{"id":"579518ac-7b92-4157-b53c-972f08d0f50c","arxiv_id":"2610.00989","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"The induced gravitational wave bispectrum formally retains the discrete Poisson source's three-point statistics, but that signature is subleading, limiting its use to distinguish a discrete source from a Gaussian white-noise source.","lead":"This paper computes gravitational waves induced when a sudden matter-to-radiation transition follows black hole evaporation, tracking the full Poisson statistics of the black holes' positions. It finds those higher-order statistics do not show up in the wave power spectrum and appear only weakly in the bispectrum, so the signature is hard to observe in practice.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"Structural selection rule is sound and the 8/15 and 4/10 partition counts check out; the load-bearing weakness is that the paper's own numbers (|B_BB/B_PPP| ~ 7×10^-3) leave the advertised bispectrum discriminator at the percent level, and the quantitative UV amplitude rests on the acknowledged…","rationale":"The paper's central claim has two parts: a structural selection rule and a quantitative assessment of the bispectrum as a discriminator. The structural part is robust. I independently checked the partition combinatorics (8 of 15 PPP, 4 of 10 BB surviving; T/PT/C6 vanish via the axisymmetric single-leg identity), and the near-resonance treatment correctly handles the non-commuting δ→0 and x1→∞ limits, so the selection rule and its generalization (b155) rest on sound, parameter-free reasoning. I also verified that the neglected non-derivative source terms are suppressed by ~(c_s z_eva)^{-2} ~ 10^-13 near the peak, so they cannot pollute the quoted BB amplitude. The genuinely load-bearing concern is the significance framing: the paper's own quantitative result puts the Poisson-specific BB channel at ~7×10^-3 of the Wick PPP piece, meaning the advertised 'additional information for distinguishing the discrete PBH scenario' is carried by a sub-percent effect, below the modeling uncertainty. The abstract's scale-dependence discriminator reflects the quadratic conversion and sudden-transition background, not Poisson discreteness, and would be reproduced by a Gaussianized source. This agrees with the reader's first rationale point and is the strongest reason the verdict cannot be ACCEPT. The reader's second concern (ingested S_Φ, linear-to-k_UV extrapolation) is partly weaker than stated: S_Φ is quoted from published work [14,16] and applied consistently as a per-leg amplitude correction (rule 12 limits its force), while the linear extrapolation is an acknowledged, literature-standard limitation. Its residual bite is that it affects exactly the BB UV amplitude, i.e., the one number the framing depends on; that residual justifies not moving to REJECT (the structural result stands) and not to ACCEPT (the framing overstates practical discrimination). UNCHANGED preserves the reader's CONDITIONAL verdict, which matches my read.","tokens_in":20377,"tokens_out":12254,"duration_ms":90892,"concrete_test":"Independently evaluate the BB integral (Eq. 62) by direct 6-dimensional Monte Carlo with the full oscillatory kernel I from Eq. (49) and the late-time form Eq. (A6), bypassing the envelope substitution (65); then repeat with the top-hat window Θ(k_UV−|p|) replaced by a smooth exponential cut exp(−|p|/k_UV) and with S_Φ softened to (2/3 yR_eva)^{-1/6}. If |B_BB/B_PPP| moves above roughly 5% under these replacements, the quoted 7×10^-3 subleading conclusion is not robust to the UV modeling and the abstract's discriminator claim weakens further; if it stays near 1%, the quantitative subleading result and the current hedged framing survive as reported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central structural result — the TT selection rule of Sec. III and Appendix B — is clean, and I verified its combinatorics by hand: of the 15 pair partitions at six-point order, 8 form valid 3-cycles across distinct tensor vertices, and of the 10 BB partitions, 4 put one leg of each vertex into each block; T, PT, and C6 vanish through the single-leg identity L^λ(k) = 0, which relies on the Poisson window W_n factoring per momentum (hence per tensor vertex). Inclusion-exclusion confirms 8 = 15 − (9 − 3 + 1), so the power-spectrum PP-only claim and the bispectrum PPP+BB claim are correct, and the near-resonance