2.1. Reverberation mapping

The core of AGN reverberation mapping is based on arguments of the virialised orbit of the broad line region about the black hole, with the SMBH mass being correlated with the estimated radius of the BLR, $R_\mathrm{BLR}$ , and its velocity dispersion $\langle{\sigma_v^2}\rangle$ as in equation (1). The complexities of the BLR’s unresolved geometry are captured in the broad uncertainties of the dimensionless ‘virial factor’, f:Footnote ^a

(1) \begin{equation} M_\textrm{BH} = f\frac{ R_\textrm{BLR} \langle{\sigma_v^2}\rangle }{G}. \end{equation}

Though conceptually simple, each term in the numerator of equation (1) is non-trivial to measure for any given source. One hurdle is that the kinematics at play are dependent on the shape and dynamics of the BLR, the nature of which is as yet unresolved (Pancoast et al. Reference Pancoast, Brewer and Treu2014a,b). This uncertainty is quantified by calibrating the population average virial factor, $\langle{f}\rangle$ from population analysis of nearby AGN, the masses of which are estimated through use of the $M-\sigma$ relation (Woo et al. Reference Woo, Yoon, Park, Park and Kim2015; Grier et al. Reference Grier2013a), or more recently through dynamical modelling via velocity resolved reverberation mapping (Villafaña et al. Reference Villafaña2023; Shen et al. Reference Shen2024). The f is only loosely constrained for any individual source, as there is an inherent scatter of $\approx0.3$ – $0.4$ dex about $\langle {f}\rangle$ , and this dominates the uncertainty in BLR RM derived masses (Woo et al. Reference Woo, Yoon, Park, Park and Kim2015; Shen et al. Reference Shen2023).

Measurements of $\langle{\sigma_v^2} \rangle$ are made from the spectroscopic broadening of the reverberating emission lines, either by their full-width half-maximum (FWHM), or their dispersion (i.e. the second moment of the line profile about its peak), with different virial factors being tuned for each (see Section 3.3 for examples of RM papers using either technique).

High redshift AGN are beyond the distances at which we have been able to angularly resolve $R_{\mathrm{BLR}}$ , and so BLR RM instead uses the timescale at which light propagates through the AGN geometry to measure this size by proxy (see Figure 1). Variations in broad band photometric luminosity are dominated by the light emanating directly from the accretion disk ‘engine’, with these variations being echoed in the driven response in the emission lines of the BLR after some delay, $\Delta t$ , typically of order days to months. The BLR spectrum exhibits atomic recombination lines that can be easily distinguished from the broad AGN continuum emission, allowing us to identify this echo through spectroscopic observations even where our photometric observations are dominated by light from the accretion disk. With sufficiently tight cadence of photometric and spectroscopic observations, the delay (or lag) between driving continuum and driven emission line response, $\Delta t$ can be observed, allowing temporal resolution of observations to substitute for the lack of spatial resolution:

(2) \begin{equation} R_{\mathrm{BLR}} = c \Delta t .\end{equation}

To perform reverberation mapping, a source must provide a clearly identifiable reverberating emission line from the BLR. At cosmological scales, the emission from different emission lines redshift in and out of typical spectroscopic wavelength ranges, and so different atomic emission lines are visible at near, moderate, and high redshift ranges.

For the nearest AGN ( $z\lesssim0.6$ ), optical RM is performed with H $\alpha$ at $6\,562\,\unicode{x00C5}$ and H $\beta$ at $4\,861 \unicode{x00C5}$ (e.g. Bentz et al. Reference Bentz2014). Beyond measuring the mean lag, some studies have even gone as far as using high signal to noise (SNR) measurements of hydrogen recombination lines to constrain the BLR structure and kinematics in further detail. In this ‘velocity resolved reverberation mapping’ (e.g. Denney et al. Reference Denney2009; Grier et al. Reference Grier2013b), the reverberating line is treated as being affected by varying degrees of Doppler broadening from different velocities at different radii within the BLR, resulting in different parts of the line profile reverberating at different times.

At higher redshifts ( $0.6\lesssim z\lesssim1.8$ ), optical RM is performed with the MgII line at $2\,798\,\unicode{x00C5}$ (e.g. Metzroth et al. Reference Metzroth, Onken and Peterson2006). optical RM masses derived from MgII are complicated by iron contamination from the adjacent FeII emission lines, which can distort estimates of the line width unless accounted for (discussed in detail in Section 4).

At high redshift ( $z\gtrsim 1.8$ ), optical RM is performed with the CIV line at $1\,549\,\unicode{x00C5}$ . CIV lag measurements have also been made by observing this line in the observer-frame UV range for low redshift and low luminosity sources (e.g. Rosa et al. Reference Rosa2015; Peterson et al. Reference Peterson2005; Metzroth et al. Reference Metzroth, Onken and Peterson2006).

The $R-L$ relationship (Kaspi et al. Reference Kaspi2000), an empirically observed correlation between the luminosity of an AGN and its physical scale as recovered from RM, is parameterised as a power-law expressed as:

(3) \begin{equation}a ,\end{equation}

where $\Delta t$ is the rest-frame lag and $\lambda L_{\lambda}$ is the monochromatic luminosity of the AGN at a line-specific wavelength: typically $5\,100\,\unicode{x00C5}$ for H $\beta$ , $3\,000\,\unicode{x00C5}$ for MgII, and $1\,350\,\unicode{x00C5}$ for CIV. Though the $R-L$ relationship for H $\beta$ is well constrained and found to have small scatter ( $\approx 0.1$ – $0.2$ dex; Bentz et al. Reference Bentz2013), recent surveys have suggested this scatter may be higher (e.g. Du et al. Reference Du2016, Reference Du2018). The historical $R-L$ relations for MgII and CIV are found to have higher scatter than H $\beta$ (Zajaček et al. Reference Zajaček2020; Kaspi et al. Reference Kaspi2007).

The strong dependence of AGN lags on luminosity has given rise to the technique of stacking (Fine et al. Reference Fine2012, Reference Fine2013; Li et al. Reference Li2017), in which sources of poor signal to noise but similar luminosity have their lag measurements combined. Through stacking, lag measurements can be made up in aggregate over many sources that would otherwise give poorly constrained results, such as done for the OzDES data in (Malik et al. Reference Malik2024).

There have been attempts to explain the diversity of lags through more expressive $R-L$ relationships, reducing the intrinsic scatter by adding predictive variables or dividing the AGN population into sub-groups based on the spectral properties (e.g. Mejía-Restrepo et al. Reference Mejía-Restrepo, Trakhtenbrot, Lira and Netzer2018, Reference Mejía-Restrepo, Trakhtenbrot, Lira, Netzer and Capellupo2016). There is, for example, some evidence that lags are lower for highly accreting sources, above some critical threshold (Du et al. Reference Du2016, Reference Du2018). This is described by the dimensionless accretion rate, $\dot{M}$ , given by equation (4), based on the thin-disk model of Shakura & Sunyaev (Reference Shakura and Sunyaev1973).

(4) \begin{equation} \dot{M} = 20.1 \left( \frac{L_{5\,100\,\unicode{x00C5}} / 10^{44} \mathrm{erg/s}}{\cos(i)} \right)^\frac{3}{2} \left( \frac{M}{10^7\,\mathrm{M_{\odot}}}\right)^{-2}. \end{equation}

Here, $\cos(i)$ is the cosine of the inclination of the AGN, taken to be $0.75$ as an average value for the quasar inclinations in which the BLR and accretion disk are both visible (Seyfert – 1 type AGN), while $L_{5\,100\,\unicode{x00C5}}$ is the bolometric luminosity at $5\,100\,\unicode{x00C5}$ in units of $10^{44}\,\mathrm{erg/s}$ and M is the RM-derived SMBH mass in units of $10^7$ solar masses. The Super-Eddington Accreting Mass Black Hole (SEAMBH) collaboration use $\dot{M}\gt3$ as a benchmark for the separation between low accretion ‘sub-Eddington’ sources and the highly accreting ‘super-Eddington’ sources (Du et al. Reference Du2015, Reference Du2016).

2.2. The aliasing problem

Though the uncertainties in $M_\mathrm{BH}$ within a single source are dominated by the $0.3$ – $0.4 $ dex population variability in f, lag-recovery presents significant issues with bias and contamination of mass estimates. Cosmological time-dilation means that distant AGN have longer timescales of variability (Lewis & Brewer Reference Lewis and Brewer2023), meaning that RM requires such sources to be observed over a baseline of multiple years to adequately capture variations in their light curves. For the ground-based observations of most distant RM-campaigns, such multi-year observations are necessarily impacted by a seasonal windowing function that imposes $\sim$ 6-month gaps in observations. These seasonal gaps give rise to the problem of ‘aliasing’ (see Figure 2), which can yield spurious lag recoveries at $n+1/2$ yearly gaps (180 d, 420 d, etc.; Penton et al. Reference Penton2021; Malik et al. Reference Malik2022).

