OzDES Reverberation Mapping Program: Stacking analysis with H$\beta$, Mg II and C IV

Reverberation mapping is the leading technique used to measure direct black hole masses outside of the local Universe. Additionally, reverberation measurements calibrate secondary mass-scaling relations used to estimate single-epoch virial black hole masses. The Australian Dark Energy Survey (OzDES) conducted one of the first multi-object reverberation mapping surveys, monitoring 735 AGN up to $z\sim4$, over 6 years. The limited temporal coverage of the OzDES data has hindered recovery of individual measurements for some classes of sources, particularly those with shorter reverberation lags or lags that fall within campaign season gaps. To alleviate this limitation, we perform a stacking analysis of the cross-correlation functions of sources with similar intrinsic properties to recover average composite reverberation lags. This analysis leads to the recovery of average lags in each redshift-luminosity bin across our sample. We present the average lags recovered for the H$\beta$, Mg II and C IV samples, as well as multi-line measurements for redshift bins where two lines are accessible. The stacking analysis is consistent with the Radius-Luminosity relations for each line. Our results for the H$\beta$ sample demonstrate that stacking has the potential to improve upon constraints on the $R-L$ relation, which have been derived only from individual source measurements until now.


INTRODUCTION
Reverberation mapping is a powerful technique that can resolve the cores of active galactic nuclei (AGN) in the time domain.The accretion disk around the central supermassive black hole (SMBH) produces intrinsically variable emission at UV-optical wavelengths.The surrounding broad-line region (BLR) is ionised by this continuum emission, which drives a reverberation response in the emission-line flux from the BLR on time-scales of weeks to months (Blandford & McKee 1982;Peterson 1993).Multi-epoch photometric and spectroscopic observations can be used to trace the variability of the continuum and the emission-line response, respectively.
The size difference between the accretion disk (light-days to weeks) and the BLR (light-weeks to months) introduces a delay in the response of the BLR to the variation in the ionising flux.The delay, i.e. reverberation lag, , can be recovered by cross-correlating the two ★ umang.malik@anu.edu.au† rob.sharp@anu.edu.aulight curves, in order to measure the radius of the BLR ( BLR = ).
The velocity dispersion of the BLR (Δ) can be estimated from the width of the broadened emission lines.The mass of the central black hole ( BH ) can then be measured using the virial theorem: where  is the virial coefficient; a dimensionless scale factor that accounts for the geometry, orientation, and kinematics of the BLR (Woo et al. 2015).
Reverberation mapping (RM) is presently the only method that can be used to directly measure SMBH masses beyond the local Universe, as other techniques are reliant on resolving the gravitational sphereof-influence of the black hole, which remains challenging even with high angular resolution instruments (e.g., Gebhardt et al. 2000Gebhardt et al. , 2011;;Kuo et al. 2011;Event Horizon Telescope Collaboration et al. 2019).However, RM is by nature observationally intensive.It requires repeated observation over the relevant variability time-scales of AGN in order to ensure the light-curve variability and reverberation lag are resolved (Horne et al. 2004).Early surveys monitored AGN on a source-by-source basis, making observations over several months to years.Lag measurements were made for dozens of sources, using the H line (e.g., Peterson et al. 1998;Kaspi et al. 2000;Peterson & Horne 2004;Bentz et al. 2009).From these measurements, a tight correlation was found between the AGN luminosity and the radius of the BLR ( −  relation; e.g., Bentz et al. 2009Bentz et al. , 2013)).Lags recovered using higher ionisation emission lines (e.g., C iv) were found to be shorter than lags recovered using lower ionisation emission lines (e.g., H), demonstrating the ionisation stratification of the BLR (Gaskell & Sparke 1986;Dietrich et al. 1993).The  −  relation is importantly used to calibrate secondary mass-scaling relations to estimate single-epoch virial BH masses (e.g., Shen et al. 2011), and has also been proposed as a way to standardise AGN for use as a cosmological distance indicators (Watson et al. 2011;Martínez-Aldama et al. 2019).
Through the advent of wide-field photometric surveys such as the Dark Energy Survey, (DES, Dark Energy Survey Collaboration et al. 2016), multi-epoch photometric data for large samples of AGN has become readily available.Concurrent observations can be made with multi-object spectrographs, however the demand for these instruments is high and therefore limits the number of epochs which can be feasibly acquired.The Australian Dark Energy Survey (OzDES) and Sloan Digital Sky Survey Reverberation Mapping (SDSS-RM) Project have conducted the first multi-object RM surveys, observing hundreds of AGN probing a wide range of AGN luminosities and redshifts (King et al. 2015;Shen et al. 2015).These programs have delivered over one hundred new lag measurements (Grier et al. 2017;Hoormann et al. 2019;Grier et al. 2019;Homayouni et al. 2020;Yu et al. 2021Yu et al. , 2023;;Malik et al. 2023, Penton et al. in prep).This allowed the Mg ii and C iv  −  relations to be constrained for the first time using statistically significant samples, however there is significant scatter in the measurements.This is mostly due to challenges with data quality (low signal-to-noise, limited sampling; see Malik et al. 2022) and lag recovery reliability (Li et al. 2019;Penton et al. 2022).These factors have limited the lag recovery efficacy of each survey to about 10-25%.
Stacking can be used to combine the cross-correlation signals of physically similar AGN to recover average lags for these objects.The technique was first applied by Fine et al. (2012Fine et al. ( , 2013) ) using only two spectroscopic epochs, which yielded a marginal result.After demonstrating the success of the technique with the Bentz et al. (2013) H𝛽 sample, Li et al. (2017) measured composite lags with H, H, He ii and Mg ii using a subset of the SDSS-RM sample.Stacked averages are not swayed by the systematic errors from individual sources.Their consistency (or not) with the measurements made for individual sources is therefore important.Additionally, with the wide redshift range covered by our sample, the gaps in the observational window function can be filled in to some extent.Therefore, stacking leverages the data in a way that cannot be done with traditional individual measurements.
We present a stacked lag analysis of the entire OzDES sample, for the H, Mg ii and C iv lines.Section 2 details the observations obtained by OzDES and the data calibration procedures.In Section 3 we describe the technique used to recover stacked lags.In Section 4 we present our average lag measurements, and comparisons with individual measurements on the respective  −  relationships for each emission line.We summarize our results and the discuss the outlook to the future in Section 5.

