The variability structure function of the highest luminosity quasars on short time-scales

ABSTRACT

The stochastic photometric variability of quasars is known to follow a random-walk phenomenology on emission time-scales of months to years. Some high-cadence rest-frame optical monitoring in the past has hinted at a suppression of variability amplitudes on shorter time-scales of a few days or weeks, opening the question of what drives the suppression and how it might scale with quasar properties. Here, we study a few thousand of the highest luminosity quasars in the sky, mostly in the luminosity range of |$L_{\rm bol}$||$=[46.4, 47.3]$| and redshift range of |$z=[0.7, 2.4]$|⁠. We use a data set from the NASA/Asteroid Terrestrial-impact Last Alert System facility with nightly cadence, weather permitting, which has been used before to quantify strong regularity in longer term rest-frame-UV variability. As we focus on a careful treatment of short time-scales across the sample, we find that a linear function is sufficient to describe the UV variability structure function. Although the result can not rule out the existence of breaks in some groups completely, a simpler model is usually favoured under this circumstance. In conclusion, the data are consistent with a single-slope random walk across rest-frame time-scales of |$\Delta t=[10, 250]$| d.

1 INTRODUCTION

In the last two decades, the field of quasar variability has made tremendous progress due to the availability of large time-domain surveys (e.g. Morganson et al. 2014; Caplar, Lilly & Trakhtenbrot 2017; Li et al. 2018; Suberlak, Ivezić & MacLeod 2021). These data sets are hoped to reveal correlations between variability behaviour and physical quasar properties to reveal the underlying physics behind the variability of quasar accretion discs (e.g. Ulrich, Maraschi & Urry 1997; Peterson 2001; Padovani et al. 2017; Cackett, Bentz & Kara 2021). The variability could be caused by the turbulence that is driven by magnetorotational instability (MRI; Balbus & Hawley 1991), and in addition the disc will respond to variable heating from the X-ray corona near the inner disc, an effect known as reprocessing or lamp-post effect (e.g. Clavel et al. 1992; Shappee et al. 2014; Hernández Santisteban et al. 2020; Cackett et al. 2021).

A common technique to study quasar variability in the UV/optical domain is structure function (SF; Hughes, Aller & Aller 1992) analysis: this is suitable for analysing observed light curves (LC) with non-uniform sampling, which is often a natural consequence of weather and other constraints on observing. The variability SF of an LC shows how observed brightness changes on average as a function of time interval. The detailed mathematical background of SF analysis is described very well by Kozłowski (2016).

If the brightness change of the quasar is truly stochastic, i.e. there is no correlation of a present change with any of the past changes, the behaviour can be described as the random walk (RW). According to the RW model, the power-law slope of the SF is expected to be 0.5 (Kelly, Bechtold & Siemiginowska 2009), indicating the source of variability is caused by disc instability (Takeuchi, Mineshige & Negoro 1995; Kawaguchi et al. 1998). This 0.5 slope is confirmed by observations (e.g. Rengstorf, Brunner & Wilhite 2006; Caplar et al. 2017; Li et al. 2018; Tang, Wolf & Tonry 2023) on time-scales |$\gtrapprox$|100 d. Likewise, studies that expressed the variability as a power spectral densities (PSDs) show a slope of |$-2$| in the PSD, which is mathematically equivalent to a slope of 0.5 in the SF (e.g. Giveon et al. 1999; Kelly et al. 2009).

However, Mushotzky et al. (2011) found the slope on time-scales of a few days seems to be significantly steeper than 0.5, indicating the existence of a break. Several studies that cover a broad range of time-scales have found breaks on short time-scales (e.g. Edelson et al. 2014; Kelly et al. 2014; Simm et al. 2016; Smith et al. 2018; Stone et al. 2022), motivating attempts to explain this feature in the SF (e.g. Sun et al. 2020; Tachibana et al. 2020) or to model quasar LCs including this effect (e.g. Kelly et al. 2014; Kasliwal, Vogeley & Richards 2017; Moreno et al. 2019; Yu et al. 2022). However, other studies that also probe short time-scales find no evidence of breaks (e.g. Morganson et al. 2014; Caplar et al. 2017), casting doubt on whether there is any slope change or not.

