Round 1

Bispectrum–Trispectrum Phase Closure for a Non-Relativistic Tilted-Ghost Collider

Idea Summary

Calculate the collapsed primordial trispectrum mediated by a massive tilted-ghost spectator and combine it with the known squeezed bispectrum to construct coupling-independent phase and amplitude consistency observables. The new object probes bilinears of the spectator’s two late-time Whittaker connection coefficients, whereas the existing bispectrum probes a different linear combination. This can determine whether propagation-induced branch asymmetry is observably distinct from an ordinary chemical potential. The calculation uses weak mixing only and therefore lies outside the excluded strong-mixing direction.

Motivation

arXiv:2605.20102 establishes that non-relativistic propagation can enhance or suppress a squeezed bispectrum and shift its clock phase, but a single bispectrum can confuse this effect with chemical-potential-like production, a vacuum-normalization choice, or a free template phase. A collapsed trispectrum supplies a second, factorized measurement of the same late-time branches. A coupling-independent bispectrum–trispectrum closure relation would therefore convert an interesting modified-dispersion signal into a falsifiable identification test. Removing the trispectrum calculation would remove this discrimination capability, not merely an alternative representation of the known bispectrum.

Calculation Plan

Model and regime. Use a slow-roll Goldstone mode $\pi$ and a principal-series scalar spectator $\sigma$, with weak interactions $\mathcal L_{\rm int}/a^3=\rho\dot\pi\sigma+c_3\dot\pi^2\sigma$ and $|\rho|/H\ll1$. The canonically normalized spectator mode obeys

with $\lambda=s_{\rm tilt}\beta/(4\gamma)$, adiabatic pure-ghost initial conditions, and the no-real-turning-point bound $\beta<2\gamma\sqrt{\mu^2+1/4}$. All interaction scales must remain below the ghost EFT cutoff.

Bounded first calculation. Compute the connected tree-level exchange trispectrum $T_\zeta(\mathbf k_1,\ldots,\mathbf k_4)$ from two $c_3\dot\pi^2\sigma$ vertices in the collapsed channel $q=|\mathbf k_1+\mathbf k_2|\ll K_{12},K_{34}$. Insert the exact Whittaker bulk propagator and separate

Derive $A_T$ as gamma functions and at most one stable numerical integral. Evaluate the phase-closure statistic $\Phi_{BT}=\arg A_T-2\arg A_B$ after removing calculable hard-vertex phases, where $A_B$ is the squeezed-bispectrum coefficient already available for the same mode. Also evaluate the normalization-independent ratio $\mathcal R_{BT}=|A_T|\,\Delta P_\zeta^{(\sigma)}/|A_B|^2$. These combinations test propagation physics while cancelling the real couplings $c_3$, $\rho$, and arbitrary normalization of $\sigma$.

Deliverable. Provide the analytic collapsed template, a numerical grid for $0.5\le\mu\le5$ and the EFT-allowed $(\lambda,\gamma)$ region, and shape cosines against relativistic/low-sound-speed and scalar-chemical-potential trispectra. Success requires either $|\Delta\Phi_{BT}|\ge0.2$ radians or cosine $<0.98$ with every comparison template over an observationally useful collapsed range $10^{-2}\le q/K\le10^{-1}$. Kill criterion: stop after the collapsed calculation if $\Phi_{BT}$ and $\mathcal R_{BT}$ reduce to the relativistic results up to amplitude and constant phase, or if all EFT-valid tilted templates have cosine $\ge0.98$ with an existing chemical-potential template after nuisance marginalization.

Novelty Checks

• Query site:arxiv.org tilted ghost spectator trispectrum cosmological collider (2026-07-26): found arXiv:2605.20102, which calculates the tilted-ghost exchange bispectrum and its squeezed signal, but no tilted-ghost trispectrum.

• Query site:arxiv.org ghost spectator collapsed trispectrum inflation modified dispersion: found arXiv:2601.14413, which treats a massive ghost spectator, bispectrum enhancement, and correlator differential equations; it does not calculate a collapsed four-point function.

