arXiv subjects

Michal Zajaček

Publications and source records attributed to Michal Zajaček.

At least 19 recordsLinked to original sources

arXiv2026

Radiation-hydrodynamics of star-disc collisions: From system parameters to outflows and lightcurves

Quasi-periodic eruptions (QPEs) are nuclear transients producing bright, repeating soft X-ray flares superimposed on quiescent emission. A promising interpretation is that they are powered by star-disc collisions, in which a star crosses an accretion disc around a supermassive black hole, drives shocks, and launches dense outflows from which radiation emerges. We present a systematic study of star-disc collisions, linking the physical parameters of the collision to the resulting outflows and emerging bolometric luminosities. We perform three-dimensional local radiation-hydrodynamics simulations, varying the disc surface density and vertical density profiles, stellar velocity and radius, and local collision angle. We focus on the regime where the star remains unperturbed by the collision. We find that variations in stellar velocity and accretion disc surface density leave the bow shock and outflow morphology largely unchanged. Faster stars produce brighter flares, while denser discs mainly increase the flare duration. Increasing the stellar radius increases the momentum of the forward outflow and produces brighter and longer flares. More centrally concentrated discs yield brighter and shorter flares because radiation escapes more efficiently through outer low-density layers. More oblique crossings reduce the momentum and luminosity asymmetry of two outflows, and lengthen the flares. We provide empirical scalings of the peak luminosity and flare duration with the individual system parameters and apply them to GSN 069. The best candidate solutions favour a star with a radius $\sim R_\odot$ on a retrograde orbit, colliding with a dense post-TDE disc with a vertically concentrated density profile. Our findings suggest that specific combinations of system parameters can reproduce characteristic flare amplitudes, durations, duty cycles, and strong-weak flare patterns observed in QPE sources.

astro-ph.HE↗

arXiv2026

Eclipses of Nearby Millimeter-bright Galactic Nuclei by Stars in Nuclear Star Clusters

It is of a general interest to look for signatures of stellar bodies orbiting supermassive black holes (SMBHs) in galactic nuclei other than the Galactic center. Previously stellar transits were analyzed in UV, optical, and X-ray domains as well as potential microlensing signatures due to more compact bodies orbiting SMBH accretion disks. Here we complement previous studies by considering nearby ($z=0.001$) millimeter-bright active galactic nuclei targeted by different facilities in the millimeter domain. At these wavelengths the jet base and the nuclear ring are sufficiently compact so that they can be occulted by large evolved stars in dense nuclear star clusters. We find that in the millimeter domain evolved stars with stellar radii of $\gtrsim 500\,R_{\odot}$ can cause eclipses with the relative depth of $\sim 10-40\%$. Typical recurrence timescales are at least 10 years and the eclipse durations are $\sim 10$ days. Towards lower frequencies the eclipse temporal profiles become shallower and broader while towards higher frequencies they are deeper and narrower. Although expected to be rare due to selection effects and evolved stars being prone to tidal disruption, eclipses of millimeter radio cores can be applied to estimate the distance of the extragalactic star from the SMBH and its stellar radius. In case the eclipse is recurring, we provide a relation for estimating the SMBH mass.

astro-ph.GA↗

arXiv2026

Traversing the Galactic Centre in space and time

The Galactic Center is often identified with its central supermassive black hole, Sgr A*. Yet the black hole governs gravitationally only the innermost few parsecs of the Milky Way, while the surrounding Nuclear Star Cluster, Nuclear Stellar Disc and Central Molecular Zone (CMZ) shape the dynamics of stars and gas on progressively larger scales. Understanding how these components interact is essential not only for reconstructing the history of our own Galaxy, but also for interpreting galactic nuclei more generally.