Figure 2.

Qualitative demonstration of the source of the aliasing problem for mock RM light curves generated with a true lag of $360\,\mathrm{d}$ , with shaded bands to demonstrate the overlap/gaps in the observations. When observational seasons of our windowing function are of similar or smaller size to the gaps, lags that give no overlap cannot be easily identified as bad fits. This creates local optima in many fitting procedures, inducing ‘aliasing peaks’ in lag recovery distributions every $\approx \! 180\,\mathrm{d}$ which can obscure the true lag.

The dangers posed by aliasing are twofold; firstly, it degrades our ability to detect true lags that fall within seasonal gaps, and more dangerously it can lead to false positives arising within these same gaps (Malik et al. Reference Malik2022). A common approach is to adopt weighting methods in combination with stringent selection criteria to remove sources with spurious lags. Such approaches reduce contamination from false positives, but at the cost of a drastically reduced sample, with typical acceptance ratios being of order $\approx10\%$ in the OzDES sample. OzDES selections are based on agreement between competing methods (e.g. JAVELIN (Zu et al. Reference Zu, Kochanek and Peterson2010) and PyCCF (Sun et al. Reference Sun, Grier and Peterson2018); discussed in detail in Section 4), the emergence of a single well constrained lag, and arguments about the physical reasonability of the overall lag posterior distribution (e.g. whether negative lags are properly excluded). More involved approaches rely on characterising the false positive rate (FPR) through simulation of sources (e.g. Yu et al. Reference Yu2023; Penton et al. Reference Penton2026).Footnote ^b

3.1. OzDES

The OzDES project (Yuan et al. Reference Yuan2015; Childress et al. Reference Childress2017; Lidman et al. Reference Lidman2020) was a parallel project of the Dark Energy Survey (DES), providing spectroscopic follow-up of sources imaged photometrically by DES. DES itself performed a 6-year imaging survey on the CTIO Blanco 4-metre Telescope in Chile. In addition to a wide-field survey, DES repeatedly imaged ten fields to search for supernovae (field coordinates given in Table 2 of Smith et al. Reference Smith2020). These supernova fields had weekly imaging (during each 6 month season) for 5 years from 2013 to 2018 and a sixth year with lower cadence in 2019 (for details see Lidman et al. Reference Lidman2020), using DECam in g, r, i and z bands (Flaugher et al. Reference Flaugher2015). Primarily, these fields were used to obtain supernova light curves, but they also provide an extensive dataset for monitoring the variability of other sources, such as AGN. For ease of access we provide the photometry relevant for AGN reverberation mapping with this paper [link to be provided upon acceptance]. The raw images and catalogues from the whole DES survey are available at https://des.ncsa.illinois.edu/releases/dr2.

To complement the DES photometry, OzDES obtained monthly spectroscopy using the AAOmega spectrograph fed by the Two Degree Field (2dF) fibre positioner (Lewis et al. Reference Lewis2002) on the Anglo-Australian Telescope (AAT; for details see Lidman et al. Reference Lidman2020). The 2dF instrument has $\sim$ 400 optical fibres that can be positioned on targets within a 2-degree diameter field of view, and about a quarter of those were placed on AGN in each exposure. Using a spectral resolution of $R=1\,400$ to $1\,700$ and a wavelength range of $3\,700\,\unicode{x00C5}$ to $8\,800\,\unicode{x00C5}$ OzDES monitored a total of 735 AGN over six years, gathering between 18 and 25 epochs on each AGN. OzDES typically observed for five months between August and January, perforce leaving large seasonal gaps in the time series data while the fields passed near to the sun. The spectra are publicly available from https://docs.datacentral.org.au/ozdes/overview/dr2/.

3.2. Light curve & spectral calibration

For reliable reverberation mapping, a multi-stage calibration procedure of the photometric and spectroscopic measurement is necessary. In OzDES, we make use of the calibration procedures outlined in Hoormann et al. (Reference Hoormann2019), with the exception of line flux and width measurements for the MgII line for which we use the procedure of Yu et al. (Reference Yu2023). In this section we briefly review this calibration pipeline, but direct the reader to the original papers for more detail.

DECam measurements provide fluxes at an irregular but approximately weekly cadence in the g, r, i, z filters, though in OzDES RM we make use of only those in the g, r and i bands. In the Hoormann et al. (Reference Hoormann2019) pipeline, spectroscopic epochs are discarded if they are of low quality (require quality flag 4 or greater, see Yuan et al. Reference Yuan2015) or lack a co-temporal calibrating magnitude. Epochs are then discarded as outliers if they differ by more than $0.2$ magnitudes from the source’s mean in the same year and same band pass, with the remaining measurements being averaged if they occur on the same night. An additional filter-dependent calibration uncertainty is also added to the variance as per Burke et al. (Reference Burke2017).

Each spectral epoch is calibrated, both converting from photon counts to flux values and correcting for systematic distortions of the spectrum, by comparing the spectra to the g, r and i photometric measurements. Synthetic photometric measurements are made by integrating over the product of the spectrum and band-pass, and a ratio taken between synthetic and measured brightness. Describing this factor as a function of wavelength, a quadratic is fit across the three measured ratios, acting as a measure of the relative throughput. The entire spectrum was warped to remove this bias (e.g. Figure 3). After warping, spectra are co-added if they share the same night.

Figure 3.

Demonstration of the spectral warping procedure from Hoormann et al. (Reference Hoormann2019). The top panel shows a smoothed version of the spectrum of AGN DES J022828.19-040044.30. The second panel shows the gri filter transmission functions, while the third shows the wavelength-dependent transmission coefficients, found by integrating the spectrum with these filters, and the quadratic fit between them, each in units of $10^{-16}\,\text{erg}\; \textrm{s}^{-1} \text{cm}^{-2} \unicode{x00C5}^{-1} \text{counts}^{-1}$ . The bottom panel shows the spectrum after correcting by these scale factors to produce a fully calibrated spectrum in units of $10^{-16} \text{erg}\; \textrm{s}^{-1} \text{cm}^{-2} \unicode{x00C5}^{-1}$ .

Once the spectrum is calibrated, line fluxes and widths are calculated by isolating a window of rest-frame wavelengths in which the reverberating line occurs. By linearly interpolating between the flux at the window boundaries, the local continuum is subtracted and the total flux, line dispersion and line full width half maximum estimated. Uncertainties in this value are found by Monte Carlo variation of the window boundaries and the flux measurements within bounds of uncertainty. The flux calibration uncertainties from this procedure is typically on the order of 5–10% across most of the visible wavelength range (See Section 2.2 and Figure 2 of Hoormann et al. Reference Hoormann2019).

Following the approach of Du et al. (Reference Du2016), we apply a correction to the $5\,100\,\unicode{x00C5}$ monochromatic luminosities of all H $\beta$ sources by way of the empirical scaling relationship provided by Equation 1 in Shen et al. (Reference Shen2011). A more thorough approach would be to decompose the spectra into host galaxy starlight and AGN activity, but is not crucial in this paper as the OzDES H $\beta$ sample’s small size means that it offers only weak constraining power to the $R-L$ parameters (see Section 6.1.1).

For the MgII sources we add an additional step to account for the iron emission lines that flank and coincide with the MgII emission line, which, if not accounted for, can contaminate measurements of line width. In their analysis of the OzDES MgII lags, Yu et al. (Reference Yu2023) account for this by fitting iron spectral templates (Tsuzuki et al. Reference Tsuzuki2006; Salviander et al. Reference Salviander, Shields, Gebhardt and Bonning2007; Vestergaard & Wilkes Reference Vestergaard and Wilkes2001) in a Bayesian fashion using the python Monte Carlo Markov Chain (MCMC) package emcee. Spectral fits for the emission lines were visually inspected and poorly fitted epochs were removed before lag-fitting with JAVELIN.

Following the line width pipeline of Hoormann et al. (Reference Hoormann2019), the measurements presented here are made using the spectrum averaged over all epochs and use line dispersion as a measure of the line width. All fluxes are converted to luminosities assuming a $\Lambda$ CDM cosmology with parameters $H_0=70$ km s $^{-1}$ Mpc $^{-1}$ , $\Omega_m = 0.3$ and $\Omega_\Lambda = 0.7$ .

