{"id":"e2c97ec0-cfc0-4e9f-8baa-97c127f5d6a7","arxiv_id":"2504.13132","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A two-stage MCMC method using the Yang relation between PRS luminosity and FRB rotation measure yields H0 = 75 ± 30 km/s/Mpc from six observed persistent radio sources, and H0 = 75 ± 15 from mock data.","lead":"Astronomers propose using persistent radio sources linked to fast radio bursts as a new way to measure how fast the universe is expanding. Their first attempt, using only six sources, gives a Hubble constant of 75 plus or minus 30 km/s/Mpc, far too broad to settle the cosmic expansion debate.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation 6 as printed does not eliminate H0 from the first-stage calibration, making the two-stage MCMC circular and unable to break the H0–ζR² degeneracy as claimed.","rationale":"I read the paper as proposing a two-stage calibration: first use DM to constrain the nuisance product ζeγth²R² without needing H0, then use that as a prior to constrain H0 from PRS luminosity. This is a clever idea if the first stage is genuinely H0-free. The printed Eq. 6 is the linchpin of that first stage, and it does not show the claimed elimination. This is a concrete, checkable internal inconsistency, and it sits upstream of the log-normal universality assumption that the reader identified as weakest. Even with a universal ζeγth²R² distribution, the method fails if the calibration stage is coupled to H0. I therefore flag this as the single most load-bearing concern. The mock test in the paper cannot detect this flaw because the same equations are used for injection and recovery, so it only checks self-consistency. The paper should provide a corrected Eq. 6, a DM-only baseline, and the data and code before the method can be accepted. This does not move the verdict: CONDITIONAL remains appropriate, so I leave the reader's verdict unchanged.","tokens_in":12445,"tokens_out":12928,"duration_ms":118436,"concrete_test":"Independently re-derive Eq. 6 by solving Eq. 2 for H0 and substituting into Eq. 5, then compare the result with the printed equation. If H0 remains, rerun the first-stage MCMC on the mock sample with H0 fixed at 67 and 73 km s−1 Mpc−1 and check whether the median of the ζeγth²(R/0.01 pc)² posterior shifts by more than its 1σ width; then run the full two-stage chain and compare the final H0 posterior with a one-stage joint fit of Eq. 2 and Eq. 5. A significant shift or disagreement would confirm that the two-stage decoupling is not H0-independent.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing defect is in the derivation of Eq. 6. The text states that Eq. 6 is obtained 'by eliminating H0 from Equation 5 using Equation 2 and substituting the result into Equation 4,' yet the printed Eq. 6 contains H0^4 inside the square root, alongside ζeγth²R²(1+z)|RM_obs|/Fν. This is not the result of elimination: after substitution the H0 dependence should cancel completely, leaving a relation among DM_ex, z, Fν, RM_obs, and ζeγth²R² only. As printed, the first-stage MCMC (Sec. 3, step 3) must either fix H0 or impose a prior on it. The posterior for ζeγth²R² is therefore H0-dependent. The second stage then feeds that posterior into Eq. 2 and fits H0 using the same data, which double-counts the data and cannot break the H0–ζR² degeneracy. The claimed independence of the calibration step is not demonstrated. If the H0^4 is a typesetting remnant of an intended substitution, the substituted expression is not shown and the manuscript is internally inconsistent as written; if it is genuine, the method is circular. In either case the central claim that the Yang relation helps to unravel the degeneracies is not supported by the printed equations.