• The same search found arXiv:1803.04412, which calculates QSFI trispectra with relativistic modes and general sound speeds. It is a useful limiting benchmark but lacks quartic dispersion, signed tilt, and propagation-induced Whittaker branch asymmetry.

• Search for nonrelativistic cosmological collider four-point ghost dispersion found arXiv:1004.1776, but that work studies the self-trispectrum of ghost inflation, where the ghost is the adiabatic sector, rather than exchange of a massive non-relativistic spectator during standard slow-roll inflation.

• Query cosmological collider chemical potential trispectrum scalar exchange found arXiv:2604.07434 and arXiv:2010.04727, both centered on scalar-chemical-potential bispectra, plus arXiv:2203.06349 on a spin-2 chemical-potential trispectrum. None implements the proposed scalar tilted-ghost collapsed trispectrum or the bispectrum–trispectrum closure statistic.

• arXiv:2512.07941 separates local and nonlocal pieces of a generic massive-scalar exchange trispectrum using an open-EFT/Keldysh description. It supplies a strong structural cross-check, but assumes the standard heavy-field setting rather than the tilted $k^2+k^4$ propagation law.

• The verified ARC domain pack contains no solved case with this mathematical object; it explicitly lists propagation-driven dispersion and higher-point observables as open axes. A separately configured shared ARC paper cache was unavailable in this runtime, so the focused check used the verified pack and arXiv searches.

Validation Checks

• Evaluate the full Schwinger–Keldysh nested integral numerically at representative points and require agreement with the collapsed asymptotic expression to better than 5% for $q/K\le0.05$.

• Verify reality, all relevant external-leg permutations, correct overall scaling dimension, and cancellation of spurious analytic terms between Schwinger–Keldysh branches.

• Check cosmological cutting/factorization: the nonanalytic collapsed coefficient must equal the product of the two on-shell production/decay wavefunction coefficients built from the same Whittaker modes.

• Take $\lambda\to0$ and recover the pure-ghost spectator result; independently replace the mode equation by a relativistic Hankel mode and recover the standard QSFI collapsed clock signal of arXiv:1803.04412.

• Confirm numerically that $\Phi_{BT}$ and $\mathcal R_{BT}$ are unchanged under real rescalings of $c_3$, $\rho$, and $\sigma$, within numerical precision.

• Run convergence tests under contour angle, time cutoff, integration precision, and momentum-grid refinement; require stable phase at the 0.05-radian level.

• Before claiming distinguishability, maximize template cosines over amplitude, mass/frequency, constant phase, sound speed, and chemical-potential parameters; apply the stated $0.98$ kill threshold.

• For every retained grid point, document the no-turning-point condition, adiabatic initial state, EFT cutoff hierarchy, weak mixing, loop perturbativity, and negligible spectator backreaction.

Risks

• Manageable: Absolute enhancement is sensitive to the ultraviolet vacuum prescription. Report normalized closure quantities and never interpret the pure-ghost normalization difference itself as a measurable chemical potential.

• Manageable: Oscillatory nested in-in integrals may be numerically stiff. Use contour rotation or exponential damping and compare against the factorized late-time result before scanning parameters.

• Manageable: The allowed small-tilt, stable EFT region may be narrow. Enforce the turning-point, cutoff, backreaction, and perturbativity bounds before computing template distinguishability.

• Scientific blocker/kill condition: Reality and factorization may force the trispectrum coefficient to contain no independent branch information beyond the bispectrum. The bounded collapsed calculation directly decides this; if closure is universal, do not proceed to a full-shape or estimator project.

• Scope risk: A full five-dimensional trispectrum template is expensive and unnecessary for the first decision. Restrict the initial deliverable to the collapsed channel and one hard-energy ratio scan.

Reviewer Feedback

The signed-tilt extension is a feasible and potentially substantive non-strong-mixing calculation, but the proposal requires major revision because its closest-prior-work claim is false and its closure observable, EFT-valid scan, and chemical-potential comparator are not sufficiently defined to support the full deliverable.

