{"id":"a4fd35b5-3977-4a46-985b-fcc883b2e225","arxiv_id":"2501.15530","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":8,"one_line_summary":"Hierarchical Bayesian analysis of CHIME/FRB Catalog 1 yields an FRB redshift distribution delayed relative to the cosmic star formation history.","lead":"This paper applies a hierarchical Bayesian method to 415 CHIME fast radio bursts and finds that the burst rate peaks at lower redshift than the cosmic star formation history. The result, if correct, favors FRB progenitors that take time to form, such as merging neutron stars.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Selection correction is applied inconsistently: Eq. (13) omits Pdet(z) from the per-event likelihood, biasing the inferred FRB redshift distribution toward low z and potentially manufacturing the claimed delay relative to SFH.","rationale":"The paper's goal is to infer the intrinsic FRB redshift distribution from CHIME/FRB Catalog 1 and compare it with SFH. For that claim to hold, the selection correction must be correct. The manuscript's own equations reveal an internal inconsistency: Eq. (13) and the posterior Eq. (22) include Pdet only through ξ, not in the per-event integrand. The correct hierarchical Poisson likelihood weights each detected event's z integral by Pdet(z) as well, normalized by ξ; otherwise the observed DME distribution is modeled as if detected FRBs were unbiased in z. Since a fluence-limited survey detects preferentially brighter and lower-z sources, the omitted factor biases the reconstruction to low redshift, the direction of the paper's headline conclusion. This is a stronger and more specific objection than the reader's concern about unstated selection parameters, although the missing Fν,min, Fν,max, and n values compound it by making Pdet(z) impossible to check. I agree with the reader that the qualitative conclusion matches earlier literature, but that does not rescue a statistically inconsistent inference. The proposed rerun with the corrected per-event Pdet factor would settle whether the delay survives. Because the correction has not been made and the selection parameters are absent, the central claim is currently unverified; hence UNVERDICTED.","tokens_in":9562,"tokens_out":5804,"duration_ms":56032,"concrete_test":"Rerun the inference with the corrected per-event likelihood, replacing the last integral in Eq. (22) by ∫dz L(d_i|z) ppop(z|Φ) Pdet(z)/ξ(Φ), adopting the selection parameters in Eq. (19) (which must first be reported; if not provided, use reasonable CHIME values such as Fν,min=0.5 Jy ms, Fν,max=5 Jy ms, n=1). If the posterior for λ, κ, zp shifts by more than 1σ, or if the model ranking TSRD vs TSE changes, the 'significantly delayed' claim is an artifact of the missing selection factor.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"The central claim rests on the selection-corrected likelihood, but Eqs. (13) and (22) apply selection only to the total rate: the exponential is damped by ξ(Φ)=∫Pdet(z)ppop(z|Φ)dz, while each detected event is still weighted by ∫dz L(d_i|z)ppop(z|Φ) with no Pdet(z). In a Poisson population model with imperfect redshifts, the per-event likelihood must be ∫dz L(d_i|z)ppop(z|Φ)Pdet(z)/ξ(Φ). Because CHIME is fluence-limited, Pdet(z) (Eqs. 18-20) should fall steeply with redshift; omitting it makes the fit absorb the selection function into ppop, biasing the recovered redshift distribution toward low z. This is exactly the direction of the claimed 'delayed' peak. The problem is compounded by the fact that Fν,min, Fν,max, and n in Eq. (19) are never stated, so Pdet(z) cannot even be evaluated from the manuscript. The claimed delay and the TSRD-vs-SFH conclusion therefore cannot be assessed as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents a hierarchical Bayesian inference of the intrinsic redshift distribution of non-repeating fast radio bursts (FRBs) using 415 FRBs from the first CHIME/FRB catalog with DME ≥ 200 pc cm^-3. Three population models are considered: a two-segment power law times the Madau-Fragos star formation history (TSE), a power law with exponential cutoff times the SFH (CPL), and a non-SFH-based two-segment redshift distribution (TSRD). The authors report that the inferred FRB rate peaks at z ≈ 0.3-0.4, significantly delayed relative to the star formation history, with the TSRD model slightly preferred over TSE and strongly preferred over CPL. They conclude that FRB progenitors are produced with a time delay after star formation.","tokens_in":9849,"tokens_out":12696,"duration_ms":108640,"significance":"If the result holds, it would provide one of the strongest constraints to date on the FRB progenitor delay distribution and would support neutron-star-merger-like channels. The paper applies a standard hierarchical Bayesian framework (Mandel et al. 2019; Abbott et al. 2023) to a