analysis (Eqs. A8–A11) properly respects the non-commuting limits. The load-bearing weakness is therefore not derivational but the framing of the bispectrum as a discriminator. The paper's own benchmark gives |B_BB/B_PPP| ~ 7×10^-3 (Eq. 70): the only topology carrying the intrinsic Poisson three-point cumulant is a roughly 0.7% effect, smaller than the intrinsic uncertainty of the sudden-transition/S_Φ modeling. Yet the abstract advertises the bispectrum's scale dependence as possibly 'distinguishing the discrete PBH scenario'; that scale dependence is carried mainly by the PPP (Wick) piece, which a Gaussianized source — or any isocurvature source in this background — reproduces, so the Poisson-specific signal is the sub-percent BB. Separately, the acknowledged linear-to-k_UV extrapolation (b211–b215) is exactly the regime where the BB amplitude and the (2/3 yR_eva)^{-1/3} S_Φ suppression act; nonlinear PBH growth modifying the source three-point cumulant near k_UV could shift BB by more than its own magnitude. These make the quantitative, not structural, claims fragile.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"This paper studies gravitational waves induced by Poisson isocurvature perturbations sourced by a discrete population of primordial black holes (PBHs) that evaporate, producing a sudden transition from an early matter-dominated to a late radiation-dominated era. Within this sudden-transition model and the linear evolution of the isocurvature mode, the authors derive a transverse-traceless (TT) selection rule for the induced tensor statistics: the tensor power spectrum receives only the two-point (PP) source contraction while the connected four-point (T) term vanishes identically; the tensor bispectrum receives the Wick (PPP) and connected three-point (BB) topologies, with 8 of 15 PPP and 4 of 10 BB partitions surviving. The BB topology is the unique channel carrying the intrinsic Poisson three-point cumulant. For a benchmark with beta = 1.2e-2 and M_PBH,f = 1e4 g, the BB contribution is numerically sub-percent (|B_BB/B_PPP| ~ 7e-3), and the total bispectrum shows an equilateral-squeezed-equilateral dominance sequence as the perimeter varies, which the authors suggest may help distinguish this mechanism from other induced-GW scenarios.","tokens_in":20794,"tokens_out":8085,"duration_ms":60865,"significance":"The structural selection rule is the paper's cleanest contribution: it rigorously establishes that higher-order source cumulants are invisible in the induced-GW power spectrum yet formally enter the tensor bispectrum. Appendix B derives it cleanly, and I verified the combinatorics by hand: 8 of 15 PPP partitions and 4 of 10 BB partitions survive, while T, PT, and C6 vanish through the single-leg identity L^lambda(k)=0, which relies on the Poisson window W_n factoring per tensor vertex. Appendix A correctly handles the non-commuting late-time and resonant limits of the three-sine kernel. The numerics are transparent, benchmark-fixed rather than fitted, and the authors honestly report that the Poisson-specific BB signal is sub-percent, which is itself a meaningful quantitative finding even though it undercuts the advertised discrimination. The main weakness is the mismatch between this sub-percent result and the abstract's advertised distinguishing power, and the dependence of the quantitative bispectrum on the acknowledged linear extrapolation to the discreteness cutoff.","major_comments":[{"comment":"The abstract (b4) and the concluding discussion advertise the bispectrum's scale dependence as possibly distinguishing the discrete PBH scenario. But the BB topology, the only channel carrying the intrinsic Poisson three-point cumulant, is sub-percent: Eq. (70) gives |B_BB/B_PPP| ~ 7e-3, and the D_amp definition of Eq. (69) shows that the BB contribution leaves the zeros and the configuration features of the total bispectrum essentially unchanged. The scale dependence that does distinguish mechanisms is carried mainly by the PPP (Wick) piece, which any Gaussianized source with the same two-point spectrum reproduces. The introduction (b30-b31) already concedes the low practical efficiency of the bispectrum as a discriminator, so the abstract and Section V should be re-scoped to present the structural selection rule as the main result and the weak-discriminator statement as the quantitative one.","section":"Abstract and