{"id":"0aec0607-f5d5-40bb-80a6-662a2b45b210","arxiv_id":"2607.23892","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"LISA and DECIGO together can probe O(10^2) stellar-mass PBH binaries with e>0.01; excluding them would improve PBH abundance limits by an order of magnitude.","lead":"Primordial black-hole binaries that stay isolated mostly circularize before ground-based detectors see them, but those that live in dense dark-matter halos can keep measurable eccentricity. Future space detectors LISA and DECIGO could see about a hundred such eccentric systems, and a non-detection would tighten PBH abundance limits by roughly ten times.","discovery_kind":"extension","skeptic_critique":{"model":"moonshotai/kimi-k3","headline":"The O(10^2) e>0.01 count is supplied almost entirely by the high-e tail of the binary-single channel, which is generated by a deterministic, always-positive eccentricity-pump term (Eq. 2, coefficient K) imported from SMBH-binary/stellar-cusp calibrations. Whether stochastic equal-mass PBH encounters","rationale":"The reader's weakest assumption (self-cited rates from Ref. [36] and the f_PBH=3.4e-3 normalization at the LVK ceiling) is a genuine but proportional risk: if rates are overstated by a factor of a few, the O(10^2) claim degrades gracefully to O(10), weakening but not collapsing the qualitative result. The concern I identify is structurally sharper for this specific headline because it targets the *shape* of the eccentricity distribution, not its normalization. The entire eccentric-detection claim is carried by the high-e tail of one channel, and that tail is generated by a monotonic deterministic eccentricity pump whose coefficients were calibrated in a different physical regime (SMBH binaries in stellar cusps) and whose continuum-drift implementation is only valid in the many-encounter limit. This is a correctness question internal to the paper's modeling choices, not a consensus disagreement. That said, the paper is otherwise careful: Peters evolution for the unperturbed channel is standard, the single-harmonic strain approximation errs toward underestimating SNR (conservative for detection counts), and the qualitative finding (isolated binaries circularize; dense-halo binaries can stay eccentric) is robust and consistent with prior literature. A discrete-encounter cross-check in one representative shell would settle whether the quantitative O(10^2) figure is solid or an artifact of the drift approximation. This strengthens the case for the reader's CONDITIONAL verdict but does not justify REJECT: the result is falsifiable, the pipeline is explicit, and the fix is well-defined. I therefore recommend keeping CONDITIONAL, with the additional requirement — beyond the reader's rate validation and multi-harmonic SNR check — that the eccentricity-excitation model be validated against stochastic three-body scattering before the order-of-magnitude abundance claim is treated as settled.","tokens_in":21597,"tokens_out":3802,"duration_ms":71812,"concrete_test":"For the inner shell of the 1.46e7 M_sun halo, compute the mean number of strong binary-single encounters per binary over its residence time using Gamma = n_env * sigma(a) * v_disp with gravitational focusing. If <N_enc> is less than a few, replace the deterministic environmental terms in Eqs. (1)-(2) with a discrete-encounter Monte Carlo that draws per-encounter (Delta a, Delta e) from published equal-mass three-body scattering distributions (e.g., Heggie/Samsing-type experiments), keep everything else in the pipeline fixed, and recompute the DECIGO e>0.01 tail of Fig. 6. If the combined LISA+DECIGO e>0.01 count drops below ~10, the O(10^2) headline and the order-of-magnitude exclusion claim do not survive.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The headline number is not \"binaries detected\" but \"binaries detected WITH e>0.01.