{"id":"f0c2dcd5-d3be-408f-a753-182542d2f765","arxiv_id":"2610.36591","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"In a marked-Poisson warm-inflation model, induced gravitational waves depend only on the bath's covariance while black-hole formation is highly sensitive to kick kurtosis, separating memory from statistics as observables.","lead":"Warm inflation produces cosmic fluctuations through random kicks from a hot radiation bath. This paper shows that primordial black holes can detect how the kick strengths are distributed while the gravitational waves they generate cannot, a split that lets two measurements tell one property of the bath apart from the other.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"No significant objection identified. The central claim is internally supported, its approximations are quantified, and the fragility of the [41] selection rule is acknowledged and correctly scoped.","rationale":"The reader's ACCEPT verdict is justified. The paper's strongest claim has a clear chain: Campbell's theorem gives all cumulants in Eq. (7); the [41] rule removes the connected four-point contribution to the leading induced-GW signal; the saddle-point tail is validated against exact contour inversion to 12.3 sigma; and the PBH-side uncertainties are tested one by one in Table III. I checked Eq. (4) for the cumulant hierarchy, Eq. (10) for the tilted profile, and Eq. (11) for the collapse statistic; they are internally consistent. The free parameter K is openly treated as a parameter, the one-point nature of beta_M is stated, and the physical calibration is described as an order-of-magnitude guide. The only genuine fragility is the applicability of the three-ingredient selection rule to real warm inflation; since the paper explicitly limits its claim to the symmetric, linear marked-Poisson family and delegates the microscopic computation to future work, this is a scope limitation rather than an error. No machine-checked proof or code is provided, but the exact contour check and the systematic robustness scan supply independent support. Therefore no verdict change is needed.","tokens_in":22685,"tokens_out":17437,"duration_ms":238948,"concrete_test":"Numerically evaluate the connected-trispectrum contribution to the induced tensor power spectrum for the marked-Poisson correlator of Eq. (7) with the log-normal kernel of Eq. (6), using the full tensor projection of [41], over a range of k around k_p. If it vanishes to numerical precision relative to the disconnected A^2 term, the central blindness claim is confirmed; if it is comparable to the disconnected contribution, the discrimination argument must be revised.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After reading the full text in good faith, I find no load-bearing internal flaw in the central argument. The statement that the leading induced-GW signal depends only on the covariance follows from Eq. (7), whose connected correlators factor into functions of individual momentum magnitudes, together with the published angular selection rule of [41]; the paper explicitly identifies the three ingredients (independent events, isotropic profiles, linear response) and states that relaxing any of them reintroduces higher-cumulant contributions [b124-b125]. The collapse statistic is admittedly one-point and not a full mass function [b42, b154], but the authors quantify the largest corrections: maxima instead of points lowers the enhancement by 0.2-0.6 in the exponent, a three-times-broader spectrum by 0.4-0.6, and profile scatter, shape-dependent threshold, kernel sign and a mild quadratic nonlinearity change it by less; none removes the effect (Table III). The tail calculation is checked against exact contour inversion to 12.3 sigma. The residual issue is not internal consistency but the transfer of the model to a specific warm-inflation bath, for which the paper explicitly calls for a microscopic calculation [b289, b307]. Because the claim is framed as a model-level comparison rather than a closed prediction for warm inflation, this acknowledged limitation does not constitute a load-bearing objection.