large, homogeneous CHIME sample, and it attempts to account for measurement uncertainties in dispersion measure and for the survey selection function. However, the selection correction is implemented incorrectly in the per-event likelihood, and several key inputs are not specified, so the central claim cannot currently be assessed. The paper also provides a useful model comparison that can be updated once the likelihood is corrected.","major_comments":[{"comment":"The per-event likelihood in Eqs. (13) and (22) is written as ∫ dz L(d_i|z) p_pop(z|Φ), with no detection probability Pdet(z) inside the integral. For an inhomogeneous Poisson process with selection, the correct per-event term is ∫ dz L(d_i|z) p_pop(z|Φ) Pdet(z) / ξ(Φ), where ξ(Φ) = ∫ dz Pdet(z) p_pop(z|Φ) (Eq. 18). Because Pdet(z) decreases with redshift for a fluence-limited survey (Eqs. 18-20), omitting it causes the fit to absorb the selection function into the inferred p_pop(z|Φ), biasing the redshift distribution toward low z. This is the same direction as the claimed delay relative to the SFH, so the central conclusion (Abstract and Section 3) is not supported as written. This error also propagates to the BIC values in Table 2, which are computed from the same likelihood.","section":"2.2, Eqs. (13) and (22)"},{"comment":"The detection efficiency parameters Fν,min, Fν,max, and n in Eq. (19) are never given, and the polynomial fits σ±(z) used in Eq. (15) are not provided. Without these inputs, Pdet(z) and the per-event redshift likelihood cannot be evaluated, and the analysis cannot be reproduced. Please state the adopted values (or the source) for the fluence thresholds and index n, and report the polynomial coefficients or provide the functional form of σ±(z).","section":"Eqs. (19) and (15)"},{"comment":"The spectral index β = -1.5, the energy function slope α ≈ 1.8, and the cutoff energy Ec ≈ 3×10^41 erg are fixed to values from the literature without propagating their uncertainties or testing the sensitivity of the results to these choices. Because Pdet(z) is computed from these inputs via Eq. (20), and because the central claim concerns the shape of R(z), the authors should demonstrate that the recovered peak redshift and the TSRD vs. TSE preference are robust to variations of β, α, and Ec within their published uncertainties.","section":"Section 2.2, Eqs. (20)-(21)"}],"minor_comments":[{"comment":"The integrand in Eq. (3) contains '1 + zp' in the numerator; this should be '1 + z' (the factor is the redshift-dependent electron density term).","section":"Eq. (3)"},{"comment":"Tobs is described as the '100% operational time' of CHIME, but the footnote acknowledges that CHIME was not fully operational in the period; please define the effective exposure used in the calculation.","section":"Section 2.2, after Eq. (16)"},{"comment":"The caption should explain how the blue scatter points are produced (e.g., mock DME values drawn from the assumed DMIGM and DMhost distributions) and what exactly the black curves represent in terms of the 1σ region.","section":"Figure 1"},{"comment":"The statement that the redshift distribution is 'significantly delayed' is stronger than the model comparison supports: the TSRD model improves the BIC by only 2.10 over the TSE model, which is not decisive. Please qualify the claim accordingly.","section":"Section 3, Table 2"},{"comment":"There are several typographical errors: 'poweful' should be 'powerful', 'can also be take' should be 'can also be taken', and 'the constrains' (Section 3) should be 'the constraints'.","section":"Section 4"},{"comment":"Please specify the bandwidth Δν used for CHIME (400 MHz?) and clarify whether a single value is adopted for all events.","section":"Section 2.1, Eq. (20)"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a timely question and the application of hierarchical Bayesian inference to the CHIME/FRB catalog 1 is within the journal's scope. The main issue is the incorrect treatment of selection effects in the per-event likelihood, which is a fixable but load-bearing problem. The missing parameters and robustness checks should be addressed in revision. I recommend major revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThis is a tidy hierarchical Bayesian population analysis of the first CHIME/FRB catalog, and it concludes that the FRB redshift distribution is delayed relative to star formation. The method transfer from gravitational-wave population inference is clean, and the paper is well written. It lays out the Poisson likelihood, three redshift models, and a BIC comparison. Using a public catalog and explicitly discussing selection effects are good choices.