\" From Fig. 7, LISA sees only Ndet=76 total and the unperturbed channel circularizes below 0.01 by the DECIGO band, so the O(10^2) eccentric detections must come from the high-e tail of the binary-single channel (Fig. 6, DECIGO Ndet=11840, tail extending to e~1). That tail is produced by Eqs. (1)-(2): environmental terms de/dt = +GHK(r,t) rho_env a / v_disp, with the hardening/eccentricity coefficients H and K taken from Quinlan (1996) and Sesana et al. (2006) — N-body calibrations for massive black hole binaries hardening in dense stellar backgrounds. Three issues compound. (1) The K term is strictly positive: while the environmental term dominates, eccentricity can only be pumped upward, for potentially ~Gyr in dense shells. Real equal-mass binary-single encounters change e stochastically in both directions (Heggie-style scattering); a monotonic secular pump is a different object and can systematically inflate the e>0.01 fraction. (2) The transfer of H, K from the SMBHB-in-stellar-cusp regime (mass ratio extreme, cusp of light stars) to equal-mass PBH-PBH encounters in a DM halo is asserted, not validated, in this paper; it is inherited from Refs. [35, 36]. (3) The continuum drift approximation (environmental properties updated every 200 Myr, (a,e) Euler-stepped at 0.1 Myr) is only valid if each binary undergoes many encounters per update interval; if the mean encounter count per binary is O(1) or less, the discrete-encounter statistics — not the drift coefficients — set the tail, and the smooth integration cannot represent them. The shell-mass weighting (Eq. 16) then suppresses exactly the inner shells where pumping is strongest, so the surviving tail is sensitive to the details of this model. If the effective eccentricity excitation is even moderately weaker than the K-term implies, the DECIGO e>0.01 tail — and hence the O(10^2) figure and the claimed order-of-magnitude abundance —","agreement_with_reader":"partial"},"referee_report":{"model":"moonshotai/kimi-k3","summary":"The manuscript computes the eccentricity distributions of stellar-mass primordial black hole (PBH) binaries at the frequencies of current and future GW detectors (LISA, DECIGO, ET, CE, aLIGO), tracking three channels: early-formed binaries evolving in isolation, early-formed binaries that enter dark matter halos and undergo binary-single interactions, and direct GW captures in halos. The authors integrate coupled semimajor-axis/eccentricity evolution equations (GW Peters–Mathews terms plus environmental hardening/eccentricity-pump terms) for samples of ~5×10^6 binaries per halo shell, compute characteristic strains via the peak harmonic, and rescale simulated histograms with channel merger rates from Ref. [36] at f_PBH = 3.4×10^-3. They find that unperturbed binaries circularize below detectability in all bands except LISA (residual e ~ 10^-2), that halo binaries retain a high-eccentricity tail, and that LISA+DECIGO together should probe O(10^2) binaries with e > 0.01, so that an exclusion of such eccentric binaries could improve PBH abundance limits by an order of magnitude.","tokens_in":22085,"tokens_out":6507,"duration_ms":325254,"significance":"If the results hold, this is a useful and timely forecast: it is, to my knowledge, the first study to follow PBH binary eccentricity self-consistently across all formation channels and all present/planned GW bands, with large Monte Carlo samples (5×10^6 binaries per shell/halo), explicit shell-resolved halo environments, and per-detector SNR/horizon calculations. The output is a falsifiable prediction — a concrete count of eccentric (e > 0.01) events for LISA and DECIGO under a stated abundance normalization — and a clear statement of what an exclusion would imply for f_PBH. The unperturbed-channel result (full circularization before ground-based bands, residual e ~ 10^-2 in LISA) is a robust, parameter-light check that agrees with prior analytic expectations and lends credibility to the pipeline. The main weaknesses are not in the GW-emission machinery (Eqs. 3–10 are standard and correctly applied) but in the imported environmental-interaction physics and in the normalization of the headline numbers.","major_comments":[{"comment":"The headline O(10^2) count of e>0.01 detections is supplied almost entirely by the high-e tail of the binary-single channel (Fig. 6: DECIGO Ndet = 11840 with a tail extending to e ~ 1, versus the unperturbed channel which is below 0.01 in the DECIGO band). That tail is generated by the environmental term de/dt = + G H K(r,t) ρ_env a / v_disp, with H and K taken from Quinlan (1996) and Sesana et al. (2006) — N-body calibrations for massive black hole binaries hardening in dense stellar cusps, i.e., an extreme mass-ratio regime with a large reservoir of light perturbers. Three compounded issues: (i) as written, the K term is strictly positive, so while the environmental term dominates, eccentricity can only be pumped upward, in some shells for ~Gyr; real equal-mass