The damped random walk (DRW; Kelly et al. 2009) model predicts that the variability at the time-scales longer than the RW regime becomes uncorrelated with the time-scale, producing a flatten slope 0 in SF. The breaking time-scale indicating this turnover is often referred to as characteristic time-scale or decorrelation time-scale. To clarify, we focus on the break shorter than this break in this work.

Previously, we have used the NASA/Asteroid Terrestrial-impact Last Alert System (ATLAS; Tonry et al. 2018) survey to study the quasar variability, as it has proven to be one of few very rich data sets. Tang et al. (2023, hereafter Paper I) performed the SF analysis on optical ATLAS LCs of |$\sim 5000$| brightest quasars and showed that the RW relation is universal in variability amplitude when time is expressed in units of orbital or thermal time-scale of the emitting portion of the accretion disc, seemingly independent of physical parameters that are not encapsulated in the disc time-scales. This has emphasized the similarity between quasar accretion discs and pointed to evidence that MRI may be related to the root cause of the variability. While that work has shown breaks on short time-scales, its analysis was focused on the longer term RW. In this work, we aim to deepen our analysis of the short time-scales of the same LCs from NASA/ATLAS. In Section 2, we provide a description of the sample and data. We outline the method employed to calculate and fit the SF in Section 3 and show the result in Section 4. Throughout our analysis, we adopt a flat Lambda cold dark matter cosmology with |$\Omega _\Lambda = 0.7$| and |$H_0 = 70$| km s|$^{-1}$| Mpc|$^{-1}$|⁠. Additionally, we use Vega magnitudes for data from Gaia and AB magnitudes for data from NASA/ATLAS and for absolute magnitude estimates.

2 DATA AND SAMPLE

We use the LCs from the analysis of the SF in Paper I, but reduce the sample somewhat to objects with good-quality spectral fitting and black holes mass estimates from Rakshit, Stalin & Kotilainen (2020). Paper I had selected 6163 spectroscopically confirmed, bright (Gaia magnitude |$G_{\rm RP}\lt 17.5$|⁠), redshift |$0.5\lt z\lt 3.5$|⁠, non-lensed¹(Lemon, Auger & McMahon 2019; Lemon et al. 2023), and isolated quasars from the Million Quasars Catalogue (MILLIQUAS v7.1 update; Flesch 2015) matched with the Gaia eDR3 catalogue (Gaia Collaboration 2021). These quasars have declination |$\delta \gt -45^\circ$| to match the sky coverage of NASA/ATLAS and Galactic foreground reddening (Schlegel, Finkbeiner & Davis 1998) of |$E(B-V)_{\rm SFD}\lt 0.15$|⁠. To avoid photometric contamination from nearby sources, the Gaia BpRp Excess Factor was limited to 1.3. Paper I obtained the LCs from the NASA/ATLAS database between 2015 to 2021 in two passbands, orange (6785 Å) and cyan (5330 Å), whose magnitudes are denoted by |$m_{\rm o}$| and |$m_{\rm c}$|⁠, respectively. They rejected quasars with noisy LCs, i.e. where the 90-percentile flux errors in the LCs were above |$\sigma _{f_\nu , {\rm o}} \gt 150\mu$| or |$\sigma _{f_\nu , {\rm c}} \gt 85\mu$|Jy.

Paper I rejected radio-loud quasars as their optical variability might be contaminated by processes outside of the accretion disc. They cross-matched the sample with catalogues from the Faint Images of the Radio Sky at Twenty-cm (FIRST; Becker, White & Helfand 1995) Survey, the NRAO VLA Sky Survey (NVSS; Condon et al. 1998), and the Sydney University Molonglo Sky Survey (SUMSS; Mauch et al. 2003). To account for the different spatial resolution of the radio surveys, they matched the radii between the Gaia coordinates and the FIRST, NVSS, and SUMSS coordinates with 3, 12, and 11 arcsec, respectively. When multiple radio sources in NVSS or SUMSS were matched to one quasar, the one with the closest separation was used. The radio-loudness criterion for matches with NVSS and SUMSS magnitudes (⁠|$t_{\rm NVSS/SUMSS}$|⁠) was |$0.4\times (m_{\rm o}-t_{\rm NVSS/SUMSS})=1.3$|⁠, below which radio-detected objects were labelled as radio-intermediate. All quasars with matches in FIRST but not in NVSS were radio-intermediate, too. After that, 5315 quasars were left over.