DATA
Our photometric data were obtained as part of the Dark Energy Survey (DES) Supernova Program, which observed 10 deep fields covering 27 deg 2 , comprising the ELAIS, XMM-LSS, Chandra deep-field South, and SDSS Stripe 82 fields (Kessler et al. 2015;Morganson et al. 2018).These fields were observed in the  filters, with the Dark Energy Camera (DECam) on the 4m Blanco telescope at Cerro Tololo Inter-American Observatory (CTIO) (Flaugher et al. 2015).The fields were observed with ∼6 day cadence over a 5-6 month season (August to January) from 2013 to 2018, with additional science verification data taken in 2012.The OzDES project (Yuan et al. 2015;Childress et al. 2017;Lidman et al. 2020) conducted follow-up spectroscopic observations with the 2dF multi-object fibre positioning system and the AAOmega spectrograph (3700-8800 Å, Sharp et al. 2006) on the 3.9m Anglo-Australian Telescope (AAT), taken with approximately monthly cadence over the same seasons, from 2013 to 2019.After the conclusion of the Supernova Program, additional DECam observations were taken monthly in the 2018-19 season, taking the baseline of our photometric light curves to 7 years.The OzDES Reverberation Mapping sample comprises 735 AGN, ranging from 0.1 <  < 4.0, with apparent magnitudes 17.2 <  AB < 22.3 (Tie et al. 2017).Of these 735 AGN, we removed 9 of the 78 AGN from the H sample due to broad-absorption lines (BALs) or incorrect classification as a Type 1 AGN, 3 of the 453 AGN from the Mg ii sample for the same reason (these sources overlapped with the H sample), and 88 of the 378 AGN from the C iv sample due to BALs.These sources were included in the initial survey selection to improve the source density on the sky and study BAL variability, but have proven challenging for reverberation analysis.The final sample we use in this analysis comprises 69 H sources, 450 Mg ii sources, and 290 C iv sources (690 AGN in total).The redshift and luminosity distribution of these targets is shown in Figure 1.
As done for the individual lag measurements made by Yu et al. (2023) and Malik et al. (2023), for the H and Mg ii samples we measure the emission-line flux from spectra obtained on different nights as separate epochs in order to maximise the cadence of our sampling for our emission-line light curves.As we have lower signal-to-noise for the C iv sample, we co-add the spectra over each observing run (typically 4-7 nights during dark time each month), as done by Hoormann et al. (2019).
We do not measure the continuum luminosity directly from the spectra due to fibre aperture effects from variable atmospheric seeing and fibre placement uncertainties.From the average -band magnitude and redshift of the AGN, we estimated the monochromatic continuum flux at rest frame 5100 Å, 3000 Å and 1350 Å using the DECam -band filter transmission curve and the SDSS quasar template (Vanden Berk et al. 2001).The template is scaled to the magnitude of the source, assuming  bol = 9   (5100 Å) (Kaspi et al. 2000).The source properties for the OzDES sample used in this work are provided in §A and complete sample characteristics for the final OzDES RM sample will be provided in a future OzDES RM Program paper.
The DES photometry is calibrated using the DES data reduction pipeline (Morganson et al. 2018;Burke et al. 2018).We perform a spectrophotometric flux calibration following Hoormann et al. (2019).For the H and C iv samples, we measure the line fluxes as done by Hoormann et al. (2019).For the continuum subtraction, the local continuum windows we use for H are 4760 to 4790 Å and 5100 to 5130 Å, and for C iv are 1450 to 1460 Å and 1780 to 1790 Å.For the Mg ii sample, the iron subtraction and line flux measurement is performed as detailed in Yu et al. (2023).The calibration uncertainties of the line flux for each line are measured using the F-star warping function method as detailed in Yu et al. (2021).