3.3. External datasets

In addition to presenting our recovered lags, we make use of a number of sources from prior RM works, both for comparison and to improve the constraining power of the $R-L$ relationship. A summary of these sources are listed in Table 1 The bulk of these lags are those recovered from the Sloan Digital Sky Survey (Shen et al. Reference Shen2023), a contemporary to OzDES at a similar ‘industrial scale’ and redshift range. In Section 4.2.4, we describe how the SDSS lags use dissimilar selection criteria to OzDES, including the use of different software as their primary detection method. For the sake of homogeneity in this paper, we adopt the JAVELIN lags for SDSS and apply an analogue of our own selection criteria.

Table 1.

A summary of the number of sources and redshift ranges for the OzDES, SDSS and other literature data used for constraining $R-L$ relationships in this work. Note that the number of sources here is the number used, and not the total number of published lags from that reference.

Table 1 Long description

A table with three rows and three columns. The table is divided into three sections: Hβ Sources, MgII Sources, and CIV Sources. Each section lists different sources, the number of active galactic nuclei (AGN), and the redshift range. The sources include OzDES, SDSS, Bentz collection, SEAMBH, LAMP, Misc, Zajacek, and Kaspi. For Hβ Sources: OzDES has 8 AGN with a redshift range of 0.127-0.332, SDSS has 26 AGN with a redshift range of 0.289-1.003, Bentz collection has 48 AGN with a redshift range of 0.002-0.292, SEAMBH has 40 AGN with a redshift range of 0.017-0.400, LAMP has 16 AGN with a redshift range of 0.029-0.078, and Misc has 6 AGN with a redshift range of 0.017-0.327. For MgII Sources: OzDES has 25 AGN with a redshift range of 0.840-1.860, SDSS has 25 AGN with a redshift range of 0.360-2.149, and Zajacek has 6 AGN with a redshift range of 0.003-1.890. For CIV Sources: OzDES has 29 AGN with a redshift range of 1.922-3.451, SDSS has 15 AGN with a redshift range of 1.675-2.453, and Kaspi has 17 AGN with a redshift range of 0.001-3.368.

Table 2.

Our final $R-L$ Relationships for H $\beta$ , MgII and CIV using a combination of multiple datasets for each. Monochromatic luminosities are measured in the rest-frame in units of erg/s, and resulting radii are in units of $\log_{10}$ light-days.

Table 2 Long description

A table with five rows and six columns comparing the R-L relationships for different spectral lines. The columns are labeled Line, Luminosity unit, Slope (α), Offset (β), Scatter (σ), and Slope-offset fit correlation (φ). The rows present data for Hβ, MgII, and CIV lines. Each row lists the luminosity unit, slope, offset, scatter, and fit correlation for the respective spectral line. The values are presented with their respective uncertainties.

For low redshift lags associated with the H $\beta$ line, we also supplement our data with the external lags used in the analysis of Malik et al. (Reference Malik2023). These include lags from: the Lick AGN Monitoring Project (LAMP) (U et al. Reference U2022), a low redshift ( $z\lt0.08$ ) study of statistically diverse AGN; the SEAMBH survey, which targets highly accreting high luminosity sources known to produce lower reverberation lags (Du et al. Reference Du2016, Reference Du2018; Hu et al. Reference Hu2021); and a collection of low redshift sources collated by Bentz et al. (Reference Bentz2013), coupled with similar sources from Fausnaugh et al. (Reference Fausnaugh2017), Bentz et al. (Reference Bentz, Peterson, Netzer, Pogge and Vestergaard2009, Reference Bentz2013, Reference Bentz2014, Reference Bentz2016a,b) and (Reference Bentz, Onken, Street and Valluri2023). Also included are a number of additional sources from Zhang et al. (Reference Zhang2019), Li et al. (Reference Li2021), Lu et al. (Reference Lu2016), Pei et al. (Reference Pei2014), Rakshit et al. (Reference Rakshit2019). Sources prior to 2019 and not associated with the SDSS collaboration are drawn from a convenient collation by Martínez-Aldama et al. (Reference Martínez-Aldama2019).

The combination of a larger sample size and good signal-to-noise yields a larger set of lags associated with the MgII line in the OzDES intermediate redshift sample than are recovered in the low redshift H $\beta$ sample (Yu et al. Reference Yu2021, Reference Yu2023). In total, 25 OzDES MgII lags are available, which we supplement and contrast with the set of SDSS MgII lags, a smaller set of lags from Zajaček et al. (Reference Zajaček2021) and a wider collection of sources presented by Zajaček et al. (Reference Zajaček2020). This collection is comprised of lags collated from Lira et al. (Reference Lira2018), Metzroth et al. (Reference Metzroth, Onken and Peterson2006), Czerny et al. (Reference Czerny2019) and earlier SDSS lags from Shen et al. (Reference Shen2016, Reference Shen2019). We use the entire set of lags as listed by Zajaček et al. (Reference Zajaček2020) when analysing the collection by itself, but when using these data in combination with other surveys we defer to the SDSS measurements when the same source has been reported twice.

For the CIV lags, we compare the OzDES results presented in Penton et al. (Reference Penton2026) with the latest SDSS CIV lags from Shen et al. (Reference Shen2023) and a diverse collection of earlier sources, which we label as as ‘The Kaspi Collection’. It is worth noting that the Kaspi Collection includes SDSS lags from Grier et al. (Reference Grier2017), Grier et al. (Reference Grier2019), Shen et al. (Reference Shen2019), as well as earlier OzDES lags from Hoormann et al. (Reference Hoormann2019). As we already analyse more recent and complete SDSS results independently, we do not include these sources in our analysis of the Kaspi Collection.

The sources of the Kaspi Collection can be broadly separated into two groups: high redshift, high luminosity AGN with the CIV line measured in the optical band by Lira et al. (Reference Lira2018) and Kaspi et al. (Reference Kaspi2021), and a number of nearby low luminosity sources from Peterson et al. (Reference Peterson2005), Metzroth et al. (Reference Metzroth, Onken and Peterson2006) and Rosa et al. (Reference Rosa2015) in which the CIV line is observed in the observer-frame UV. We separate these two groups and label them as ‘Kaspi High-Z’ and ‘Kaspi Low-Z’. As we discuss further in Section 6.1.3, these two sub-samples have different statistical properties, and treating them as a single group runs afoul of tensions when constraining the $R-L$ relationship.

When considering the velocity in equation (1), there are two decisions about how this is interpreted from the quasar spectra: whether to use the dispersion/variance of the line profile (disp) or its full-width half maximum (FWHM), and whether to make these measurements from the spectrum as time-averaged over all epochs (mean-spec) or by its root-mean-squared over all epochs (RMS-spec). Different examples of prior RM work occupy all four quadrants of this decision tree. Note that the disp/FWHM decision represents a different choice of physical measurement, and so leads to a correspondingly different virial factor, while the mean-spec/rms-spec decision is purely about which gives the clearer measurement.

The dispersion approach tends to be the preferred approach in more modern surveys (e.g. Shen et al. Reference Shen2023; Rakshit et al. Reference Rakshit2019; Zhang et al. Reference Zhang2019), while both mean-spec (e.g. Li et al. Reference Li2021) and RMS-spec (e.g. Lu et al. Reference Lu2019) approaches are still in active use. The RMS spectrum is preferred for high SNR sources to isolate the time-varying component of the signals (e.g. Bentz et al. Reference Bentz2014; Pei et al. Reference Pei2014), but many surveys will measure line widths in all, or at least multiple, ways (e.g. Shen et al. Reference Shen2023; Fausnaugh et al. Reference Fausnaugh2017).

In this work, as with past OzDES RM papers, we use the dispersion-based definition of line width, as measured on the mean spectrum, as our measure of velocity rather than the FWHM, and default to this measurement for literature measurements where available in the source paper. In making the decision between mean-spec and RMS-spec, we defer to the preferred method of the source paper, e.g. mean spectrum for sources from Bentz et al. (Reference Bentz2013), RMS spectra for sources from Shen et al. (Reference Shen2023) etc. For source-papers in which only the FWHM line widths are provided, we follow the lead of Kaspi et al. (Reference Kaspi2021) and use $\textrm{disp}=\textrm{FWHM}/2.35$ based on an assumption of a Gaussian line profile.

4.1. Lag recovery

The central element of reverberation mapping is identifying and constraining the delay between two light curves, a task that is complicated by the stochastic variability in the underlying signal, and a host of obstacles arising from the windowing function. These issues have led to a wide range of competing techniques and software for computing the lag in AGN-RM. These can be broadly sorted into two categories: the first containing non-parametric techniques that assume no underlying statistical behaviour and fit to only the light curves, and the second being those that model the AGN variability as a Gaussian process.