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arXiv2026

Low-angular-momentum accretion shocks can power weak-to-moderate X-ray flares from SgrA*

X-ray flares from Sgr~A* span broad ranges in duration, fluence, and luminosity, but their origin remains unsettled. We test whether standing shocks in low-angular-momentum magnetized accretion flows can provide a viable energy reservoir to account for these events. Using semi-analytic trans-magnetosonic shock solutions, we estimate the kinetic energy available in the downstream post-shock flow and compare it with the 25-year \textit{Chandra} X-ray flare catalog. For each theoretical solution, we compute the required efficiency $ε=E_{\rm data}/E_{\rm sh}$, where $E_{\rm data}$ is the observed radiated energy for each flare and $E_{\rm sh}$ is the available energy in the shocked flow. Notably, the weak flares require $ε\sim10^{-3}$--$10^{-2}$, while moderate flares require a few percent. We perform the analyses for weakly and highly spinning cases, and the resulting weak-to-moderate flare energy budget remains unchanged. For the fiducial accretion rate and bolometric correction, the kinetic energy reservoir in standing shock is sufficient for the observed weak-to-moderate flare population, whereas the strongest events likely require higher efficiency which can be mediated via additional magnetic energy dissipation channel.

astro-ph.HE↗

arXiv2026

What is the reason for the lack of cool evolved stars near the center of the Galaxy?

The observed deficit of bright late-type giants in the central parsec of the Milky Way remains an open problem. We investigate whether repeated passages of red giants (RGs) through a past jet from Sgr$~$A$^*$ can modify their envelopes and apparent spectral types. Three-dimensional hydrodynamical simulations follow a $1\,M_\odot$, $100\,R_\odot$ RG through up to ten jet crossings at $10^{-3}\,\text{pc}$, using jet kinetic luminosities of $10^{42}$, $10^{44}$, and $10^{48}\,\text{erg}\,\text{s}^{-1}$. Each passage produces shocks, envelope ablation, and an asymmetric downstream tail. For jet luminosities up to $10^{44}\,\text{erg}\,\text{s}^{-1}$, the cumulative ablated mass evolves approximately as $ΔM\propto t^{1/2}$ and reaches about $10^{-4}\,M_\odot$ over a $10^5\,\text{yr}$ active phase. Repeated heating also raises the surface temperature from about $3600$ to $8500\,\text{K}$ during the first ten passages, potentially making an M-type giant appear as an A-type source. Jet feedback may therefore contribute to the apparent depletion of cool giants and the excess of hot stars in galactic nuclei (GN).

astro-ph.HE↗

arXiv2026

Observations of X-ray quasi-periodic eruptions

Quasi-periodic eruptions (QPEs) are a novel class of repeating nuclear transients, discovered exclusively in the X-ray band to date. Since their initial discovery in 2019, the QPE sample has grown to 13 sources, exhibiting large amplitude, quasi-regular eruptive variability patterns that are distinct from previously known modes of massive black hole variability. In this chapter, we provide a comprehensive overview of their observational characteristics. We review the X-ray spectral and timing properties of QPE eruptions, their long-term evolution, as well as the underlying quiescent emission, which is well described by thermally dominated, compact accretion disks. We discuss population-level emerging trends and selection biases, and present an updated census of their host galaxy properties. We also highlight the growing body of evidence pointing to strong connections between QPEs and tidal disruption events. Finally, we briefly summarize the key observational constraints on proposed QPE model interpretations, before looking ahead to the observational challenges and opportunities that will shape future progress in understanding this emerging population of nuclear transients.

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arXiv2026

HALO II: Constraining Hubble constant $H_{0}$ through continuum delay fitting of Fairall 9

The Hubble tension remains one of the most significant unresolved problems in modern cosmology. A key question is whether it may arise from underestimated systematic uncertainties in the different measurement techniques. In this context, new independent methods are of exceptional importance. We therefore pursue a novel approach to determining the Hubble constant, $H_{0}$ based on continuum time delay and spectral energy distribution (SED) modeling in active galactic nuclei (AGNs). Unlike conventional techniques, this method is entirely independent of the cosmic distance ladder and does not require cross-calibration against other distance indicators. As a result, it enables a direct determination of $H_{0}$, free from the arbitrary normalizations that often affect indirect measurements. We conducted a dedicated monitoring campaign of the Seyfert galaxy Fairall 9 and further developed the {\tt H0RIZON-AGN} model to interpret the resulting observations. The model incorporates the effects of radiation reprocessing in the surrounding cold accretion disk, enabling a more realistic description of the observed continuum delays. Through the simultaneous modeling of the continuum lag-spectrum and the broadband SED of Fairall 9, we derived a Hubble constant of $H_{0}=72.4_{-3.7}^{+3.4} \, \rm km \, s^{-1} \, Mpc^{-1}$. Achieving a measurement precision of approximately 5% from a single source demonstrates the considerable potential of this method for independent determinations of the Hubble constant. Our determination of $H_{0}$ is broadly consistent, within the current uncertainties, with both early- and late-Universe measurements. Future applications of the method to larger datasets, particularly those provided by the Vera Rubin Observatory, are expected to reduce the uncertainty to below 1%, thereby establishing this approach as a powerful independent probe of the Hubble tension.

