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REVIEW 4 major objections 4 minor 55 references

On the Reliability of Quasars as Cosmological Distance Indicators

T0 review · 4 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Using a model-independent calibration of the X-ray–ultraviolet luminosity relation for 2038 quasars, this paper finds that quasars currently fail as reliable cosmological distance indicators when added to supernova, BAO, and CMB data.

desk verdict The 'quasars fail' claim is likely an artifact of an understated error budget and an unvalidated Bezier extrapolation, though the paper reports the analysis honestly and deserves a revision, not a desk rejection. read the letter →

arxiv 2509.08983 v2 pith:CWQKTPNA submitted 2025-09-10 astro-ph.CO

Ariadna Montiel , Sofia Samario-Nava , Juan Carlos Hidalgo , Jose Ignacio Cabrera This is my paper · ORCID
classification astro-ph.CO
keywords quasarsstandardcandlesdarkenergydistanceindicatorscosmologicalcalibrationX-ray/UVluminosityrelationcosmicchronometersHubblediagram
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to decide whether quasars—objects visible out to z≈7.5—can be trusted as high-redshift distance indicators. It calibrates the empirical correlation between quasars' X-ray and ultraviolet luminosities without assuming a cosmology, anchoring distances to chronometer H(z) measurements at z≤1.43, then pushes the resulting Hubble diagram to z≈7.5. The calibration parameters are stable and match earlier work, but when the calibrated quasars are combined with Type Ia supernovae, baryon acoustic oscillations, and CMB distance priors, the inferred matter density and dark-energy equation of state move far from the values those probes give on their own. The paper's conclusion is that, with current data and this calibration route, quasars fail as reliable distance indicators. It also notes the result is not definitive: a future-work passage explicitly lists Bézier-degree cross-validation, alternative smoothing methods, and mock-catalog calibration checks as still to be done.

What carries the argument

The load-bearing object is a degree-2 Bézier polynomial, a smooth polynomial curve used to represent the Hubble parameter H(z). It is fitted to 28 chronometer H(z) measurements at z≤1.43, fixing H0 as its zeroth coefficient. That polynomial is then integrated to produce calibrated luminosity distances out to z≈7.5, and those distances are inserted into the flux form of the log-LX–log-LUV relation to obtain quasar distance moduli. The same relation's slope and intercept, fitted with a Bayesian regression that handles measurement errors and intrinsic scatter, carry the calibration. The entire high-redshift quasar signal flows through the extrapolation of this quadratic curve beyond the redshif

What would settle it

Re-run the full analysis with the H(z) reconstruction replaced by a nonparametric smoother or a higher-order Bézier curve, and with the calibration tested on mock quasar catalogs of known cosmology. If Ω_m and ω0 return to standard values, the failure is an artifact of the quadratic extrapolation; if they stay shifted, quasars genuinely fail.

Watch

Extended reading notes

Core claim

The central discovery is negative. After calibrating the log-LX–log-LUV relation with a Bézier polynomial fit to 28 cosmic-chronometer Hubble-parameter measurements at z≤1.43, the authors build distance moduli for the full 2038-quasar sample and fit flat ΛCDM and ωCDM models. Adding the quasars to otherwise standard combinations shifts Ω_m from about 0.28–0.33 to about 0.52–0.72 and, in the ωCDM case, moves the dark-energy equation of state from near −1 to values like −0.7 to −0.9, depending on the dataset. Because the luminosity-relation slope and intercept remain statistically consistent with earlier determinations, the paper attributes the failure to the calibration-to-cosmology step rath

Load-bearing premise

The claim rests on the assumption that the quadratic Bézier curve fitted to H(z) data at z≤1.43 remains the true expansion history when integrated out to z≈7.5; if that extrapolation is wrong, the high-z quasar distances, the calibration parameters, and every cosmological constraint built on them are biased.

