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REVIEW 4 major objections 5 minor 1 cited by

All Tree-Level Massive Cosmological Correlators via Spectral Gluing

T0 review · 4 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read This paper claims that every tree-level massive cosmological correlator can be built from universal type-C Lauricella vertex functions glued by a spectral algorithm governed solely by the graph's incidence and path matrices, and that after

desk verdict Real algorithm, honest gaps: the spectral gluing construction is likely correct, but the universal claim outruns the analytic-continuation proof, which is demonstrated only for 2F1. read the letter →

arxiv 2607.18223 v1 pith:JX7PVAJH submitted 2026-07-20 hep-th astro-ph.COgr-qc

Jonathan Gräfe , Denis Werth This is my paper · ORCID
classification hep-thastro-ph.COgr-qc MSC 33C6533C7081T2083F05
keywords cosmologicalcorrelatorsdeSitterspaceLauricellafunctionshypergeometricspectralrepresentationtree-levelgraphsgraphcombinatoricsmagicalidentities
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper sets out to prove that the hypergeometric complexity of massive cosmological correlators at tree level is not intrinsic: every tree graph decomposes into universal single-time-integral 'vertex functions' that are type-C Lauricella functions, and the full correlator is reconstructed by a spectral gluing algorithm whose only input is elementary graph data—the incidence matrix of the rooted tree and its inverse path matrix. The intended payoff is practical and conceptual: any tree-level correlator of scalar fields with generic masses receives an explicit convergent multiple-series representation in which internal energies are resummed, and these series solve the correlator's boundary differential equations as expansions in eigenfunctions. The central claim is that applying graph annihilators—differential operators that strip away the dynamical propagators—yields 'magical identities' in which the remaining hypergeometric kinematic dependence collapses to a rational function (Eq. 6.28), so the apparent transcendentality of massive correlators is kinematical rather than dynamical. A sympathetic reader would care because this turns a zoo of intractable multivariable special functions into graph combinatorics and suggests a geometric principle behind de Sitter observables.

What carries the argument

The central objects are the vertex functions V^{(n)}: single-time integrals over de Sitter time of the external plane wave and n massive Hankel mode functions. Their Mellin–Barnes evaluation expresses each as a sum over ±iμ_j frequency modes of the regularized type-C Lauricella function F_C^{(n)} (multivariable generalizations of the hypergeometric 2F1 function; Eqs. 3.40–3.41). These are the universal building blocks. The spectral propagator (2.14) represents the time-ordered bulk-to-bulk propagator as an integral over off-shell mass ν with density ρ_ν(μ)=1/(ν²–μ²)_{iϵ}, factorizing time integrals and making gluing a residue-collection problem. The combinatorial core is the pair (Q, P): the

What would settle it

Compute, by direct numerical time integration, the two-massive-leg vertex V^{(2)}_{+,µ1,µ2}(u1,u2;p) deep inside region (IV), for example along u1=u2=0.8 with p=1.5, and compare against the paper's reduction-formula series: the claimed continuation must match the integral to numerical precision. Separately, evaluate the spectral gluing integral for two vertices in the regime u_ij u_ji > 4; the paper claims the equal-frequency pieces vanish after analytic continuation, so any nonzero residue series there would falsify the algorithm. For marginal vertices, numerically integrate W^{(3)}_{+,µ}(u;p

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Extended reading notes

Core claim

At the paper's core is the claim that every tree-level massive correlator is governed by one rigid structure. The authors define a vertex function V^{(n)}_{+,µ}(u;p) as a single time integral of n Hankel functions, show via Mellin–Barnes that it is a sum of 2^n frequency modes each given by a regularized type-C Lauricella function F_C^{(n)} (Eqs. 3.40–3.41). They then introduce a spectral representation of the time-ordered bulk-to-bulk propagator that factorizes time integrals and trades nesting for spectral integrals; the spectral gluing algorithm evaluates these by residues. The algebraic heart is the off-shell ansatz (4.13): for a rooted tree with incidence matrix Q, the on-shell conditio

Load-bearing premise

The construction stands or falls on the assumption that the analytic-continuation and pole-cancellation machinery is globally valid in every kinematic region—specifically that the bI^P_{±±} spectral integrals vanish for general multivariable Lauricella functions (shown only for 2F1), that the regularization of the divergent series correctly continues the two-massive-leg vertex in region (IV), and that marginal vertices, for which the authors state no analytic continuation of

Editorial extensions

If this is right

  • For every tree graph with principal-series massive internal lines, the maximally-nested analytic contribution is written as a multiple series over m∈N^{V-1} with terms fixed by the path matrix; no nested time integral needs to be evaluated.
  • The same differential operators D_{ij} that define the graph's differential system also contract edges: applying all of them maps any tree-level massive correlator to a single rational function of the external energies, so the correlator satisfies an explicit hierarchy of differential identities.
  • Because the magical identities survive with Lauricella factors removed, the rational soft-limit expression fixes the full hypergeometric series; this gives an infinite family of non-trivial identities among generalized hypergeometric functions that hold for general twists and internal masses.
  • If the algorithm's convergence statements hold, changing the root vertex yields distinct but valid representations for disjoint kinematic regions, so any given correlator can be evaluated by patching finitely many root expansions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the magical identities are as universal as claimed, they imply a form of analytic bootstrap: the full hypergeometric dependence of a massive graph is fixed by its rational contraction at the total-energy pole, so one could try to prove the identities by matching discontinuities around the singular loci rather than by direct summation.
  • The incidence/path-matrix mechanism points to a natural loop-level extension: for graphs with cycles, the analogue of the path matrix would be a cycle-space or matroid object; the tree-level rooted-decoration construction is the first step toward a purely combinatorial loop integrand.
  • The marginal-vertex gap (Sec. 3.5) is the cleanest place to stress-test the framework: one could numerically integrate W^{(3)} in its physical region and compare to the F_C^{(2)} series; if the series diverges or mismatches, the algorithm needs a new continuation prescription before it can handle folded vertices such as those in the triple-exchange bispectrum.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper develops a spectral-gluing framework for tree-level massive cosmological correlators. It first defines universal one-time-integral "vertex functions" and evaluates them as type-C Lauricella functions F_C^(n) (Sec. 3, Eqs. (3.40)-(3.41)). It then proposes a combinatorial algorithm (Sec. 4) that evaluates the spectral integrals of nested propagators by collecting residues fixed by the incidence matrix Q and path matrix P=Q^{-1} of a rooted tree (Eqs. (4.13)-(4.18)). Explicit series are given for two-site, three-site, N-site chain and N-site star graphs (Sec. 5). Acting with graph annihilators is shown to remove propagator factors, leading to "magical identities" that reduce products/sums of Lauricella functions to rational functions (Sec. 6, Eq. (6.28)). The paper claims that the hypergeometric complexity of massive correlators is governed entirely by graph topology.