Sec. V; Eq. (70)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The quantitative headline (BB subleading, hence a weak discriminator) rests on Eqs. (34)-(35) and on extrapolating the linear evolution of the Poisson isocurvature mode up to the discreteness cutoff k_UV. The authors transparently acknowledge (b211-b215) that the PBH density contrast can reach order unity before evaporation and that k_NL may lie below k_UV. Since the BB topology carries an additional momentum integration that samples the high-k part of the source bispectrum, nonlinear small-scale corrections could plausibly change its amplitude relative to PPP, and the distinguishing features (the near-cutoff peaks of Fig. 2 and the equilateral-squeezed switch of Fig. 3 for perimeter C ~ 0.5-2) sit partly in this UV regime. Because the sub-percent conclusion is exactly the quantity this limitation bears on, a short sensitivity discussion (for example, the effect of truncating modes above k_NL on B_BB/B_PPP) would clarify the robustness of the headline claim.","section":"Sec. V; Sec. II.C","authors_acknowledge":"Following the linear treatment adopted in previous analyses, we nevertheless extrapolate the Poisson scalar statistics up to the discreteness cutoff kUV [15, 16].Since the corresponding nonlinear scale kNL may lie below the discreteness cutoff kUV, small scale scalar fluctuations may receive corrections from nonlin- ear PBH evolution and clustering.","authors_acknowledge_p":0.89},{"comment":"The paper states that 'cancellations in the convolution kernel' leave the BB contribution subleading, but does not identify the origin of the ~7e-3 suppression. One paragraph distinguishing the kinematic/angular suppression from the effect of the Poisson amplitude AP in the connected bispectrum BS of Eq. (7), together with the extra phase-space integration in Eq. (62), would make the sub-percent result interpretable rather than merely numerical.","section":"Sec. IV, Eq. (70)","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"Typographic ligature artifacts ('efficiency', 'difficult', 'sufficiently' rendered with visible ff/fi ligature breaks) appear through the text and should be cleaned in the production draft.","section":"Throughout","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The short-wavelength amplitude prescription S_Phi(y) is joined to S_Phi = 1 by an unspecified smooth interpolation; stating the form of the interpolation would improve the reproducibility of the UV bispectrum shapes in Section IV.","section":"Sec. II.C, Eq. (34)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The caption says the figure shows the total bispectrum but does not state which polarization channel is plotted; the surrounding text emphasizes the RRR channel for the discriminator assessment, so the captions should specify the polarization for consistency.","section":"Fig. 3","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The choice of the late-time observational surface k_UV * tau_out = 1.5 R_eva is stated, but not justified; a sentence explaining that this surface lies well after the oscillatory transient has decayed for the relevant modes would help.","section":"Sec. II.D, before Eq. (46)","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"major_revision","confidential_remarks":"The paper is honest and well-executed; the TT selection rule is a sound structural result worth publishing, and the combinatorics survive independent checking. My main concern is the novelty framing: the advertised discriminator claim is undercut by the paper's own ~0.7% BB result, and the quantitative bispectrum rests on the acknowledged linear-to-k_UV extrapolation and on the unspecified interpolation in Eq. (34), which hurts reproducibility of the UV shapes more than the rest of the paper. I would recommend acceptance once the abstract and Section V are re-scoped to lead with the structural result and to temper the distinguishing-power language."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"The structural core is genuinely new and it holds. The claim: for induced GWs from a Poisson isocurvature source, the TT projection kills the connected T, PT, and C6 contractions through a single-leg integral identity, leaving only PP in the power spectrum and only PPP + BB in the bispectrum. I checked the combinatorics independently — 8 of 15 PPP pairings and 4 of 10 BB partitions survive, and the vanishing classes all reduce to L^lambda(k) = 0. Appendix B proves this cleanly, and Appendix A handles the three-sine kernel with the right care for the non-commuting late-time and resonant limits. That is a real contribution, and the authors correctly note it extends beyond monochromatic unclustered PBHs as long as the connected blocks carry no mixed angular dependence.