binary-single encounters change e stochastically in both directions (Heggie-style scattering), and a monotonic secular pump is","section":"§IIA, Eqs. (1)–(2)"},{"comment":"The SNR integral is taken over the overlap of the binary's full detector-frame frequency range [f_bin,min, f_bin,max] with the detector band, and t_obs enters only as a linear multiplier in Eq. (15). For a 30+30 M_sun binary, the inspiral time from 10^-3 Hz to merger is ~10^4 yr, and from 10^-2 Hz is ~20 yr — both far longer than the assumed t_obs = 5 yr (LISA) and 3 yr (DECIGO). If f_bin,max is the ISCO frequency rather than the frequency reached after t_obs of observation, the SNR, and hence z_max and N_det, are overestimated for the space detectors. The text does not state what f_bin,max is. Please clarify, and if the finite observation time is not accounted for, recompute z_max and N_det with f_start set by t_obs before merger (as in standard multiband treatments, e.g., Ref. [75]). This is load-bearing because the LISA numbers (N_det = 8 and 68 in the two channels) sit close to thres","section":"§IIC, Eq. (12)"},{"comment":"The characteristic strain is approximated by the single peak harmonic, dropping the quadrature sum of Eq. (11). At moderate eccentricities the power is spread over several harmonics around n_peak, so this approximation misestimates h_c by a factor that varies with e; since z_max scales with h_c and N_det roughly with z_max^3 at low z, even a ~20–30% strain error is non-negligible for the event counts. The error could be quantified cheaply on a subsample by comparing Eq. (11) with the peak-harmonic approximation. Relatedly, the orientation-averaging factor F in Eq. (12) is introduced but its numerical value is never given; please state it.","section":"§IIB, Eqs. (9)–(11)"},{"comment":"The two headline numbers — O(10^2) binaries with e>0.01 and the 'order of magnitude' improvement in f_PBH limits — are linear rescalings of the channel merger rates R_channel(z) imported from Ref. [36] (a companion preprint by the same authors) at the single normalization point f_PBH = 3.4×10^-3, f_PBH binaries = 0.5, m = 30 M_sun. This is not circular in a logical sense, but the manuscript should be more explicit that both numbers scale linearly with the assumed rate normalization, and should tabulate the central result directly: N_det(e>0.01) and N_det(e>0.1) per detector per channel. At present the O(10^2) figure can only be extracted by eye from the Fig. 7 histogram, which is unsatisfactory for the paper's main quantitative claim. In addition, DECIGO's N_det is dominated by the volume out to the (capped) z_max = 1000, where the halo mass function and the Ref. [36] rates are extrapola","section":"§IIIB and §IV, Eq. (15) and Fig. 7"}],"minor_comments":[{"comment":"The legends quote 'DECIGO (10^-2 Hz, z ≤ 12)' while Table I gives z_max = 1000 for DECIGO. Presumably the z ≤ 12 reflects that halo-channel mergers only occur after halo formation; please make the legends consistent with Table I or explain the cut explicitly.","section":"Fig. 4 captions/legends"},{"comment":"The sentence 'The fact that N1/N8 is much larger than one for DECIGO, ET, and CE, but less than one, is due to...' is incomplete — 'but less than one [for LISA and aLIGO]' appears to be intended.","section":"§IIIB, paragraph after Fig. 4"},{"comment":"Each panel carries a duplicated halo-mass label (e.g., 'M(z=0) = 1.68×10^4 M⊙ M(z=0) = 1.68×10^4 M⊙').","section":"Figs. 8 and 9"},{"comment":"Typo: 'withihn' → 'within' (paragraph above Eq. 16 reference to Fig. 5 bottom panel).","section":"§IIIB"},{"comment":"The switch criterion |ȧ|env/|ȧ|GW ≤ 0.5 (and the same for ė) means the environmental term is still a ~50% correction at the handoff time t_GW; a brief note on the sensitivity of the final (a_GW, e_GW) to the choice 0.5 would be useful.","section":"§IIA"},{"comment":"The noise curves are taken from GWplotter [74]; please specify which curve configuration is used for each instrument (e.g., LISA mission duration/arm length assumption, DECIGO vs B-DECIGO, ET-D, CE 40 km), since z_max values in Table I depend on these choices.","section":"§IIC"},{"comment":"The statement that alternative (f_PBH, f_PBH binaries) combinations consistent with LVK give 'the same number of detected events as in Table I' would benefit from one sentence of explanation, since the unperturbed rate generally scales as a non-trivial power of