Significance. If the main results hold, this is a substantial step: it provides a systematic, graph-topological construction of tree-level massive correlators, with explicit partially resummed series, and it uncovers a new family of identities connecting multivariable Lauricella functions to rational functions. The paper is commendably concrete: it includes numerical cross-checks, recovers the known two-site result of Ref. [46], credits Ref. [66] for the two-site magical identity, and argues for uniform transcendental weight. The construction is parameter-free in the sense that the series coefficients are fixed by graph incidence data, not fitted. However, the advertised universality currently outruns the proof: the central pole-cancellation step is justified only for F_C^(1)=2F1, the analytic continuation for multivariable Lauricella functions is incomplete, and marginal vertices are explicitly excluded. These gaps prevent the paper from fully establishing the claim that all tree-level massive correlators are governed by the asserted rigid structure.

major comments (4)
  1. [Sec. 4.1, Eqs. (4.5)-(4.10)] The reduction of the gluing integral to the mixed ±∓ residues depends on the assertion that bI^P_{++}=bI^P_{--}=0 for arbitrary vertex functions. The text states that for u_{ij}u_{ji}>4 one can see this using a connection formula (A.11), but adds explicitly: 'For the case of F_C^{(1)}=2F1'. In generic trees, internal vertices have degree ≥2, so the relevant vertex functions are F_C^{(n)} with n≥2 (e.g. Appell F4 for a two-massive-leg vertex). No analytic continuation of these multivariable Lauricella functions in the required kinematic region is supplied. Since the off-shell ansatz (4.13) and the path-matrix on-shell condition (4.17)-(4.18) depend on this cancellation, the algorithmic output is not established to equal the true correlator for generic graphs. This is a load-bearing gap in the universal claim.
  2. [Sec. 3.5, Eq. (3.74); Sec. 7] The title and abstract claim 'all tree-level massive cosmological correlators', but marginal vertices (X=0) are not covered. Eq. (3.74) places the physical region for marginal vertices outside the domain in which type-C Lauricella analytic continuations are available, and the text explicitly says 'We will discuss this in a future work.' Marginal vertices occur in legitimate tree-level diagrams (e.g. any internal vertex with no external conformally-coupled leg). The precise class of graphs covered should be stated, and the headline claim should be softened accordingly, or the marginal-vertex continuation must be provided.
  3. [Sec. 3.4, Proposition 1] The proof of sufficiency of the inequalities (3.63) is flawed. The exhibited rank-one construction does not generally satisfy 1^T G1≤1: the step 'the last step uses (3.63) for i=j*' is invalid when u_{j*} - sum_{i≠j*} u_i is negative with large magnitude. For example, u=(0.9,1,1) satisfies (3.63) but the proposed collinear configuration gives 1^T G1=(0.9-2)^2=1.21>1. The proposition may still be true (for instance by arranging non-collinear vectors so the resultant vanishes when no single vector dominates), but the proof as written needs repair. Since this proposition underlies the characterization of the Euclidean kinematic region, a correct proof should be supplied.
  4. [Sec. 3.2, Eq. (3.34) and Fig. 4] The analytic continuation into region (IV) of the two-massive-leg vertex relies on a Levin u-transform (3.34) applied to a formally divergent series. This is a numerical extrapolation technique, but no theorem or rigorous bound is given showing that the transformed sequence converges to the value of the original integral (3.19). The vertex functions are subsequently used as analytic building blocks in the spectral gluing algorithm, so this is not a merely cosmetic issue. A rigorous convergence statement, or a reformulation in which the integral definition is primary and the series is only an auxiliary representation, is needed.
minor comments (5)
  1. [Sec. 3.2, Eq. (3.31)] In the factorized reduction formula for p=1/2, the upper parameters of the two 2F1 factors appear to contain a spurious '+k'; since (2p-1)/2 = 0, the F4 sum collapses to k=0, so these '+k' should be absent. Please check.
  2. [Sec. 5.4, Fig. 7(d)] The central vertex of the N-site star is drawn with no external line, yet the formulas use a nonzero X0. Please clarify whether the computation assumes X0>0, and whether the X0=0 marginal case is deferred to Sec. 3.5.
  3. [Appendix A.3, Lemma A.1] The Lauricella reflection formula is proven by formal manipulation of the defining series, without stating the domain of z or justifying the interchange of the limit c→1-m with the infinite sum. A statement of uniform convergence on compact subsets of the Lauricella convergence domain should be added.
  4. [Sec. 4.2.3] The claimed absolute convergence of the on-shell series is justified only by the soft-limit ('geometric type series') heuristic. The full series contains Lauricella functions whose growth in the summation indices m_i is not controlled. A direct convergence argument or an explicit bound is needed.
  5. [General notation] The heading 'Reflexion formula' should be 'Reflection formula'. Also, the denominator Γ[+iν,-iν] in Eq. (4.6) is notationally confusing because the measure factor Nν has already been displayed; clarifying the cancellation would help the reader.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the central gluing algorithm is a parameter-free derivation; noted analytic-continuation gaps and self-cited inputs affect completeness, not circularity.