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"The paper studies a marked Poisson-process model for the random force in warm inflation. Each independent event contributes a signed kick of random strength to the curvature through an isotropic linear kernel, chosen so that the power spectrum is a narrow log-normal peak. Campbell's theorem gives all connected correlators in closed form, and because each correlator factorizes into a product of functions of individual momentum magnitudes, the angular selection rule of Garcia-Saenz et al. [41] removes the connected trispectrum from the leading induced gravitational-wave signal. The Gaussian reference and the non-Gaussian bath therefore share the same induced-wave amplitude at fixed covariance. The paper then computes the one-point tail of the curvature distribution with saddle-point and Lugannani-Rice approximations, checked against contour inversion to 12.3 sigma, and feeds it into a mass-weighted collapse statistic based on the compaction function, critical collapse, and a tilted mean profile. The central quantitative result is that an excess kurtosis K=0.1-0.3 enhances this statistic by 10^0.9-10^6.2 at fixed induced-wave amplitude, or equivalently allows the same collapse statistic with induced waves up to about five times weaker. A robustness table examines threshold prescription, profile scatter, maxima instead of points, kernel sign, spectrum width, time profile, and a quadratic nonlinearity; the enhancement survives in all cases.","tokens_in":22998,"tokens_out":22695,"duration_ms":184051,"significance":"The result is significant as a model-level proof of principle: it cleanly separates memory (covariance) from kick statistics (higher cumulants) and shows that the joint use of PBHs and induced gravitational waves can in principle break a degeneracy that neither observable alone resolves. The paper's strengths are its closed-form cumulant hierarchy, the explicit numerical check of the tail approximation, the quantitative robustness table, and the candid statement of the model's scope and limitations. The main conclusions are internally coherent: the leading-order blindness of the induced waves follows from the factorization in Eq. (7) together with the published selection rule of [41], and the authors identify the three ingredients whose relaxation would restore the higher-cumulant contributions. The quantitative factors apply to a one-point collapse statistic rather than a full PBH mass function, and the model inputs are not yet derived from a specific warm-inflation bath; these limitations are acknowledged in the text. If transferred to a fully derived bath model, the framework would provide a quantitative warm-inflation prediction.","major_comments":[],"minor_comments":[{"comment":"The headline factors 'by factors of ten to a million' and 'about five times weaker' refer to the mass-weighted collapse statistic beta_M defined in Eq. (11), not to a PBH abundance. The paper states this limitation clearly in Sec. IV, but the abstract and the conclusions should carry the same qualifier, for example 'collapse statistic' or 'within the one-point model', so that readers do not read the numbers as f_PBH predictions.","section":"Abstract and Sec. IX","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The displayed expression for kappa_n/sigma^n in Eq. (4) is garbled in the manuscript ('mn m n/2-22m n/2-14 Rn') and needs proper typesetting; the same passage contains the typos 'conviencence' and 'developement', and Sec. VIII contains 'advcated'. Please proofread these passages.","section":"Sec. II, Eq. (4)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The maxima correction is obtained by evaluating the density of maxima with the tilted covariance of the gradient and Hessian, which is exact only for Gaussian fields. Since the tilted field is strongly non-Gaussian at the thresholds considered, please state explicitly that this row is an estimate and, if possible, provide a rough error estimate or a cross-check.","section":"Sec. VI, maxima row of Table III","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"Ref. [42] is cited in its arXiv form; because the paper uses it only as an independent analogue of the [41] result, please update it to the published version if one becomes available before publication.","section":"References","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"minor_revision","confidential_remarks":"None beyond the report. The manuscript is internally consistent, the model-level framing is stated honestly, and the central derivation and numerical checks support the claimed separation between covariance and kick statistics. The revision request concerns presentation and qualification of the headline numbers rather than the technical content."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"Read this one. It is a well-executed model calculation with an honest scope statement, and the authors earn their central claim rather than assuming it.\n\nWhat is actually new: a marked-Poisson source for the warm-inflation bath in which the covariance and the kick statistics can be varied independently; the closed-form cumulant hierarchy of Eq. (4); a saddle-point evaluation of the one-point tail checked against exact contour inversion out to 12.3 sigma (agreement within about 2%); and, most usefully, a direct comparison of a mass-weighted PBH collapse statistic with the leading induced-GW amplitude, with every approximation in the collapse prescription tested in Table III.