\n\nThe paper is also honest about its own numbers, and that deserves credit. They report D_amp ~ 4e-3 and |B_BB/B_PPP| ~ 7e-3, and the conclusion states plainly that the bispectrum mainly reflects the nonlinearity of GW generation rather than the intrinsic source non-Gaussianity. The negative result — that the Poisson-specific channel stays subleading after full phase-space normalization — is itself useful and not obvious.\n\nThe soft spot is framing, not derivation. The abstract says the nontrivial scale dependence \"may provide additional information for distinguishing the discrete PBH scenario,\" but their own numbers undercut that. The BB topology is the only carrier of the intrinsic three-point cumulant, and it is a sub-percent effect, smaller than the modeling uncertainty. The scale dependence they highlight lives mostly in the PPP (Wick) piece, which any Gaussianized source in the same background reproduces. The abstract should hedge hard; the body already does.\n\nThe UV quantitative amplitude also rests on the linear-to-kUV extrapolation, which the authors flag explicitly in Sec. V (kNL may lie below kUV), and on the imported S_Phi prescription from [16]. This is fully acknowledged, so it limits confidence in the numerical amplitude and cutoff behavior rather than in the structural result. That is exactly why the conditional reading is right.\n\nFor the PBH / induced-GW community this is a solid analytic result with an honest numerical pass. It deserves a serious referee. I would accept for review with the expectation of a revised abstract, not a reworked calculation.","headline":"Clean new structural result on which contractions survive TT projection, honestly reported numerical limits, but the abstract oversells a bispectrum discriminator that is percent-level.","tokens_in":21465,"tokens_out":2506,"would_cite":true,"duration_ms":18051,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":["98.80.-k","04.30.-w"],"model":"deepseek/deepseek-v4-flash","headline":"Evaporating primordial black holes leave a Poisson shot-noise source whose non-Gaussianity is invisible in the induced gravitational-wave power spectrum and first appears in the tensor bispectrum, where it is weak.","keywords":["induced gravitational waves","primordial black holes","Poisson isocurvature perturbations","tensor bispectrum","non-Gaussianity","sudden evaporation transition","early matter-dominated era","transverse-traceless selection rule"],"falsifier":"Run an N-body (or lattice) simulation of the PBH-dominated era starting from the initial Poisson field, and measure the connected three-point function of the matter-density perturbation down to the mean-separation scale $k_{\\rm UV}$; if $\\langle S^3\\rangle_c$ deviates from the ideal Poisson value $A_P^2$ at any scale below the nonlinear scale $k_{\\rm NL}$, the $B B$ amplitude and the ultraviolet bispectrum shape reported here change quantitatively. A complementary check is to replace the imported amplitude prescription $S_\\Phi(y)$ from Eq. (34) with one derived from a finite-duration evaporation calculation and compare the resulting bispectrum.","tokens_in":21908,"feed_emoji":"🕳️","tokens_out":23753,"duration_ms":243760,"temperature":0.7,"pith_summary":"This paper studies gravitational waves (GWs) generated by density fluctuations of primordial black holes (PBHs) when the early matter-dominated era ends abruptly through PBH evaporation, and asks how the discreteness of the black-hole population imprints itself on the induced tensor spectrum. Because PBHs are discrete and unclustered, their number-density fluctuations form an ideal Poisson process with nonzero connected correlations at every order, so the source is intrinsically non-Gaussian. The paper shows that the transverse-traceless projection of the tensor modes filters out all source correlations beyond the two-point level: the induced GW power spectrum depends only on the source's white-noise two-point spectrum, while the intrinsic three-point cumulant can enter only through the tensor bispectrum, via the 'BB' contraction topology