full rationale

The paper's derivation chain is not circular: the spectral factorization (2.14) is a stated mathematical input (attributed to [46,70,68]) and is not a restatement of the final series; the off-shell ansatz (4.13), on-shell condition (4.18), and path-matrix algorithm are then derived and cross-checked numerically (Figs. 4, 8, 9) and against the independent differential system (6.8)/(App. C). The two-site magical identity (6.23) is explicitly attributed to the authors' prior [66] rather than claimed as new, and the generalized identity (6.28) is obtained by applying graph annihilators to the series and comparing to the independently contracted single-vertex graph, so it is a derived identity rather than an input renamed. The paper's own caveats are limitations, not circularity: Sec. 4.1 proves the key pole-cancellation for F_C^(1)=2F1 only ('For the case of F_C^(1)=2F1 one can see this by using a connection formula (A.11)') and asserts it for general n; Sec. 3.5 states that 'analytic continuations of type-C Lauricella functions are currently not available' for marginal vertices. These are completeness/correctness gaps in the universality claim, not self-referential reductions. I assign 2 only for the minor self-citation content in the imported spectral input; the central combinatorial content is independent.

Assumptions & free parameters 2 free parameters · 6 assumptions · 2 invented entities

The paper introduces no new physical entities and fits no physical parameters to data: masses µ_j, twists p_i and energies are inputs from the physical setup. The free-parameter ledger therefore contains only a complex-branch choice in the F4 reduction formula (matched to numerics) and Levin-transform truncation parameters. The load-bearing external inputs are the spectral representation (2.14) from the authors' own prior work, the differential system (6.8) from [48], and the standard special-function and graph-theory toolkit. The invented entities are new mathematical constructs (Lauricella reflection formula, graph annihilators), both falsifiable by direct evaluation.

free parameters (2)
  • Branch choice for ξ1, ξ2 in the Appell F4 reduction formula = smaller root selected in Eq. (3.30)
    Retaining the smaller root is justified by 'the analytic continuation matches the numerical evaluation of the vertex function,' i.e. matched to numerics rather than derived; a wrong branch would alter the analytic continuation in region (IV) of Fig. 3.
  • Levin u-transform truncation parameters = N=20, n=10 (Fig. 4 right panel)
    Technical parameters of the nonlinear sequence transformation used to assign values to a formally divergent series in region (IV); chosen ad hoc to stabilize the numerics, with residual oscillations acknowledged.
assumptions (6)
  • domain assumption Spectral representation of time-ordered bulk-to-bulk propagator, Eq. (2.14), with density ρ_ν(µ) and measure [dν]
    Load-bearing factorization that turns nested time integrals into spectral integrals; cited to the authors' own prior work [46, 68, 70] rather than re-derived in this paper.
  • domain assumption Differential system for tree graphs, Eq. (6.8): D_ij G = C_ij[G] with the contracted graph as source
    Basis of the magical identities in Sec. 6.3; taken from [48, 53, 58]. The general identity (6.28) is exactly as strong as this external system, and only the two-site case is numerically checked here.
  • standard math Distributional split (2.18) of the spectral density into on-shell δ and Cauchy principal-value parts
    Standard distribution identity in this context; underlies the two-channel (on-shell vs analytic) decomposition of the gluing algorithm.
  • standard math Mellin-Barnes representation of Hankel functions (3.2) and the z-integral identity (3.4) with analytic continuation
    Standard special-function machinery used for all vertex-function computations in Sec. 3.
  • standard math Graph-theoretic facts: reduced incidence matrix of a rooted tree is unimodular and its inverse is the path matrix (Props. 2-3)
    Standard algebraic graph theory (Kirchhoff / matrix-tree theorem); used to solve the on-shell linear system (4.17)-(4.18).
  • domain assumption Physical setup: Bunch-Davies vacuum, principal-series masses µ_j ∈ R_{>0}, conformally coupled external scalars, weak coupling, tree level, IR-finite twists
    Restricts the scope of 'all' correlators; complementary series is only claimed via analytic continuation, and dS-breaking dispersions are stated as straightforward extensions without analysis.
invented entities (2)
  • Lauricella reflection formula (Lemma A.1 / Cor. A.1.1) independent evidence
    purpose: Shows apparent poles of vertex functions at µ_j = ±im cancel, enabling the gluing algorithm to discard the ±± residue series (Eq. 4.9).
    A new identity claimed 'to our knowledge, was not known before'; falsifiable by evaluating both sides at small truncation order for specific parameters — such checks are not shown in the paper.
  • Graph annihilators D_ij (Eq. 6.8) independent evidence
    purpose: Differential operators that remove propagators from series solutions and contract internal edges; they generate the magical identities of Sec. 6.3.
    New differential-operator construction; its key property (eigenvalue x^2+µ^2 on the off-shell ansatz) is derived in (6.13)-(6.15), and its consequences (magical identities) are numerically testable, as done for the two-site case.

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Cite this review

Pith. "Pith review of All Tree-Level Massive Cosmological Correlators via Spectral Gluing." pith.science (2026). https://pith.science/paper/JX7PVAJH

@misc{pith2026260718223,
  author       = {Pith},
  title        = {Pith review of: All Tree-Level Massive Cosmological Correlators via Spectral Gluing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JX7PVAJH}},
  note         = {Machine review of arXiv:2607.18223}
}
read the original abstract

Massive cosmological correlators exhibit a rich hypergeometric structure already at tree level, reflecting the distorted propagation of particles in de Sitter spacetime. In this paper, we reveal that this apparent complexity conceals a remarkably simple underlying mathematical structure. Using the spectral representation, we compute arbitrary tree-level correlators of scalar fields with generic masses and show that they are constructed from fundamental building blocks belonging to the family of Lauricella generalised hypergeometric functions, glued together by spectral integrals. We develop a spectral gluing algorithm that evaluates these integrals through elementary graph combinatorics, yielding explicit series representations that resum the dependence on internal energies away from soft limits. This algorithm naturally generates solutions to the differential equations satisfied by massive correlators as expansions in the corresponding eigenfunctions. Acting with a set of graph annihilators, we uncover a new class of magical identities among generalised hypergeometric functions, revealing an unexpected simplification: once the dynamical propagators are stripped away, the remaining hypergeometric kinematic dependence collapses to rational functions. Our results expose a hidden simplicity in the rigid hypergeometric analytic structure dictated by graph combinatorics, and hint at an intrinsic geometric principle from which properties of massive correlators naturally emerge.

