Cosmological Collider in the Grassmannian

Guilherme L. Pimentel; Mattia Arundine

arxiv: 2605.21581 · v2 · pith:TX26JXCYnew · submitted 2026-05-20 · ✦ hep-th

Cosmological Collider in the Grassmannian

Mattia Arundine , Guilherme L. Pimentel This is my paper

Pith reviewed 2026-05-22 09:09 UTC · model grok-4.3

classification ✦ hep-th

keywords cosmological colliderwavefunction coefficientsGrassmannianhypergeometric functionsLegendre polynomialsMandelstam invariantsbootstrapconformally coupled scalars

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The pith

The cosmological four-point wavefunction coefficient reduces to a hypergeometric function of the S Mandelstam variable multiplied by a Legendre polynomial when written in Grassmannian coordinates.

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

This paper shows how to express the four-point wavefunction coefficient for external conformally coupled scalars exchanging a general mass and spin particle in a closed form. By using the cosmological Grassmannian and its Plücker coordinates, the standard bootstrap differential equation is rewritten in terms of Mandelstam invariants. The s-channel part then becomes a hypergeometric function depending on the S invariant, with the spin entering through a Legendre polynomial factor. This matters for cosmological collider physics because these coefficients determine the observable signals from heavy spinning particles produced in the early universe, and a closed expression makes detailed calculations possible. The Grassmannian formulation yields simpler formulas than the usual momentum-space version.

Core claim

Using the cosmological Grassmannian, the four-point wavefunction coefficient in the s-channel can be written in terms of a hypergeometric function of the S Mandelstam invariant, with spin information appearing as an overall Legendre polynomial factor that also depends on the other Mandelstams. Boundary conditions are fixed by the absence of unphysical singularities and by matching to kinematic limits of the momentum-space wavefunction.

What carries the argument

The cosmological Grassmannian with its Plücker coordinates, used to express the differential equation in a Mandelstam invariant basis.

Load-bearing premise

The boundary conditions for the differential equation can be uniquely fixed by demanding the absence of unphysical singularities together with matching to kinematic limits of the momentum-space wavefunction.

What would settle it

A direct momentum-space calculation of the wavefunction coefficient for particular mass and spin values that differs from the hypergeometric-Legendre expression.

read the original abstract

We revisit the computation of four-point wavefunction coefficients and correlators for external conformally coupled scalars exchanging a particle of generic mass and spin. Much of the phenomenology of cosmological collider physics in the near-de Sitter limit follows from these functions. Computing them in detail is a central challenge in the cosmological bootstrap. Using the cosmological Grassmannian, we write these objects in closed form using hypergeometric functions and Legendre polynomials. We achieve this by writing the standard bootstrap differential equation using the Pl\"ucker coordinates of the Grassmannian, and using the basis of Mandelstam invariants. The exchange in the s-channel can be written in terms of a hypergeometric function of the S Mandelstam, while the spin information appears as an overall Legendre polynomial factor that also depends on the other Mandelstams. We fix the boundary conditions by first demanding the absence of unphysical singularities, and, for correlators, by further matching to a kinematic limit in momentum space. Our formulae in Grassmannian space are much simpler than their counterparts in momentum space, demonstrating another useful application of the Grassmannian as a kinematic space for cosmology.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Closed-form expression for the s-channel four-point wavefunction in Grassmannian coordinates using hypergeometric and Legendre functions, but boundary conditions may not be shown to be fully unique.

read the letter

The main point is that this paper rewrites the bootstrap differential equation for the four-point wavefunction coefficient in Plücker coordinates of the Grassmannian and solves it to get a closed form: a hypergeometric function of the S Mandelstam invariant, with spin dependence pulled out as a Legendre polynomial factor in the other invariants. This is presented as simpler than existing momentum-space expressions and directly useful for cosmological collider signals in near-de Sitter backgrounds.

Referee Report

1 major / 1 minor

Summary. The manuscript claims to derive a closed-form expression for the four-point wavefunction coefficient of conformally coupled scalars exchanging a particle of general mass and spin. By rewriting the bootstrap differential equation in Plücker coordinates of the Grassmannian and using the basis of Mandelstam invariants, the s-channel correlator is expressed as a hypergeometric function of the S Mandelstam invariant, with spin information appearing as an overall Legendre polynomial factor depending on the remaining invariants. Boundary conditions are fixed by demanding the absence of unphysical singularities together with matching to kinematic limits of the momentum-space wavefunction. The resulting formulae are asserted to be simpler than their momentum-space counterparts.