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arXiv2026

Evolution of Quasi-Periodic Eruptions in the post-TDE Accretion Disk Perturbed by an Orbiting Star

Quasi-periodic eruptions (QPEs) are a recently discovered class of highly variable X-ray bursts originating in galactic nuclei. These high-amplitude bursts exhibit periodicity ranging from tens of minutes to several days. QPEs are also characterized by variable peak amplitudes that can vary by a factor of few. While multiple physical models have been proposed to explain QPE light curves, none can fully account for all the observed features. A possible connection between QPEs and tidal disruption events (TDEs) has been suggested, particularly due to the past optical/UV outbursts that can be traced back for several sources, the long-term decay in the continuum luminosity, and the soft, thermal-dominated X-ray spectrum. Our primary goal is to verify whether the long-term decrease in eruption amplitudes detected for some QPE sources is consistent with the accretion disk being formed following a TDE. In this work, we adopt a simplified extreme mass ratio inspiral (EMRI) scenario, where a Solar-type star orbits a supermassive black hole (SMBH) and collides with an accretion disk twice per orbit, generating eruptions. We assume a post-TDE disk that follows a temporal power-law decline in mass accretion ($\propto t^{-p}$, $p>0$). As our aim is to develop a toy-model scenario, we have used purely analytical methods without considering all intervening processes in their full generality. Indications are that (i) the observed long-term decline in QPE amplitudes can be reproduced if the first monitored epoch occurs years to a few decades after the tidal disruption, (ii) stellar mass loss caused by ablation can play an important role in the evolution of QPE amplitudes in systems with heavy main-sequence (MS) stars.

astro-ph.HE↗

arXiv2026

Tidal disruption of a low-mass star in an active galactic nucleus as the origin of the PS16dtm outburst

The event PS16dtm, which occured in the center of the Narrow Line Seyfert 1 (NLS1) galaxy SDSS J015804.75-005221.8 (z = 0.080440), is one of the few candidates for a tidal disruption event in an already-acretting active galactic nucleus (AGN). We aim to shed light on the character of the tidal disruption event in this source since it exhibits unusual peculiarities, such as the double-peak optical/UV light curve and a low blackbody temperature with a lack of X-ray emission. We perform spectral analysis of the source before and during the event. We model the time evolution of the luminosity profile using a numerical code that describes the viscous evolution of the flow. From the combined spectral and timing studies, we interpret the event as the disruption of a $\sim 0.3 M_{\odot}$ main-sequence star, or gradual partial disruption of the low-mass giant star. The star is likely on a circular orbit, embedded in the accretion disc. The discussion of the evolution of the star rather suggests that the orbit is counter-rotating. We observe the system at a sufficiently large viewing angle that the actual disruption process is not directly observed. The disrupted star and inner disc are shielded from the observer by a gaseous envelope. Further observations of the system returning to the previous NLS1 state, particularly in the X-ray band, are needed to confirm the proposed scenario and to put constraints on the return to a regular NLS1 state.

astro-ph.HE↗

arXiv2026

Supernova interactions with aspherical circumstellar material I: calculations of light curves, AB magnitudes, spectra, and polarisation

We present an upgraded detailed numerical calculations of supernova (SN) interactions with significantly aspherical circumstellar matter (CSM), primarily formed as a disc or bipolar lobes. The circumstellar disc can arise as a result of, for example, mass transfer in a binary, while bipolar lobes can be the result of a violent pre-explosive ejection of matter, similar to the iconic cases of luminous blue variable stars (LBVs). We numerically simulate the radiation-hydrodynamic (RHD) behaviour of interaction processes using a 2D cylindrical version of the RHD code CASTRO. We then calculate light curves, spectral patterns, and polarisation profiles, all up to a relatively long time of two years after an SN shock breakout and from different directions, using the multidimensional Monte Carlo radiation transfer (MC-RT) codes SEDONA and SIROCCO. We calculated a total of five models for the two aforementioned configurations of the surrounding CSM, for stratified density levels, comparing the simulated hydrodynamic behaviour and differences in their observable properties. RHD models exhibit similar behaviour to previous adiabatic models, but with a significantly slower expansion velocity. The calculated light curves show a relatively smooth evolution in SN-disc interaction, and declines and brightening in SN-lobes interaction. Comparing models with real events with a presumed similar physical process provides guidance for selecting a more accurate CSM configuration when simulating real situations.