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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Reference graph

Works this paper leans on

132 extracted references · 83 linked inside Pith · cited by 1 Pith paper

  1. [46]

    Werth,Spectral representation of cosmological correlators,JHEP12(2024) 017 [2409.02072]

    D. Werth,Spectral representation of cosmological correlators,JHEP12(2024) 017 [2409.02072]

    arXiv 2024

  2. [66]

    Gr¨ afe and I

    J. Gr¨ afe and I. Sachs,Split Representations and Bubble Resummation for Massive de Sitter Correlators,2602.09977

    arXiv

  3. [1]

    Hillman,Symbol Recursion for the dS Wave Function,1912.09450

    A. Hillman,Symbol Recursion for the dS Wave Function,1912.09450

    arXiv 1912

  4. [2]

    De and A

    S. De and A. Pokraka,Cosmology meets cohomology,JHEP03(2024) 156 [2308.03753]

    arXiv 2024

  5. [3]

    Arkani-Hamed, D

    N. Arkani-Hamed, D. Baumann, A. Hillman, A. Joyce, H. Lee and G. L. Pimentel, Differential equations for cosmological correlators,JHEP09(2025) 009 [2312.05303]

    arXiv 2025

  6. [4]

    Baumann, H

    D. Baumann, H. Goodhew and H. Lee,Kinematic flow for cosmological loop integrands,JHEP07(2025) 131 [2410.17994]

    arXiv 2025

  7. [5]

    T. W. Grimm, A. Hoefnagels and M. van Vliet,Structure and complexity of cosmological correlators,Phys. Rev. D110(2024) 123531 [2404.03716]

    arXiv 2024

  8. [6]

    S. He, X. Jiang, J. Liu, Q. Yang and Y.-Q. Zhang,Differential equations and recursive solutions for cosmological amplitudes,JHEP01(2025) 001 [2407.17715]

    arXiv 2025

Show all 132 references
  1. [7]

    T. W. Grimm and A. Hoefnagels,Reductions of GKZ systems and applications to cosmological correlators,JHEP04(2025) 196 [2409.13815]

    2025 arXiv

  2. [8]

    Hang and C

    Y. Hang and C. Shen,A note on kinematic flow and differential equations for two-site one-loop graph in FR W spacetime,JHEP09(2025) 209 [2410.17192]

    2025

  3. [9]

    De and A

    S. De and A. Pokraka,A physical basis for cosmological correlators from cuts,JHEP 03(2025) 040 [2411.09695]

    2025 arXiv

  4. [10]

    Baumann, H

    D. Baumann, H. Goodhew, A. Joyce, H. Lee, G. L. Pimentel and T. Westerdijk, Geometry of kinematic flow,JHEP05(2026) 211 [2504.14890]

    2026 arXiv

  5. [11]

    Capuano, L

    M. Capuano, L. Ferro, T. Lukowski and A. Palazio,Canonical Differential Equations for Cosmology from Positive Geometries,2505.14609

  6. [12]

    Glew and A

    R. Glew and A. Pokraka,Kinematic flow from the flow of cuts,JHEP06(2026) 158 [2508.11568]

    2026 arXiv

  7. [13]

    McLeod, A

    A. McLeod, A. Pokraka and L. Ren,A Graphical Coaction for FR W Wavefunction Coefficients,2603.25703

  8. [14]

    A. J. McLeod, A. Pokraka and L. Ren,A Graphical Coaction for FR W Integrals from Partial/Relative Twisted (Co)homology,2606.13627. – 102 –

    arXiv

  9. [15]

    Fevola, G

    C. Fevola, G. L. Pimentel, A.-L. Sattelberger and T. Westerdijk,Algebraic approaches to cosmological integrals,Matematiche80(2025) 303 [2410.14757]

    2025 arXiv

  10. [16]

    Capuano, L

    M. Capuano, L. Ferro, T. Lukowski and A. Palazio,Cosmology meets cluster algebra, 2512.14859

  11. [17]

    Mazloumi and X

    P. Mazloumi and X. Xu,Cluster algebras for cosmological correlators,JHEP03 (2026) 256 [2512.14854]

    2026

  12. [18]

    Paranjape, M

    S. Paranjape, M. Skowronek, M. Spradlin, A. Volovich and H.-C. Weng,Cluster Bootstrap for Cosmological Correlators,2603.08670

  13. [19]

    Capuano, L

    M. Capuano, L. Ferro, T. Lukowski, A. Palazio and Y.-Q. Zhang,Generalised Cluster Adjacency for Cosmology,2603.09965

  14. [20]

    Ferro, T

    L. Ferro, T. Lukowski, L. Ren, M. Spradlin, A. Volovich, H.-C. Weng et al.,de Sitter Wavefunction from Quadrangular Polylogarithms: Chain Graphs,2605.06542

    arXiv

  15. [21]

    Fan and Z.-Z

    B. Fan and Z.-Z. Xianyu,Cosmological amplitudes in power-law FR W universe, JHEP12(2024) 042 [2403.07050]

    2024 arXiv

  16. [22]

    Fan and Z.-Z

    B. Fan and Z.-Z. Xianyu,Anatomy of family trees in cosmological correlators,JHEP 12(2025) 179 [2509.02684]

    2025 arXiv

  17. [23]