• Correct the literature record and reproduce arXiv:2601.14413 at λ=0 as the first validation limit.
• Replace the dimensionally incomplete expression for Tζ with a precise reduced-trispectrum convention. The pure-ghost seed has collapsed dependence √(uv)(uv)^{±iμ}, where u=q/K12 and v=q/K34, before external prefactors.
• Derive coupling and normalization cancellation algebraically and generalize the phase closure to unequal hard energies using a fixed logarithmic pivot.
• Separate the ready formal collapsed calculation from the later EFT-certified scan and chemical-template comparison, which require additional inputs.
• Add a detectability or forecast condition; a low shape cosine alone does not establish that both correlators can be measured.

Marks

{ "confidence_of_novelty": 5, "novelty": 8, "planning": 9, "problem_well_definedness": 8, "scientific_value": 8, "total_score": 61, "user_intent_relevance": 23 }

Round 2

Signed-Tilt Ghost Trispectrum: A Factorization-First Bispectrum–Trispectrum Closure Test

Idea Summary

Compute the $\lambda\neq0$ signed-tilt deformation of the already-known pure-ghost collapsed trispectrum and test a precisely normalized bispectrum–trispectrum consistency ratio. The central question is deliberately narrower than “calculate a ghost trispectrum”: does the tilted Whittaker propagator contain higher-point branch information not already fixed by its squeezed bispectrum and mixing-induced power correction? A factorization-first calculation can answer this before any expensive full-shape scan. The work uses weak mixing and is therefore outside the excluded strong-mixing direction.

Motivation

The pure-ghost spectator trispectrum is already known, while arXiv:2605.20102 shows that a signed quadratic tilt changes the bispectrum’s Whittaker branch weights and phase but leaves its EFT embedding and higher-point extension open. A single bispectrum cannot cleanly distinguish propagation-induced asymmetry from vacuum normalization or a chemical-potential-like template. A coupling- and pivot-independent closure observable would provide that discrimination if non-universal; a proof that Schwinger–Keldysh factorization makes it universal would instead prevent an unproductive trispectrum-template program. Either outcome resolves a concrete ambiguity left by the recent literature.

Calculation Plan

Model and regime. Start from the canonically normalized decoupling-limit action

with $m>3H/2$, $\lambda=s\beta/(4\gamma)$, and interactions

Here $[\rho]=1$, $[g_3]=0$, and $[\Lambda_3]=1$. Initially impose $|\rho|/H\ll1$, $|\lambda|\ll1$, and $\beta<2\gamma\sqrt{\mu^2+1/4}$, where $\mu=\sqrt{m^2/H^2-9/4}$. Before calling any point physical, complete the symmetry-required cubic and quartic EFT, derive its strong-coupling scale by power counting around $\omega^2=p^4/M^2+s\beta p^2+m^2$, and require $H,m,p_*\sim\sqrt{HM},\rho\ll\Lambda_{\rm sc}$, perturbative vertices, and spectator energy density $\ll\epsilon M_{\rm Pl}^2H^2$. No parameter grid is promised before this audit.

Observable definitions. Let $P_\zeta(k)=2\pi^2\Delta_\zeta^2/k^3$, $K=k_1+k_2$, $L=k_3+k_4$, and $q=|\mathbf k_1+\mathbf k_2|$. Define dimensionless reduced correlators

Using one positive pivot $k_*$ and the $+i\mu$ branch fixed by the pure-ghost WKB prescription, extract

For $p_\sigma\equiv\Delta P_\zeta^{(\sigma)}/P_\zeta$, form

This gives $\Phi_{BT}=\arg A_T-\arg A_B(K)-\arg A_B(L)$; $-2\arg A_B$ is used only when $K=L$. A pivot change adds $2\mu\log(k_*'/k_*)$ to $\arg A_T$ and $\mu\log(k_*'/k_*)$ to each bispectrum phase, so $\mathcal Q_{BT}$ is invariant. Under branch conjugation report $\mathcal Q_{BT}\leftrightarrow\mathcal Q_{BT}^*$, or the unambiguous $\cos\Phi_{BT}$.