Editorial extensions

If this is right

  • If quasars are unreliable at these high redshifts, then reported high-z quasar tensions with standard cosmology should not be interpreted as evidence for new physics until the calibration anchor is checked.
  • Low-redshift quasar parameters remain stable, so the sample can still serve as a consistency check, but not as a primary distance indicator.
  • Jointly sampling calibration and cosmological parameters propagates uncertainties more fully than fixing the calibration first; the stepwise route should be treated as a diagnostic, not as final.
  • Reducing intrinsic scatter—through better X-ray/UV data or sample cuts—is a prerequisite for quasars to constrain dark energy.
  • Cosmological fits that include quasars remain sensitive to the choice of H(z) anchor, so the anchor must be varied as a systematic.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the high-z shift persists when the H(z) reconstruction is replaced by a more flexible smoother, then the failure is intrinsic to the calibration rather than an artifact of the quadratic extrapolation; if it disappears, the paper's strong conclusion weakens.
  • A mock-catalog test with known input cosmology would quantify whether the stepwise route is biased and whether the joint fit actually recovers the input parameters; the paper lists this as future work.
  • The abstract's softer framing—self-consistent constraints with limitations driven by scatter—sits uneasily beside the full-text conclusion that quasars fail. Readers should treat the strong claim as conditional on the Bézier anchor.
  • The same calibration machinery could be applied to other high-redshift standardizable candles to see whether the systematics are shared across different astrophysical sources.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper calibrates the non-linear X-ray/UV luminosity relation of 2038 quasars using cosmic-chronometer H(z) data at z≤1.43 through a degree-2 Bézier fit, and then builds a quasar Hubble diagram extending to z≈7.5. The calibrated sample is combined with Pantheon+SH0ES supernovae, DESI DR2 BAO, and Planck compressed CMB data to fit flat ΛCDM and ωCDM models. The authors report that low-redshift calibration parameters agree with previous work, but that adding quasars shifts cosmological parameters strongly (e.g., Ωm≈0.63 and w0≈−0.70 in Tables 5–7), leading them to conclude that quasars fail as reliable distance indicators. The full text does not present the joint QSO+SN calibration–cosmology framework advertised in the arXiv abstract; it mentions such a simultaneous fit only as future work.

Significance. The question addressed is important: quasars are the main extragalactic probes of the z≈2–7 expansion history, and a robust reliability assessment would be a valuable contribution. The calibration is externally anchored to cosmic-chronometer H(z) data, so the analysis is not circular by construction; the paper also uses recent PPS, DESI DR2, and compressed Planck likelihoods, compares BCES and Linmix regressions, and is candid that its conclusion is not definitive. However, the central negative claim is not established by the analysis as presented. Two load-bearing problems—unvalidated high-redshift extrapolation of the Bézier H(z) fit and omission of the estimated intrinsic scatter from the quasar distance-modulus variance—directly affect the results in Tables 5–8. These issues are fixable, but the conclusion cannot be accepted until they are addressed.

major comments (4)
  1. [Section 2.2, Eqs. (5)–(8)] The degree-2 Bézier polynomial H2(z) is fitted to 28 cosmic-chronometer H(z) measurements at z≤1.43, but Eq. (8) integrates H2(z) to z≈7.5 to obtain quasar luminosity distances. The Bernstein basis is positive only on [0, z_m]; outside that interval the quadratic extrapolation is uncontrolled and is never validated against alternative smoothers or mocks. The high-z part of the Hubble diagram and the full-sample γ,β calibration therefore rest on an unvalidated functional choice, so the conclusion that quasars fail cannot be separated from this extrapolation. Section 6 lists cross-validation with splines/GPs as future work, confirming that this test is currently missing.
  2. [Section 5.3, Eq. (21), Tables 5–8] The quasar distance-modulus variance in Eq. (21) omits the intrinsic scatter δ estimated by Linmix. Table 3 gives δ≈0.237 for the low-z sample and δ≈0.231 for the full sample. Through Eq. (4) this contributes |5/[2(γ−1)]|δ ≈ 1.5 mag, which is an order of magnitude larger than the flux and parameter terms included in Eq. (21). The likelihood therefore overweights quasars relative to their actual scatter; the large parameter shifts in Tables 5–7 (e.g., Ωm=0.630 with PPS+DESI+low-z quasars) are likely an artifact of this understated error budget. The claim that quasars fail as distance indicators cannot be assessed until δ is added in quadrature or marginalized over consistently.
  3. [Section 2.2 and Eq. (21)] The covariance matrix of the Bézier coefficients β0, β1, β2 is displayed after Eq. (7) but is never propagated into the quasar distance moduli. The calibrated luminosity distance in Eq. (8) is used to determine γ and β, so the H(z) anchor uncertainty should enter σ_μ; Eq. (21) contains only γ, β, and flux uncertainties. This is a second omission in the error budget and weakens the claim of a fully model-independent calibration with reliably propagated uncertainties.
  4. [Abstract vs Sections 5–6] The abstract at the head of the arXiv record advertises a joint QSO+SN calibration–cosmology framework that yields self-consistent constraints, while the full text's abstract and conclusions state that quasars fail as distance indicators. The body does not present the joint framework; it is only mentioned as a future direction in Section 6. The central claim of the paper is therefore ambiguous, and the reader cannot tell which analysis is being reported. The abstract and main text must be reconciled, and the actual analysis presented in Sections 5–6 should be stated as the paper's result.
minor comments (4)
  1. [Section 2.1] Typo: "we binning the sample" should be "we bin the sample"; also "discrepances" in Section 5.1 should be "discrepancies."
  2. [Section 5.2, Eq. (20)] Equation (20) is written as a forward relation for F_X^cal given d_L^cal, but the text says the calibrated luminosity distance is obtained from it. Please clarify the inversion used to compute d_L^cal from observed fluxes.
  3. [Figure 2] The caption refers to "binned quasar DM" but DM is not defined; use the full phrase "distance modulus."
  4. [Section 5.3, Table 5] The text says H0 cannot be constrained once quasars are included, but Tables 5–7 leave H0 blank. Consider adding an explicit dash or statement in the table caption so the reader knows H0 is fixed by the calibration.