Paper I cleaned the NASA/ATLAS LCs by excluding all observations with large errors of |$\log (\sigma _{f_\nu , {\rm c}})\gt -4.17-0.10\times (m_{\rm c}-16.5)$| and |$\log (\sigma _{f_\nu , {\rm o}})\gt -3.94-0.12\times (m_{\rm o}-16.5)$|⁠; and further removed outliers by comparing each measurement with other observations within |$\pm$|7 d and then using a |$2\sigma$|-clipping technique to reject spurious outliers. Observations were retained if they are isolated within |$\pm$|7 d. Altogether, 11  per cent of the data points are removed by the cleaning. They found that data at rest-frame wavelengths |$\lambda _{\rm rf}$||$\gtrapprox 3000$| Å do not follow the simple mean relation, and they also avoided wavelengths that were short enough to be contaminated by Ly|$\alpha$| emission. After discarding LCs from affected passbands, they were left with 4724 quasars.

Here, we match the 4724 quasars with the Rakshit et al. (2020) catalogue to obtain quasar properties derived from the SDSS DR14 spectral fitting. Using a separation radius within 1 arcsec, we find 3253 matches. We adopt the black hole mass, |$M_{\rm BH}$|⁠, measured from the Mg ii line and calibrated with the Vestergaard & Osmer (2009) relation in the Rakshit et al. (2020) catalogue. There are 2750 quasars with good |$M_{\rm BH}$| estimates, i.e. LOG_MBH_MGII_VO09 > 0 and QUALITY_MBH = 0. Among this sample, 13 of them have their cyan data contaminated by the strong Ly |$\alpha$| line. After excluding them, we keep 2737 quasars for further analysis.

We adopt luminosities at rest-frame 3000 Å (⁠|$L_{3000}$|⁠) calculated in Paper I. They used Gaia|$G_{\rm BP}$| and |$G_{\rm RP}$| magnitudes to interpolate |$L_{3000}$| although at redshift |$z\gt 2$| this becomes a mild extrapolation beyond the coverage of the |$G_{\rm RP}$| bandpass. They also correct for Milky Way (Schlegel et al. 1998) foreground dust extinction with bandpass coefficients |$R_{\rm G_{BP}}=3.378$| and |$R_{\rm G_{RP}}=2.035$| (Casagrande & VandenBerg 2018). They finally apply an offset of −0.07 dex after comparing their |$L_{3000}$| with the values in Rakshit et al. (2020) that are derived from spectral decomposition into continuum and emission-line contributions. Bolometric luminosities, |$L_{\rm bol}$|⁠, were derived using |$\log {({L_{\rm bol}}/{L_{3000}})}=\log {(f_{\rm {\small BC}} \times \lambda)}=\log {(5.15\times 3000{{\mathring{\rm A}}})}=4.189$|⁠, where |$f_{\rm {\small BC}}$| is the bolometric correction factor (Richards et al. 2006).

3 METHODS

3.1 Variability structure functions

The SF analysis has been developed to quantify the observed variability in an LC, especially for the common cases of uneven sampling (Hughes et al. 1992). In this work, we adopt the noise-corrected definition of variability amplitude from di Clemente et al. (1996):

where |$m_{\rm i}$| and |$m_{\rm j}$| are any two observed apparent magnitudes and |$\sigma$| is the noise due to magnitude error.