LAG RECOVERY METHOD
As reverberation lags should be dependent on the intrinsic AGN luminosity alone (at least to first order), we bin the sources by their continuum luminosity.Further details are provided for each emissionline subsample in §4.
For each source in a redshift-luminosity bin, we covert the observation dates to the rest frame of the source by dividing by (1+).We use the PyCCF code to perform the interpolated cross-correlation function method for each individual source (ICCF; Gaskell & Peterson 1987;Sun et al. 2018).The continuum and emission-line light curves of each source are linearly interpolated to a grid spacing of 3 days.The interpolated light curves are cross-correlated as a function of time-lag.We then average the cross-correlation functions (CCF) of each source in the bin to obtain the stacked CCF.We search for lags over a (rest-frame) lag range of [−100, 300] days for H and C iv, and [−100, 500] days for Mg ii as it has a longer expected lag for our sample.
We measure the average lag and its uncertainties by bootstrapping the sample in each bin.We perform the above procedure to calculate the stacked CCF's for each bootstrapped re-sample of the original binned sample.We repeat this 1000 times, and record the centroid of each CCF ( max ) to build the bootstrap distribution, from which we adopt the median and 16th and 84th percentiles of this distribution as the recovered average stacked lag, , and lower and upper uncertainties,   , for the bin.
Following Li et al. (2017), we shuffle the spectroscopic epochs and repeat the stacking for 100 Monte Carlo realisations, and compare the stacked CCF produced by these uncorrelated light curves to the original stacked CCF.This is similar to the null hypothesis test used by Malik et al. (2023) to check that the lag recovery is not simply a product of the interaction of the window function with underlying red-noise correlation in the photometric light curves, considering the relatively low sampling density and modest signal-to-noise of our light curves.

RESULTS
We present the results of our stacking analysis for each emission line sample in the OzDES RM sample, and a multi-line analysis of the subsamples for which two emission-lines are present.With the basic assumption that the lag of a source is dependent on the intrinsic AGN luminosity alone, by using the stacking method we are assuming the lags of each source within a bin are similar (in the rest-frame) so that we recover a representative average lag for these AGN.Since the binned sources are at different redshifts, when the light curve data is converted to the rest-frame, different variability and reverberation time-scales are probed by each source through their unique restframe observational window function.By stacking we can partially circumvent the usual impact of the sparse sampling (particularly the 7-month seasonal gaps) in the light curves of any one source, as we are combining the CCF's for all sources in a bin.We compare the average lag measurements to the sample of existing individual lag measurements on the respective  −  relations for each line.