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proposes a new method to measure the Hubble constant using persistent radio sources (PRSs) associated with fast radio bursts. The method exploits the Yang relation Lν∝|RM|, which gives the PRS luminosity from its rotation measure up to a nuisance product ζeγth²R². The authors combine this with dispersion-measure data: a first-stage MCMC fits the combination of DM_host+DM_src and ζeγth²R² from an equation that is supposed to be H0-free, and a second-stage MCMC uses the resulting posterior as a prior in the luminosity-distance relation to infer H0. The pipeline is tested on ~50 mock PRSs and then applied to six observed systems (four confirmed plus two candidate PRSs), giving H0=75±30 km/s/Mpc. The paper also presents a mock population synthesis and discusses future improvements.","tokens_in":12707,"tokens_out":23987,"duration_ms":213571,"significance":"The proposed probe is timely and, if the derivation is correct, would provide an independent H0 measurement that does not rely on Type Ia supernovae or the CMB. The authors make several good practical choices: they treat DM_host+DM_src as a free parameter rather than fixing it at ~100 pc/cm³, they check insensitivity to the Milky Way electron-density model, and they explicitly acknowledge the remaining H0–ζR² degeneracy. The main weakness is that the printed central equations are inconsistent: Equation (6) claims to have eliminated H0 but still contains H0^4 inside the square root, and Equation (2) has a suspicious coefficient. Because the mock test is generated with the same equations, it cannot validate the physical calibration. The idea is worth pursuing, but the manuscript needs a corrected derivation and a rerun of the numerical results before the claims can be accepted.","major_comments":[{"comment":"The printed Equation (6) is not the result of eliminating H0 from Equation (5) using Equation (2), because H0^4 still appears inside the square root. After a correct substitution the Hubble constant should cancel completely, leaving a relation among DM_ex, z, Fν, RM_obs, ζeγth²R², Ωm (with Ωb h² and fIGM fixed). As written, Sec. 3, step 3 fits Equation (6) without any prior or fixed value for H0, so the first-stage posterior for ζeγth²R² is either H0-dependent or undefined. The claimed H0-independent calibration of ζeγth²R², which is the basis of the two-stage decoupling, is therefore not demonstrated. If the H0 in Equation (6) is meant to be a shorthand for the right-hand side of Equation (2), that substitution must be displayed explicitly and the resulting H0-free equation used in the MCMC.","section":"2.1, Eq. (6)"},{"comment":"Equation (2) appears to have a dimensional error in its coefficient. Combining Equation (1) with Lν = 4πDL²Fν/(1+z) and DL = c(1+z)/H0 ∫ dz/E gives |RM_obs| = 27 m_e c^4 Fν [∫ dz/E]^2 / [16π² ζeγth²R² H0²(1+z)], i.e., a factor m_e c^4 in the numerator, not 1/m_e as printed. Because the mock sample is generated using the same Equation (2), the mock test cannot detect this normalization error, while the quoted H0=75±30 from the observed sample depends directly on it. Please verify the coefficients in Equations (2) and (6) and rerun the analysis if the factor is corrected.","section":"2.1, Eq. (2)"},{"comment":"The mock test is an internal consistency check rather than a validation of the method. The mock sample is generated from the same Yang relation, the same DM decomposition, the same log-normal distributions for ζeγth²(R/0.01pc)² and DM_host+DM_src, and the same fiducial H0=73.04 that are used in the recovery, so obtaining H0=75±15 only shows that the pipeline can invert its own input. The key physical assumption that ζeγth²R² is a universal log-normal variable with fixed scatter is not tested by this procedure. The paper should state this limitation explicitly and, if possible, include a stress test in which the assumed ζeγth²R² distribution is incorrect (e.g., redshift evolution or environmental dependence) to quantify the resulting bias on H0.","section":"3, steps 1-4"},{"comment":"The observational constraint is not reproducible as printed because the paper gives no table of the six systems with their z, DM, Fν, and RM values. In addition, two of the six are candidate PRSs from Ibik et al. (2024) whose association with FRBs is not yet confirmed; including them without a separate treatment or an association-probability model can bias the central value H0=75±30. Please include the