This work focuses on short-term variability, where an accurate handling of the noise is critical. Thus, we will be testing two different methods to estimate the mean error |$\langle \sigma ^2 \rangle$|⁠. The first method follows the procedure in Paper I. Their |$\langle \sigma ^2 \rangle$| was determined as a function of observed magnitude, |$m_{\rm obs}$|⁠, such that the SF for |$\Delta t \lt 1$| d becomes |$A\approx 0$| (Kozłowski 2016). This strategy was chosen because the typical amplitude of intraday variability in radio-quiet quasars is so small that it is within the uncertainty of the noise itself and will not noticeably affect the analysis of the RW portion in the SF. They fitted a noise model to the ATLAS data using

for |$\Delta t \lt 1$| d of their sample and found |$(n_0, n_1)= (-12.411, 0.585)$| and |$(-12.428, 0.573)$| for the orange and cyan passbands, respectively. Applying the same model for the 2750 quasars in this work, we find |$(n_0, n_1)=(-12.683, 0.600)$| and |$(-12.360, 0.569)$| for the orange and cyan passbands, respectively. For reference, the left panel of Fig. 1 shows the median magnitude error, |$\sigma _{\rm mag,med}$|⁠, of those 2750 quasars. For |$m_{\rm obs}=(16, 17.5)$|⁠, |$\sigma _{\rm mag,med}$||$=(0.021, 0.059)$| for orange passband and |$(0.016, 0.043)$| for cyan passband. They are comparable to the first definition of |$\langle \sigma ^2 \rangle$| after taking the squared value times 2. After applying the noise subtraction with these parameters, we find that the term in the square root of equation (1) is negative for about two-thirds of the intraday magnitude pairs instead of the 50 per cent expected for pure noise without true variability. Thus, we choose a second definition of |$\langle \sigma ^2 \rangle$| following again the concept of setting the SF for |$\Delta t \lt 1$| d to |$A\approx 0$|⁠; however, this time, we do not try to fit a global magnitude-dependent function of the form above to the whole sample. Instead, we choose |$\langle \sigma ^2 \rangle$| to be the median value of the |$(\pi /2)$||$\langle \Delta m \rangle ^2$| for |$\Delta t \lt 1$| d in each subsample group defined in Section 3.2, in order to force set the recovered variability amplitude to zero at short times. In the right panel of Fig. 1, we show the comparison between the |$\langle \sigma ^2 \rangle$| defined using these two methods. If this approach was inappropriate because of true intraday variability being present and inflating the signal, it would bias the inferred amplitude lower especially at lowest amplitudes, i.e. at shortest time-scales; hence, such an effect would enhance an apparent suppression of short-term amplitude in the SF, but it could not cause a bias towards reducing any suppression we find.

3.2 Binning

We split the 2737 quasars within the redshift range of |$0.698 \lt z \lt 2.4$| into five redshift groups with roughly the same size in number of quasars. Within each z bin, we further divide the quasars into three |$L_{3000}$| groups. Moreover, within each |$L_{3000}$| bin, we divide the quasars into three |$M_{\rm BH}$| bins. Each (z, |$L_{3000}$|⁠, |$M_{\rm BH}$|⁠) group contains a similar number of quasars. The boundaries of z, |$L_{3000}$|⁠, and |$M_{\rm BH}$| of each bin is shown in Table 1. The SF is evaluated as a function of the intrinsic (rest-frame) time intervals, |$\Delta t$|⁠, and represented as |$\log {A(\Delta t)}$|⁠. Within each (z, |$L_{3000}$|⁠, |$M_{\rm BH}$|⁠) group as well as each z group of quasars, we divide the observational pairs into 50 |$\Delta t$| bins. The shortest bin with |$\Delta t\lt 1$| is used for noise analysis and thus forced to |$A=0$|⁠; the remaining bins are balanced in terms of the number of pairs. This operation is carried out separately for the orange and cyan data. Overall, there are a total of |$2\, 250$| bins in (z, |$L_{3000}$|⁠, |$M_{\rm BH}$|⁠, |$\Delta t$|⁠) and 300 bins in (z, |$\Delta t$|⁠) for each passbands. We represent the centre of each |$\Delta t$| bin by the average |$\log {\Delta t}$| among the pairs in each bin. In total, there are 1.0 and 0.08 billion observational pairs for the orange and cyan passband, respectively, used in the following analysis.

