Cosmological Cutting Rules

• Cosmological Cutting Rules are algebraic and diagrammatic identities that dictate the analytic and factorization structure of wavefunction coefficients in FLRW and de Sitter spacetimes.
• They extend flat-space Cutkosky rules to cosmology, enabling recursive bootstrapping and efficient computation of both tree and loop-level correlators without standard energy conservation.
• These rules underpin practical methods for bootstrap, loop reduction, and phenomenological constraints, crucial for analyzing primordial signals and EFT bounds in cosmology.

The "cosmological cutting rules" are a set of algebraic and diagrammatic identities that constrain the analytic and factorization properties of cosmological correlators and wavefunction coefficients in Lorentzian FLRW spacetimes, derived as direct consequences of perturbative unitarity of time evolution and, in certain cases, causality and analyticity. These rules generalize the familiar Cutkosky rules of flat-space quantum field theory to cosmological backgrounds, encoding unitarity in the absence of energy conservation or a standard S-matrix, and providing powerful recursion, bootstrap and geometric tools for both tree and loop-level computations of primordial correlation functions.

1. Formal Statement and Mathematical Structure

Let $\Psi[\bar\sigma(\bfk),\eta_0]$ be the wavefunction of the universe, with expansion coefficients $\psi'_n(\bfk_1,\dots,\bfk_n)$ at conformal time $\eta_0$ for fields of arbitrary mass and spin, generally in a (quasi-)de Sitter or FLRW background. The cosmological cutting rules take the schematic form, for any Feynman–Witten diagram $D$ with internal line set $I$ (Melville et al., 2021):

$$i\,\mathrm{Disc}_I\left[i\,\psi^{(D)}\right] = \sum_{\emptyset\neq C\subseteq I} \left[\prod_{a\in C} \int d^3 q_a\, P_{q_a}\right] \prod_{j=1}^r \left(-i\right)\, \mathrm{Disc}_{I_j \cup \{q_a\}}\left[i\,\psi^{(D_C^{(j)})}\right]$$

Here, the discontinuity operator $\mathrm{Disc}$ is defined by analytic continuation of energy variables (or the corresponding wavefunction coefficients) across their branch cuts induced by the Bunch-Davies $i\epsilon$-prescription. Each nontrivial cut $C$ splits the diagram into disconnected pieces $D_C^{(j)}$, each inheriting their own restricted discontinuity.

Special cases:

• At tree-level, cutting a single internal line (the "single-cut rule") yields a factorization of the $\psi'_n(\bfk_1,\dots,\bfk_n)$0-point discontinuity into lower-point functions (Goodhew et al., 2021, Das et al., 23 Dec 2025):

$\psi'_n(\bfk_1,\dots,\bfk_n)$1

• At loop-level, each cut reduces the loop-count by at least one, and nested application expresses any discontinuity as a sum over products of tree-level data (Agui-Salcedo et al., 2023).

A crucial ingredient is the Hermitian analyticity of the bulk-to-boundary propagators, enforced by the choice of initial Bunch-Davies (or more general Bogoliubov) vacuum (Ghosh et al., 2024, Ghosh et al., 8 Feb 2025).

2. Derivation: Unitarity, Analyticity, and Causality

The rules follow from unitary time evolution, $\psi'_n(\bfk_1,\dots,\bfk_n)$2, in the functional Schrödinger representation or the Schwinger–Keldysh in-in path integral (Goodhew et al., 2020, Melville et al., 2021, Colipí-Marchant et al., 27 Dec 2025). Diagrammatically, for a given wavefunction coefficient or correlator, the operator identity from unitarity enforces

$\psi'_n(\bfk_1,\dots,\bfk_n)$3

where the right-hand side corresponds to a sum over factorized configurations where a set of internal lines are replaced by on-shell intermediate states, analogous to the cutting of lines in Cutkosky's rules in Minkowski QFT.