f_PBH.","section":"§IIIA"}],"recommendation":"major_revision","confidential_remarks":"The quantitative forecasts rest on merger rates and the f_PBH normalization taken from Refs. [36] and [16], both recent preprints by the same authors that (as of this writing) have not themselves completed peer review. The present manuscript is therefore part of a self-referential chain: the rates that set the headline O(10^2) and the order-of-magnitude improvement claim cannot be independently verified from this submission. This is not disqualifying — the eccentricity evolution itself is newly simulated here — but the editor may wish to consider the two companion papers' status when weighing the strength of the forecast claims."},"author_rebuttal":{"model":"moonshotai/kimi-k3","summary":"We thank the referee for a careful and constructive report. We agree that the main weaknesses lie in the imported environmental-interaction physics and in the presentation/normalization of the headline numbers, rather than in the GW machinery. We will make four substantive revisions: (i) clarify the role and limitations of the Quinlan/Sesana hardening and eccentricity-pump coefficients and quantify the sensitivity of the high-e tail to the sign stochasticity of K; (ii) define f_bin,max explicitly and restrict the SNR integral to the portion of the inspiral observable within t_obs for LISA and DECIGO, recomputing z_max and N_det; (iii) quantify the error of the peak-harmonic approximation on a subsample and state the numerical value of F; and (iv) tabulate N_det(e>0.01) and N_det(e>0.1) per detector per channel, with explicit statements of the linear scaling with the rate normalization and the extrapolation involved at high z. We address each comment below.","responses":[{"response":"We agree this is the weakest part of the modeling, and we thank the referee for pressing on it. Two defenses and one concession. First, the use of continuum hardening/eccentricity coefficients as effective orbit-averaged rates is a standard shortcut (e.g. Refs. [62–64, 73]), and our calibration in Ref. [36] — where the same formalism was benchmarked against direct N-body-style encounter rates for PBH halos — reproduces the binary-single merger-rate enhancement reasonably well. Second, the mass regime is less extreme than for MBH binaries in the sense that all perturbers have the same mass as the binary members, so the cusp-reservoir issue does not arise; the H and K coefficients enter only as an effective encounter-rate efficiency. However, we concede the referee's central physical point: our K is a mean positive drift, while Heggie-style equal-mass scattering randomizes e with a thermally-biased distribution, and a strictly monotonic pump over Gyr likely overproduces the extreme tail (e → 1). In the revision we will (a) state this limitation explicitly in §IIA, (b) perform a sensitivity test in which the environmental ė term is treated as a stochastic kick with zero mean drift but the same variance (or capped at the thermal-distribution expectation), and report how N_det(e>0.01) for DECIGO changes, and (c) if the O(10^2) headline count shifts materially, we will revise it and the abstract accordingly. We cannot, within this work, replace the effective equations with full few-body scattering experiments; we will say so plainly.","revision_made":"partial","referee_comment":"[§IIA] The high-e tail driving the O(10^2) count comes from the environmental term de/dt = + G H K(r,t) ρ_env a / v_disp with H, K from Quinlan (1996)/Sesana et al. (2006), calibrated on massive BH binaries in stellar cusps; K is strictly positive, so eccentricity is pumped monotonically for ~Gyr, whereas real equal-mass binary-single encounters change e stochastically in both directions."},{"response":"The referee is correct that the text is ambiguous, and we apologize for the omission. In the submitted version f_bin,max is set by the frequency at ISCO; the only account of t_obs is the linear multiplier in Eq. (15). This indeed overestimates the accumulated SNR for LISA and DECIGO, whose sources dwell in band far longer than the mission duration. In the revision we will (i) define f_bin,min and f_bin,max explicitly, and (ii) restrict the integration in Eq. (12) for the space detectors to the window [f(t_ISCO − t_obs), f_ISCO] (equivalently the harmonics radiated during the observed segment), following the standard multiband treatment of Ref. [75], and recompute z_max and