    Chowdhury, A

    C. Chowdhury, A. Lipstein, J. Mei, I. Sachs and P. Vanhove,The subtle simplicity of cosmological correlators,JHEP03(2025) 007 [2312.13803]

    2025 arXiv

  18. [24]

    Chowdhury, A

    C. Chowdhury, A. Lipstein, J. Marshall, J. Mei and I. Sachs,Cosmological dressing rules,JHEP03(2026) 076 [2503.10598]

    2026

  19. [25]

    Chowdhury, S

    C. Chowdhury, S. Jazayeri, A. Lipstein, J. Marshall, J. Mei and I. Sachs, Cosmological Correlator Discontinuities from Scattering Amplitudes,2602.03841

  20. [26]

    Chowdhury, S

    C. Chowdhury, S. He, Y.-X. Su and D. Yang,On the simplicity of de Sitter correlators,2604.26421

    arXiv

  21. [27]

    S. Das, D. Karan, B. Khatun and N. Kundu,Reconstructing cosmological correlators via dispersion: from cutting to dressing rules,2602.05546

  22. [28]

    Arkani-Hamed, P

    N. Arkani-Hamed, P. Benincasa and A. Postnikov,Cosmological Polytopes and the Wavefunction of the Universe,1709.02813

    arXiv

  23. [29]

    Arkani-Hamed and P

    N. Arkani-Hamed and P. Benincasa,On the Emergence of Lorentz Invariance and Unitarity from the Scattering Facet of Cosmological Polytopes,1811.01125

    arXiv

  24. [30]

    Benincasa,From the flat-space S-matrix to the Wavefunction of the Universe, 1811.02515

    P. Benincasa,From the flat-space S-matrix to the Wavefunction of the Universe, 1811.02515

    arXiv

  25. [31]

    Benincasa,Cosmological Polytopes and the Wavefuncton of the Universe for Light States,1909.02517

    P. Benincasa,Cosmological Polytopes and the Wavefuncton of the Universe for Light States,1909.02517

    1909 arXiv

  26. [32]

    Benincasa, A

    P. Benincasa, A. J. McLeod and C. Vergu,Steinmann Relations and the Wavefunction of the Universe,Phys. Rev. D102(2020) 125004 [2009.03047]. – 103 –

    2020 arXiv

  27. [33]

    Albayrak, P

    S. Albayrak, P. Benincasa and C. Duaso Pueyo,Perturbative unitarity and the wavefunction of the Universe,SciPost Phys.16(2024) 157 [2305.19686]

    2024 arXiv

  28. [34]

    Benincasa,Amplitudes meet Cosmology: A (Scalar) Primer,2203.15330

    P. Benincasa,Amplitudes meet Cosmology: A (Scalar) Primer,2203.15330

    arXiv

  29. [35]

    Arkani-Hamed, C

    N. Arkani-Hamed, C. Figueiredo and F. Vaz˜ ao,Cosmohedra,JHEP11(2025) 029 [2412.19881]

    2025

  30. [36]

    Glew and T

    R. Glew and T. Lukowski,Amplitubes: graph cosmohedra,JHEP09(2025) 074 [2502.17564]

    2025 arXiv

  31. [37]

    Figueiredo and F

    C. Figueiredo and F. Vaz˜ ao,Correlator polytopes,Phys. Rev. D113(2026) 025005 [2506.19907]

    2026

  32. [38]

    Forcey, R

    S. Forcey, R. Glew and H. Kim,Generalised wavefunction coefficients and acyclonesto-cosmohedra,J. Phys. A58(2025) 465403 [2507.09736]

    2025

  33. [39]

    Ardila-Mantilla, N

    F. Ardila-Mantilla, N. Arkani-Hamed, C. Figueiredo and F. Vaz˜ ao,Combinatorics of the Cosmohedron,2603.03425

  34. [40]

    Telen,Positive Geometry of Polytopes and Polypols,2506.05510

    S. Telen,Positive Geometry of Polytopes and Polypols,2506.05510

    arXiv

  35. [41]

    S. De, S. Paranjape, A. Pokraka, M. Spradlin and A. Volovich,Hidden zeros of the cosmological wavefunction,JHEP07(2025) 174 [2503.23579]

    2025 arXiv

  36. [42]

    Li and L

    Y. Li and L. Rodina,Cosmological Wavefunctions as Amplitudes: Dual Shuffle Factorization and Uniqueness from New Hidden Zeros,2604.01133

  37. [43]

    Arkani-Hamed, D

    N. Arkani-Hamed, D. Baumann, H. Lee and G. L. Pimentel,The Cosmological Bootstrap: Inflationary Correlators from Symmetries and Singularities,JHEP04 (2020) 105 [1811.00024]

    2020 arXiv

  38. [44]

    Qin and Z.-Z

    Z. Qin and Z.-Z. Xianyu,Helical inflation correlators: partial Mellin-Barnes and bootstrap equations,JHEP04(2023) 059 [2208.13790]

    2023 arXiv

  39. [45]

    Qin and Z.-Z

    Z. Qin and Z.-Z. Xianyu,Closed-form formulae for inflation correlators,JHEP07 (2023) 001 [2301.07047]

    2023 arXiv

  40. [47]

    Xianyu and J

    Z.-Z. Xianyu and J. Zang,Inflation correlators with multiple massive exchanges, JHEP03(2024) 070 [2309.10849]

    2024 arXiv

  41. [48]

    Liu and Z.-Z

    H. Liu and Z.-Z. Xianyu,Massive inflationary amplitudes: differential equations and complete solutions for general trees,JHEP09(2025) 183 [2412.07843]

    2025 arXiv

  42. [49]

    Arkani-Hamed and J

    N. Arkani-Hamed and J. Maldacena,Cosmological Collider Physics,1503.08043

    arXiv

  43. [50]

    G. L. Pimentel and D.-G. Wang,Boostless cosmological collider bootstrap,JHEP10 (2022) 177 [2205.00013]

    2022 arXiv

  44. [51]