alongside the Wick-type contractions that any scalar source generates. Numerically this new channel is subleading, changing the normalized bispectrum amplitude by at most about $4\\times10^{-3}$, so in practice a discrete Poisson source and a 'Gaussianized' source with the same two-point spectrum remain nearly indistinguishable.","feed_headline":"Black-hole shot noise shows up only in the GW bispectrum","feed_subtitle":"Power spectra cannot tell a discrete black-hole source from Gaussian noise; only the bispectrum can, and weakly.","key_machinery":"The load-bearing machinery is the connected-block decomposition of the Poisson source correlators together with a transverse-traceless (TT) selection rule. An ideal Poisson process fixes every connected $N$-point function as $(2\\pi)^3\\delta_D(\\sum_a \\mathbf{k}_a)\\,A_P^{N-1} W_N$, so the cumulant hierarchy carries no free parameters once the mean separation $k_{\\rm UV}$ is chosen. The selection rule says that any contraction forming a 'self-loop' — pairing the two source legs of one tensor vertex — factorizes into a single-leg integral $L^\\lambda(\\mathbf{k})=\\int d^3p\\,\\Pi^\\lambda(\\mathbf{k},\\mathbf{p})F(p,\\hat{\\mathbf{k}}\\cdot\\hat{\\mathbf{p}})$ of the form $e^\\lambda_{ij}(\\hat{\\mathbf{k}})(A\\delta_{ij}+B\\hat{k}_i\\hat{k}_j)$, which vanishes by $e^\\lambda_{ii}=0$ and $e^\\lambda_{ij}\\hat{k}_j=0$. This reduces the tensor power spectrum to the $P P$ term and the tensor bispectrum to $P P P + B B$. The quantitative evaluation then rests on the sudden-evaporation time kernel $I_{\\rm lRD}$, whose acoustic resonance at $u+v=c_s^{-1}$ produces the logarithmic GW enhancement, and on the short-wavelength amplitude prescription $S_\\Phi(y)$ imported from earlier studies to set the ultraviolet response.","core_discovery":"The paper's central claim is that the full hierarchy of intrinsic Poisson statistics of the source enters the induced tensor correlators according to a sharp selection rule. Whenever a contraction pairs the two scalar legs attached to the same tensor vertex, the surviving integral is axisymmetric around the external tensor momentum and is annihilated by the transverse-traceless (TT) polarization tensor, so the term vanishes. Applied to the Poisson cumulant hierarchy, the rule leaves only the $P P$ (Wick) term in the tensor power spectrum, and in the tensor bispectrum it leaves only the $P P P$ topology (8 of 15 pairings) and the $B B$ topology (4 of 10 pairings); the connected four-point term $T$, the $P T$ mixtures, and the fully connected six-point term $C_6$ all vanish. The $B B$ channel is the sole survivor carrying the intrinsic Poisson three-point cumulant $B_S = A_P^2 W_3$, and it therefore formally lifts the degeneracy between a discrete Poisson source and a Gaussianized source with the same white-noise power spectrum. The numerical evaluation for the sudden-evaporation benchmark ($M_{\\rm PBH,f}=10^4$ g, $R_{\\rm eva}=2.3\\times10^6$) shows that the lift is weak in practice: the symmetric normalized amplitude difference between the full bispectrum and the pure $P P P$ part is $D_{\\rm amp}\\sim4\\times10^{-3}$, and the $B B$ term leaves the zeros and configuration features of the bispectrum essentially unchanged. The paper also reports a scale-dependent ordering of bispectrum configurations — equilateral dominance at long wavelengths and near the cutoff, squeezed dominance in between — that differs from other induced-GW and primordial-tensor scenarios.","pith_inferences":["Because the self-loop vanishing is purely a property of the transverse-traceless projector, the same filtering should recur at the tensor trispectrum and higher orders: the connected source trispectrum $T_S$ and six-point spectrum $C_{6,S}$ would enter only through surviving non-separable topologies, so the tensor order at which intrinsic source non-Gaussianity can first appear likely rises with e","The small size of the $B B$ channel is tied to the sudden-evaporation kernel and its resonance band; in a different background the phase-space cancellation that suppresses $B B$ could be less severe, so the same formal selection rule might yield a much larger discreteness signal in other induced-GW setups.","The equilateral–squeezed crossover happens near the perimeter $\\bar{C}\\sim k_{\\rm UV}$, suggesting that the bispectrum shape carries information about the discreteness scale itself; if the linear