\n\nThe punchline: at leading order the induced GWs see only the covariance of the bath, because the connected correlators factor into functions of individual momenta and the angular selection rule of Garcia-Saenz et al. kills the trispectrum contribution. PBH formation is not blind to the kick distribution, so two baths with the same covariance can give the same GW signal while differing in collapse statistic by factors of 10^0.9 to 10^6.2 for kurtosis 0.1-0.3. Within the stated model this is supported.\n\nThe soft spots are real but acknowledged. The connection to warm inflation is motivational, not derived: the event density, mark law and response kernel are inputs, and K is treated as a free parameter between 10^-2 and 0.3. The authors say plainly that a microscopic bath calculation is the natural next step. The induced-wave blindness is a leading-order property of the symmetric, linear family; a quadratic nonlinearity feeds GWs at second order, as they note. And the collapse statistic is one-point, not a mass function, with a degeneracy between K and the collapse threshold that they quantify (K=0.1 mimics a 6% threshold change). None of these is hidden, and Table III shows none removes the enhancement.\n\nThe paper does not over-claim: it is a model-level comparison, not a prediction for warm inflation. For people interpreting LISA or PTA limits in terms of PBHs, the memory-vs-statistics separation is a useful conceptual tool. I would send it to a serious referee and would cite it. Minor: the prose has a few typos, but nothing that affects the science.","headline":"A careful, honest model calculation showing induced GWs are blind to kick statistics at leading order while PBH abundance is not; worth refereeing despite the warm-inflation link being motivational rather than derived.","tokens_in":23605,"tokens_out":3403,"would_cite":true,"duration_ms":35221,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek/deepseek-v4-flash","headline":"The paper claims that induced gravitational waves see only the bath's covariance while primordial black holes read its rare kicks, so identical wave signals can hide black-hole abundances differing by up to a millionfold.","keywords":["warm inflation","primordial black holes","induced gravitational waves","primordial non-Gaussianity","marked Poisson process","kurtosis","curvature perturbation","collapse threshold"],"falsifier":"A direct test would evaluate the full tensor power spectrum at next order for a source with the same covariance but non-Gaussian marks: if the connected (trispectrum) contribution to the induced waves does not vanish when the kernel is anisotropic or the response is nonlinear, the blindness claim fails in that regime. Concretely, a microscopic warm-inflation bath calculation that delivers non-independent or anisotropic kicks, or a future measurement in which the inferred gravitational-wave peak amplitude correlates with the bispectrum shape at fixed power spectrum, would falsify the clean covariance-versus-statistics separation.","tokens_in":24067,"feed_emoji":"🕳️","tokens_out":8555,"duration_ms":66557,"temperature":0.7,"pith_summary":"Warm inflation replaces an abstract inflationary vacuum with a physical source: the curvature perturbation is built from random kicks delivered by a hot radiation bath. This paper asks whether primordial black holes and the gravitational waves they induce can tell apart two properties of those kicks—how long the bath remembers them (the covariance, which sets the power spectrum) and how the kick strengths are spread (the higher cumulants, which set the rare tail). Modelling the force as a marked Poisson process, the authors show that at leading order the induced waves depend only on the covariance, while black-hole formation is controlled by rare large kicks: a curvature kurtosis of 0.1 to 0.3 raises the mass-weighted collapse statistic by factors of $10^{0.9}$ to $10^{6.2}$ at fixed wave amplitude. Two baths can therefore emit identical gravitational-wave signals while forming black holes at very different rates. The practical consequence is that an upper limit on the induced background becomes a limit on black-hole abundance only once the source statistics are fixed, and a joint measurement of both observables can in principle read the kurtosis of the bath.","feed_headline":"Same gravitational-wave signal, black-hole rates a millionfold apart","feed_subtitle":"Rare strong kicks leave the waves untouched but multiply primordial black-hole formation enormously.","key_machinery":"The load-bearing object is the marked Poisson process for the random force: independent events at positions $X_j$ carry signed strengths $x_j$ drawn from a symmetric bounded power law, and the curvature responds linearly, $\\zeta(x)=\\sum_j x_j\\,g(|x-X_j|)$, through an isotropic