data table and either exclude the candidates from the main result or model their unknown association probability.","section":"3, observational sample"}],"minor_comments":[{"comment":"The text 'RM_src = (1+z)^2 RM_obs^2' should read 'RM_src = (1+z)^2 RM_obs'; the square on RM_obs is either a typo or dimensionally incorrect.","section":"2.1, below Eq. (1)"},{"comment":"The citation for σ_MW should be to the Galactic electron density models (Cordes & Lazio 2002; Yao et al. 2017), not to Manchester et al. (2005), which is a pulsar catalogue.","section":"2.1, Eq. (7)"},{"comment":"The claim that the Yang relation 'unravels degeneracies among H0, Ωb, and fIGM' is stronger than what is implemented, since Ωb h² and fIGM are fixed in the fits; please rephrase to describe the degeneracy that is actually broken (H0 versus ζR²).","section":"4, second bullet"},{"comment":"The first-stage posterior is marginalized over ζeγth²R² before being used as a prior in the second stage; this discards correlations with Ωm and DM_host+DM_src. Please either propagate the full joint posterior or discuss why the correlation is negligible.","section":"3, second-stage MCMC"},{"comment":"The title contains a typo: 'F ast Radio Bursts' should be 'Fast Radio Bursts'.","section":"Title"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the stress-test concern is confirmed by the manuscript text. Eq. (6) indeed retains H0^4 inside the square root, contradicting the claim that H0 was eliminated; this makes the first-stage calibration H0-dependent and the two-stage inference circular as printed. I also find a likely factor error in Eq. (2). These are central, but they appear fixable: the authors need to derive the correct H0-free relation, propagate the correct coefficients, rerun the MCMC, and present the data table. Hence my recommendation is major revision rather than rejection. A short statement on the unvalidated universality of ζeγth²R² should also be added."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Fair warning before you read: the paper has a nice idea and a serious flaw. The central derivation is inconsistent as printed, and the claimed H0 constraint is far too weak to discriminate between Planck and Riess values. But the underlying idea—using the Yang relation between PRS luminosity and RM to break the DM degeneracy—is worth taking seriously.\n\nWhat's new: the two-stage MCMC, first calibrating ζe γth^2 R^2 from DM and then using that posterior as a prior for H0, is a natural extension of Yang+20. They also give the first H0 estimate from six PRS systems, H0=75±30. The mock generation and population model (star-forming, L^-1.3 luminosity function) are reasonable and clearly described. They test both YMW16 and NE2001 Galactic electron models and find insensitivity.\n\nThe soft spot is not small. Equation 6 as printed contains H0^4 inside the square root even though the text says H0 was eliminated. If you follow the derivation, DM_IGM should depend on H0 linearly, and substituting from Eq.2 should remove it entirely. As printed, the first-stage MCMC must fix H0 or put a prior on it, which makes the second-stage fit of H0 using the same data circular. This is load-bearing: the paper's whole claim to break the degeneracy rests on the calibration being H0-independent. If it's a typographical remnant, the authors need to show the substituted expression; if not, the method is circular. The paper's own discussion admits a degeneracy remains between ζeγth^2R^2 and H0 in Figure 3, which further undercuts the main claim.\n\nAlso worth noting: the mock test only recovers injected parameters—that's a consistency check, not validation. The assumption that ζe γth^2 R^2 follows a single universal log-normal distribution across all PRS systems is asserted from a handful of sources, and there's no demonstration that evolution with redshift or host environment doesn't bias it. No code or data table is provided, though the six observed values are in the literature.