model is trusted, a future measurement of the GW bispectrum shape could in principle extract $k_{\\rm UV}$ and hence the evaporating PBH mass.","An independent handle on the PBH population — from merger rates, microlensing, or the stochastic-background amplitude — could be cross-correlated with the predicted bispectrum shape, sharpening a test that the raw amplitude difference of order $10^{-3}$ cannot deliver by itself."],"forward_implications":["The induced tensor power spectrum cannot distinguish a discrete Poisson isocurvature source from any other source with the same white-noise two-point spectrum, because higher-order source cumulants are filtered out at second order.","The tensor bispectrum decomposes into a Wick-type part ($P P P$) that any scalar source generates and a $B B$ part that uniquely carries the intrinsic Poisson three-point cumulant, so the bispectrum breaks the two-point degeneracy in principle.","Quantitatively the $B B$ channel is subleading: the normalized amplitude difference against the pure $P P P$ part is about $4\\times10^{-3}$, so the practical discrimination efficiency between a discrete and a Gaussianized source in this setup is very low.","The relative amplitude of the equilateral and squeezed bispectrum configurations follows an equilateral–squeezed–equilateral sequence as the momentum perimeter grows, a scale dependence distinct from the poltergeist model, standard radiation-era scalar-induced GWs, and the primordial tensor bispectrum of single-field inflation.","The selection rule extends beyond monochromatic, unclustered PBHs to any statistically homogeneous source with magnitude-only correlators and no preferred direction in the time kernel, including rare discrete objects such as oscillons and Q-balls that decay after matter domination."],"supporting_citations":[{"why":"Supplies the sudden-transition setup: the junction conditions, the post-evaporation acoustic oscillations of the scalar potential, and the dominant late-radiation time kernel used in the numerical evaluation.","marker":"[16]"},{"why":"Origin of the short-wavelength amplitude prescription $S_\\Phi(y)$ (Eq. 34) that sets the ultraviolet amplitude and resonance structure of the induced tensor modes.","marker":"[14]"},{"why":"Gives the probability generating functional of the ideal Poisson process that fixes the full cumulant hierarchy $\\langle S^N\\rangle_c \\propto A_P^{N-1}$ used throughout.","marker":"[39]"},{"why":"Provides the linear treatment of PBH isocurvature perturbations during evaporation whose transfer-function conventions the paper adopts.","marker":"[15]"},{"why":"Documents nonlinear growth and clustering of PBH density fluctuations, delimiting the validity of extrapolating linear Poisson statistics up to $k_{\\rm UV}$.","marker":"[17]"},{"why":"Shows that the $k^3$ infrared scaling of the dimensionless spectrum is generic for causal sources, motivating the need for higher-order statistics to identify discreteness.","marker":"[18]"},{"why":"Reference case for the radiation-era scalar-induced GW bispectrum dominated by the equilateral configuration, against which the paper's scale dependence is compared.","marker":"[23]"},{"why":"Reference for the poltergeist abrupt-transition model whose bispectrum behavior differs from the Poisson-isocurvature case.","marker":"[25]"},{"why":"Reference for the squeezed-enhanced primordial tensor bispectrum in single-field inflation, used as another comparison case in the conclusion.","marker":"[41]"}],"fun_headline_variants":["Poisson black-hole noise hides in GW bispectrum","Only the bispectrum sees black-hole shot noise","Black-hole discreteness skips power spectrum, lands in bispectrum","GW power spectrum blind to Poisson black holes; bispectrum catches them","Black-hole shot noise shows up only in tensor bispectrum"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The load-bearing premise is that the Poisson density fluctuation evolves linearly all the way down to the discreteness cutoff $k_{\\rm UV}$ — the authors acknowledge small-scale PBH fluctuations may enter the nonlinear regime before evaporation — together with the imported short-wavelength amplitude prescription $S_\\Phi(y)$ of Eq. (34); if the small-scale three-point statistics deviate from the ideal Poisson values, or if that amplitude profile is not the right one, the $B B$ amplitude and the ultraviolet shape of the bispectrum would