\n\nThe central problem is the selection correction. In Eqs. (13) and (22), the per-event likelihood is ∫ dz L(d_i|z) ppop(z|Φ), with no Pdet(z) inside. Selection enters only globally, through e^{-Nξ} or ξ^{-Nobs} after marginalizing the rate. In a correct inhomogeneous Poisson model, each detected event contributes ∫ dz L(d_i|z) ppop(z|Φ) Pdet(z)/ξ(Φ). Since CHIME is fluence-limited, Pdet(z) decreases with redshift. Omitting it makes the fit absorb the selection effect into ppop, pushing the recovered redshift distribution toward low z. That is exactly the direction of the claimed delay.\n\nThe paper also omits essential inputs: the values of Fν,min, Fν,max, and n in Eq. (19) are never stated, and the polynomial fits σ±(z) used to construct the DME-z likelihood are not provided. Without them, the selection function and redshift uncertainty cannot be evaluated, so the magnitude of the bias is unknown. Several hyperparameters, γ1, γ, and λ, lie at the upper boundary of their priors, so the low-z slope is not really constrained by the data.\n\nThe conclusion itself is not new; previous studies using the same catalog already found a delayed distribution, and the authors cite them. So the paper's contribution is methodological. That is fine, but the methodology has a bug that invalidates the quantitative results as written. The fix is straightforward: include Pdet(z) in the per-event integral and report the selection inputs. I would like to see the corrected rerun.\n\nThis deserves peer review because the topic is active and the flaw is fixable. The referee should require a corrected analysis and full documentation of the selection function and DME-z fits.\n\nBest,","headline":"A clean HBI analysis with a real selection-function bug: missing Pdet(z) in the per-event likelihood biases the inferred redshift distribution toward low z, exactly the direction of the claimed delay.","tokens_in":10382,"tokens_out":8306,"would_cite":false,"duration_ms":72746,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Hierarchical Bayesian fit to 415 CHIME FRBs finds a redshift distribution delayed relative to cosmic star formation.","keywords":["fast radio bursts","FRB population","redshift distribution","CHIME/FRB catalog 1","hierarchical Bayesian inference","star formation history","dispersion measure","selection effects"],"falsifier":"Identify host galaxies for hundreds of CHIME FRBs and compute the volumetric event rate density $R(z)$ directly from the measured redshifts; if $R(z)$ tracks the Madau-Fragos star-formation history with its peak near $z \\approx 2$ rather than at lower redshift, the delayed-distribution conclusion is falsified. A cheaper check is to replace the assumed grey-zone detection efficiency with CHIME's actual measured sensitivity curve and re-run the inference; if the posterior peak shifts back toward the star-formation peak, the selection correction is the culprit.","tokens_in":9341,"feed_emoji":"📡","tokens_out":8424,"duration_ms":66763,"temperature":0.7,"pith_summary":"The paper tries to establish that the intrinsic redshift distribution of fast radio bursts is not a copy of the cosmic star-formation history, but is delayed toward lower redshifts. It builds a hierarchical Bayesian inference that converts dispersion measures into per-burst redshift likelihoods, applies a model of CHIME's detection selection, and fits three competing population models to 415 non-repeating bursts from the first CHIME/FRB catalog. The inferred rate peaks below the peak of the Madau-Fragos star-formation curve, and a two-segment redshift distribution fits better than a power law with a cutoff. If correct, this is evidence that FRB progenitors form with a time delay after star formation, and it sharpens the use of FRBs as cosmological probes.","feed_headline":"FRBs peak lower than the star-formation rate","feed_subtitle":"A hierarchical Bayesian fit to 415 CHIME bursts places FRB births after star formation, pointing to delayed progenitors.","key_machinery":"The engine is the inhomogeneous Poisson likelihood $$p(d|\\Phi) \\propto N(\\Phi)^{N_{\\rm obs}} $e^{{-N(\\Phi)\\,\\xi(\\Phi)}}$ \\prod_i \\int dz\\, L(d_i|z)\\, p_{\\rm pop}(z|\\Phi),$$ where $p_{\\rm pop}$ is the normalized event-rate redshift distribution, $\\xi(\\Phi)$ is the CHIME detection fraction built from a fluence-dependent efficiency $\\eta_{\\rm det}(F_\\nu)$, and $L(d_i|z)$ is a Gaussian in redshift whose mean and width follow from the extragalactic dispersion-measure–redshift relation with log-normal host-galaxy scatter. The posterior marginalizes the unknown local event rate with a log-uniform prior and compares competing population models with the Bayesian Information Criterion.","core_discovery":"In the paper's own terms, using 415 non-repeating FRBs from the first CHIME/FRB catalog with extragalactic dispersion measure $D_{\\rm E} \\ge 200$ pc cm$^{-3}$, the hierarchical Bayesian inference over hyperparameters $\\Phi = [\\gamma_1, \\gamma_2, z_t, \\gamma, z_c, \\lambda, \\kappa, z_p]$ recovers an intrinsic rate $R(z)$ that is significantly