Factorization-first calculation. Vertex counting proves at tree level that $A_T\propto(g_3/\Lambda_3)^2$, $A_B\propto(g_3/\Lambda_3)\rho$, and $p_\sigma\propto\rho^2$; hence $\mathcal Q_{BT}$ cancels both real couplings and the normalization of $\sigma$. Insert the exact Whittaker propagator into the collapsed Schwinger–Keldysh integral with two $\dot\pi_c^2\sigma$ vertices. Before evaluating a full template, use its late-time four-branch decomposition to test whether cutting/factorization forces $A_Tp_\sigma=A_B(K)A_B(L)\times C(\mu)$, independent of $\lambda$. If it does, derive the corresponding universality theorem and stop. If it does not, derive $A_T$ as gamma functions plus at most one stable one-dimensional integral and evaluate $\mathcal Q_{BT}(\lambda)/\mathcal Q_{BT}(0)$ at two EFT-audited benchmark points.

Deliverable and decision rule. Deliver the reduced collapsed trispectrum, a proof or counterexample to universal closure, the coupling/pivot invariance derivation, and an EFT benchmark table. Continue to shape discrimination only if the residual signed-tilt dependence exceeds either $0.1$ rad in phase or 10% in modulus. At that stage construct a scalar-chemical-potential comparator using the same external mixing and $\dot\pi_c^2\sigma$ operators before computing any cosine. Kill criterion: stop if closure is $\lambda$-independent up to the known pure-ghost result, if no controlled EFT benchmark survives, or if the operator-matched comparator absorbs the residual deformation after nuisance marginalization.

Novelty Checks

• Opened arXiv:2601.14413v2, §4.2: it explicitly computes the massive pure-ghost spectator exchange trispectrum and its collapsed nonlocal term. Therefore the $\lambda=0$ trispectrum is a benchmark, not the novelty claimed here.

• Opened arXiv:2605.20102: it derives the signed-tilt Whittaker mode and squeezed bispectrum; its conclusions explicitly identify both a complete EFT embedding and the collapsed trispectrum as future work. This directly supports the narrowly defined $\lambda\neq0$ calculation.

• Queries site:arxiv.org tilted ghost trispectrum signed tilt Whittaker and cosmological collider bispectrum-trispectrum phase relation (2026-07-26) found the two ghost papers above but no direct calculation of the signed-tilt closure quantity. The search also returned unrelated late-time matter consistency relations, so it does not establish an exhaustive absence result.

• Query scalar chemical potential collapsed trispectrum cosmological collider found arXiv:2010.04727 and arXiv:2604.07434, both centered on scalar-chemical-potential bispectra, plus arXiv:2203.06349, whose chemical-potential trispectrum is spin-2 and has different external operators. No ready-made scalar comparator matched to the proposed vertices was found.

• arXiv:2512.07941 separates local and nonlocal parts of a generic massive-scalar exchange trispectrum using Keldysh/open-EFT methods. It is a strong factorization check but does not contain the signed $k^2+k^4$ Whittaker propagation law or the proposed reduced closure ratio.

• arXiv:1803.04412 supplies the relativistic QSFI collapsed-clock benchmark, but not the pure-ghost ultraviolet prescription or signed-tilt deformation.

• The verified domain pack contains propagation-driven dispersion and higher-point observables as an open intersection. A separately configured shared ARC paper cache was unavailable in this runtime, so novelty checking used the verified pack and focused arXiv/web queries.

Validation Checks

• Recover the collapsed pure-ghost coefficient of arXiv:2601.14413 as $\lambda\to0$, including its $q^{3\pm2i\mu}$ scaling and phase convention.