Circularity Check

2 steps flagged · score 5.0 of 10

Partial circularity: quasar distances are calibrated and tested on the same sample; the paper defers the circularity problem to future work.

  1. fitted input called prediction [Sections 5.1–5.3, Eq. (20), Tables 3 and 5–8]
    "The cosmological analysis is conducted only after this fitting procedure and relies exclusively on the low-redshift quasar sample and the full dataset."

    The distance moduli used in the cosmological fits (Tables 5–8) are computed from the same quasar sample and the same best-fit gamma, beta that were obtained by fitting that sample (Table 3, Eq. 20). Thus the quasar 'data' are not independent predictions: they are in-sample reconstructions from the fitted LX-LUV relation. Using the same objects to calibrate and then to constrain cosmology double-counts the quasar fluxes and can force the cosmological parameters to absorb calibration residuals. The authors implicitly concede this by deferring a strategy to 'tackle the circularity problem' to a follow-up study.

  2. other [Section 6, Conclusions]
    "We shall perform these alternative routes and compare strategies to tackle the circularity problem in a follow-up study."

    This is an explicit acknowledgment that the present analysis contains a circularity that is not resolved here. In the context of the paper, the circularity is the reuse of the same quasar data for calibration and cosmological inference (Step 1). The admission supports the finding of partial circularity: the paper's central claim that quasars fail as distance indicators is not established until this circularity is addressed.

full rationale

The calibration is anchored to external cosmic-chronometer H(z) data (Capozziello et al. 2018), which is independent of the quasar fluxes and of the cosmological models tested, so there is no self-definitional circularity at the level of the distance scale. The main circularity is that the same 2038 quasars are used first to fit gamma and beta (Table 3) and then the fitted relation is inverted to assign distance moduli to those same objects (Eq. 20), which are then used as data in cosmological fits (Tables 5–8). This is an in-sample calibration: the predicted distances are not independent of the calibration, and the quasar likelihood can be artificially overconfident. The paper explicitly defers a strategy to tackle the circularity problem to a follow-up study. The omission of the intrinsic scatter delta (estimated ~0.237 dex for the low-z sample) from the distance-modulus variance in Eq. (21) is a serious statistical issue that amplifies the impact of the in-sample calibration, but it is not itself a circularity; it is an error-budget problem. The stepwise results supporting the 'quasars fail' conclusion also conflict with the arXiv abstract, which reports that the joint calibration-cosmology framework gives self-consistent constraints; this inconsistency weakens the reliability of the central negative claim, but it is not a circularity. The self-citation to Montiel et al. (2020) for the Bezier calibration method is load-bearing for the functional form, but that prior work applied the method to GRBs and is externally falsifiable, so it does not by itself make the derivation circular. Overall, the derivation has partial circularity because the quasar data used for cosmological inference are not independent of the calibration fit, but the external H(z) anchor provides independent content, preventing a higher score.

Assumptions & free parameters 8 free parameters · 3 assumptions · 0 invented entities

The calibration depends on three Bezier coefficients fitted to cosmic-chronometer data and on two LX-LUV parameters fitted to quasars, plus an intrinsic-scatter estimate that is not used. The main assumptions are constant LX-LUV parameters with no redshift evolution, a degree-2 Bezier shape that is extrapolated far beyond its data range, and spatial flatness. No new physical entities are introduced.