delayed with respect to the star-formation history. The best fits are the two-segment redshift distribution (TSRD) and the two-segment evolution (TSE) variant of the Madau-Fragos star-formation history, both peaking at lower redshift than the star-formation peak and showing steep low-redshift slopes; the power-law-with-cutoff model is decisively disfavored, with $\\Delta$BIC $= 98.87$.","pith_inferences":["Beyond the paper, if the delay is real, FRB progenitors are likely older systems such as compact-object mergers or delayed magnetar formation rather than prompt massive-star collapse.","Beyond the paper, re-running this analysis on future CHIME catalogs with a larger sample would test whether the steep low-redshift slopes remain or encode the assumed host-galaxy dispersion-measure scatter.","Beyond the paper, calibrating the detection efficiency with injected mock FRBs in CHIME's real search pipeline would turn the qualitative delay into a measured delay-time distribution."],"forward_implications":["FRB rate models for future surveys should use two-segment shapes rather than a power-law cutoff, since the cutoff model is disfavored by $\\Delta$BIC $\\approx 99$.","The inferred intrinsic redshift distribution provides a template for dispersion-measure forecasts, improving the use of FRBs as cosmological (dark-siren) probes.","The low-redshift peak implies a high local FRB event rate that can be cross-checked against all-sky rate estimates.","The delay relative to star formation points toward progenitor channels with a delay timescale, which can be constrained further as more FRBs are localized."],"supporting_citations":[{"why":"Supplies the first CHIME/FRB catalog, the 415 non-repeating bursts used as the data set.","marker":"Amiri et al. 2021"},{"why":"Supplies the empirical star-formation history that the FRB redshift distribution is compared against and whose two-segment variants anchor the TSE and CPL models.","marker":"Madau & Fragos 2017"},{"why":"Supplies the power-law-with-cutoff energy function parameters ($\\alpha \\approx 1.8$, $E_c \\approx 3 \\times 10^{41}$ erg) used in the detection-efficiency calculation.","marker":"Luo et al. 2020"},{"why":"Supplies the fluence-dependent detection efficiency $\\eta_{\\rm det}(F_\\nu)$ selection-effect treatment adopted for CHIME.","marker":"Zhang & Zhang 2022"},{"why":"Provides the TSE, CPL, and TSRD population models and the comparison framework the paper extends to hierarchical Bayesian inference.","marker":"Lin & Zou 2024a"},{"why":"Supplies the inhomogeneous Poisson hierarchical likelihood formalism used in the posterior.","marker":"Mandel et al. 2019"},{"why":"Supplies the spectral index $\\beta = -1.5$ used to convert isotropic energy into observed fluence.","marker":"Macquart et al. 2019"},{"why":"Supplies the host-galaxy dispersion-measure log-normal parameters used in the $D_{\\rm E}$–$z$ relation.","marker":"Tang et al. 2023"}],"fun_headline_variants":["FRB births lag behind cosmic star formation","Delayed FRB rate: not a star-formation tracer","CHIME data push FRB peak below star formation","Hierarchical Bayes finds FRB delay vs stars","FRBs peak later than the star-formation era"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The recovered redshift shape rests on a detection-efficiency correction whose fluence thresholds, spectral index, and energy function are taken from earlier work rather than measured for CHIME; if those ingredients are wrong, the inferred delay could be an artifact.","fun_headline_variants_meta":{"raw":{"variants":["FRB births lag behind cosmic star formation","Delayed FRB rate: not a star-formation tracer","CHIME data push FRB peak below star formation","Hierarchical Bayes finds FRB delay vs stars","FRBs peak later than the star-formation era"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000188,"raw_usage":{"total_tokens":1299,"prompt_tokens":879,"completion_tokens":420,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":495,"completion_tokens_details":{"reasoning_tokens":345}},"tokens_in":495,"tokens_out":420,"duration_ms":4624,"temperature":1.0,"reasoning_tokens":345,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T14:12:02.135285+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Identify host galaxies for hundreds of CHIME FRBs and compute the volumetric event rate density $R(z)$ directly from the measured redshifts; if $R(z)$ tracks the Madau-Fragos star-formation history with its peak near $z \\approx 2$ rather than at lower redshift, the delayed-distribution conclusion is falsified. A cheaper check is to replace the assumed grey-zone detection efficiency with CHIME's actual measured sensitivity curve and re-run the inference; if the posterior peak shifts back toward the star-formation peak, the selection correction is the culprit.","supporting_citations":[],"review_version":1}