\n\nOn balance: the idea is publishable after major revision, but not in this form. The authors should fix Eq.6, add a DM-only baseline to show the Yang relation actually helps, and be clearer about what the 30 km/s uncertainty includes. A serious referee could make this a solid paper.","headline":"Nice idea, but Eq. 6 doesn't eliminate H0 as claimed, making the two-stage calibration circular; the paper needs a real fix and a baseline before the result means anything.","tokens_in":13282,"tokens_out":4108,"would_cite":false,"duration_ms":33353,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A proposed method uses the Yang relation between FRB rotation measure and persistent radio source luminosity to break the H0–Ωb–fIGM degeneracy, yielding H0 = 75 ± 30 km/s/Mpc from six observed systems.","keywords":["fast radio bursts","persistent radio sources","Hubble constant","Yang relation","rotation measure","dispersion measure","cosmological parameters","MCMC analysis"],"falsifier":"Measure the nebular radius $R$ directly with very long baseline interferometry for a dozen or more PRSs, compute the implied product $\\zeta_e \\gamma_{\\rm th}^2 R^2$ for each, and compare the spread and redshift trend against the single log-normal assumed here; a redshift drift or a scatter much larger than the assumed width would falsify the calibration.","tokens_in":12167,"feed_emoji":"📡","tokens_out":7292,"duration_ms":67220,"temperature":0.7,"pith_summary":"This paper proposes a way to measure the Hubble constant $H_0$ by pairing fast radio bursts with their persistent radio sources. The key input is the Yang relation, $L_\\nu \\propto |\\mathrm{RM}|$, which lets the observed rotation measure of an FRB stand in for the luminosity of its persistent radio counterpart. The authors show that this relation breaks the degeneracy among $H_0$, $\\Omega_b$, and $f_{\\mathrm{IGM}}$ that normally plagues dispersion-measure-only analyses, and they extract $H_0 = 75\\pm 30~\\mathrm{km\\,s^{-1}\\,Mpc^{-1}}$ from six observed systems. The result matters because it offers an independent, late-universe probe of the Hubble tension.","feed_headline":"Six radio sources put the Hubble constant at 75 km/s/Mpc","feed_subtitle":"Persistent sources tied to fast radio bursts yield H0 = 75 ± 30 km/s/Mpc, a new independent probe.","key_machinery":"The load-bearing object is the Yang relation $$L_\\nu = \\frac{64\\$pi^{3}$}{27 m_e $c^{2}$} \\zeta_e \\gamma_{\\rm th}^2 $R^{2}$ |\\mathrm{RM}_{\\rm src}|,$$ which links the synchrotron luminosity of a persistent radio source to the rotation measure of its FRB. The argument runs through two coupled equations: Equation 6 uses dispersion measure to calibrate the nuisance product $\\zeta_e \\gamma_{\\rm th}^2 R^2$, and Equation 2 then converts RM and flux into a distance that depends on $H_0$. The two-stage MCMC chains these together so that the DM-based calibration supplies the prior that breaks the otherwise circular $H_0$–$\\zeta_e \\gamma_{\\rm th}^2 R^2$ degeneracy.","core_discovery":"On the paper's own terms, the central claim is that the Yang relation turns persistent radio sources into standard candles: once the product $\\zeta_e \\gamma_{\\rm th}^2 R^2$, the factor connecting PRS luminosity to rotation measure, is calibrated from dispersion measures, the same relation gives a distance and hence $H_0$ from flux, redshift, and RM alone. A two-stage MCMC first fits Equation 6 to DM data to constrain the product, then uses that posterior as a prior in Equation 2 to infer $H_0$. With a mock sample the method recovers the injected value ($75\\pm 15$), and with the six observed PRSs it gives $75\\pm 30~\\mathrm{km\\,s^{-1}\\,Mpc^{-1}}$, insensitive to the choice of Galactic electron density model.","pith_inferences":["Beyond the paper, the same two-stage design should also constrain $\\Omega_b$ and $f_{\\mathrm{IGM}}$ simultaneously, since the first-stage fit to Equation 6 depends on both; a large PRS sample could turn the method into a baryon census rather than just an $H_0$ probe.","Beyond the paper, if the product $\\zeta_e \\gamma_{\\rm th}^2 R^2$ correlates with host-galaxy star formation, that correlation could be measured and used as an additional distance calibrator