H𝛽
We divide our sample into five luminosity bins of equal size, as shown in Figure 2. The size of the bins were chosen as to maximise the number of sources in each bin while avoiding introducing a broad underlying distribution in the expected lags.The highest luminosity source was excluded from the analysis for this reason.Although there are a relatively small number of sources stacked in each bin, particularly when compared to stacking done by Fine et al. (2012Fine et al. ( , 2013)), the signal-to-noise of our stacked CCF's are sufficiently high as we have much more light curve data.
The stacked CCF's for each bin of the H sample are shown in Figure 3, alongside the total number of overlapping light curve epochs as a function of time-lag for all sources stacked within the bin.As the observation dates of each source in a bin are converted to the rest-frame of the source, and there is a distribution of source redshifts within each bin, the total light curve sampling of the stacked sample begins to in-fill the rest frame observational gap imposed by  the observed frame 7-month seasonal gap present in the individual observed light curves.
Comparing the stacked CCF, and its scatter measured from bootstrapping, to the stacked CCF's after light curve randomisation, we see significant correlation signal present.This implies that the signal is not dominated by the correlation of any individual source.In all cases there is one major peak present, however, the lowest luminosity bin (blue) has a flatter CCF.There is limited but non-zero data overlap around the expected mean lags for the two highest luminosity bins, as they coincide with the first seasonal gap in our light curves.However, we recover significant average lags in each of these bins.This demonstrates the ability of stacking to overcome the limitations imposed by sparse sampling, which impede lag recovery for individual sources.
We plot the recovered average lags from each luminosity bin on the  −  relation, as shown in Figure 4. Our stacked measurements are consistent with the Bentz et al. (2013) slope, which agrees with the physically motivated slope of ∼0.5.The uncertainty in the average lags are consistent to the uncertainties in the eight lags recovered for individual sources (Malik et al. 2023), and are inconsistent with the distribution of the SDSS-RM measurements (Grier et al. 2017;Li et al. 2017), for which shorter lags are recovered.Given the similarities of our programs and sample selection, the reason for this discrepancy is unclear.Although the main difference between the surveys is the baseline and cadence of the data, simulations by Fonseca Alvarez et al. (2020) and Malik et al. (2022) find that this does not bias the lag recovery of SDSS-RM to shorter lags, or OzDES to longer lags.For our composite lags we bin across our entire H sample, and do not reject any sources based on light curve signal-tonoise, or any other criteria.As a test, we repeated the stacking after excluding all the objects with individual recovered lags (Malik et al. 2023).We show the results of this test in §B.Although the average lag uncertainties increase after the exclusion, the lags remain in close agreement with those from the original analysis.

Mg ii
The three luminosity bins used for the Mg ii sample are shown in Figure 5.The lowest and highest luminosity bins are slightly wider to include sources close to the edge of the bin.Ten sources were excluded from the analysis to optimise the bin densities and reduced smoothing of the stacked CCF's.Larger luminosity bins were required to achieve adequate signal-to-noise in the stacked CCF's for this sample.
We present the stacked CCF's in Figure 6.The strength of the correlation signals in each bin are lower than for the H sample, however, comparing to the CCF's produced using randomised light curves we can see there is significant signal present.The signal-tonoise of the Mg ii light-curve input data is lower than for the H sample, as the Mg ii sample is fainter and requires subtraction of Fe ii from the emission-line (Yu et al. 2021(Yu et al. , 2023)).However, we have 450 AGN in the Mg ii sample, and are therefore stacking many more sources in each bin.Since sample size is not the limiting factor in this case, the lower signal-to-noise of the Mg ii line light curves must be producing weaker stacked cross correlation signals.In the lowest luminosity bin, the signal is flat and no clear peak is present.For the other two bins a dominant peak is present, and the bootstrap distributions are adequately constrained to recover average lags.
We plot the recovered average lags from each luminosity bin on the Mg ii  −  relation in Figure 7, along with the 25 individual measurements made with this sample by Yu et al. (2023).The average lags are in agreement with the recent individual measurements from Homayouni et al. (2020) and Yu et al. (2023).There is no clear progression to longer lags with higher luminosities given that the average lags for the lowest and intermediate luminosity bins are not particularly well constrained.As shown in Figure 6, there is limited data coverage over the expected mean lags for the lowest luminosity bin (blue), which coincides with the first seasonal gap in our light curves.The time-dilation distribution over the binned sample is not sufficient to 'fill in' the short timescales, but it is able to sufficiently bridge the second seasonal gap.This could explain why the uncertainty on the recovered average lag for the lowest luminosity bin is larger than the uncertainties for the intermediate and highest luminosity bins.The average lags for the three luminosity bins are formally consistent with both the steeper  −  relation measured by Trakhtenbrot & Netzer (2012), and the shallower relation recently constrained by Yu et al. (2023), however, the better constrained mean lag for the highest luminosity bin is only consistent with the shallower relation.