kernel. Campbell's theorem then supplies every cumulant in closed form, $\\kappa_n=\\bar{n}\\langle x^n\\rangle S_n$, and—crucially—makes every connected correlator a product of functions of the individual momentum magnitudes. That factorization triggers an azimuthal selection rule that kills the scalar-trispectrum contribution to the induced gravitational waves at leading order, while the one-point cumulant generating function $C(s)=\\bar{n}\\int d^3r\\,[\\cosh(s\\,x g(r)/\\sigma)-1]$ gives an accurately computable tail from which the collapse statistic $\\beta_M$ is built via the compaction function, critical-collapse scaling and the mean tilted profile of a rare peak. The kurtosis formula $K=(m_4/m_2^2)/(S_4\\bar{n}/S_2^2)$ makes the physics explicit: the kurtosis of the curvature is a single-kick fourth-moment ratio divided by the effective number of kicks, which is why a modest $K$ is reachable with thousands of events per bath cell.","core_discovery":"On the paper's own terms, the central discovery is a clean division of labour between two observables. For the marked-Poisson source with a linear, isotropic response, every connected correlator of the curvature factors into a product of functions of the individual momentum magnitudes, and a known angular selection rule for tensor-induced gravitational waves then removes the connected trispectrum from the leading tensor power spectrum: the waves are blind to how kick strengths are distributed. The tail of the one-point curvature distribution, computed by saddle-point methods and verified against exact contour inversion out to $12.3\\,\\sigma$, carries the higher cumulants, and the excess kurtosis $K$ enhances the mass-weighted collapse statistic by $10^{0.9}$ to $10^{6.2}$ at fixed induced-wave amplitude for $K$ between $0.1$ and $0.3$ (Table I); equivalently, the same black-hole abundance can be reached with induced waves up to about five times weaker. The enhancement survives every refinement tried: counting maxima instead of points, wider spectra, profile scatter, shape-dependent thresholds, kernel sign flips and a mild quadratic nonlinearity trim the exponent by at most roughly $0.6$ but never remove it (Table III).","pith_inferences":["The factorization argument is a property of the source class rather than of warm inflation itself: any curvature source whose connected correlators factor by individual momenta—Poisson isocurvature, particle-production bursts, other point-process models—should inherit the same leading-order blindness of the induced waves, so the wave-versus-black-hole diagnostic transfers to those settings.","If microscopic bath calculations deliver anisotropic kicks, non-independent events, or a response that is nonlinear below the collapse threshold, the leading-order blind spot closes; the paper's own two-observable plane would then need a third axis, such as the bispectrum shape, to recover the discrimination.","The bounded power-law marks with $\\mu<2$ produce a one-point tail that decays faster than exponential but slower than Gaussian; this is a concrete, testable template that a full mass-function calculation—the paper's own stated next step—could convert into a predicted mass spectrum and merger-rate signature."],"forward_implications":["An upper limit on the induced gravitational-wave background constrains the primordial black-hole abundance only after the source statistics are fixed; the same wave amplitude is compatible with collapse statistics spanning up to six orders of magnitude.","A joint measurement of the induced-wave amplitude and the black-hole abundance can separate a change in the covariance amplitude (for which bath memory is one route) from a change in the curvature kurtosis, and can pin down $K$ if the collapse threshold is known to a few per cent ($K=0.1$ acts like a $6.2\\%$ threshold shift at the reference covariance).","A non-Gaussian bath forms the same black holes with induced waves up to about five times weaker than a Gaussian bath, so searches that assume Gaussian statistics could misread the amplitude of a detected background.","At an asteroid-mass benchmark ($M_H\\simeq10^{-12}M_\\odot$), the induced background peaks in the LISA band well above the projected sensitivity: the task would be interpreting the amplitude, not detecting it."],"supporting_citations":[{"why":"Supplies the angular selection rule that removes the trispectrum contribution from leading-order induced waves, the load-bearing ingredient for the waves' blindness to kick statistics.","marker":"[41]"},{"why":"The parallel Poisson-isocurvature result, corroborating that factorized correlators drop out of the leading tensor power spectrum.","marker":"[42]"},{"why":"Provides the radiation-era kernel used to compute the induced gravitational-wave spectrum from the log-normal