shift.","fun_headline_variants_meta":{"raw":{"variants":["Poisson black-hole noise hides in GW bispectrum","Only the bispectrum sees black-hole shot noise","Black-hole discreteness skips power spectrum, lands in bispectrum","GW power spectrum blind to Poisson black holes; bispectrum catches them","Black-hole shot noise shows up only in tensor bispectrum"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.000188,"raw_usage":{"cost":0.000188468,"is_byok":false,"cost_details":{"upstream_inference_cost":0.000188468,"upstream_inference_prompt_cost":0.000119868,"upstream_inference_completions_cost":6.86e-05},"total_tokens":1306,"prompt_tokens":1061,"completion_tokens":245,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":256,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":171}},"tokens_in":1061,"tokens_out":245,"cost_source":"openrouter_usage","duration_ms":3071,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":171,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-02T20:45:03.810822+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"Run an N-body (or lattice) simulation of the PBH-dominated era starting from the initial Poisson field, and measure the connected three-point function of the matter-density perturbation down to the mean-separation scale $k_{\\rm UV}$; if $\\langle S^3\\rangle_c$ deviates from the ideal Poisson value $A_P^2$ at any scale below the nonlinear scale $k_{\\rm NL}$, the $B B$ amplitude and the ultraviolet bispectrum shape reported here change quantitatively. A complementary check is to replace the imported amplitude prescription $S_\\Phi(y)$ from Eq. (34) with one derived from a finite-duration evaporation calculation and compare the resulting bispectrum.","supporting_citations":[{"cited_title":"Gravitational wave constraints on the primordial black hole dominated early universe","cited_arxiv_id":"2012.08151","evidence_quote":"Supplies the sudden-transition setup: the junction conditions, the post-evaporation acoustic oscillations of the scalar potential, and the dominant late-radiation time kernel used in the numerical evaluation."},{"cited_title":"Gravitational Wave Production right after a Primordial Black Hole Evaporation","cited_arxiv_id":"2003.10455","evidence_quote":"Origin of the short-wavelength amplitude prescription $S_\\Phi(y)$ (Eq. 34) that sets the ultraviolet amplitude and resonance structure of the induced tensor modes."},{"cited_title":"Last and M","cited_arxiv_id":null,"evidence_quote":"Gives the probability generating functional of the ideal Poisson process that fixes the full cumulant hierarchy $\\langle S^N\\rangle_c \\propto A_P^{N-1}$ used throughout."},{"cited_title":"Gravitational waves from a universe filled with primordial black holes","cited_arxiv_id":"2010.11573","evidence_quote":"Provides the linear treatment of PBH isocurvature perturbations during evaporation whose transfer-function conventions the paper adopts."},{"cited_title":"From formation to evaporation: Induced gravitational wave probes of the primordial black hole reheating scenario","cited_arxiv_id":"2409.12125","evidence_quote":"Documents nonlinear growth and clustering of PBH density fluctuations, delimiting the validity of extrapolating linear Poisson statistics up to $k_{\\rm UV}$."},{"cited_title":"Causal gravitational waves as a probe of free streaming particles and the expansion of the Universe","cited_arxiv_id":"2010.03568","evidence_quote":"Shows that the $k^3$ infrared scaling of the dimensionless spectrum is generic for causal sources, motivating the need for higher-order statistics to identify discreteness."},{"cited_title":"Testing Primordial Black Holes as Dark Matter through LISA","cited_arxiv_id":"1810.12224","evidence_quote":"Reference case for the radiation-era scalar-induced GW bispectrum dominated by the equilateral configuration, against which the paper's scale dependence is compared."},{"cited_title":"Hu, C.-J","cited_arxiv_id":null,"evidence_quote":"Reference for the poltergeist abrupt-transition model whose bispectrum behavior differs from the Poisson-isocurvature case."},{"cited_title":"Full bispectra from primordial scalar and tensor perturbations in the most general single-field inflation model","cited_arxiv_id":"1207.0588","evidence_quote":"Reference for the squeezed-enhanced primordial tensor bispectrum in single-field inflation, used as another comparison case in the conclusion."}],"review_version":1}