C iv
As for the previous samples, we present the luminosity bins for the C iv sample in Figure 8, and the stacked CCF results in Figure 9.As required for Mg ii, larger luminosity bins were necessary to achieve sufficient signal-to-noise after stacking, and nine sources were excluded from the analysis to avoid overly wide bins in luminosity.
The OzDES C iv sample has lower signal-to-noise than the H and Mg ii samples, as the AGN are faint.The bootstrap distributions are not well constrained for the lowest or highest luminosity bins.However, Li et al. (2017) found that the mean recovered lag remains stable when the light curve signal-to-noise is degraded, although the uncertainty increases proportionally with the decline.Therefore we continue to recover the average lags and compare with individual source measurements.
We plot the stacked average lags from the C iv sample alongside existing measurements from the literature in Figure 10.The average lags for the intermediate and highest luminosity bins are in agreement with the  −  relations constrained by Hoormann et al. (2019) and Grier et al. (2019), although the uncertainty on the average lag for the highest luminosity bin is large.As discussed for the Mg ii sample, the time dilation over the binned samples does not sufficiently bridge the first of the 7-month seasonal gaps in our light curves.With the  shorter expected lags for C iv, the average lags for the lowest and intermediate luminosity bins coincides with this gap.In addition to the lower signal-to-noise of the light curves, this may be contributing to the poorer quality of the stacked CCF's.It is unclear why the average lag recovered for the lowest luminosity bin is much longer than expected.

Multi-line measurements
Both the H and Mg ii lines are visible for 13 AGN, and Mg ii and C iv for 106 AGN (see Figure 1).We attempt to recover average lags independently with each line, in order to compare the lag ratios to investigate the ionisation stratification of the BLR.We repeated the stacking procedure for the sample of 13 AGN with the H and Mg ii light curves.As there are few sources we do not bin them by luminosity.We present the stacked CCF's for each line in Figure 11.We measure an average lag of 75 +14 −16 days for the H sample, and an average lag of 79 +22 −53 days for the Mg ii sample.We formally recover a Mg ii to H lag ratio of 1.05±0.54,however the inherent uncertainty is significant due to the large uncertainty of the Mg ii average lag.This result is broadly consistent with the expectation that these two BLRs are approximately cospatial.This ratio is consistent with previous multi-line measurements made by Homayouni et al. (2020), who found that Mg ii is emitted from a similar or slightly larger region than H in several individual sources, as well as Clavel et al. (1991) and Czerny et al. (2019).
We repeated this for the sample of 106 AGN with Mg ii and C iv, and present the stacked CCF's in Figure 12.The signal-to-noise was insufficient to divide the sample into two luminosity bins.We measure an average Mg ii lag of 182 +128 −37 days, and an average C iv lag of 64 +165 −45 days.We formally recover a Mg ii to C iv lag ratio of 2.84±4.84,however this is poorly (if at all) constrained given that the average lags are not well constrained (particularly for the CIV sample).The average lags are broadly consistent with the BLR stratification model, and the multi-line comparison made for a single source by Homayouni et al. (2020).
We include the average lags recovered from the multi-line samples on the respective  −  plots presented in Figure 4, Figure 7 and Figure 10.We also provide all average lags recovered in this work from each multi-line and emission line sample in Table 1.