Modelling the AGN light curve as a Gaussian process is physically motivated, as they are known to show consistent patterns in their variability (MacLeod et al. Reference MacLeod2010; Zu et al. Reference Zu, Kochanek, Kozłowski and Udalski2013; Kozłowski Reference Kozłowski2016), but the associated fitting procedures are computationally expensive and fraught with numerical and statistical obstacles (Penton et al. Reference Penton2021; Malik et al. Reference Malik2022; Read et al. Reference Read, Smith, Jarvis and Gürkan2019). As a result, non-parametric methods like PyCCF (Sun et al. Reference Sun, Grier and Peterson2018) and PyROA (Donnan Reference Donnan2021) are still in use as more robust, if less precise, alternatives to full Bayesian modelling. The SDSS team uses PyROA for their ‘primary’ lag recovery method, while OzDES use a combination of the GP-based JAVELIN and the non-parametric PyCCF for validation and source selection cuts. For the interested reader, in Appendix A we provide an introduction to these fitting methods, their statistical basis, and numerical shortfalls.

4.2. Post-recovery quality cuts

The stochastic nature of AGN light curves means that non-physical false positive lags are a possibility even in the presence of arbitrarily good measurements. Coupled with matters of measurement noise, calibration error and, in multi-year surveys such as OzDES, the impacts of aliasing from the seasonal windowing function, the contamination rate from such false positives can outstrip physically meaningful lag recoveries by near to an order of magnitude if unaccounted for.

Bespoke analysis can be used to assess reliability of a handful of sources at a time, but this approach is untenable in industrial-scale studies like OzDES. Instead, it is necessary to use a general, widely applicable criteria for selecting and post-processing the lag recoveries of the sample sources at large to remove spurious results (Penton et al. Reference Penton2021; Malik et al. Reference Malik2022).

The methods for suppressing the impacts of aliasing through quality cuts varies significantly between different studies. In this section we outline the quality cuts used in each of the previous OzDES RM papers as well their underlying motivation, and present also a set of selection criteria to apply to the SDSS RM sample of Shen et al. (Reference Shen2023) aimed at drawing from their data a set of sources with selection effects analogous to those of OzDES.

4.2.2. Quality cuts for MgII lags

The majority of OzDES AGN sources are observed in the redshift range that allow for visibility of the MgII line, with 453 of its 753 target AGN being in the $z\in[0.65, 1.92]$ redshift range where the MgII line is visible in range of OzDES data. These MgII sources exist over a wide redshift range, and the range of observed lags significantly overlaps with the $n+1/2$ year ‘danger-zones’ for aliasing effects. These high redshift AGN lags require strict and principled quality cuts to avoid contamination of the $R-L$ relationship. To this end, Yu et al. (Reference Yu2023) applies two distinct sets of selection criteria, one based on the posterior distribution weighting approach of Grier et al. (Reference Grier2019), and another based on the simulation based false positive rate estimation of Penton et al. (Reference Penton2021).

In the SDSS approach, the marginalised lag posterior distribution $P(\Delta t)$ , as recovered by PyROA, JAVELIN or PyCCF, is attenuated by a weighting function that down-weights lags that correspond to the poorly constrained seasonal gaps. This weighting function is the convolution of two components. The first term, based on the fraction of observations that overlap between the two light curves after shifting by some lag, down-weights ‘off-season’ lags in seasonal gaps where this overlap is small. The second term is the auto-correlation function of the continuum signal (ACF $_\textrm{Cont}$ ), which smooths and widens the ‘on-season’ peaks of the weighting function to account for the continuous nature of the light curve. The entire posterior is then convolved with Gaussian smoothing kernel of width 15 d (equation 5):

(5) \begin{equation} P^\prime(\Delta t) = \left( P(\Delta t) \times \left[ \left(\frac{N(\Delta t)}{N(0)} \right) ^ 2 \circledast \textrm{ACF}_{\mathrm{Cont}}(\Delta t) \right] \right) \circledast \mathcal{N} \left(\frac{\Delta t}{15\textrm{d}} \right) , \end{equation}

where $\circledast$ represents the convolution operator. The square bracketed terms represent the weighting function while the convolved normal distribution represents the final smoothing.

This ad-hoc approach suppresses lags that are in danger of being the result of aliasing, retaining only those that give an extremely high likelihood. Different quality cuts have been used in past analysis based on this weighting, including discarding sources in which the weighting reduces the posterior evidence by too large of a fraction (e.g. Grier et al. Reference Grier2019).

In Yu et al. (Reference Yu2023), the OzDES analysis reports lags from the un-weighted distribution as produced by JAVELIN, but uses the weighted distribution to identify the ‘primary’ peak as being the highest likelihood mode in the weighted-distribution $P^\prime(\Delta t)$ . Sources are retained only if the unweighted distribution has:

1. 1. More than 60% of its evidence (posterior distribution integral) contained within the primary peak,

2. 2. The width of this peak, as measured between the 16th and 84th percentiles, be less than 110 d, and

3. 3. This peak be in agreement with the unweighted PyCCF distribution to within $2 \sigma$ .

Yu et al. (Reference Yu2023) also includes quality cuts modelled after the more rigorous simulation-work of Penton et al. (Reference Penton2021), in which simulations of mock DRW signals are used to characterise false positive rates, and quality cuts tuned to remove suspicious sources. These cuts, designed specifically for unweighted JAVELIN posterior distributions and OzDES-like observations, require:

1. 1. That the standard deviation of the JAVELIN posterior distribution be less than 110 d,

2. 2. That the separation of the JAVELIN median lag & peak likelihood lag be within 110 d of one another, and

3. 3. That the lags from PyCCF and JAVELIN agree to within 110 d.

Of the 453 available MgII sources, 25 pass both sets of quality cuts and produce reliable lags, $\approx\! 5.5 \%$ of the initial sample, with an estimated false-positive rate in this post-cut sample of $\approx \! 4 \%$ (i.e. 1 false positive out of the 25).

Figure 4.

Examples of the lag and scale-corrected light curves (maximum a posteriori estimate) for two OzDES CIV sources: DES J022620.86-045946.48 (top, gold quality recovery) and DES J032703.62-274425.27 (bottom, bronze quality recovery). These are adapted from Penton et al. (Reference Penton2026), the initial paper for these lag recoveries. For a list of the criteria used to classify sources into these grades, see Section 4.2.3.

Figure 4. Long description

Two line graphs depict the relationship between flux and time for two different sources, showing photometry and spectroscopy data. Panel A: The top graph shows the flux in arbitrary units on the vertical axis and the Modified Julian Date (MJD) in days on the horizontal axis. The data points are represented by blue dots for photometry and orange lines for spectroscopy shifted and scaled. The flux values range from 0.5 to 2.5 arbitrary units, and the MJD ranges from 56000 to 58500 days. Panel B: The bottom graph shows the flux in arbitrary units on the vertical axis and the Modified Julian Date (MJD) in days on the horizontal axis. The data points are represented by blue dots for photometry and orange lines for spectroscopy shifted and scaled. The flux values range from 0 to 15 arbitrary units, and the MJD ranges from 56000 to 58500 days.

4.2.4. SDSS sub-sampling criteria

In the final data release for the Sloan Digital Sky Survey’s 7-yr RM campaign, Shen et al. (Reference Shen2023) constrain and select lags in a fundamentally different way to OzDES. Rather than using simulation-based estimates of the false positive rate, SDSS instead allow each source to fit for both positive and negative lags, and then apply the aliasing mitigation weighting strategy outlined in Section 4.2.2. Under the rationale that any lag recovered at $\Delta t\lt0$ must necessarily be a false positive, they use this as an estimate of the population level false positive rate, using this to tune their selection criteria. To this end, they make use of only one quality cut: that the ICCF r value (equation A1) must be $\gt0.4$ as evaluated at the lag recovered by their chosen primary lag recovery method PyROA. This is similar to the selection criteria of the OzDES H $\beta$ analysis of Malik et al. (Reference Malik2023), which requires $r\gt0.6$ at the lag as recovered by JAVELIN.

By contrast to SDSS, OzDES treats this lag search range as a true prior, examining only the physically reasonable positive lag domain and applying stringent cuts to minimise the effect of aliasing on each individual source. This produces a more restricted sample with fundamentally different statistical properties, which can be seen in the tension between the SDSS and OzDES $R-L$ constraints. To make joint fits to the $R-L$ relation using both the SDSS and OzDES datasets, we required a set of lags with relatively homogeneous selection properties. To that end, we derive an alternate set of lags from the published SDSS data of Shen et al. (Reference Shen2023) that differ from their sample in two ways: firstly, in that we use the lags as recovered from JAVELIN instead of their choice of PyROA, and secondly that we apply a number of quality cuts to this alternate sample to bring it more in line with the rationale of the OzDES selections.