peak.","marker":"[2]"},{"why":"Provides the analytic collapse-threshold prescription used for the compaction criterion ($C_{\\rm th}=0.59$).","marker":"[43]"},{"why":"Supplies the nonlinear relation between curvature and density that the compaction function encodes.","marker":"[45]"},{"why":"Critical-collapse scaling ($\\gamma\\simeq0.36$) that weights the collapse statistic by black-hole mass.","marker":"[46, 47]"},{"why":"Supplies the mean profile of rare peaks and the maxima-counting correction used in the robustness checks.","marker":"[70]"},{"why":"Saddle-point tail approximations (Lugannani–Rice formula) used to compute the one-point density far into the tail.","marker":"[72, 73]"}],"fun_headline_variants":["Same gravitational-wave signal, black-hole rates a millionfold apart","Rare kick statistics change black-hole formation but not gravitational waves","Kurtosis 0.1–0.3 boosts black holes by up to a million at fixed wave signal","Warm inflation: kick spread hidden from gravitational waves, visible in black holes"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The whole separation story rests on one leading-order premise: that the bath's kicks arrive independently with isotropic profiles and a strictly linear response all the way to the collapse threshold, and relaxing any of those three lets the higher cumulants leak back into the induced gravitational waves, which the paper itself acknowledges.","fun_headline_variants_meta":{"raw":{"variants":["Same gravitational-wave signal, black-hole rates a millionfold apart","Rare kick statistics change black-hole formation but not gravitational waves","Kurtosis 0.1–0.3 boosts black holes by up to a million at fixed wave signal","Warm inflation: kick spread hidden from gravitational waves, visible in black holes"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":6.5e-05,"raw_usage":{"cost":6.54108e-05,"is_byok":false,"cost_details":{"upstream_inference_cost":6.54108e-05,"upstream_inference_prompt_cost":2.88288e-05,"upstream_inference_completions_cost":3.6582e-05},"total_tokens":1301,"prompt_tokens":1100,"completion_tokens":201,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":979,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":127}},"tokens_in":1100,"tokens_out":201,"cost_source":"openrouter_usage","duration_ms":5265,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":127,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-28T02:49:54.024787+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"A direct test would evaluate the full tensor power spectrum at next order for a source with the same covariance but non-Gaussian marks: if the connected (trispectrum) contribution to the induced waves does not vanish when the kernel is anisotropic or the response is nonlinear, the blindness claim fails in that regime. Concretely, a microscopic warm-inflation bath calculation that delivers non-independent or anisotropic kicks, or a future measurement in which the inferred gravitational-wave peak amplitude correlates with the bispectrum shape at fixed power spectrum, would falsify the clean covariance-versus-statistics separation.","supporting_citations":[{"cited_title":"No-go Theorem for Scalar-Trispectrum-Induced Gravitational Waves","cited_arxiv_id":"2207.14267","evidence_quote":"Supplies the angular selection rule that removes the trispectrum contribution from leading-order induced waves, the load-bearing ingredient for the waves' blindness to kick statistics."},{"cited_title":"Effects of Poisson Isocurvature Perturbations on Induced Gravitational Waves","cited_arxiv_id":"2610.00989","evidence_quote":"The parallel Poisson-isocurvature result, corroborating that factorized correlators drop out of the leading tensor power spectrum."},{"cited_title":"Semianalytic Calculation of Gravitational Wave Spectrum Nonlinearly Induced from Primordial Curvature Perturbations","cited_arxiv_id":"1804.08577","evidence_quote":"Provides the radiation-era kernel used to compute the induced gravitational-wave spectrum from the log-normal peak."},{"cited_title":"The Threshold for Primordial Black Hole Formation: a Simple Analytic Prescription","cited_arxiv_id":"2011.03014","evidence_quote":"Provides the analytic collapse-threshold prescription used for the compaction criterion ($C_{\\rm th}=0.59$)."},{"cited_title":"Primordial black hole formation and abundance: contribution from the non-linear relation between the density and curvature perturbation","cited_arxiv_id":"1904.00984","evidence_quote":"Supplies the nonlinear relation between curvature and density that the compaction function encodes."},{"cited_title":"The statistics of peaks of Gaussian random fields,","cited_arxiv_id":null,"evidence_quote":"Supplies the mean profile of rare peaks and the maxima-counting correction used in the robustness checks."}],"review_version":1}