N_det for LISA and DECIGO. For ground-based detectors the inspiral through the band is much shorter than t_obs, so those numbers are unaffected. We note that the affected LISA counts (N_det = 8 and 68) will decrease, and the DECIGO counts will as well; we will update Tables I–II, Figs. 3, 6, 7, the abstract, and the O(10^2) claim accordingly. Since the high-e tail of the binary-single channel resides at the lower-frequency (earlier) portion of the band, the eccentric fraction within t_obs should be less suppressed than the total, but we will report the recomputed numbers rather than assert this.","revision_made":"yes","referee_comment":"[§IIC, Eq. (12)] f_bin,max is not defined. If it is the ISCO frequency rather than the frequency reached after t_obs, then for space detectors (inspiral times of 10^4 yr from 1e-3 Hz) the SNR, z_max, and N_det are overestimated. Please clarify and recompute with f_start set by t_obs before merger, as in Ref. [75]. This is load-bearing for the LISA counts."},{"response":"We agree with both points. On the first: for the eccentricities relevant to the detectable populations (e ≲ 0.1 in the LISA/DECIGO bands for nearly all sources, as the high-e tail merges quickly after entering band), g(n,e) is strongly concentrated at n_peak, so we expect the quadrature correction to be modest; but 'we expect' is not a quantification. We will compute both Eq. (11) and the peak-harmonic approximation for a representative subsample spanning the simulated (a, e) grid and report the fractional difference in h_c, z_max, and N_det as a function of e; if the error exceeds ~10% anywhere in the population that dominates N_det, we will switch to the full sum, which is computationally cheap at our sample sizes. On the second: the value of F was omitted in error. We use F = 1/5 (sky- and orientation-averaged response for a single interferometer, following Refs. [68, 69]); we will state this explicitly in §IIC and note the rescaling of SNR had a different convention been adopted.","revision_made":"yes","referee_comment":"[§IIB, Eqs. (9)–(11)] The characteristic strain uses only the peak harmonic, dropping the quadrature sum of Eq. (11); at moderate e the power is spread over several harmonics, and since z_max ∝ h_c and N_det ∝ z_max^3, even 20–30% strain errors matter. Quantify the error on a subsample. Also, the value of the orientation-averaging factor F in Eq. (12) is never given."},{"response":"We agree on all three requests. (i) We will add an explicit statement that N_det and the resulting f_PBH-improvement factor scale linearly with the assumed channel rate normalization, and give the simple rescaling formula so readers can translate to other (f_PBH, f_PBH binaries, mass) choices. (ii) We will add a table giving N_det(e>0.01) and N_det(e>0.1) per detector, per channel (unperturbed, binary-single, direct capture), replacing the current need to read Fig. 7 by eye — this is clearly the right way to present the paper's main quantitative claim. (iii) We will state that DECIGO's z_max = 1000 is a cap, that the Press–Schechter halo mass function and the Ref. [36] rates are extrapolations beyond their calibrated range at high z, and we will report a bounded variant — e.g. N_det computed with the integral truncated at z = 20 or 30 where the halo mass function is better constrained — so that the sensitivity of the DECIGO total to the high-z extrapolation is transparent. We note the eccentric (e>0.01) subset is less dominated by the highest redshifts than the total, since high-e sources are preferentially nearby, but the table will show this explicitly.","revision_made":"yes","referee_comment":"[§IIIB, §IV] Both headline numbers are linear rescalings of channel merger rates from Ref. [36] at one normalization point (f_PBH = 3.4e-3, f_PBH binaries = 0.5, 30+30 M_sun). Be explicit about the linear scaling; tabulate N_det(e>0.01) and N_det(e>0.1) per detector per channel; and note that DECIGO's counts are dominated by z up to the capped z_max = 1000, where the halo mass function and rates are extrapolated."}],"tokens_in":22033,"tokens_out":2128,"duration_ms":729330,"standing_objections":["The referee's concern about the Quinlan/Sesana K-coefficient monotonic eccentricity pump cannot be fully resolved within the effective-equation framework; a definitive fix would require direct few-body scattering experiments for PBH-mass binaries in halo conditions, which is beyond the scope of this revision. We will quantify the sensitivity and state the limitation, but the high-e tail retains a model dependence we cannot eliminate."]