Our sub-sampling criteria for the SDSS sources are:

1. 1. That recovered lags must be positive, $\Delta t\gt0$ at $1\sigma$ .

2. 2. That results should be robust across methods, such that JAVELIN, ICCF and PyROA all agree in peak lag to within $2\sigma$ of the JAVELIN uncertainty,

3. 3. That the fit should be statistically significant, as measured by setting $r\gt0.6$ evaluated at the JAVELIN modal lag,

4. 4. That the anti-aliasing down-weighting weighting scheme must not overly distort the lag probability distribution. We require that the posterior evidence rejected (fraction of MCMC chain samples in their terminology) by the SDSS anti-aliasing weighting must be $\lt60 \%$ .

For step 2, in cases where the JAVELIN lag peak is strongly constrained, we replace the JAVELIN confidence interval in this test with 30 d for H $\beta$ or 110 d for CIV and MgII, following a similar reasoning to the cuts of Penton et al. (Reference Penton2021). After applying these cuts, we retain a sub-sample with 26 H $\beta$ lags, 25 MgII lags and 15 CIV lags from the SDSS sample. JAVELIN ’s MCMC sampler is known to exhibit strong aliasing artefacts in seasonal signals (see McDougall et al. Reference McDougall, Pope and Davis2026), meaning that its use by itself can give misleading results. However, JAVELIN ’s use in concert with specifically tuned quality cuts produces a set of reliable lags with a low FPR, as validated by the simulations of Penton et al. (Reference Penton2021). We use SDSS’s JAVELIN lags rather than their PyROA lags, as the OzDES quality cuts are tuned specifically for JAVELIN ’s systematics.

The motivation of this sub-sampling is to sacrifice much of the SDSS sample’s statistical power by cutting many sources, including discarding what is likely a large fraction of true positives, to yield a smaller sample that has OzDES’s high reliability on a per-source basis. This SDSS- JAVELIN subsample is broadly similar to the published SDSS sample for the H $\beta$ and CIV lags, but prefers a markedly for lower scatter in the MgII $R-L$ relationship. For the MgII lags, these selection criteria drastically reduce the scatter of the lags about their best-fit $R-L$ relationship, and marginally decrease its preferred slope (see Figure 7 for a comparison).

Figure 5.

A summary of all OzDES reverberation mapping (circles) and single epoch findings (squares) as well as comparison with literature RM results (plus signs). The top row plots measured and estimated lags (left, estimates from the R-L relationship) and accretion rates (right) against redshift, while the bottom row shows SBMH mass plotted against redshift (left) and bolometric luminosity (right). In each plot, the top panel colours sources by emission line, while the bottom colours by data source. On the mass vs luminosity plot, we also overlay power laws of index $0.5, 1.0$ and $2.0$ as a way to illustrate the slope of the relation.

5.1. Lag measurements

In total, the OzDES reverberation mapping program has produced 62 high quality AGN lags, consisting of 8 H $\beta$ lags (Malik et al. Reference Malik2023), 25 MgII lags (Yu et al. Reference Yu2021, Reference Yu2023) and 29 CIV lags (Penton et al. Reference Penton2026). For each of these sources, we calculate the mass using equation (1) and accretion rate using equation (4), where the $5\,100\,\unicode{x00C5}$ monochromatic luminosity uses bolometric corrections of Runnoe et al. (Reference Runnoe, Brotherton and Shang2012) (Table 2 in their paper) and we use the virial factor of $\log_{10}(f) = 0.62 \pm 0.07$ from Shen et al. (Reference Shen2023), as well as their $0.31$ dex inherent scatter. Rest-frame lags range from $16^{+2.7}_{-2.7} \; \mathrm{d}$ Footnote ^c to $335^{+2.5}_{-16.0} \; \mathrm{d}$ , sitting within the range of lags recovered by other works. We find that the distribution of lags over redshift is reasonably consistent with existing reverberation mapping results (Figure 5, upper left). We note that the distribution of lags changes abruptly as one moves between lines, most notably between the MgII and CIV lines, indicating different lags, and thus radii, for different ions. This is quantified in Section 7. These RM results, as well as the redshifts and luminosities for the relevant sources, are listed in Table B2 for our H $\beta$ sources, Table B2 for MgII sources, and Table B3 for CIV sources.

5.2. Black hole masses from direct RM measurements

In total, we present a compilation of the 62 SMBH masses from OzDES reverberation mapping, ranging from $25.3_{-6.8}^{+8.5}\times 10^6\,\mathrm{M_{\odot}}$ to $4.0_{-1.7}^{+1.8} \times 10^9\,\mathrm{M_{\odot}}$ . The relevant measurements are listed, along with estimates of the corresponding accretion rates, in Tables B1, B2 and B3. Accretion rates are calculated per equation (4), correcting luminosities to the $L_{5\,100\,\unicode{x00C5}}$ using the bolometric corrections of Runnoe et al. (Reference Runnoe, Brotherton and Shang2012). The calculation uses full error propagation by varying the lag, luminosity and line width within their respective measurement uncertainties. Results are reported as the media, with uncertainties drawn from the 16th and 84th percentiles.

We broadly see an increase in mass with redshift (Figure 5 lower left), which is expected, as the AGN at higher redshifts are selected to be more luminous (see, for example, Figure 1 in Malik et al. Reference Malik2024), and this selection effect will dominate any evolutionary effects. We similarly see the expected general increase of more luminous AGN being more massive, but are unable to infer a clear power-law relationship (Figure 5 lower right).

It should be noted also that the estimated accretion rates for high redshift CIV sources can be extremely high (Figure 5 upper right), with most direct RM samples sitting above the $\dot{M}=3$ threshold used by Du et al. (Reference Du2015) as the division between low and high accretion sources. In Section 6.1.3 we note an apparent flattening of the CIV $R-L$ curve at high luminosity, and, given that SEAMBH associate high accretion rates with low lags (U et al. Reference U2022), this offers a possible physical explanation. Within the OzDES sample, there is a weak trend for more luminous, higher-redshift sources to accrete more strongly (black data points, Figure 5 upper right).

6.1. Comparison of $\boldsymbol{R - L}$ parameter constraints

In this section examine what constraints can be places on the $R-L$ relationship for all lines using OzDES data alone, and also using OzDES data in combination with data from other sources. Though OzDES and SDSS have similar physical target populations, the same can not be said for the wealth of prior sources. Further still, different surveys have different lag measurement methods and quality cut criteria, meaning we cannot assume them all to trace the same population in the $R-L$ plane. We resolve some of this issue by using the SDSS data products that most closely match the published OzDES population (see Section 4.2.4), but for other surveys we take a high level approach of only allowing datasets to be used in combination if there is some reasonable combination of $R-L$ parameters that satisfy all surveys in the grouping simultaneously. The rationale here is that every survey will have complex multi-layered selection effects, but we are here concerned only if these impose enough of a difference in the $R-L$ plane that they are significantly in tension with one another. A more complex treatment would require modelling of every survey, their selection effects and pipelines and the AGN population, while the approach taken here is a natural extension of our comparison of the surveys in the $R-L$ plane. In Table 2 we list the $R-L$ parameters for our chosen ‘primary’ datasets, while the following sub-sections elaborate on alternate $R-L$ relationships with different combinations of literature data and quality cuts.

For consistency, we refit the $R-L$ relationship with the same model for all data, while making use of the published lags and monochromatic luminosities from previous papers.Footnote ^d We model the $R-L$ relationship as being linear in log-log space, with slope $\alpha$ , offset $\beta$ , and an intrinsic scatter $\sigma$ such that the a given luminosity $\lambda L_{\lambda,i}$ predicts a lag $\Delta t_{\mathrm{RL},i}$ by:

(6) \begin{equation} \log_{10}{\Delta t_{\mathrm{RL},i}} = \alpha (\!\log_{10}(\lambda L_{\lambda,i}) - \log_{10}(\lambda L_0)) + \beta \pm \sigma.\end{equation}

Here, $\log_{10}(\lambda L_0))$ represents the choice of luminosity units, typically $\lambda L_0=10^{44}$ or $10^{45}$ erg/s, and is chosen to minimise the covariance between offset $\beta$ and slope $\alpha$ in the posterior distribution. $\beta$ varies with the choice of time units for the lag, and we here follow the convention of all lags being in days.

