A large archival analysis of quasars found a weak positive association between the X-ray photon index, the study's measure of the X-ray spectrum, and accretion state. The link appeared when researchers compared each object with itself across repeated XMM-Newton observations. After translating the X-ray response into an Eddington-ratio response, the estimated slope ranged from 0.14 to 0.26, depending on the bolometric scaling used. The result is positive but shallow, and it does not account for most of the observed variety in quasar X-ray spectra.
The one-off picture changes with the estimator
The researchers began with a cleaned cross-match containing 7,835 quasars and 11,455 XMM-Newton detections. They divided the objects by the single-epoch mass indicator available from H-beta, Mg II or C IV: 834 in H-beta, 5,495 in Mg II and 1,506 in C IV. To measure their X-ray spectra, they used fluxes in the 1 to 2 keV and 2 to 4.5 keV bands to estimate the photon index and the rest-frame 2-keV monochromatic luminosity.
For the one-off population comparison, the team modeled photon index against the logarithm of the Eddington ratio separately for each mass indicator and bolometric correction. The resulting slope changed with the recipe: H-beta masses gave 0.10 to 0.29; Mg II masses gave negative 0.05 to negative 0.01 when X-ray routes were used, but 0.07 to 0.18 with optical and ultraviolet corrections; C IV masses gave negative 0.11 to negative 0.03. The sign changes show that the whole-sample answer was strongly estimator-dependent.
Forward-modelled mock samples offered an explanation for that instability. The observed spread in log Eddington ratio was 0.33 to 0.54 dex, similar to the approximately 0.4 dex systematic uncertainty in single-epoch virial masses. When the researchers injected slopes from 0 to 0.4 into the mocks, the regressions recovered values only from negative 0.02 to positive 0.03. The exercise shows how mass noise can hide a population relationship.
Following quasars through time
Researchers then turned to repeated observations. Of 1,701 objects with at least two XMM-Newton observations, 893 were retained after a variability cut required the scatter in each object's log X-ray luminosity to exceed its typical measurement uncertainty, with a ratio greater than 1. This selected a subset where changes were large enough to analyze.
For every retained object, they fitted the photon index against log X-ray luminosity and combined the resulting per-object posterior distributions into population estimates. The population mean slope was 0.35 plus or minus 0.03. It was 0.33 plus or minus 0.04 among 164 objects with at least five epochs, and 0.34 plus or minus 0.08 among 45 with at least 10. The spread of individual slopes was approximately 0.28, while most individual posteriors remained broad.
To express the result in the study's accretion measure, the researchers converted the X-ray-luminosity slope using an assumed relation between X-ray and ultraviolet luminosity and a bolometric scaling. The reference correction gave an Eddington-ratio slope of 0.21 plus or minus 0.02; a thin-disc scaling gave 0.14 plus or minus 0.01. Across the plausible choices examined, the value ran from 0.14 to 0.26. That conversion is conditional, because it assumes population-level scalings also describe individual sources.
An even stricter quality cut made the relation look stronger. Keeping only epochs with photon-index signal-to-noise above 10 left 528 objects. Their mean X-ray-luminosity slope was 0.43 plus or minus 0.04, corresponding to Eddington-ratio slopes of 0.25 plus or minus 0.02 under the paper's K13 prescription and 0.17 plus or minus 0.01 under SS. The authors caution that this selection can favor brighter states and brighter objects, so the higher estimate carries a sample-selection caveat.
A weak clue, not a master key
The positive association still has a narrow reach. The sample's mean photon index was 2.02, with an observed dispersion of 0.27 and an estimated intrinsic dispersion of 0.148 plus or minus 0.003. Given an intrinsic spread in log Eddington ratio of at most 0.15 dex, the analysis attributed only about one-fifth of the intrinsic photon-index dispersion to Eddington ratio. The authors therefore do not treat it as a complete explanation for why quasars show different X-ray spectra.
A check on the X-ray measurement compared photometric and spectroscopic photon indices in 90 randomly selected objects. Their mean offset was negative 0.04, and the mean uncertainty fell from 0.21 for the photometric values to 0.13 for the spectroscopic ones; none of the spectra required intrinsic absorption. The comparison found consistency, while spectroscopic estimates were more precise.
The document is a preprint, identified as arXiv:2609.37607v1 and dated 29 September 2026. Its observational design means the findings describe associations and estimated within-object responses, not a demonstrated causal effect of changing Eddington ratio on photon index. Taken together, the results show that repeated X-ray data can recover a weak population response that single-epoch mass regressions may obscure.
Paper data and sources
Original title: Fooled by mass? Recovering the $Γ-λ_{Edd}$ relation of quasars from their X-ray variability
Authors: Matilde Signorini, Andrea Sacchi, Bartolomeo Trefoloni et al.
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-09-29
DOI: Not available
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