    Jazayeri and S

    S. Jazayeri and S. Renaux-Petel,Cosmological bootstrap in slow motion,JHEP12 (2022) 137 [2205.10340]. – 104 –

    2022 arXiv

  45. [52]

    Chen and B

    J. Chen and B. Feng,Towards systematic evaluation of de Sitter correlators via Generalized Integration-By-Parts relations,JHEP06(2024) 199 [2401.00129]

    2024 arXiv

  46. [53]

    Gasparotto, P

    F. Gasparotto, P. Mazloumi and X. Xu,Differential equations for tree-level cosmological correlators with massive states,JHEP09(2025) 043 [2411.05632]

    2025 arXiv

  47. [54]

    J. Chen, B. Feng and Y.-X. Tao,Multivariate hypergeometric solutions of cosmological (dS) correlators by d log-form differential equations,JHEP03(2025) 075 [2411.03088]

    2025 arXiv

  48. [55]

    S. Aoki, L. Pinol, F. Sano, M. Yamaguchi and Y. Zhu,Cosmological correlators with double massive exchanges: bootstrap equation and phenomenology,JHEP09(2024) 176 [2404.09547]

    2024 arXiv

  49. [56]

    Xianyu and J

    Z.-Z. Xianyu and J. Zang,Massive inflationary amplitudes: new representations and degenerate limits,JHEP03(2026) 122 [2511.08677]

    2026

  50. [57]

    J. Chen, B. Feng, Z. Qin and Y.-X. Tao,Loop integrals in de Sitter spacetime: The parity-split IBP system andd log-form differential equations,2604.14549

    arXiv

  51. [58]

    Baumann, A

    D. Baumann, A. Joyce, H. Lee and K. Salehi Vaziri,Differential Equations for Massive Correlators,2604.08658

    arXiv

  52. [59]

    Sleight,A Mellin Space Approach to Cosmological Correlators,JHEP01(2020) 090 [1906.12302]

    C. Sleight,A Mellin Space Approach to Cosmological Correlators,JHEP01(2020) 090 [1906.12302]

    2020 arXiv

  53. [60]

    Sleight and M

    C. Sleight and M. Taronna,Bootstrapping Inflationary Correlators in Mellin Space, JHEP02(2020) 098 [1907.01143]

    2020 arXiv

  54. [61]

    Sleight and M

    C. Sleight and M. Taronna,From dS to AdS and back,JHEP12(2021) 074 [2109.02725]

    2021 arXiv

  55. [62]

    Qin and Z.-Z

    Z. Qin and Z.-Z. Xianyu,Phase information in cosmological collider signals,JHEP 10(2022) 192 [2205.01692]

    2022 arXiv

  56. [63]

    Xianyu and H

    Z.-Z. Xianyu and H. Zhang,Bootstrapping one-loop inflation correlators with the spectral decomposition,JHEP04(2023) 103 [2211.03810]

    2023 arXiv

  57. [64]

    H. Liu, Z. Qin and Z.-Z. Xianyu,Dispersive bootstrap of massive inflation correlators,JHEP02(2025) 101 [2407.12299]

    2025 arXiv

  58. [65]

    H. Liu, Z. Qin, J. Wu, Z.-Z. Xianyu and H. Zhang,On-Shell Bootstrap of Loop Inflation Correlators with Spectral Dispersion,2606.02686

    arXiv

  59. [67]

    Belrhali, A

    N. Belrhali, A. Poisson, S. Renaux-Petel and D. Werth,De Sitter Momentum Space, 2601.15228

    arXiv

  60. [68]

    Belrhali, A

    N. Belrhali, A. Poisson, S. Renaux-Petel and D. Werth,Kontorovich-Lebedev-Fourier Space for de Sitter Correlators,2604.15251

    arXiv

  61. [69]

    Melville and G

    S. Melville and G. L. Pimentel,de Sitter S matrix for the masses,Phys. Rev. D110 (2024) 103530 [2309.07092]. – 105 –

    2024 arXiv

  62. [70]

    Melville and G

    S. Melville and G. L. Pimentel,A de Sitter S-matrix from amputated cosmological correlators,JHEP08(2024) 211 [2404.05712]

    2024 arXiv

  63. [71]

    Arundine, D

    M. Arundine, D. Baumann, M. H. G. Lee, G. L. Pimentel and F. Rost,The Cosmological Grassmannian,2602.07117

    arXiv

  64. [72]

    Arundine and G

    M. Arundine and G. L. Pimentel,Cosmological Collider in the Grassmannian, 2605.21581

    arXiv

  65. [73]

    Melville and E

    S. Melville and E. Pajer,Cosmological Cutting Rules,JHEP05(2021) 249 [2103.09832]

    2021 arXiv

  66. [74]

    C´ espedes, A.-C

    S. C´ espedes, A.-C. Davis and S. Melville,On the time evolution of cosmological correlators,JHEP02(2021) 012 [2009.07874]

    2021 arXiv

  67. [75]

    Agui Salcedo and S

    S. Agui Salcedo and S. Melville,The cosmological tree theorem,JHEP12(2023) 076 [2308.00680]

    2023 arXiv

  68. [76]

    Goodhew, S

    H. Goodhew, S. Jazayeri and E. Pajer,The Cosmological Optical Theorem,JCAP04 (2021) 021 [2009.02898]

    2021 arXiv

  69. [77]

    S. Das, D. Karan, B. Khatun and N. Kundu,A single-cut discontinuity for cosmological correlators from unitarity and analyticity,2512.20720

  70. [78]

    X. Chen, Y. Wang and Z.-Z. Xianyu,Schwinger-Keldysh Diagrammatics for Primordial Perturbations,JCAP12(2017) 006 [1703.10166]

    2017 arXiv

  71. [79]

    Baumann, C

    D. Baumann, C. Duaso Pueyo, A. Joyce, H. Lee and G. L. Pimentel,The cosmological bootstrap: weight-shifting operators and scalar seeds,JHEP12(2020) 204 [1910.14051]

