects and the order of inference

At a late-time cutoff $\eta_0$, write a pure-state wavefunctional schematically as

$$\Psi_{\eta_0}[\varphi] =\exp\!\left[ -\frac12\int_{\mathbf k}\psi_2(k)\varphi_{\mathbf k}\varphi_{-\mathbf k} -\sum_{n\geq3}\frac1{n!} \int_{\mathbf k_1\cdots\mathbf k_n} (2\pi)^3\delta^3\!\left(\sum_a\mathbf k_a\right) \psi_n\,\prod_a\varphi_{\mathbf k_a} \right].$$

The coefficients $\psi_n$ depend on the field normalization, cutoff, state, branch prescription, and local boundary counterterms. Equal-time in-in correlators follow from the probability weight $\Psi\Psi^\ast$—or from the diagonal of a density matrix—not from $\psi_n$ alone. For the Gaussian term,

$$\langle\varphi_{\mathbf k}\varphi_{\mathbf k'}\rangle =(2\pi)^3\delta^3(\mathbf k+\mathbf k') \frac1{2\,\operatorname{Re}\psi_2(k)}.$$

At leading order, connected higher correlators involve $2\operatorname{Re}\psi_n$ together with inverse quadratic kernels and lower-order contractions. Purely imaginary local phases can change $\psi_n$ while leaving the equal-time probability unchanged.

The safe order is:

1. define the state, branch, cutoff, field basis, and analytic object;
2. impose the relevant exact or softly broken Ward identities;
3. construct contact and exchange seeds in that convention;
4. identify total- and partial-energy singularities and their sheets;
5. apply state-appropriate factorization and cutting identities;
6. quotient local boundary terms and field-redefinition equivalence;
7. state growth bounds and subtractions before writing a dispersion relation;
8. reconstruct only modulo the surviving homogeneous and local data.

Arkani-Hamed, Baumann, Lee, and Pimentel demonstrate how symmetry and singularity data can determine broad classes of tree-level inflationary correlators under Bunch–Davies, locality, and analytic assumptions (Arkani-Hamed et al. 2020, §§ 2–5). Those results motivate the method; they do not make every cosmological boundary object uniquely reconstructible.

The structure map shows this order. Inspect the separate paths from a wavefunction coefficient to an in-in correlator and from a total-energy singularity to a flat-space amplitude.

The cosmological analytic-data chain from object definition to qualified reconstruction. The diagram is schematic and not to scale; branch, state, normalization, contact, and subtraction data accompany every arrow.

The failure map supplies stopping rules. Inspect where direct coefficient–correlator identification, unqualified flat-space factorization, state-independent cuts, and subtraction-free positivity claims fail.

Failure conditions for cosmological analyticity and bootstrap arguments. The diagram is schematic and not to scale; a failed later inference leaves earlier, correctly defined analytic data intact.

Route through the chapter

Order	Page and task
1	Wavefunction and correlator object dictionary: fix normalization, branches, state, cutoff, and the probability inversion.
2	Cosmological symmetry and Ward identities: solve exact and softly broken momentum-space constraints while retaining homogeneous solutions.
3	Contact, exchange, and seed solutions: build explicit tree-level kernels and normalize their boundary behavior.
4	Total-energy and factorization singularities: separate the flat-space total-energy residue from partial-energy exchange data and analytic contacts.
5	Cosmological cutting identities: formulate perturbative unitarity for a specified state, contour, and pair of wavefunction branches.
6	Locality and field-redefinition equivalence: quotient integration-by-parts, equation-of-motion, boundary, and local contact freedom.
7	Dispersion relations and qualified positivity: declare analytic domains, growth bounds, infrared treatment, and subtraction constants.
8	Reconstruction from singularity data: combine constraints and enumerate the remaining contact basis.
9	Loops, initial states, validity, and handoffs: test which tree-level standard-state statements survive broader physics.

Domain and failure conditions

This is the chapter’s canonical comparison table. “Determines” always means within the hypotheses shown; local and contact data are retained unless an additional condition fixes them.

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

What counts as a reconstruction

A useful reconstruction statement has the form

$$\psi_n =\psi_n^{\mathrm{nonlocal}} +\sum_{\alpha=1}^{N_{\mathrm{loc}}} c_\alpha\,\psi_{n,\alpha}^{\mathrm{local}},$$

where the first term is fixed by declared residues, cuts, and boundary conditions, and the finite local basis is bounded by derivative order or an EFT power counting. The coefficients $c_\alpha$ are not errors. They are Wilsonian data, boundary-scheme choices, or field-basis representatives until another observable or ultraviolet assumption fixes them.

Melville and Pajer derive cosmological cutting rules from perturbative unitarity for standard-state wavefunction coefficients (Melville and Pajer 2021, §§ 2–4). Maldacena’s inflationary three-point calculation demonstrates both the physical information in nonlocal shapes and the local terms moved by nonlinear field redefinitions (Maldacena 2003, §§ 3–4).

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.
• Maldacena, J. “Non-Gaussian Features of Primordial Fluctuations in Single Field Inflationary Models.” Journal of High Energy Physics 2003, no. 05 (2003): 013. DOI. Open PDF.
• Melville, S., and E. Pajer. “Cosmological Cutting Rules.” Journal of High Energy Physics 2021, no. 05 (2021): 249. DOI. Open PDF.