The observed RM lag, $\Delta t_{\mathrm{RM}}$ , has an additional scatter from observational uncertainty. We describe this uncertainty in log-space, i.e. $\log_{10}(\Delta t_{\mathrm{RM},i}) = \log_{10}(\Delta t_{\mathrm{RL},i}) \pm E_{\mathrm{log},i}$ . Under this modelling, the resulting likelihood for fit parameters $\theta={\alpha,\beta,\sigma}$ is:

(7) \begin{equation} \mathcal{L} = \prod_i{ \frac{1} {\sqrt{2\pi \big(E_{\log,i}^2+\sigma^2\big)}} \exp{ \frac{ -(\!\log_{10}{\Delta t_{\mathrm{RM},i}} - \log_{10}{\Delta t_{\mathrm{RL},i}})^2 } { 2 \big(E_{\log,i}^2+\sigma^2\big) } } }.\end{equation}

We approximate the measurement uncertainty, $E_{\mathrm{log}}$ , as an asymmetric ‘split Gaussian’ with uncertainties found from the positive and negative uncertainties of the lag measurement. I.e. if a lag measurement $\Delta t_{\mathrm{RM},i}$ is constrained within upper and lower bounds $\Delta t_{\mathrm{+RM},i}$ and $\Delta t_{\mathrm{-RM},i}$ , then the log-space measurement uncertainty is:

(8) \begin{align} E_{\log,i} = \begin{cases} \log_{10}\left( \frac{\Delta t_{\mathrm{+RM,},i}}{\Delta t_{\mathrm{RM},i}}\right)\!, \quad \Delta t_{\mathrm{RL},i}\gt\Delta t_{\mathrm{RM},i} \\[6pt] \log_{10}\left( \frac{\Delta t_{\mathrm{RM},i}}{\Delta t_{\mathrm{-RM},i}}\right)\!, \quad \Delta t_{\mathrm{RL},i}\lt\Delta t_{\mathrm{RM},i} \end{cases}\!\!\!\!\!\!.\end{align}

For similar consistency reasons, we use the same procedure to re-fit the $R-L$ relation of literature sources and combinations thereof. We treat all published lags and their uncertainties as describing the cumulative summary statistics of the lag posterior distribution (i.e. median and 16th, 84th percentiles).

All fits are performed using the Bayesian analysis tool NumPyro (Phan et al. Reference Phan, Pradhan and Jankowiak2019), specifically its No U-Turn (NUTS) sampler, a Hamiltonian Monte Carlo (HMC) method that is well suited to unimodal distributions. We note that this fitting is relatively insensitive to the choice of sampler, as it is unimodal and produces approximately Gaussian contours (e.g. Figure 7).

6.1.1. H $\beta$ $\textit{R - L}$ relationship

OzDES is designed primarily for higher redshifts, and so captures only a small number of sources in the H $\beta$ redshift range. Coupled with the high rejection rate, this yields only 8 high quality lag recoveries for the OzDES H $\beta$ $R-L$ relationship (Malik et al. Reference Malik2023). In past OzDES analyses, the limited number of sources has been bolstered by lags from other RM projects, namely the 5-year SDSS lags (Grier et al. Reference Grier2017), a wide range of low redshift lags collated by Bentz et al. (Reference Bentz, Peterson, Netzer, Pogge and Vestergaard2009) as well as later additions to this ‘Bentz Collection’ (Bentz et al. Reference Bentz2013, Reference Bentz2014, Reference Bentz2016a,Reference Bentzb, Reference Bentz, Onken, Street and Valluri2023), lags from the SEAMBH project, lags from the LAMP collaboration (U et al. Reference U2022), and a small number of lags from miscellaneous sources (Li et al. Reference Li2021; Lu et al. Reference Lu2016; Zhang et al. Reference Zhang2019; Rakshit et al. Reference Rakshit2019). We make use of these same sources, in addition to the final SDSS lags of Shen et al. (Reference Shen2023), but re-fit each dataset individually to more fairly measure tensions between them. Considering surveys to be in tension if they differ by more than $2\sigma$ , the groups of surveys with mutually consistent constraints are listed in Table 3. In total, 8 OzDES sources and 26 SDSS sources are used.

Figure 7(a) shows $R-L$ relationship parameters for data from OzDES, SDSS, the Bentz Collection, the SEAMBH Survey, the LAMP survey, and a number of other miscellaneous sources. Shown also are the contours for SDSS lags as recovered from JAVELIN instead of PyROA, and selected according to the criteria outlined in Section 4.2.4. The constraining power is highest for surveys with a high source-count, like SDSS & SEAMBH, or a wide range of luminosities as in the Bentz Collection.

Table 3.

Consistency between data sources for H $\beta$ $R-L$ relation parameters. If two data sources are statistically consistent (T), their recovered parameters for slope, offset and scatter are consistent to within $2\sigma$ . Otherwise they are visibly in tension (F). No result is listed for the main SDSS results and the sub-sampled JAVELIN SDSS results, as they are drawn from the same survey. These tensions yield four distinct sub-groups of mutually consistent data sources.

Table 3 Long description

A matrix showing consistency between different data sources for H beta R-L relation parameters. The matrix has six rows and six columns, with row and column labels including OzDES, SDSS, SDSS-JAV, LAMP, Bentz, SEAMBH, and Misc. The matrix is symmetrical with the diagonal elements marked as consistent (T). Cells are color-coded, with green indicating consistency (T) and red indicating tension (F). Notable trends include clusters of consistent data sources forming distinct sub-groups. The matrix provides a visual representation of the statistical consistency of parameters recovered from different data sources.

Due to the low sample count, the OzDES and miscellaneous sources are poorly constrained and are consistent with all other surveys. The largest tension exists between the slope and offset of the Bentz Collection and SEAMBH data, with this tension being the deciding factor about which surveys are mutually consistent. The SEAMBH data are known to produce consistently lower lags from their highly accreting AGN, and this is borne out by the lower values for $\alpha$ and $\beta$ in collections including SEAMBH sources.

The SDSS results are consistent with these SEAMBH values in slope and scatter, but show systematically higher scatter than any other survey with strong constraints. This lower offset and higher scatter puts the SDSS data in tension with the low redshift Bentz Collection. Limiting the SDSS lags to only those that pass our cuts in Section 4.2.4 (the SDSS- JAVELIN contour), the constraints become weak enough to overcome this inconsistency, but the preferred values for $\alpha$ , $\beta$ and $\sigma$ do not significantly change. The LAMP survey lags sit the lowest of any dataset, and their comparatively narrow range of luminosities mean they do not strongly constrain the $R-L$ slope by themselves.

Figure 7(b) shows the constraints on $R-L$ relationship parameters when combining all datasets that are not in tension with one another. Four such groups are possible:

1. 1. OzDES combined with the Bentz Collection, miscellaneous sources and the sub-sampled SDSS data,

2. 2. As for 1, but with SEAMBH lags in place of Bentz,

3. 3. As for 2 but using the full SDSS sample,

4. 4. OzDES data combined with miscellaneous sources, the SEAMBH lags and the lags from the LAMP survey.

Where we use a ‘primary’ H $\beta$ grouping elsewhere in the paper, it refers to the first of these groupings. We make this choice of grouping as it combines the data of OzDES and SDSS with a broad set of literature lags, while excluding the lags of the SEAMBH survey that are, by the survey’s design, deliberately intended to be atypical of the broader AGN population.

6.1.2. MgII $\textit{R - L}$ relationship

Until this recent generation of high redshift surveys, the bulk of reverberation mapping lags have been measured using H $\beta$ , with a far more sparse set of lags in MgII. Rather than dividing these few sources by survey, we compare the OzDES and SDSS results to the convenient collection by Zajaček et al. (Reference Zajaček2020), consisting of 5 lags from Lira et al. (Reference Lira2018), Czerny et al. (Reference Czerny2019) and Metzroth et al. (Reference Metzroth, Onken and Peterson2006), and a recent lag from Zajaček et al. (Reference Zajaček2021). We exclude the 6 lags from Shen et al. (Reference Shen2016) and (2019), as they are already included in the SDSS results sample. From the OzDES and SDSS data releases we use 25 sources each.

Figure 6.

Rest-frame lags and monochromatic luminosities for all data sources from all lines, shown on a log-log scale to show the linear trend that forms the basis for the $R-L$ relationship. Sources marked with a circle contribute to the constraint of the shown $R-L$ relationship, while sources marked with a cross do not. Sub-plots from top to bottom are for H $\beta$ , MgII and CIV lags with their ‘best fit’ $R-L$ relationships overlaid (see Table 2). The monochromatic luminosity is measured at $5\,100\,\unicode{x00C5}$ , $3\,000\,\unicode{x00C5}$ and $1\,350\,\unicode{x00C5}$ from top to bottom.