free parameters (8)
  • beta0 (H0 anchor) = 70.81 km/s/Mpc
    Zero-order Bezier coefficient fitted to 28 cosmic-chronometer H(z) points; sets the distance scale for the whole calibration.
  • beta1 = 81.99
    Linear Bezier coefficient from the cosmic-chronometer fit; controls the low-z shape of H(z).
  • beta2 = 179.02
    Quadratic Bezier coefficient from the cosmic-chronometer fit; controls curvature and dominates the extrapolated high-z behavior.
  • gamma (low-z) = 0.618 +/- 0.017
    Slope of the LX-LUV relation fitted with Linmix to the z < 1.43 quasar subsample; used to compute distance moduli.
  • beta (low-z intercept) = 7.824 +/- 0.522
    Intercept of the LX-LUV relation from the same low-z Linmix fit.
  • gamma (full sample) = 0.688 +/- 0.009
    Slope fitted to the full 2038-quasar sample; used for the full-sample distance moduli.
  • beta (full sample intercept) = 5.747 +/- 0.264
    Intercept fitted to the full quasar sample.
  • delta (intrinsic scatter) = 0.237 +/- 0.006 (low-z)
    Intrinsic dispersion estimated by Linmix in the LX-LUV relation. Estimated but not propagated into sigma_mu in Eq. (21), which is a key soundness problem.
assumptions (3)
  • domain assumption The LX-LUV relation has no redshift evolution; gamma and beta are constants.
    Stated in Sections 1 and 2.1 as a first-order choice supported by Lusso et al. (2025). If false, high-z distance moduli are biased and the high-z cosmological constraints are invalid.
  • ad hoc to paper H(z) is well described by a degree-2 Bezier polynomial over the full calibration and extrapolation range.
    The polynomial is fitted to CC data at z <= 1.43 but used in Eq. (8) up to z about 7.5. No independent justification for n=2 or for the extrapolation is given.
  • domain assumption Spatial flatness, Omega_K = 0, holds for the calibration and for all cosmological models.
    Used in Eq. (8) and in the Friedmann equations. Motivated by Planck constraints, but it restricts the generality of the 'model-independent' calibration.

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Pith. "Pith review of On the Reliability of Quasars as Cosmological Distance Indicators." pith.science (2026). https://pith.science/paper/CWQKTPNA

@misc{pith2026250908983,
  author       = {Pith},
  title        = {Pith review of: On the Reliability of Quasars as Cosmological Distance Indicators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CWQKTPNA}},
  note         = {Machine review of arXiv:2509.08983}
}
abstract

We assess the viability of quasars as cosmological distance indicators based on the non-linear $L_X$--$L_{\rm UV}$ relation. We calibrate this relation in a model-independent way by anchoring quasar luminosity distances to cosmic-chronometer $H(z)$ measurements over $z\leq 1.43$, and construct a quasar Hubble diagram extending up to $z\simeq 7.5$. We compare a traditional stepwise approach, in which the calibration is fixed before cosmological inference, with a joint QSO+SN calibration--cosmology framework where calibration and cosmological parameters are sampled simultaneously. The stepwise analysis, supplemented with DESI DR2 BAO measurements and Planck compressed CMB distance priors, is used as a diagnostic benchmark, while the joint framework, with and without the SH0ES $H_0$ information, provides our main cosmological results. We selected a low-$z$ quasar subsample ($z<1.43$), matching the redshift range of the cosmic-chronometer calibration, and found calibration parameters consistent with previous studies. However, the stepwise cosmological constraints can become unstable once quasars are included, reflecting the incomplete propagation of calibration uncertainties and calibration--cosmology degeneracies. In contrast, the joint analysis yields self-consistent constraints because the quasar calibration parameters are fitted simultaneously with the cosmological parameters, allowing these uncertainties and degeneracies to be propagated into the final posteriors. Our results indicate that the current limitations of quasar cosmology are driven mainly by intrinsic scatter and possible sample-dependent effects in the $L_X$--$L_{\rm UV}$ relation, rather than by a fundamental inconsistency with standard cosmology.

Figures

Figures reproduced from arXiv: 2509.08983 by the authors.

Figure 1
Figure 1. Rest-frame monochromatic luminosities log(𝐿𝑋 ) against log(𝐿𝑈𝑉 ) for the final sample of 2038 quasars (blue circles) as described in Sect. 2.1. The results from the BCES fit (dashed red line) and from the Linmix fit (black solid line) are also reported. On the other hand, we further clarify that the binned results are not employed in the cosmological analysis, since binning can smooth out intrinsic scatter and poten… view at source ↗
Figure 2
Figure 2. Hubble diagram of Pantheon+ supernovae (orange points) and quasars in the redshift range 0.009 < 𝑧 < 7.541 (blue points). The black points are the binned quasar DM. The dotted line in red is the flat ΛCDM model with Ω𝑚 = 0.315 and 𝐻0 = 67.66 km/s/Mpc [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.