3.3 Simple bin average structure function

For each |$\Delta t$| bin of each quasar, we calculate an SF amplitude, A, with equation (1) using all available pairs. We calculate the magnitude differences, |$\langle \Delta m \rangle$|⁠, using the 3|$\sigma$|-clipped mean among the pairs, and the noise, |$\langle \sigma ^2 \rangle$|⁠, using the second definition in Section 3.1. Then, for each (z, |$L_{3000}$|⁠, |$M_{\rm BH}$|⁠, |$\Delta t$|⁠) bin and each (z, |$\Delta t$|⁠) bin, we calculate the median squared amplitude, |$A^2$|⁠, and the standard error of the median among quasars, which ensures that quasars with negative noise-corrected amplitudes |$A^2$|⁠, due to oversubtraction by the |$\langle \sigma ^2 \rangle$|⁠, are still included in the statistics. Finally, both the amplitude and its error are converted into log space for the following fitting.

3.4 Fitting

We conduct Levenberg–Marquardt least-squares fits (Markwardt 2009) to the |$A(\Delta t)$| of each (z, |$L_{3000}$|⁠, |$M_{\rm BH}$|⁠) and each z group of quasars for two models to examine whether there is a break or not.

The first model is a linear equation:

The |$A_{0,\rm lin}$| is the constant and |$\gamma _{\rm lin}$| is the slope. We perform two fits under this model, one with a free slope |$\gamma _{\rm lin}$|⁠, and one with a fixed RW slope of |$\gamma _{\rm lin}$||$=\gamma _{\rm RW}=0.5$|⁠.

The second model is a smoothed piecewise function:

The |$A_{\rm 0,pw}$| is the constant, the |$t_{\rm brk}$| is where the |$\Delta t$| having the break, the |$\gamma _{\rm 1,pw}$| and |$\gamma _{\rm 2,pw}$| are the slopes at shorter and longer time-scales, respectively, and the |$\alpha _{\rm pw}$| is the smoothing parameter. To simplify the comparison with the first model, we fix |$t_{\rm brk}$||$=30$| d, |$\gamma _{\rm 1,pw}$||$=0.75$|⁠, |$\gamma _{\rm 2,pw}$||$=\gamma _{\rm RW}=0.5$|⁠, and |$\alpha _{\rm pw}=5$| in the fitting of the second model.

The shorter time-scale end of the fitting range is decided based on the purpose, while the longer time-scale end is chosen to exclude possible window effects.

4 RESULTS AND DISCUSSION

4.1 Break

We show the SF result under (z, |$L_{3000}$|⁠, |$M_{\rm BH}$|⁠) grouping with their fits of orange and cyan data in Figs 2 and 3, respectively. We first look at the SFs across this variety of group to make sure that we are not missing any breaks that may depend on mass or luminosity and get washed out in an overall SF for the whole sample. To determine whether the break exists, we compare the fixed slope linear fit and the smoothed piecewise fit under each (z, |$L_{3000}$|⁠, |$M_{\rm BH}$|⁠) group. The fitting range is |$0.2\lt \log {(\Delta t)}\lt 2.4$| so that their performance on the shorter time-scale can be considered. Since the two models are nested, i.e. one model can be obtained simply by fixing parameters in the other model (if |$\gamma _{\rm 1,pw}$||$=0.5$| in the smoothed piecewise model, it becomes identical with the fixed slope linear model), we can judge which model is a better one by comparing their |$\chi _\nu ^2$| values. The model with a significantly smaller |$\chi _\nu ^2$| is certainly a better fit. If they are comparable, then they are probably equally good. The result shows that most of the groups have comparable |$\chi _\nu ^2$| values between those two fits in the orange SF, while more than half of the groups have significant better |$\chi _\nu ^2$| value for the fixed linear model fit in the cyan SF. None of the groups show a significantly better |$\chi _\nu ^2$| value for the smoothed piecewise model fit. Although this result cannot rule out the smoothed piecewise model completely, a simpler model is usually favoured under this circumstance. Therefore, we think the linear function should be sufficient to describe the SF behaviour in our data and no break is needed.