Figure 6. Long description

Three scatter plots depict the relationship between measured rest-frame lags and monochromatic luminosity for different emission lines in active galactic nuclei. Panel A: The top scatter plot shows the relationship for the H beta emission line. The x-axis represents monochromatic luminosity in log scale, measured at 5100 angstroms, and the y-axis represents measured rest-frame lags in days, also in log scale. Various data sources are represented by different symbols and colors, with a blue dashed line indicating the best fit relationship. Panel B: The middle scatter plot shows the relationship for the Mg II emission line. The x-axis represents monochromatic luminosity in log scale, measured at 3000 angstroms, and the y-axis represents measured rest-frame lags in days, in log scale. Different data sources are again represented by various symbols and colors, with a red dashed line indicating the best fit relationship. Panel C: The bottom scatter plot shows the relationship for the C IV emission line. The x-axis represents monochromatic luminosity in log scale, measured at 1350 angstroms, and the y-axis represents measured rest-frame lags in days, in log scale. Different data sources are represented by various symbols and colors, with a green dashed line indicating the best fit relationship. Each plot includes a legend explaining the symbols and colors used to represent different data sources.

Comparing $R-L$ relationship parameters for SDSS and OzDES, we find that they broadly agree in terms of the slope and offset of the fit, but that SDSS lags exhibit significantly higher scatter by roughly $0.2$ dex. The low number of sources in the Zajacek Collection means that its fit is poorly constrained, and so is not in tension with OzDES or SDSS. Restricting the SDSS results to the SDSS-JAV sample drastically reduces the number and constraining power, but also decreases the scatter significantly. This decrease in scatter removes the tension with OzDES, allowing these two datasets to be combined.

This yields two mutually consistent groupings, making use of the full SDSS dataset and excluding OzDES or including OzDES but restricting the SDSS data. Owing to the complementary correlations of the datasets, both choices have near identical results for the main properties of the $R-L$ relation, but with a slightly higher scatter for the SDSS-based results (Figure 7(c)). We proceed with the OzDES + SDSS- JAVELIN + Zajacek fit as our primary MgII $R-L$ relation as it is derived from the most diverse set of sources while still showing very low scatter about the mean relationship, suggestive of a lower false positive rate.

Figure 7.

Constraints on $R-L$ relationship parameters for (from top to bottom) H $\beta$ , MgII and CIV. The left column shows constraints for each individual survey/data source, while the right column shows constraints for mutually consistent data sources, as listed in Table 3. To minimise covariance between $\alpha$ and $\beta$ , fitting is performed using units of $10^{44} \mathrm{erg/s}$ for H $\beta$ and units of $10^{45} \mathrm{erg/s}$ in equation 6 for MgII and CIV.

6.1.3. CIV $\textit{R - L}$ relationship

Like MgII, there are only a small number of CIV lags outside of large-scale high redshift surveys. For these supplemental lags, we draw sources from the collation by Kaspi et al. (Reference Kaspi2021), including 6 low redshift AGN in which the $1\,350\,\unicode{x00C5}$ CIV line is observed in the observer-frame ultra-violet range from Peterson et al. (Reference Peterson2005), Rosa et al. (Reference Rosa2015) and Metzroth et al. (Reference Metzroth, Onken and Peterson2006), as well as 6 high redshift sources from Lira et al. (Reference Lira2018) and 3 high redshift sources new to the Kaspi paper itself. As with the collation of existing MgII lags, we exclude the Kaspi sources from Grier et al. (Reference Grier2019) and Hoormann et al. (Reference Hoormann2019) as they are already included in the SDSS and OzDES data.Footnote ^e We use 29 high quality lags from OzDES and 15 from SDSS.

Unlike the H $\beta$ and MgII $R-L$ relationship fits, it is much harder to find consistent agreement between different data sources. The lags from prior works sit systematically lower than the trends set by either OzDES or SDSS, and there is consistently greater scatter than exhibited for either H $\beta$ or MgII. Some of this discrepancy can be attributed to selection effects (see below); some may be due to CIV being contaminated by a non-virial wind component (Denney Reference Denney2012). We attempt to account for selection effects in this section, but leave more detailed study of contamination for future work.

The high redshift surveys of SDSS and OzDES are highly consistent with one another except for the markedly higher scatter of the SDSS data, much as is seen for H $\beta$ and MgII. The OzDES-like SDSS-JAV sample has lower scatter, removing the tension with the OzDES sample, though partially through a loss of constraining power from the reduction in source count.

The literature lags sit noticeably lower than either SDSS or OzDES lags, even at high redshift. Though it is not statistically objectionable to combine both high redshift surveys with either the Kaspi Collection’s high redshift lags or its low redshift anchor, this gap means that it is not reasonable to do both at once. This allows for five possible groupings if taking the standard approach to the fitting of the $R-L$ relationship:

1. 1. The high redshift literature lags with the combined OzDES and SDSS-JAV sample,

2. 2. As above but with the low redshift anchor literature sources,

3. 3. As per 1. but with the full SDSS and no OzDES sources,

4. 4. As above but with the low redshift anchor literature sources,

5. 5. The low and high redshift sources of the Kaspi Collection with no new sources.

Though still exhibiting a high scatter, the above groupings that fit for the entire luminosity range (groups 1, 3 and 5) give slopes roughly consistent with those of H $\beta$ and MgII, while the flattened branch of the high-luminosity only groupings (2 and 4) have flat relationships between reverberation lag and AGN luminosity. This apparent disagreement between sources means we are unable to choose a primary dataset for fitting the CIV $R-L$ relationship.

The tension between the slope of the high and low redshift is manifest in an apparent ‘levelling out’ of the $R-L$ relation at high redshift (Figure 6, bottom panel). This is best illustrated when examining the relationship between the residuals and the ‘high-z’ fit (our ‘grouping 1)’). As shown in Figure 8, high luminosity/redshift sources give unmistakably and consistently higher lags than the lower luminosity/redshift data.

Figure 8.

Residuals of CIV lag measurements for all RM sources about the best fit high redshift $R-L$ relationship, coloured by accretion rate $\log_{10}(\dot{M})$ . There is a clear trend of the low redshift/luminosity sources (most of which are drawn from the Low-Z Kaspi sample) sitting below the fit. Though the high-accretion rate sources sit below the $R-L$ fit, there is no independent correlation with either accretion rate or emission line velocity dispersion (bottom panel, discussed further in Section 6.2).

Figure 8. Long description

Three scatter plots depict residuals of CIV lag measurements for various RM sources, colored by accretion rate. Panel A: The scatter plot shows residuals about the best fit high redshift relationship on the y-axis and redshift on the x-axis. The data points are colored based on the dimensionless accretion rate. Panel B: The scatter plot shows residuals about the best fit high redshift relationship on the y-axis and monochromatic luminosity on the x-axis. The data points are colored based on the dimensionless accretion rate. Panel C: The scatter plot shows residuals about the best fit high redshift relationship on the y-axis and rest frame line width velocity on the x-axis. The data points are colored based on the dimensionless accretion rate. The black dashed lines represent the R-L relationship confidence interval.

This nonlinear $R-L$ relationship for CIV may have a physical basis, such as distant AGN having higher accretion rates, which have been associated with lower lags (e.g. Du et al. Reference Du2015; Hu et al. Reference Hu2021). A simpler explanation lies in describing this as a selection effect: more luminous AGN have longer rest-frame lags, stretched further at high redshift by time dilation, and are therefore more likely to have lags that exceed the length of the survey. This induces a selection effect in which high redshift/luminosity sources are biased downwards from the true population distribution as the finite survey length imposes a ceiling on what observer-frame lags can be recovered. CIV sources are often the most distant, and consequently the most luminous and associated with the longest observer-frame lags, and so their observations are the most impacted by this effect.

Coupled with recovery rates and quality cuts, the exact effects of this under-sampling of high lag sources is difficult to characterise, as it interacts with the entire observational and analysis pipeline. A complete treatment would require modelling the underlying source population, survey selection, and the entire analysis chain including quality cuts. Though not intractable, this is an involved task, particularly when trying to unify multiple surveys and studies.