astro-ph.SR↗

arXiv2026

The complex relationships between AGN, bars and bulges

Context. Via scaling relations, it is well-known that active galactic nuclei (AGN) and bulges are linked. This link was thought to be driven by mergers, but recent studies show that secular processes are the dominant mechanism of supermassive black hole growth. One such secular mechanism is gas inflow driven by large-scale bars. Since bulges can also grow via these bars, there is likely some common process between these three features. Aims. We investigate whether the observed correlation between AGN and bars is real or arises as a result of correlations between bars and bulges. Methods. Using a catalogue of AGN identifications and galaxy morphologies in the DESI Legacy Survey at $z\leq0.1$, we control for mass and colour and investigate the AGN fraction variation with bulge prominence and bar strength. Results. We first show that the variation in AGN fraction between strongly barred, weakly barred and unbarred galaxies does not qualitatively change if we additionally control for bulge prominence. Second, we find that in fixed bins of bulge prominence, the AGN fraction increases with increasing bar strength. In subsamples split by bar strength, the AGN fraction increases with bulge prominence, indicating that AGN presence correlates with both bar strength and bulge prominence simultaneously.

astro-ph.GA↗

arXiv2025

HALO I: Photometric continuum reverberation mapping of Fairall 9

We investigate the origin of inter-band continuum time delays in active galactic nuclei (AGNs) to study the structure and properties of their accretion disks. We aim to measure the inter-band continuum time delays through photometric monitoring of Seyfert galaxy Fairall 9 to construct the lag-spectrum. Additionally, we explain the observed features in the Fairall 9 lag-spectrum and discuss the potential drivers behind them, based on our newly collected data from the Obserwatorium Cerro Murphy (OCM) telescope. We initiated a long-term, continuous AGN photometric monitoring program in 2024, titled 'Hubble constant constraints through AGN Light curve Observations' (HALO) using intermediate and broad band filters. Here, we present the first results from HALO, focusing on photometric light curves and continuum time-delay measurements for Fairall 9. To complement these observations and extend the wavelength coverage of the lag-spectrum, we also reanalyzed archival Swift light curves and spectroscopic data available in the literature. Using HALO and Swift light curves, we measured inter-band continuum delays to construct the lag-spectrum of Fairall 9. Excess lags appear in the $u$ and $U$ bands (Balmer continuum contamination) and in the $I$ band (Paschen jump/dust emission from the torus). Overall, the lag-spectrum deviates significantly from standard disk model predictions. We find that inter-band delays deviate from the power-law, $τ_λ \propto λ^β$ due to BLR scattering, reprocessing, and dust contributions at longer wavelengths. Power-law fits are therefore not well suited for characterizing the nature of the time delays.

astro-ph.GA↗

arXiv2025

Standardizing reverberation-mapped H$β$ active galactic nuclei using time-averaged radius$-$luminosity relations with 5100\,Å\,, broad H$β$, and narrow \OIII\ luminosities

Active galactic nuclei (AGN) have been studied as alternate probes in cosmology due to their large and stable luminosities and broad redshift range. Previously it was shown that higher-redshift AGN that were reverberation-mapped (RM) using broad Mg\,\textsc{ii} and C\,\textsc{iv} lines are standardizable and yield weak cosmological constraints that are consistent with those from better-established probes. In contrast, lower-redshift AGN that were reverberation-mapped using the broad H$β$ line exhibited tensions with the standard cosmological model, in particular they preferred currently decelerating cosmological expansion. Here we study the standardizability of a homogeneous RM H$β$ sample of $\sim 100$ AGN (over redshifts $0.00308 \leq z \leq 0.8429$), whose H$β$ time delays and three luminosity tracers (at 5100\,Å\,, broad H$β$, and narrow [O\,\textsc{iii}]) are averaged over several epochs. We find that this averaged sample is standardizable using three $R-L$ relations. While for luminosities corresponding to 5100\,Å\, and the broad H$β$ line the cosmological constraints prefer currently decelerating cosmological expansion, the cosmological parameters for the narrow [O\,\textsc{iii}] luminosity are more consistent with those from better-established probes and they are in agreement with currently accelerating cosmological expansion. This demonstrates for the first time that narrow-line region [O\,\textsc{iii}] can be utilized for AGN standardization and cosmological constraints. Selecting proper photoionizing flux proxies for the broad-line region is thus crucial in studies of RM AGN standardizability.