We then try to reduce the noise by constructing fewer groups with more objects by marginalizing over luminosity and mass, and examine the SF with grouping only in redshift, i.e. rest-frame wavelength range, in Fig. 4. The fitting range here is |$1.4\lt \log {(\Delta t)}\lt 2.4$| to capture the main RW portion of the SF and exclude any possible short-term suppression that has been reported previously. The dashed line at shorter time-scales extrapolates the fit and appears very consistent with the data, suggesting an RW relation can describe the data without hints of a break over the full range of |$\Delta t=[10;250]$| d. Below that time-scale, our results are consistent with a continued RW of the same slope, but the noise prevents us from claiming more detail. Fig. 3 of Paper I had shown breaks on the short time-scale, which we now argue are most likely due to oversubtraction of the noise. However, the noise subtraction and short-|$\Delta t$| behaviour do not affect their main scientific result on longer time-scales (⁠|$\gt 1$| month).

Early works suggesting the |$\gamma$| slope change from 0.5 to |$\gt 1$| with breaks between 5 to 50 d were carried out on high-cadence LCs from the Kepler mission for a handful of individual quasars (e.g. Edelson et al. 2014; Smith et al. 2018). Further works using ground-based telescopes to obtain LCs for |$\sim 100$| to |$\sim 200$| quasars found breaks between 30 to 300 d (e.g. Simm et al. 2016; Stone et al. 2022). We note, that the decorrelation time-scale in the DRW model should produce a break that is distinct from a short-time-scale break we study here. In theory, it is expected at a much longer time-scale and be a transition from an RW slope to a potentially flat slope with a saturated variability amplitude instead of a transition from a power-law slope to a steep slope going to shorter time intervals. However, if the SF does not feature sharp breaks but appears gradually curved, the interpretation of breaks becomes ambiguous (e.g. Kasliwal, Vogeley & Richards 2015; Arévalo et al. 2024). Our result shows neither breaks nor much curvature, but is consistent with other studies using samples of over |$10\, 000$| quasars (e.g. Morganson et al. 2014; Caplar et al. 2017), which show no strong signs of short-term suppression in a break on time-scales above a few days. Other than a possible oversubtraction of noise in the SF or PSD leading to short-term breaks, Moreno et al. (2021) discovered an unidentified instrumental issue in Kepler data that may affect measurements of stochastic variability beyond repair. Hence, we do not know for sure at this stage where to trust or re-analyse Kepler data on quasar variability. The jury thus seems to be out on where in the active galactic nucleus parameter space and why breaks in the SF due to a short-term amplitude suppression occur.

Of course, past observations of breaks on short time-scale prompted theoretical speculations of their origin. For example, Tachibana et al. (2020) proposed that an amplitude suppression of short-term variability could be caused by a kernel filtering effect. They convolved the PSD with different kinds of kernel functions and successfully reproduced breaks on short time-scales. If variability in a quasar disc is synchronized within any given annulus of the disc, and the disc is inclined relative to the line of sight, the broad range of light traveltime around a given annulus could act as the physical source of the filtering effect; this would then result in a correlation between the breaks and accretion disc size (projected onto the line of sight) for different quasars. However, an absence of breaks as in our observations does not imply limits on the sizes of discs, because it is not clear how synchronized any variability is and how strong the expected effect would be.

There is strong evidence for breaks on a time-scale of a few days in X-ray variability (e.g. Paolillo et al. 2023). While the X-ray emitting region is small enough so that light traveltime filtering is not to be blamed, the lamp-post model advocates that UV/optical variability is caused by X-ray variability, suggesting that an X-ray break would propagate into a UV break as well. However, recent magnetohydrodynamic simulations by Secunda, Jiang & Greene (2024) demonstrated that X-ray variability is not the principal driver of the UV/optical variability, and the PSD shape of the UV/optical variability is hardly affected by the X-ray behaviour.