    2020 arXiv

  72. [80]

    Z. Qin, S. Renaux-Petel, X. Tong, D. Werth and Y. Zhu,The exact and approximate tales of boost-breaking cosmological correlators,JCAP09(2025) 058 [2506.01555]

    2025 arXiv

  73. [81]

    J. M. Maldacena and G. L. Pimentel,On graviton non-Gaussianities during inflation,JHEP09(2011) 045 [1104.2846]

    2011 arXiv

  74. [82]

    Raju,New Recursion Relations and a Flat Space Limit for AdS/CFT Correlators, Phys

    S. Raju,New Recursion Relations and a Flat Space Limit for AdS/CFT Correlators, Phys. Rev. D85(2012) 126009 [1201.6449]

    2012 arXiv

  75. [83]

    X. Tong, Y. Wang and Y. Zhu,Cutting rule for cosmological collider signals: a bulk evolution perspective,JHEP03(2022) 181 [2112.03448]

    2022 arXiv

  76. [84]

    Stienstra,GKZ hypergeometric structures, inInstanbul 2005: CIMPA Summer School on Arithmetic and Geometry Around Hypergeometric Functions, 11, 2005, math/0511351

    J. Stienstra,GKZ hypergeometric structures, inInstanbul 2005: CIMPA Summer School on Arithmetic and Geometry Around Hypergeometric Functions, 11, 2005, math/0511351

    2005 arXiv

  77. [85]

    Cattani,Three lectures on hypergeometric functions, 2006, https://api.semanticscholar.org/CorpusID:13450961

    E. Cattani,Three lectures on hypergeometric functions, 2006, https://api.semanticscholar.org/CorpusID:13450961

    2006

  78. [86]

    F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark,NIST Handbook of Mathematical Functions. 2010

    2010

  79. [87]

    J. L. Burchnall and T. W. Chaundy,Expansions of appell’s double hypergeometric functions,Quarterly Journal of Mathematics1(1940) 249. – 106 –

    1940

  80. [88]

    Nakagawa,Appell-lauricella hypergeometric functions over finite fields and algebraic varieties, 2023

    A. Nakagawa,Appell-lauricella hypergeometric functions over finite fields and algebraic varieties, 2023

    2023

  81. [89]

    Clingher, C

    A. Clingher, C. F. Doran and A. Malmendier,Special function identities from superelliptic kummer varieties,Asian Journal of Mathematics21(2017) 909–952

    2017

  82. [90]

    Alim and F

    M. Alim and F. La Mantia,Modularity of Feynman Integrals and Factorization of Appell F2 Systems,2605.07431

    arXiv

  83. [91]

    W. N. Bailey,A reducible case of the fourth type of Appell’s hypergeometric functions of two variables,The Quarterly Journal of Mathematics (Oxford Series)os-4(1933) 305–308

    1933

  84. [92]

    Chen and Y

    X. Chen and Y. Wang,Quasi-Single Field Inflation and Non-Gaussianities,JCAP 04(2010) 027 [0911.3380]

    2010 arXiv

  85. [93]

    Creminelli,Conformal invariance of scalar perturbations in inflation,Phys

    P. Creminelli,Conformal invariance of scalar perturbations in inflation,Phys. Rev. D85(2012) 041302 [1108.0874]

    2012 arXiv

  86. [94]

    Nakagawa,Appell-lauricella hypergeometric functions over finite fields and algebraic varieties,Hokkaido Mathematical Journal53(2024) 307

    A. Nakagawa,Appell-lauricella hypergeometric functions over finite fields and algebraic varieties,Hokkaido Mathematical Journal53(2024) 307

    2024

  87. [95]

    Matsumoto,Appell and Lauricella Hypergeometric Functions

    K. Matsumoto,Appell and Lauricella Hypergeometric Functions. Cambridge University Press, 2020

    2020

  88. [96]

    L. J. Slater,Generalized Hypergeometric Functions. Cambridge University Press, 1966

    1966

  89. [97]

    R. J. Eden, P. V. Landshoff, D. I. Olive and J. C. Polkinghorne,The analytic S-matrix. Cambridge Univ. Press, Cambridge, 1966

    1966

  90. [98]

    V. C. Cortes, H. Frost and B. Sturmfels,Kinematic Stratifications,2503.09571

  91. [99]

    Kontorovich and N

    M. Kontorovich and N. Lebedev,On the method of solution of certain boundary value problems of the wave equation,Doklady Akademii Nauk SSSR19(1938) 441

    1938

  92. [100]

    Lebedev,Sur une formule d’inversion,Doklady Akademii Nauk SSSR52(1946) 655

    N. Lebedev,Sur une formule d’inversion,Doklady Akademii Nauk SSSR52(1946) 655

    1946

  93. [101]

    Biggs,Algebraic Graph Theory, Cambridge Mathematical Library

    N. Biggs,Algebraic Graph Theory, Cambridge Mathematical Library. Cambridge University Press, 2 ed., 1974

    1974

  94. [102]

    S. J. Parke and T. R. Taylor,An Amplitude for n Gluon Scattering,Phys. Rev. Lett. 56(1986) 2459

    1986

  95. [103]

    J. M. Drummond, J. Henn, G. P. Korchemsky and E. Sokatchev,Dual superconformal symmetry of scattering amplitudes in N=4 super-Yang-Mills theory, Nucl. Phys. B828(2010) 317 [0807.1095]

    2010 arXiv

  96. [104]

    Arkani-Hamed and J

    N. Arkani-Hamed and J. Trnka,The Amplituhedron,JHEP10(2014) 030 [1312.2007]

    2014 arXiv

  97. [105]

    Bzowski, P

    A. Bzowski, P. McFadden and K. Skenderis,Implications of conformal invariance in momentum space,JHEP03(2014) 111 [1304.7760]. – 107 –

    2014 arXiv

  98. [106]