astro-ph.GA↗

arXiv2025

Compact stellar systems hosting an intermediate mass black hole: magnetohydrodynamic study of inflow-outflow dynamics

Intermediate-mass black holes (IMBHs) are a missing link in black hole demographics, with only tentative observational evidence to date. Dense stellar clusters such as IRS 13E near the Galactic Center are promising IMBH hosts, where accretion is likely driven by winds from nearby Wolf-Rayet (WR) stars. Yet, the dynamics of such wind-fed systems remain largely unexplored. We investigate how high-velocity stellar winds, magnetic fields, and metallicity-dependent radiative cooling influence gas dynamics and black hole accretion in compact WR clusters. Using three-dimensional (magneto)hydrodynamic simulations, we model each WR star as a source of mass, momentum, energy, and magnetic flux, and include a cooling function that depends on chemical abundance. We compare isotropic versus disk-like stellar distributions to explore the impact of cluster geometry. Across all models, we find that the accretion rate onto the IMBH is suppressed by up to five orders of magnitude relative to the total stellar mass-loss rate. Turbulent, shock-heated outflows driven by wind-wind collisions dominate the flow, expelling most injected gas. While enhanced cooling in high-metallicity runs promotes the formation of dense clumps, these structures are typically unable to reach the black hole. The system's integrated X-ray luminosity is dominated by colliding WR winds, masking the IMBH's radiative signature. Accretion occurs in short-lived, quasi-periodic episodes triggered by close stellar passages, but even these flares remain difficult to detect against the luminous wind background. Our results naturally explain the low detectability of IMBHs in compact WR clusters and provide theoretical predictions to guide future X-ray and infrared observational strategies.

astro-ph.HE↗

arXiv2025

High-cadence observations of galactic nuclei by the future two-band UV-photometry mission QUVIK

The Quick Ultra-VIolet Kilonova surveyor (QUVIK), a two-band UV space telescope approved for funding as a Czech national science and technology mission, will focus on detecting early UV light of kilonovae (Werner et al., 2024). In addition, it will study the UV emission of stars and stellar systems (Krtička et al., 2024) as well as the intense and variable emission of active galactic nuclei (AGN) or galactic nuclei activated by tidal disruption events (Zajaček et al., 2024). In this contribution, we describe the role of this small ($\sim 30$-cm diameter) UV telescope for studying bright, nearby AGN. With its NUV and FUV bands, the telescope will perform high-cadence ($\sim 0.1$-$1$ day) two-band photometric monitoring of nearby AGN ($z<1$), which will allow us to probe accretion disk sizes/temperature profiles via photometric reverberation mapping. Thanks to its versatility, QUVIK will be able to perform a moderately fast repointing ($<20$ min) to target candidates for tidal disruption events (TDEs). Early detection of the UV emission following a TDE optical flare, in combination with the subsequent two-band UV monitoring performed simultaneously with other observatories, will enable us to infer the time delay (or its lack of) between the optical, UV, and X-ray emission. In combination with theoretical models, it will be possible to shed more light on the origin of the UV/optical emission of TDEs. Furthermore, the two-band monitoring of nuclear transients will be beneficial in distinguishing between TDEs (nearly constant blue colour) and supernovae (progressive reddening) in the era of intensive wide-field surveys.