Penton et al. (Reference Penton2026) estimates that, for the 6-year baseline of OzDES, a lag-recovery selection effect becomes significant for CIV sources at observer-frame lags of $\approx 1\,000\,\mathrm{d}$ and continues until lags above $1\,500\,\mathrm{d}$ are almost impossible to recover. We make the coarse approximation of treating this $1\,250\,\mathrm{d}$ limit as a hard cutoff, altering our Bayesian model so that that OzDES cannot detect lags above this limit. This is tantamount to altering the normal distribution of equation (7) to be a truncated normal distribution with a cutoff at $\Delta t_i = \frac{1\,250 \; \mathrm{d}}{1+z_i}$ . To a rough first order approximation, we scale this limit for each survey proportional to the length of their spectroscopic campaign as compared to OzDES, giving nominal lag ‘cutoffs’ in Table 4.

Table 4.

Maximum observer-frame lags for each CIV survey, along with their estimated maximum recoverable lag cutoff.

Table 4 Long description

A table with four rows and three columns. The columns are labeled Highest observer frame lag (days), Spectroscopic baseline (yrs), and Nominal lag cutoff (days). The rows are labeled with different surveys: OzDES, SDSS-JAVELIN, and Kaspi High-Z. Row 1: OzDES, 934.1 plus 31.1 minus 31.1, 6, 1250. Row 2: SDSS-JAVELIN, 1298.6 plus 14.7 minus 12.0, 7, 1460. Row 3: Kaspi High-Z, 1629.9 plus 252.1 minus 326.7, 9.5, 1980.

This model significantly relaxes the constraints from each survey, introducing the freedom for an un-observed fraction of sources to exist above the lag cutoff. This relaxes the tension between the surveys, creating a single parameter-space region of overlap between all four data sources (Figure 9, top panel). The high luminosity literature sources, drawn from the high-z Kaspi sample, are dominated by this observational window effect, loosening their constraining power such that this fit matches the constraints of the more generic $R-L$ fit achieved when excluding the high-z literature lags all together (Figure 9, bottom panel). It is this fit, with all CIV data accounted for, that we use as our primary $R-L$ fit for single-epoch mass estimates in Section 8.

6.2. Accretion rate & line width as predictors of RM lag

It is matter falling on to the accretion disk circling the SMBH that drives AGN activity, and there is evidence of a separation in behaviour between low and high accretion AGN. Highly accreting ‘super-Eddington’ AGN are associated with lower lags than their ‘sub-Eddington’ counterparts at the same luminosity (Du et al. Reference Du2015). It is also evident that high accretion sources tend to sit below our best fit $R-L$ relation (e.g. Figure 8), while low accretion sources sit above. We are then motivated to see if observations other than luminosity are predictors of AGN lag as a means to reduce the scatter of our single epoch methods, improve the predictive constraints of our single epoch methods and as a probe of the underlying AGN behaviour.

To examine this, we examine a new functional form for the single epoch lag prediction in which we add emission line width as a variable in the $R-L$ relation, i.e. an $R-L-V$ relation:

(9) \begin{equation} \log_{10}(\Delta t) = \alpha \log_{10}(\lambda L_{\lambda}) + \gamma \log_{10}\big(\big\langle{\sigma_v^2} \big\rangle\big)+ \beta, \end{equation}

where $\gamma$ , the slope of log-lag against log velocity dispersion, represents the predictive power of line-width independent of luminosity.

Our estimates for mass and accretion rate (equations 1 and 4, respectively) both follow power laws, making them linear for the logarithmic space that we fit our single epoch relations in. In this way, fitting for line width and luminosity also covers the fitting for accretion rate, as it depends entirely on these two other measurements.

Performing this $R-L-V$ fit for all lines, using our primary data groupings from Section 6.1, we find no significant improvement in predictive power (Table 5), and constraints on the lag-velocity dependence are broadly consistent with zero, i.e. no dependence of lag on line width. For MgII we see a slight preference towards $\gamma\gt0$ , but this is only weakly constrained and there is still no marked decrease in scatter about the model.

Table 5.

Constraints on line-width velocity as a supplementary predictor of lag, and comparison of the scatter about this model compared to the luminosity-only $R-L$ relationship. In all cases we fail to see a reduction in the scatter. Similarly, all lines are consistent with zero log luminosity/velocity ( $\gamma=0$ in equation 9), i.e. no lag-velocity dependence, though with MgII preferring a positive relation.

Table 5 Long description

A table with three rows and three columns comparing constraints on line-width velocity as a supplementary predictor of lag. The columns are labeled Line type, γ, σ for R - L, and σ for R - L - V. The row labels are H β, MgII, and CIV. Row 1: H β, -0.04+0.13-0.13, 0.25+0.02-0.02, 0.25+0.03-0.02. Row 2: MgII, 0.59+0.30-0.23, 0.23+0.02-0.02, 0.22+0.03-0.02. Row 3: CIV, 0.04+0.38-0.34, 0.34+0.04-0.03, 0.35+0.04-0.04.

Figure 9.

Constraints on the CIV $R-L$ relationship parameters after accounting for a maximum observable lag due to survey lengths. The top panel shows constraints for each individual dataset, and the bottom panel compares the combined data. These are based on the same data groupings as used in Figure 7, but with a model incorporating the cutoffs in Table 4.

Figure 9. Long description

The image contains multiple graphs depicting constraints on the CIV relationship parameters after accounting for a maximum observable lag due to survey lengths. Panel A: The top panel shows constraints for each individual dataset. It includes three contour plots and three histograms. The contour plots show the relationships between parameters alpha, beta, and sigma. The histograms display the distribution of these parameters. Panel B: The bottom panel compares the combined data. It includes three contour plots and three histograms. The contour plots show the relationships between parameters alpha, beta, and sigma. The histograms display the distribution of these parameters. The graphs use different colors to represent different datasets: OzDES CIV, SDSS CIV, Kaspi Low Z, and Kaspi High Z. The combined data graph uses a single color to represent the combined dataset.

The apparent trend between accretion rate and the residuals about our best fit $R-L$ relationships is a projection of the luminosity dependence onto the log-accretion rate axis, rather than an independent axis of variability. For a fixed luminosity, higher than average lags will give higher masses, which will give lower accretion rates, producing an apparent anti-correlation between the two.

There is the possibility that the lack of predictive power comes from a strong correlation between luminosity and velocity dispersion in our sample: if we only probe a narrow band of $L-V$ space, the two parameters become degenerate in our fitting. We can rule this out by examining the correlation between luminosity and velocity (Figure 10). We find that this correlation is weak, and is not consistent between lines. Table 6 notes an extremely shallow scaling index between luminosity and velocity dispersion for all lines, indicating that our sampling of the two parameters is sufficiently uncorrelated.

Table 6.

Constraints on the scaling index and scatter between AGN luminosity and velocity dispersion for all emission lines, per equation (9). Constraints are for sources in our ‘primary’ datasets as outlined in Section 6.1.

Table 6 Long description

A table with four rows and three columns. The columns are labeled ‘Line type’, ‘Log-Vel/Log-Lum slope’, and ‘Scatter (dex)’. The row labels are H beta, MgII, CIV, and All sources. The values in the table are as follows: Row 1: H beta, Log-Vel/Log-Lum slope: 0.08^{+0.02}_{-0.03}, Scatter (dex): 0.22^{+0.02}_{-0.02}. Row 2: MgII, Log-Vel/Log-Lum slope: -0.06^{+0.02}_{-0.02}, Scatter (dex): 0.12^{+0.01}_{-0.01}. Row 3: CIV, Log-Vel/Log-Lum slope: -0.07^{+0.02}_{-0.03}, Scatter (dex): 0.14^{+0.02}_{-0.02}. Row 4: All sources, Log-Vel/Log-Lum slope: 0.09^{+0.01}_{-0.01}, Scatter (dex): 0.19^{+0.01}_{-0.01}.

Figure 10.

Luminosity/velocity scatter plot for our ‘primary’ datasets, coloured by rest frame lag. Under-laid are fits for the correlation between lag and velocity, modelled as a power law. The colouring shows how lag evolves strongly over the luminosity axis, demonstrating it is a better predictor of lag compared to velocity.

We note that this lack of predictive power in velocity is in contrast to the findings of the SEAMBH collaboration, (Du et al. Reference Du2016, Reference Du2018; Hu et al. Reference Hu2021) who find that higher accretion rates (derived from higher velocities) is associated with lower lags. Our results here do not necessarily contradict their findings, this analysis is for our primary data group which does not include their data specifically because their highly accreting sources produce statistically lower H $\beta$ lags (see Section 6.1.1), and we use a different functional form for the accretion dependence in place of their use of a critical accretion rate threshold. Our fit only demonstrates that line width/accretion rate does not serve to explain the residuals about our best fits. It is also worth noting that different data sources have non-uniform approaches to measuring these line widths, and unlike the lag parameter we make no attempt to account for these differences in analysis.