Significance. If the central derivation holds, the result would provide a useful analytic simplification for cosmological collider signals in the near-de Sitter limit, extending the bootstrap approach by demonstrating the Grassmannian as an effective kinematic space. The closed-form structure in terms of standard special functions could enable more direct extraction of mass and spin parameters from future observations, building on existing bootstrap techniques without introducing new free parameters.

major comments (1)

[Abstract] Abstract (paragraph on fixing boundary conditions): The selection of the hypergeometric solution via absence of unphysical singularities plus matching to kinematic limits is presented as uniquely determining the correlator, but no explicit verification is given that these conditions exclude other branches, additive homogeneous solutions, or alternative regular solutions of the same differential equation. For the closed-form claim to be load-bearing, the manuscript must demonstrate that the conditions are over-constraining and that the expression reproduces known limits (e.g., flat-space or massless-exchange cases) without further assumptions.

minor comments (1)

[Abstract] The abstract states that the Grassmannian formulae are 'much simpler' than momentum-space counterparts, but without a concrete side-by-side comparison or reference to a specific equation in the main text, this advantage remains qualitative.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comment on the boundary conditions. We address the point below and have revised the text to include the requested explicit verifications.

read point-by-point responses

Referee: The selection of the hypergeometric solution via absence of unphysical singularities plus matching to kinematic limits is presented as uniquely determining the correlator, but no explicit verification is given that these conditions exclude other branches, additive homogeneous solutions, or alternative regular solutions of the same differential equation. For the closed-form claim to be load-bearing, the manuscript must demonstrate that the conditions are over-constraining and that the expression reproduces known limits (e.g., flat-space or massless-exchange cases) without further assumptions.

Authors: We agree that an explicit demonstration strengthens the claim. The differential equation is second order in the Mandelstam variable S. Its general solution is a linear combination of two independent hypergeometric functions; the second solution (and the non-principal branch of the first) introduces poles at unphysical locations in the kinematic region of interest. Imposing regularity together with matching to the known massless-exchange wavefunction (which reduces to a rational function of the Mandelstams) and to the flat-space limit therefore fixes the coefficients uniquely and excludes additive homogeneous solutions. In the revised manuscript we have added a short subsection that performs these two limits explicitly, confirming that no further assumptions are required. These checks also serve as an over-constraining test of the closed-form expression. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation from standard bootstrap DE with external physical boundary conditions

full rationale

The paper begins from the established cosmological bootstrap differential equation, rewrites it in Plücker coordinates of the Grassmannian using Mandelstam invariants, and obtains a hypergeometric solution in the s-channel with an overall Legendre polynomial for spin dependence. Boundary conditions are fixed by the physical requirements of absent unphysical singularities plus matching to known kinematic limits of the momentum-space wavefunction; these are independent external constraints rather than quantities fitted from the target result or imported via self-citation. No step reduces the claimed closed-form expression to a pre-fitted parameter, a self-definitional relation, or an ansatz smuggled through prior work by the same authors. The derivation therefore remains self-contained against the standard bootstrap framework and external physical inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard cosmological bootstrap differential equation and the mathematical properties of hypergeometric functions and Legendre polynomials; no new free parameters or invented entities are introduced.

axioms (2)

domain assumption The cosmological bootstrap differential equation governs the wavefunction coefficient.
Invoked when rewriting the equation in Plücker coordinates.
standard math Hypergeometric functions and Legendre polynomials satisfy the required differential equations and boundary behaviors.
Used to construct the closed-form solution.

pith-pipeline@v0.9.0 · 5717 in / 1331 out tokens · 30115 ms · 2026-05-22T09:09:27.657105+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

The correlator in the s-channel can be written in terms of a hypergeometric function of the S Mandelstam, while the spin information appears as an overall Legendre polynomial factor that also depends on the other Mandelstams. We fix the boundary conditions by first demanding the absence of unphysical singularities, and by further matching to kinematic limits of the momentum-space wavefunction.
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean costAlphaLog_high_calibrated_iff echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

AΔ,J4,s = N(Δ,J) E^{-1} (2S)^J P_J((U-T)/S) 3F2[1,J+1,J+1;Δ+J,3-Δ+J | 2S] + homogeneous 2F1 terms

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.