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arXiv2025

Standardizing a larger, higher-quality, homogeneous sample of reverberation-mapped H$β$ active galactic nuclei using the broad-line region radius-luminosity relation

We present a high-quality, homogeneous sample of 157 H$β$ reverberation-mapped active galactic nuclei (RM AGNs) spanning redshifts $0.00308 \leq z \leq 0.8429$, which is approximately 3.8 times larger than the previously available high-quality homogeneous sample. Using the broad-line region radius$-$luminosity relation ($R-L$), which involves the broad H$β$ line time delay and the monochromatic luminosity at 5100\,Å\,, we show that the sample is standardizable by using six spatially flat and nonflat cosmological models. The inferred cosmological model parameters are consistent within 2$σ$ uncertainties with those from better established baryon acoustic oscillation and Hubble parameter measurements, with the exception of two nonflat models that are ruled out by other data. The $R-L$ relation slope is found to be flatter ($γ=0.428 \pm 0.025$ in the flat $Λ$CDM model) than the slope expected from a simple photoionization model as well as the slope found previously for the smaller homogeneous sample. In addition, we find a mild dependence of H$β$ $R-L$ relation parameters as well as its intrinsic scatter on the Eddington ratio by comparing the $R-L$ relations for low- and high-accreting equal-sized subsamples. A future analysis of a larger homogeneous sample containing a broader range of luminosities and Eddington ratios is necessary to confirm the standardizability of H$β$ AGNs.

astro-ph.GA↗

arXiv2025

Red giant - jet collisions in galactic nuclei I: 3D hydrodynamical model of a few stellar orbits

Several models have been proposed to explain missing red giants (RGs) near the Galactic centre. Recently, a scenario has been suggested that predicts, among other processes, a long-term ablation of the surface layers of RGs during their repetitive passages through the Galactic jet (Zajaček et al., 2020). In this study, we perform detailed three-dimensional numerical modelling of this phenomenon. We calculate the ablation rate of the surface layers of a RG orbiting the supermassive black hole (SMBH) as it passes through the nuclear jet. In particular, we model the jet-star interaction for approximately 10 passages for the closer orbital distance of $10^{-3}\,\text{pc}$ and 2 passages for $10^{-2}\,\text{pc}$. We find that the mass loss due to ablation by the jet behaves with time as $ΔM_{\star}\propto \sqrt{t}$ and the total ablated mass during a single active galactic nucleus (AGN) phase ($10^5$ years) is $\sim 10^{-4}\,M_{\odot}$. We arrive at similar rates of the stellar ablation for the relatively smaller jet luminosity $10^{42}\,\text{erg}\,\text{s}^{-1}$ as in the previous analytical calculations. For larger jet luminosities of $10^{44}$ and $10^{48}\,\text{erg}\,\text{s}^{-1}$, the ablation rates inferred from $\sim 10$ interactions as well as extrapolated power-law fits are significantly lower than analytical values. Overall, the mass ablation rate per interaction and the extrapolated cumulative mass loss during the jet activity are comparable to the stellar-wind mass loss. For the smallest orbital distance of $10^{-3}\,\text{pc}$, we also track the thermal behaviour of the stellar surface layer, whose temperature appears to grow rapidly during the first 10 passages from $\sim 3600\,{\rm K}$ (spectral type M) to $\sim 8500\,{\rm K}$ (spectral type A). RG-jet interactions can thus lead to observable changes in the nuclear stellar population during the jet existence.

astro-ph.HE↗

arXiv2025

Standardizing reverberation-mapped H$α$ and H$β$ active galactic nuclei using radius--luminosity relations involving monochromatic and broad H$α$ luminosities

We test the standardizability of a homogeneous sample of 41 lower-redshift ($0.00415\leq z \leq 0.474$) active galactic nuclei (AGNs) reverberation-mapped (RM) using the broad H$α$ and H$β$ emission lines. We find that these sources can be standardized using four radius$-$luminosity ($R-L$) relations incorporating H$α$ and H$β$ time delays and monochromatic and broad H$α$ luminosities. Although the $R-L$ relation parameters are well constrained and independent of the six cosmological models considered, the resulting cosmological constraints are weak. The measured $R-L$ relations exhibit slightly steeper slopes than predicted by a simple photoionization model and steeper than those from previous higher-redshift H$β$ analyses based on larger datasets. These differences likely reflect the absence of high-accreting sources in our smaller, lower-redshift sample, which primarily comprises lower-accreting AGNs. The inferred cosmological parameters are consistent within 2$σ$ (or better) with those from better-established cosmological probes. This contrasts with our earlier findings using a larger, heterogeneous sample of 118 H$β$ AGNs, which yielded cosmological constraints differing by $\gtrsim 2σ$ from better-established cosmological probes. Our analysis demonstrates that sample homogeneity$-$specifically, the use of a consistent time-lag determination method$-$is crucial for developing RM AGNs as a cosmological probe.

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