In the geometric picture, this identity is encoded in a non-convex "optical polytope" whose various triangulations realize different cutting rules. The i$\psi'_n(\bfk_1,\dots,\bfk_n)$4-prescription and orientation ensures correct analytic continuation (Albayrak et al., 2023). Causality, specifically the existence of retarded Green's functions, provides additional constraints and unique determination of the wavefunction coefficients from their cuts and tree-level data (Agui-Salcedo et al., 2023).

3. Diagrammatic and Computation Rules

The cutting rules can be applied directly to diagrams either in the wavefunction (Schrödinger) formalism or at the level of in-in correlators (Schwinger–Keldysh) (Colipí-Marchant et al., 27 Dec 2025, Ema et al., 2024). The essential diagrammatic steps are:

• Single-cut: Replace a chosen internal propagator $\psi'_n(\bfk_1,\dots,\bfk_n)$5 by its imaginary part, $\psi'_n(\bfk_1,\dots,\bfk_n)$6, factorizing the nested time-integral into a product over two subdiagrams (Goodhew et al., 2021).
• Multi-cut / Loop: For any subset of cut internal lines, sum over all such partitions; the resulting factorized components may themselves require further cuts.
• Each cut is equivalent to forcing the internal energy variable onto its on-shell (branch-cut) value, matching the structure of unitarity delta functions in the flat-space S-matrix after analytic continuation (Ansari et al., 13 Jan 2026, Chowdhury et al., 3 Feb 2026).
• For wavefunction diagrams in Bogoliubov initial states, two independent discontinuity operators (corresponding to distinct Hermitian analyticity properties) must be considered (Ghosh et al., 2024, Ghosh et al., 8 Feb 2025).

At the level of in-in correlators, the cutting rule can be encoded in the Keldysh $\psi'_n(\bfk_1,\dots,\bfk_n)$7 basis, yielding a decomposition into products of fully retarded correlators and cut (Wightman) propagators, with the non-analytic branch cut components arising exclusively from cut lines (Ema et al., 2024).

4. Applications: Bootstrap, Factorization, and Phenomenology

These rules provide:

• Recursion and Bootstrap: Entire families of $\psi'_n(\bfk_1,\dots,\bfk_n)$8-point correlators (or wavefunction coefficients) can be recursively bootstrapped using only knowledge of contact diagrams and their analytic structures, together with symmetry and analyticity requirements (Goodhew et al., 2020, Melville et al., 2021, Baumann et al., 2021). All singularities (total-energy and partial-energy poles) are fixed by lower-point data.
• Loop Reduction: Loop-level cosmological diagrams can be systematically decomposed into momentum integrals over products of tree-level (cut) diagrams, eliminating the need for nested time integrations (Agui-Salcedo et al., 2023).
• UV Structure and Renormalization: The analytic "background" pieces in loop diagrams are fully local and can be subtracted by local counterterms, with nonanalytic (branch-cut) piece determined by the cutting rules (Qin, 2024).
• Cosmological Collider Signals: All non-local (oscillatory) cosmological collider signals in massive exchange diagrams necessarily arise from cut-propagators; the factorized time integral structure allows efficient analytic extraction of these signals (Tong et al., 2021, Ema et al., 2024).
• Phenomenological Constraints: The rules yield precise perturbative unitarity bounds on EFT Wilson coefficients and constrain the allowed analytic structure (e.g., Steinmann-type relations), crucial for bootstrapping inflationary correlators (Melville et al., 2021).

5. Geometric and Algebraic Structures

Recent work has established that the cutting and residue structure of FRW and de Sitter integrals is naturally encoded in the language of (twisted) cohomology, polytopes, and positive geometry. Tree-level correlators and wavefunction coefficients can be organized by a "cut basis" of logarithmic differential forms, each corresponding to an independent acyclic minor and, at loop level, associated to graphical zonotopes (De et al., 2024, Glew et al., 15 Aug 2025). The intersection theory framework elegantly determines which cut sequences correspond to physical (factorization) cuts and provides canonical differential equations for the analytic structure of cosmological observables.