4.2 Slope

Finally, in Fig. 5 we look for trends in the best-fitting slope |$\gamma$| with quasar properties (Table 1). The fitting range is chosen to be |$1.4\lt \log {(\Delta t)}\lt 2.4$| to avoid any possible short-term suppression. Although most of them are consistent with each other in Figs 2–4, we notice that the orange SF generally has slightly steeper slopes than the cyan data. The mean slopes and their errors for orange and cyan data are |$\gamma _{\rm o}=0.502\pm 0.007$| and |$\gamma _{\rm c}=0.412\pm 0.005$|⁠. Given that we took care to subtract the noise level such that we forced the result to be zero variability on a time-scale of rest-frame hours, it is hard to see how this could be due to wrong noise treatment. However, it is the case that groups with more massive |$M_{\rm BH}$| seem to have steeper slopes in Fig. 5 and also have lower noise levels than groups with less massive black holes. In line with that, no significant differences exist among the noise levels of different z or |$\lambda _{\rm rf}$| groups, and no strong dependence of the slope is seen with these parameters either. The challenge for a physical interpretation of different power-law slopes in the two passbands is that a fixed rest-frame wavelength appears in one band for one redshift and the other band for another redshift. This suggests that the slope is affected by an instrumental aspect in the SF analysis even though the noise level should have only a minor effect on a linear scale. This prevents us from further interpreting the slope with any physical properties of quasars. Note, that the slope differences observed here are modest and still statistically consistent with an RW slope of 0.5, which is supported by that fact that fixed |$\gamma _{\rm RW}=0.5$| fits in Figs 2–4 are consistent with the data points themselves.

5 CONCLUSIONS

We use five years of NASA/ATLAS data to study the short time-scales quasar variability in two optical passbands, cyan and orange. In our sample, we have more than 2700 quasars with the physical properties of z, |$L_{3000}$|⁠, and |$M_{\rm BH}$|⁠. According to these properties, we group the quasars into 45 (z, |$L_{3000}$|⁠, |$M_{\rm BH}$|⁠) groups and five z groups to perform the SF analysis. Using an improved definition of the noise level, we show that the model without breaks is sufficient all the way to a short time-scale of |$\sim$| 10 d, contradicting some of the previous observations and theoretical predictions. Although we cannot rule out a piecewise model completely, a simpler model is usually favoured under this circumstance. Our result is consistent with the RW prediction, suggested by the disc instability model.

ACKNOWLEDGEMENTS

We thank an anonymous referee for suggestions improving the manuscript. JJT was supported by the Taiwan Australian National University PhD scholarship, the Australian Research Council (ARC) through Discovery Project DP190100252, the National Science and Technology Council (MOST 111-2112-M-002-015-MY3), the Ministry of Education, Taiwan (MOE Yushan Young Scholar grant NTU-110VV007, NTU-110VV007-2, and NTU-110VV007-3), the National Taiwan University research grant (NTU-CC-111L894806, NTU-CC-112L894806, and NTU-CC-113L894806), and also acknowledges support by the Institute of Astronomy and Astrophysics, Academia Sinica (ASIAA). JT has been funded in part by the Stromlo Distinguished Visitor Program at RSAA. We thank I-Non Chiu and Jennifer I-Hsiu Li for suggestions improving the manuscript. This research has made use of idl.

This work uses data from the University of Hawaii’s ATLAS project, funded through NASA grants NN12AR55G, 80NSSC18K0284, and 80NSSC18K1575, with contributions from the Queen’s University Belfast, STScI, the South African Astronomical Observatory, and the Millennium Institute of Astrophysics, Chile.

This work has made use of SDSS spectroscopic data. Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support and resources from the Center for High Performance Computing at the University of Utah. The SDSS website is http://www.sdss.org. SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, Center for Astrophysics| Harvard and Smithsonian, the Chilean Participation Group, the French Participation Group, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU)/University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observatário Nacional/MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.

This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.

DATA AVAILABILITY

The data underlying this article will be shared on reasonable request to the corresponding author.

Footnotes

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