SUMMARY
We use the stacking technique developed by Fine et al. (2012Fine et al. ( , 2013) ) to measure average lags in luminosity bins for the H, Mg ii and C iv samples from the OzDES Reverberation Mapping Program.By utilising the bulk of our sample to recover composite lags, we avoid the potential selection biases in the lag recovery for individual sources.We successfully recover significant cross-correlation signals for each emission-line sample: • The average lags from each luminosity bin of the H sample are consistent with the  −  relation constrained by Bentz et al. (2013), and the size of the uncertainties on the average lags are on par with that of individual measurements, despite the relatively small number of sources stacked in each bin.This provides confidence in the individual measurements, and demonstrates the potential for stacked RM analyses to improve upon constraints which have thus far been made with individual source measurements alone.
• For Mg ii and C iv, the stacked cross-correlations are weaker, but still present above the correlation signal generated using randomised light curves.Further data or larger samples are required to recover significant average lags for these samples across each luminosity bin.
• From our multi-line analysis, we measure a Mg ii to H lag ratio that is consistent with earlier findings that the size of each of these line-emitting regions is similar.Our average lags for the Mg ii and C iv multi-line sample are not well-constrained due to the limited sample size, however, the lag ratio we recover is largely consistent with the BLR ionisation stratification model.
Stacking can be applied beyond RM-specific surveys.It can be done with just a few spectroscopic epochs, using large AGN samples, provided the continuum behaviour is well sampled.With LSST forthcoming, high cadence photometry of the deep-drilling fields will yield quality continuum light curves for tens of thousands of AGN.The SDSS-V Black Hole Mapper (BHM) will be spectroscopically following these fields.In addition to their dedicated RM survey of ∼ 1, 000 AGN, SDSS-V BHM will monitor 25,000 AGN over multiple epochs, which will be combined with earlier SDSS spectra (Kollmeier et al. 2017).The Time Domain Extragalactic Survey (TiDES) will also be following up these fields to conduct an RM survey of ∼ 700 AGN up to  ∼ 2.5 (Swann et al. 2019).Stacking analyses of these future, large samples has promise to significantly extend the reverberation mapping results from these projects.Although the improved signal-to-noise and survey sampling of these future programs are expected to yield an increased number of higher quality individual lag measurements than the first generation of multi-object RM surveys, these datasets can benefit from the ability of stacking to alleviate the impact of the unavoidable seasonal gaps on lag recovery (Malik et al. 2022).Stacking also presents an opportunity to combine all large time-domain datasets to recover average lags that are poten-tially more robust than individual lags, which remain challenging to recover reliably, particularly at high redshift.the traditional owners of the land on which the AAT stands.We pay our respects to their elders past and present.This analysis used NumPy (Harris et al. 2020), Astropy (Astropy Collaboration et al. 2013, 2018), and SciPy (Virtanen et al. 2020).Plots were made using Matplotlib (Hunter 2007).This work has made use of the SAO/NASA Astrophysics Data System Bibliographic Services.
This paper has gone through internal review by the DES collaboration.Funding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, the Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University, Financiadora de Estudos e Projetos, Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Científico e Tecnológico and the Ministério da Ciência, Tecnologia e Inovação, the Deutsche Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy Survey.
The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cam-    4).

Figure 1 .
Figure 1.Distribution of redshifts and monochromatic luminosity at 5100 Å for the 690 AGN in the OzDES RM sample.The H sample extends to  = 0.75, with 13 sources overlapping with our Mg ii sample.The Mg ii sample extends to  = 1.92, with 106 sources overlapping with our C iv sample.

Figure 2 .
Figure 2. The luminosity bins for the H sample, labelled with the number of binned sources.Within each bin, the standard deviation of the expected lags for the individual sources (measured using the source luminosity and the Bentz et al. (2013)  −  relation for H) is ∼10% of the expected mean lag for the binned sample (measured as the mean of the expected lags for the individual sources).