• Evaluate the full Schwinger–Keldysh integral at representative $(\mu,\lambda,\gamma,K/L)$ and require agreement with the collapsed asymptotic expression within 5% for $q/\min(K,L)\le0.05$.

• Verify cosmological cutting explicitly: compare the nonanalytic trispectrum coefficient with products of the two hard decay integrals and the same Whittaker late-time connection coefficients used in the bispectrum and power correction.

• Rescale $\rho$, $g_3/\Lambda_3$, the spectator field normalization, and the shared pivot independently; require $\mathcal Q_{BT}$ to remain unchanged within numerical precision.

• For unequal hard energies verify $\Phi_{BT}(K,L)=\arg A_T-\arg A_B(K)-\arg A_B(L)$; separately confirm the equal-energy reduction and conjugate-branch behavior.

• Check reality, all exchange-channel permutations, overall scaling dimension, and cancellation of spurious analytic terms between Schwinger–Keldysh polarities.

• Perform contour-angle, early-time cutoff, precision, and momentum-ratio convergence tests; require phase stability better than 0.03 radians.

• Audit every physical benchmark against $M=H/\gamma$, the derived $\Lambda_{\rm sc}$, quartic freeze-out momentum, no-real-turning-point condition, weak mixing, loop perturbativity, and negligible backreaction.

• If closure is non-universal, build the chemical-potential comparator with identical external operators and maximize overlaps over amplitude, mass, constant phase, hard-energy ratio, and chemical-potential parameters before claiming distinguishability.

Risks

• Scientific blocker: Schwinger–Keldysh cutting may force $\mathcal Q_{BT}$ to depend only on the spectator two-point normalization, eliminating all signed-tilt information. This is tested first; a universality theorem is the terminal result.

• EFT blocker: The phenomenological Whittaker mode may not admit a weakly coupled benchmark once its symmetry completion, strong-coupling scale, turning points, perturbativity, and backreaction are imposed. No observational grid or enhancement claim is allowed before that audit.

• Manageable: The mixing-induced power correction must be defined with the same branch, pivot, and subtraction convention as the correlators. Compute it in the identical Schwinger–Keldysh scheme and verify cancellation under field rescaling.

• Manageable: Oscillatory nested integrals can be stiff. Use contour rotation and compare the exact integral with late-time factorization at fixed benchmarks before scanning anything.

• Comparator risk: A scalar chemical-potential theory with exactly matching external operators may require an explicit two-real-scalar construction. If it cannot be made healthy and operator-matched, report that limitation and do not claim a template-cosine distinction.

• Scope risk: The surviving deformation may be mathematically real but too small to matter observationally. The 0.1-radian/10% threshold prevents escalation to a full trispectrum pipeline for an incremental effect.

Reviewer Feedback

The revised proposal now has a consequential target-domain question, precise reduced observables, a bounded factorization-first calculation, validation checks, and explicit kill criteria. The remaining EFT-completion, hierarchy, cutting-rule, comparator, and detectability issues are named and manageable rather than demonstrated feasibility blockers, so another proposer round is not required.

• Write the controlled hierarchy as p_~sqrt(mM), with H, m, p_, rho and relevant interaction energies below Lambda_sc; do not equate H itself with sqrt(HM).
• Specify the symmetry realization and finite operator basis used to generate the cubic and quartic completion, including which coefficients are independent and which are symmetry-related.
• Represent the collapsed nonanalytic sector as a two-branch Keldysh coefficient matrix. The closure becomes a rank or Gram-determinant question involving the hard-decay and mixing response vectors, providing a sharper universality test than a generic appeal to cutting.
• State explicitly that A_B and A_T are leading soft coefficients after subtraction of analytic/contact contributions and at fixed collapsed hard-pair kinematics.
• If the project reaches template discrimination, add an amplitude or forecast criterion alongside the phase/modulus deformation threshold.

Marks

{ "confidence_of_novelty": 10, "novelty": 9, "planning": 11, "problem_well_definedness": 11, "scientific_value": 8, "total_score": 73, "user_intent_relevance": 24 }