6. Extensions and Generalizations

• Bogoliubov Initial States: The cutting structure admits a nontrivial extension to general Gaussian initial states, with modifications to discontinuity operators reflecting the mixing of positive/negative frequencies and changing the analytic structure of tree- and loop-level diagrams (Ghosh et al., 2024, Ghosh et al., 8 Feb 2025).
• Spinning Fields and EFTs: The rules are valid for arbitrary mass and integer spin, with additional indices and prefactors entering in the power spectra and cubic vertices. The generality extends to slow-roll inflation, power-law backgrounds, and interactions with arbitrary derivative content (Melville et al., 2021, Goodhew et al., 2021).
• Dispersion Relations and Dressing Rules: The full correlator can be reconstructed from its discontinuities by nested momentum-space dispersion integrals. This directly connects the cosmological cutting rules to "dressing rules" that uplift flat-space amplitudes to curved backgrounds (Das et al., 5 Feb 2026, Chowdhury et al., 3 Feb 2026).
• Sum Rules and Analytic Constraints: Cutting all auxiliary lines in the dressed representation enforces exact sum rules for cosmological correlators, ensuring vanishing linear combinations off-shell and further constraining the space of allowed solutions (Chowdhury et al., 3 Feb 2026).

7. Comparison to Flat-Space and Conceptual Significance

While the Cutkosky rules in flat-space QFT constrain S-matrix elements via energy-conserving delta functions, cosmological cutting rules account for broken time-translation invariance: their discontinuities occur in external and partial energies, manifest as 1/E singularities or branch cuts. Nevertheless, in the flat-space limit, the cosmological cutting rules morph into standard S-matrix unitarity constraints, and all dynamical information about particle production, decay, and signal nonlocality is encoded in their residue and geometric structure (Baumann et al., 2021, Albayrak et al., 2023). This provides a unifying framework for the analytic bootstrap, cosmological collider physics, and the study of unitarity and microcausality in quantum fields on curved backgrounds.

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References

• (Goodhew et al., 2020) Goodhew, Turiaci, Baumann: The Cosmological Optical Theorem
• (Melville et al., 2021) Melville, Pajer: Cosmological Cutting Rules
• (Goodhew et al., 2021) Goodhew, Melville, Pajer: Cutting Cosmological Correlators
• (Ema et al., 2024) Ema, Mukaida: Cutting rule for in-in correlators and cosmological collider
• (Agui-Salcedo et al., 2023) Henry: The Cosmological Tree Theorem
• (Ghosh et al., 2024) Ghosh, Pajer, Ullah: Cosmological cutting rules for Bogoliubov initial states
• (Ghosh et al., 8 Feb 2025) Pajer, Zhang: Cosmological cutting rules for Bogoliubov initial states: any mass and spin
• (Colipí-Marchant et al., 27 Dec 2025) Goodhew, Kokomani, McFadden: Schwinger-Keldysh Cosmological Cutting Rules
• (Ansari et al., 13 Jan 2026) Ansari, Jain, Mazumdar: Cosmological Cutting Rules from Flat-Space Unitarity via Dressing
• (Das et al., 5 Feb 2026) Das, Karan, Khatun, Kundu: Reconstructing cosmological correlators via dispersion: from cutting to dressing rules
• (Das et al., 23 Dec 2025) Das et al.: A single-cut discontinuity for cosmological correlators from unitarity and analyticity
• (Baumann et al., 2021) Meltzer, Perlmutter: Linking the Singularities of Cosmological Correlators
• (Tong et al., 2021) Tong, Wang, Zhu: Cutting Rule for Cosmological Collider Signals: A Bulk Evolution Perspective
• (De et al., 2024) Bailey, Melnikov, Perlmutter: A physical basis for cosmological correlators from cuts
• (Glew et al., 15 Aug 2025) Kologlu, Mathews, Perlmutter: Kinematic flow from the flow of cuts
• (Qin, 2024) Bi, Jin, Li, Wang: Cosmological Correlators at the Loop Level
• (Albayrak et al., 2023) Cramer, Trnka, Zhang: Perturbative Unitarity and the Wavefunction of the Universe