Figure 3 .
Figure 3. Left column:The coloured solid lines are the stacked cross correlation functions (CCF) for each of the five luminosity bins for the H sample.The colours correspond to the respective bins in Figure2.The 1 scatter of the bootstrapped CCF's is shown by the coloured shaded region.The vertical dashed and dotted lines indicate the recovered average lag and its uncertainty, as measured from the bootstrap distribution (coloured histogram).The black solid line and grey shaded area show the mean and 1 scatter of the CCF's generated using the randomised spectroscopic light curves following the procedure described in §3.Right column: The number of overlapping spectroscopic and photometric epochs as a function of time lag, in total for each source in the corresponding bin.The expected mean lag for the bin is indicated by the black dashed line.

Figure 5 .
Figure 5.The luminosity bins for the Mg ii sample, labelled with the number of binned sources.Within each bin, the standard deviation of the expected lags for the individual sources (measured using the source luminosity and the Trakhtenbrot & Netzer (2012)  −  relation for Mg ii) is ∼20% of the expected mean lag for the binned sample.

Figure 6 .
Figure 6.Left column:The coloured solid lines are the stacked cross correlation functions (CCF) for each of the three luminosity bins for the Mg ii sample.The colours correspond to the respective bins in Figure5.The 1 scatter of the bootstrapped CCF's is shown by the coloured shaded region.The vertical dashed and dotted lines indicate the recovered average lag and its uncertainty, as measured from the bootstrap distribution (coloured histogram).The black solid line and grey shaded area show the mean and 1 scatter of the CCF's generated using the randomised spectroscopic light curves following the procedure described in the text.Right column: The number of overlapping spectroscopic and photometric epochs as a function of time lag, in total for each source in the corresponding bin.The expected mean lag for the bin is indicated by the black dashed line.

Figure 9 .
Figure 9. Left column:The coloured solid lines are the stacked cross correlation functions (CCF) for each of the three luminosity bins for the C iv sample.The colours correspond to the respective bins in Figure8.The 1 scatter of the bootstrapped CCF's is shown by the coloured shaded region.The vertical dashed and dotted lines indicate the recovered average lag and its uncertainty, as measured from the bootstrap distribution (coloured histogram).The black solid line and grey shaded area show the mean and 1 scatter of the CCF's generated using the randomised spectroscopic light curves following the procedure described in the text.Right column: The number of overlapping spectroscopic and photometric epochs as a function of time lag, in total for each source in the corresponding bin.The expected mean lag for the bin is indicated by the black dashed line.

Figure B2 .
Figure B2.The same as Figure 3 after excluding sources with individual lag recoveries from the stacked sample.

Table 1 .
The average lags for each luminosity bin or multi-line sample, for each emission-line sample from the OzDES RM Program.Luminosities are given in erg s −1 , and average lags are in the rest frame.NOIRLab, the University of Nottingham, The Ohio State University, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, Texas A&M University, and the OzDES Membership Consortium.Based in part on observations at Cerro Tololo Inter-American Observatory at NSF's NOIRLab (NOIRLab Prop.ID 2012B-0001; PI: J. Frieman), which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation.The DES data management system is supported by the National Science Foundation under Grant Numbers AST-1138766 and AST-1536171.The DES participants from Spanish institutions are partially supported by MICINN under grants ESP2017-89838, PGC2018-094773, PGC2018-102021,SEV-2016-0588, SEV-2016- 0597, and MDM-2015-0509, someof which include ERDF funds from the European Union.IFAE is partially funded by the CERCA program of the Generalitat de Catalunya.Research leading to these re-sults has received funding from the European Research Council under the European Union's Seventh Framework Program (FP7/2007-2013) including ERC grant agreements 240672, 291329, and 306478.We acknowledge support from the Brazilian Instituto Nacional de Ciência e Tecnologia (INCT) do e-Universo (CNPq grant 465376/2014-2).This manuscript has been authored by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics.

Table A1 .
Properties for our OzDES H stacking sample.Columns left to right: DES name (J2000), redshift, -band apparent AB magnitude, monochromatic luminosity at 5100Å.The superscript  flags sources which also have Mg ii data.