    Corian` o and M

    C. Corian` o and M. M. Maglio,On Some Hypergeometric Solutions of the Conformal Ward Identities of Scalar 4-point Functions in Momentum Space,JHEP09(2019) 107 [1903.05047]

    2019 arXiv

  99. [107]

    Bzowski, P

    A. Bzowski, P. McFadden and K. Skenderis,Conformaln-point functions in momentum space,Phys. Rev. Lett.124(2020) 131602 [1910.10162]

    2020 arXiv

  100. [108]

    Bzowski, P

    A. Bzowski, P. McFadden and K. Skenderis,Conformal correlators as simplex integrals in momentum space,JHEP01(2021) 192 [2008.07543]

    2021 arXiv

  101. [109]

    Gonzalez and V

    I. Gonzalez and V. H. Moll,Definite integrals by the method of brackets. part 1, 2008

    2008

  102. [110]

    Gonzalez, V

    I. Gonzalez, V. H. Moll and A. Straub,The method of brackets. part 2: examples and applications, 2010

    2010

  103. [111]

    Raman and Q

    P. Raman and Q. Yang,Mastering cosmological amplitudes using generalized Ramanujan’s theorem,JHEP04(2026) 036 [2508.13126]

    2026 arXiv

  104. [112]

    I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky,Generalized Euler integrals and A-hypergeometric functions,Adv. Math.84(1990) 255

    1990

  105. [113]

    I. M. Gel’fand, A. V. Zelevinskii and M. Kapranov,Hypergeometric functions and toral manifolds,Functional Analysis and Its Applications23(1989) 94

    1989

  106. [114]

    I. M. Gelfand, M. Kapranov and A. Zelevinsky,Hypergeometric functions, toric varieties and newton polyhedra, 1991, https://api.semanticscholar.org/CorpusID:123539504

    1991

  107. [115]

    I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky,Discriminants, Resultants, and Multidimensional Determinants, Modern Birkh¨ auser Classics. Birkh¨ auser Boston, Boston, MA, 1st ed. 1994. ed., 1994

    1994

  108. [116]

    Caloro and P

    F. Caloro and P. McFadden,A-hypergeometric functions and creation operators for Feynman and Witten diagrams,2309.15895

    arXiv

  109. [117]

    T. W. Grimm, A. Hoefnagels and M. van Vliet,A reduction algorithm for cosmological correlators: cuts, contractions, and complexity,JHEP03(2026) 208 [2503.05866]

    2026 arXiv

  110. [118]

    Arkani-Hamed, Y

    N. Arkani-Hamed, Y. Bai, S. He and G. Yan,Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet,JHEP05(2018) 096 [1711.09102]

    2018 arXiv

  111. [119]

    Chen,Algebras of iterated path integrals and fundamental groups,Transactions of the American Mathematical Society156(1971) 359

    K.-T. Chen,Algebras of iterated path integrals and fundamental groups,Transactions of the American Mathematical Society156(1971) 359

    1971

  112. [120]

    Hain,Iterated integrals and algebraic cycles: Examples and prospects, 2001

    R. Hain,Iterated integrals and algebraic cycles: Examples and prospects, 2001

    2001

  113. [121]

    Brown,Iterated integrals in quantum field theory, in6th Summer School on Geometric and Topological Methods for Quantum Field Theory, pp

    F. Brown,Iterated integrals in quantum field theory, in6th Summer School on Geometric and Topological Methods for Quantum Field Theory, pp. 188–240, 2013, DOI

    2013

  114. [122]

    Bogner and F

    C. Bogner and F. Brown,Symbolic integration and multiple polylogarithms,PoS LL2012(2012) 053 [1209.6524]. – 108 –

    2012 arXiv

  115. [123]

    H. M. Srivastava and P. W. Karlsson,Multiple gaussian hypergeometric series, 1985, https://api.semanticscholar.org/CorpusID:117895978

    1985

  116. [124]

    Exton,On the system of partial differential equations associated with appell’s function f4,Journal of Physics A: Mathematical and General28(1995) 631

    H. Exton,On the system of partial differential equations associated with appell’s function f4,Journal of Physics A: Mathematical and General28(1995) 631

    1995

  117. [125]

    Appell and J

    P. Appell and J. K. de F´ eriet,Fonctions hyperg´ eom´ etriques et hypersph´ eriques : polynomes d’hermite, https://api.semanticscholar.org/CorpusID:190138654

  118. [126]

    Bateman and B

    H. Bateman and B. M. Project,Higher Transcendental Functions [Volumes I-III]. McGraw-Hill Book Company, 1953

    1953

  119. [127]

    Alkofer, M

    R. Alkofer, M. Q. Huber and K. Schwenzer,Infrared Behavior of Three-Point Functions in Landau Gauge Yang-Mills Theory,Eur. Phys. J. C62(2009) 761 [0812.4045]

    2009 arXiv

  120. [128]

    Ananthanarayan, S

    B. Ananthanarayan, S. Friot, S. Ghosh and A. Hurier,New analytic continuations for the AppellF 4 series from quadratic transformations of the Gauss 2F1 function, 2005.07170

    2005 arXiv

  121. [129]

    Lauricella,Sulle funzioni ipergeometriche a piu variabili,Rendiconti del Circolo Matematico di Palermo7(1893) 111

    G. Lauricella,Sulle funzioni ipergeometriche a piu variabili,Rendiconti del Circolo Matematico di Palermo7(1893) 111

  122. [130]

    Goto and J

    Y. Goto and J. Kaneko,The fundamental group of the complement of the singular locus of lauricella’sf c, 2017

    2017

  123. [131]

    Goto,Appell-Lauricella’s hypergeometric functions and intersection theory,PoS MA2019(2022) 008

    Y. Goto,Appell-Lauricella’s hypergeometric functions and intersection theory,PoS MA2019(2022) 008

    2022

  124. [132]

    Belrhali, A

    N. Belrhali, A. Poisson and S. Renaux-Petel,Cosmological Correlators in KLF and the Double-Exchange,2607.05327. – 109 –

    arXiv

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