},"desk_editor":{"model":"grok-4.5","letter":"The new piece is end-to-end orbital tracking of stellar-mass PBH binaries through all three channels, delivered as detector-frame eccentricity histograms and N_det forecasts for LISA, DECIGO, ET, CE, and aLIGO. That multi-band distribution was missing; prior work had rates and qualitative circularization statements. They run large Monte Carlos (5e6 per shell), use standard Peters GW terms plus environmental drift, apply explicit z_max cuts, and show cleanly that isolated binaries circularize below ~10^{-2} by DECIGO while dense-halo binaries can retain e ≳ 0.01 into the space bands. Direct captures are negligible under current f_PBH. That qualitative split is robust and useful.\n\nThe quantitative headline—O(10^2) systems with e>0.01 for LISA+DECIGO, and a possible order-of-magnitude tightening of f_PBH if none appear—scales linearly with the authors’ own R_channel(z) from Ref. [36] and an f_PBH choice that saturates the LVK ceiling. If those rates move, the counts move with them. More importantly, the high-e tail that supplies almost all of the eccentric detections is generated by the always-positive K term in Eq. (2), taken from Quinlan/Sesana SMBHB-in-stellar-cusp calibrations. Real equal-mass PBH–PBH scatterings change e stochastically both ways; a secular pump that only increases e for Gyr in dense shells can inflate the tail. The continuum Euler stepping (0.1 Myr) plus 200 Myr environment updates also assumes many encounters per update; if the mean encounter count is O(1), discrete statistics set the tail instead. Shell-mass weighting then suppresses the very inner shells where pumping is strongest, so the surviving tail is model-sensitive. The single-peak-harmonic strain approximation is a smaller, secondary issue.\n\nMath and citations are otherwise standard and consistent; self-citation is expected given the series. This is for people already working PBH merger rates or multi-band GW forecasts. It deserves a serious referee who will press on the H/K transfer and ask for a stochastic-encounter cross-check or public code. I would engage, cite the qualitative circularization results, and treat the absolute O(10^2) and 10x claims as provisional until the eccentricity excitation is validated for equal-mass PBHs.","headline":"Solid multi-band eccentricity maps for PBH binaries; the O(10^2) e>0.01 forecast and 10x abundance claim ride on inherited rates and a monotonic K-term that may overstate the high-e tail.","tokens_in":23003,"tokens_out":621,"would_cite":true,"duration_ms":12398,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"LISA and DECIGO together can probe about a hundred stellar-mass primordial black hole binaries with eccentricity above 0.01; excluding them would tighten PBH abundance limits by an order of magnitude.","keywords":["primordial black holes","orbital eccentricity","gravitational waves","binary-single interactions","dark matter halos","LISA","DECIGO","PBH merger rates"],"falsifier":"A multi-year LISA plus DECIGO search that finds zero stellar-mass black-hole binaries with measured eccentricity above 0.01, or that finds a substantially different number than the O(100) predicted at the current abundance ceiling, would directly test the claim.","tokens_in":22571,"feed_emoji":"🕳️","tokens_out":1029,"duration_ms":26579,"temperature":0.7,"pith_summary":"Primordial black hole binaries form with very high orbital eccentricities. This paper tracks large simulated samples through every formation channel and evolutionary path—isolated early binaries, binaries that fall into dark matter halos and suffer binary-single encounters, and late direct captures—and asks what eccentricity remains when their gravitational waves enter the bands of LISA, DECIGO, ET, CE, and aLIGO. Isolated binaries fully circularize before ground-based detectors, retaining only residual eccentricities of order 0.01 in LISA. Halo binaries can keep higher eccentricity into late inspiral because dense environments repeatedly pump eccentricity. Under current abundance limits, the paper predicts that LISA and DECIGO together should see roughly a hundred such binaries with e greater than 0.01. A null result would improve the bound on the PBH dark-matter fraction by about a factor of ten, turning