instead of being treated as scatter.","Beyond the paper, a concrete test is to generate mock PRS catalogs with redshift-evolving $\\zeta_e \\gamma_{\\rm th}^2 R^2$ and check how many sources are needed to detect the evolution; with the current six sources, the universal log-normal prior is essentially unconstrained."],"forward_implications":["If the Yang relation holds, PRS systems become standard candles: RM, flux, and redshift give a distance without relying on the DM-only degeneracy.","With the six observed PRSs the method returns $H_0 = 75 \\pm 30~\\mathrm{km\\,s^{-1}\\,Mpc^{-1}}$, and the mock-sample recovery ($75 \\pm 15$) shows the pipeline is unbiased at least for the assumed population.","A larger PRS sample, together with a full Bayesian treatment of all DM components, will tighten the constraint substantially.","Calibrating $\\zeta_e \\gamma_{\\rm th}^2 R^2$ against an independent distance anchor in the same host galaxy—analogous to Type Ia supernova calibration—would remove the DM-based steps entirely.","The method extends to any persistent radio source with the same emission mechanism and a measured RM, not just sources already tied to FRBs."],"supporting_citations":[{"why":"This paper derives the Lν ∝ |RM| Yang relation that the entire method uses as a standard candle.","marker":"Yang et al. 2020"},{"why":"This paper reports the observed value ζeγth²(R/0.01 pc)² ≈ 1 that anchors the product prior.","marker":"Bruni et al. 2025"},{"why":"This paper introduces the IGM dispersion-measure formula and the baryon-tracing framework for FRBs.","marker":"Deng & Zhang 2014"},{"why":"This paper supplies the Macquart relation and the IGM DM scatter used in Equation 5.","marker":"Macquart et al. 2020"},{"why":"This paper provides the numerical IGM DM distribution used to set σIGM(z).","marker":"McQuinn 2014"},{"why":"This paper fixes the baryon density Ωb h² ≈ 0.0223 from quasar absorption, removing that degeneracy.","marker":"Cooke et al. 2018"},{"why":"This paper gives a previous FRB-based H0 constraint that motivates breaking the DM-only degeneracy.","marker":"Wu et al. 2022"},{"why":"This paper provides the cosmic star formation history adopted for the mock PRS redshift distribution.","marker":"Madau & Dickinson 2014"}],"fun_headline_variants":["FRB radio sources measure H0 = 75 ± 30 km/s/Mpc","Six persistent sources peg Hubble constant at 75","New H0 probe: persistent radio sources of FRBs","Yang relation turns FRB sources into Hubble rulers","FRB persistent sources give H0 = 75 ± 30"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The method assumes that the product of the electron fraction, the square of the thermal Lorentz factor, and the square of the source radius is drawn from one universal log-normal distribution across all persistent radio sources; if this product changes with redshift or host environment, the dispersion-measure calibration is biased and the Hubble constant inherits that bias.","fun_headline_variants_meta":{"raw":{"variants":["FRB radio sources measure H0 = 75 ± 30 km/s/Mpc","Six persistent sources peg Hubble constant at 75","New H0 probe: persistent radio sources of FRBs","Yang relation turns FRB sources into Hubble rulers","FRB persistent sources give H0 = 75 ± 30"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000266,"raw_usage":{"total_tokens":1609,"prompt_tokens":941,"completion_tokens":668,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":557,"completion_tokens_details":{"reasoning_tokens":583}},"tokens_in":557,"tokens_out":668,"duration_ms":6770,"temperature":1.0,"reasoning_tokens":583,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T12:14:03.306166+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the nebular radius $R$ directly with very long baseline interferometry for a dozen or more PRSs, compute the implied product $\\zeta_e \\gamma_{\\rm th}^2 R^2$ for each, and compare the spread and redshift trend against the single log-normal assumed here; a redshift drift or a scatter much larger than the assumed width would falsify the calibration.","supporting_citations":[],"review_version":1}