eccentricity into a practical abundance probe.","feed_headline":"Space detectors could catch 100 eccentric PBH binaries","feed_subtitle":"A null result from LISA and DECIGO would tighten stellar-mass PBH abundance limits tenfold","key_machinery":"Full orbital evolution of semi-major axis and eccentricity under the coupled equations that include both Peters–Mathews gravitational-wave emission and environmental binary-single terms (density and velocity dispersion inside growing dark-matter halo shells), followed by peak-harmonic characteristic-strain and SNR calculations that map each track onto detector bands and yield rescaled eccentricity histograms.","core_discovery":"Given present limits on stellar-mass PBH abundance, LISA and DECIGO together are expected to detect O(10^2) PBH binaries that still have orbital eccentricity e > 0.01 when their waves enter the detector band; if those observatories can exclude such eccentric systems, the upper limit on the PBH abundance can be strengthened by an order of magnitude. Isolated early binaries circularize completely except for residual O(10^{-2}) eccentricities in LISA, while binaries that experience binary-single interactions inside dense dark-matter halos retain higher eccentricity even at late inspiral.","pith_inferences":["Eccentricity at millihertz-to-decihertz frequencies becomes a practical abundance diagnostic rather than only a formation-channel tag, linking space-based GW catalogs directly to dark-matter fraction limits.","The same halo-environment machinery implies that any future tightening of LVK merger-rate bounds will linearly rescale the expected eccentric counts for LISA and DECIGO.","Multi-band detections that catch the same binary first in LISA/DECIGO (still mildly eccentric) and later in ET/CE (circular) would furnish an independent check of the circularization timescale assumed here."],"forward_implications":["Ground-based detectors (aLIGO, ET, CE) will see essentially circular PBH binaries, so eccentricity cannot separate them from ordinary astrophysical black-hole binaries in those bands.","LISA will retain residual eccentricities of order 0.01 for isolated PBH binaries, giving a low-frequency eccentricity window even without halo interactions.","Dense inner halo shells can leave a few systems with e > 0.1 still visible to space-based detectors.","A clean null result on e > 0.01 binaries from LISA and DECIGO would tighten the stellar-mass PBH dark-matter fraction by roughly a factor of ten.","Direct-capture binaries contribute negligibly to the eccentric sample once current abundance limits are imposed."],"fun_headline_variants":["LISA+DECIGO could see ~100 eccentric PBH binaries","Halo PBH binaries keep e>0.01 into late inspiral","Null LISA/DECIGO result tightens PBH limits tenfold","Isolated PBHs circularize; halo ones stay eccentric","Space GW detectors may probe residual PBH eccentricity"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The predicted detection counts rest on rescaling the simulated eccentricity histograms with the authors’ own earlier merger-rate calculations, normalized to a PBH abundance that saturates current LIGO-Virgo-KAGRA limits; if those rates or that abundance are too high, the O(100) forecast shrinks in proportion.","fun_headline_variants_meta":{"raw":{"variants":["LISA+DECIGO could see ~100 eccentric PBH binaries","Halo PBH binaries keep e>0.01 into late inspiral","Null LISA/DECIGO result tightens PBH limits tenfold","Isolated PBHs circularize; halo ones stay eccentric","Space GW detectors may probe residual PBH eccentricity"]},"model":"grok-4.5","effort":"low","cost_usd":0.003578,"raw_usage":{"total_tokens":1216,"prompt_tokens":875,"num_sources_used":0,"completion_tokens":74,"cost_in_usd_ticks":35784000,"prompt_tokens_details":{"text_tokens":875,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":267,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":875,"tokens_out":74,"duration_ms":5452,"temperature":1.0,"reasoning_tokens":267,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-30T14:30:51.225927+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"A multi-year LISA plus DECIGO search that finds zero stellar-mass black-hole binaries with measured eccentricity above 0.01, or that finds a substantially different number than the O(100) predicted at the current abundance ceiling, would directly test the claim.","supporting_citations":[],"review_version":2}