The characteristic timescale at which the variability of active galactic nuclei AGNs turns from red noise to white noise can probe the accretion physics around supermassive black holes SMBHs. A number of works have studied the characteristic timescale of quasars and obtained quite different scaling relations between the timescale and quasar physical properties. One possible reason for the discrepancies is that the characteristic timescale can be easily underestimated if the light curves are not long enough. In this work we construct welldefined AGN samples to observationally test the relationships between the characteristic timescale and AGN properties obtained by previous works. Our samples eliminate the effects of insufficient lightcurve lengths. We confirm that the timescale predictions of the Corona Heated Accretion disk Reprocessing model are consistent with our timescale measurements. The timescale predictions by empirical relations are systematically smaller than our measured ones. Our results provide further evidence that AGN variability is driven by thermal fluctuations in SMBH accretion disks. Future flagship timedomain surveys can critically test our conclusions and reveal the physical nature of AGN variability.
['Accretion', 'Light curves', 'Quasars', 'Supermassive black holes', '14', '918', '1319', '1663', 'Astrophysics - Astrophysics of Galaxies', 'Astrophysics - High Energy Astrophysical Phenomena']
How Long Will the Quasar UVOptical Flickering Be Damped II. The Observational Test
{'How Long Will the Quasar UV/Optical Flickering Be Damped? II. the Observational Test': '<!-- image -->  \n1  \nDepartment of Astronomy, Xiamen University, Xiamen, Fujian 361005, China; msun88@xmu.edu.cn 2 Department of Astronomy, University of Science and Technology of China, Hefei 230026, China 3 School of Astronomy and Space Science, University of Science and Technology of China, Hefei 230026, China', 'ABSTRACT': 'The characteristic timescale at which the variability of active galactic nuclei (AGNs) turns from red noise to white noise can probe the accretion physics around supermassive black holes (SMBHs). A number of works have studied the characteristic timescale of quasars and obtained quite different scaling relations between the timescale and quasar physical properties. One possible reason for the discrepancies is that the characteristic timescale can be easily underestimated if the light curves are not long enough. In this work, we construct well-defined AGN samples to observationally test the relationships between the characteristic timescale and AGN properties obtained by previous works. Our samples eliminate the effects of insufficient light-curve lengths. We confirm that the timescale predictions (Zhou et al. 2024) of the Corona Heated Accretion disk Reprocessing model are consistent with our timescale measurements. The timescale predictions by empirically relations (e.g., Kelly et al. 2009) are systematically smaller than our measured ones. Our results provide further evidence that AGN variability is driven by thermal fluctuations in SMBH accretion disks. Future flagship timedomain surveys can critically test our conclusions and reveal the physical nature of AGN variability.  \nKeywords: Accretion (14); Light curves (918); Quasars (1319); Supermassive black holes (1663)', '1. INTRODUCTION': "Active Galactic Nuclei (AGNs) are the brightest persistent sources in the Universe. They emit non-stellar continuum emission in a broad range of electromagnetic spectrum from radio to X-ray and γ -ray (for related reviews, see, e.g., Urry & Padovani 1995; Ulrich et al. 1997). The strong continuum emission, which varies significantly on timescales of minutes, hours, days, and months, is usually believed to originate in the central supermassive black hole (SMBH) accretion disk. The central engine of an AGN is often too compact to resolve directly because of its small physical scale and high redshift. The multi-band inherent and complex variations (i.e., light curves) play an essential role in probing the properties of the central engine, and the physical processes of AGNs (e.g., Cackett et al. 2021).  \nThe study of AGN multi-band light curves provides valuable information about the AGN structure and  \nmsun88@xmu.edu.cn  \nphysical properties. By measuring the time lags between light curves at different wavelengths, the sizes of the accretion disk, broad line region, and dusty torus, and even the black hole mass of the SMBH can be measured using the reverberation mapping (RM) technique (e.g., Blandford & McKee 1982; Peterson 1993; Cackett et al. 2021). The black hole masses of more than 100 AGNs have been measured by the RM technique in many previous works (e.g., Kaspi et al. 2000; Peterson et al. 2004; Kaspi et al. 2007; Denney et al. 2010; Xiao et al. 2011; Grier et al. 2012; Bentz et al. 2013; Barth et al. 2015; Edelson et al. 2015; Shen et al. 2015; Fausnaugh et al. 2016; Grier et al. 2017; Du et al. 2018; Lu et al. 2022; Pandey et al. 2022; Chen et al. 2023; Malik et al. 2023; Zastrocky et al. 2024). Some AGNs are extremely variable on scales of months or years. These highly variable AGNs may be recognized as changing-look AGNs (CLAGNs) and challenge our understanding of the SMBH accretion (e.g., LaMassa et al. 2015; Yang et al. 2018; MacLeod et al. 2019; Guo et al. 2020; Wang et al. 2024; Zhu et al. 2024).  \nAlthough many previous works studied AGN UV/optical variability (e.g., MacLeod et al. 2010; Gard-  \nner & Done 2017; Lawrence 2018; Hagen et al. 2024; Kammoun et al. 2024), the physical mechanisms that cause AGN UV/optical variations are not yet fully clear. A number of theoretical models are proposed. For instance, Dexter & Agol (2011) proposed the strongly inhomogeneous quasar accretion disk model and assumed that the temperatures vary independently around the corresponding effective temperature of the standard thin accretion disk (hereafter SSD; Novikov & Thorne 1973; Shakura & Sunyaev 1973) in each independent disk zone. This model can explain the observed half-light radii from quasar microlensing observations and the observed variability amplitude. Cai et al. (2018) proposed that the local independent temperature fluctuations are affected by the large-scale temperature fluctuation. This model can reproduce the inter-band correlation and time lag of the UV/optical bands and the timescale-dependent bluer-when-brighter color variation. Krolik et al. (1991) proposed the X-ray reprocessing model (i.e., the lamppost model) to explain the time lags of multi-band light curves of AGNs. The X-ray reprocessing model has been adopted to explain continuum RM (e.g., Cameron et al. 2012; Shappee et al. 2014; Noda et al. 2016; Pahari et al. 2020; Fian et al. 2024). Sun et al. (2020) proposed the Corona Heated Accretion disk Reprocessing (CHAR) model to describe the AGN UV/optical variability. The CHAR model can yield the dependence of UV/optical variability on AGN luminosity, black hole mass, and wavelength.  \nThe light curves of quasars can be studied by spectral techniques that include power spectra (e.g., Kelly et al. 2009). The stochastic variability (i.e., the light curves) of quasars can be well fitted by the damped random walk (DRW) model (e.g., Kelly et al. 2009; Kozglyph[suppress]lowski et al. 2010; MacLeod et al. 2010; Dexter & Agol 2011; Zu et al. 2013; Suberlak et al. 2021), albeit some optical light curves of AGNs diverge from this model (e.g., Mushotzky et al. 2011; Graham et al. 2014; Kelly et al. 2014). According to the DRW model, the power spectral density (PSD) of an optical light curve is well described as a power law function P ( f ) = f -2 at the high-frequency end, and white noise at the lowfrequency end. The transition frequency between the high-frequency end and the low-frequency end is f 0 = 1 / (2 πτ ), where τ represents the theoretical/intrinsic damping timescale that characterizes the light curve. Alternative models (e.g., the damped harmonic oscillator, a.k.a., DHO; Moreno et al. 2019) are proposed to fit quasar UV/optical light curves. Given the data quality, it is often true that the model parameters cannot be simultaneously well constrained in models that are more complicated than the DRW model. Further-  \ne, the damping timescale of the DRW model is related to the timescales of the DHO model (e.g., Yu et al. 2022). Hence, there are still observational efforts desired to measure the damping timescale.  \nSeveral previous works obtained the relationships between the theoretical/intrinsic damping timescale ( τ ) in the rest frame and several physical parameters of AGNs, including the wavelength ( λ ), monochromatic luminosity ( L λ ), and SMBH mass ( M BH ). Kelly et al. (2009) analyzed 100 quasars' optical light curves, and found that,  \nτ days =(80 . 4 +66 . 9 -35 . 8 )( λL λ 10 45 erg s -1 ) -0 . 42 ± 0 . 28 × ( M BH 10 8 M ⊙ ) 1 . 03 ± 0 . 38 . (1)  \nMacLeod et al. (2010) obtained τ for ∼ 9000 spectroscopically confirmed quasars' 10-year light curves in Sloan Digital Sky Survey (SDSS) Stripe 82 and reported the relationship between τ and λ , and M BH , i.e., τ ∝ λ 0 . 17 M 0 . 21 BH . Suberlak et al. (2021) analyzed 9248 quasars' 15-year light curves by combining the Panoramic Survey Telescope and Rapid Response System 1 Survey data and SDSS Stripe 82 light curves, and reported the correlations between τ and several AGN physical parameters, including M BH , λ , and the absolute i -band magnitude ( M i ), i.e., log 10 ( τ/ days) = 2 . 59 +0 . 02 -0 . 02 + 0 . 17 +0 . 02 -0 . 02 log 10 ( λ rest / 4000 ˚ A) + 0 . 035 +0 . 007 -0 . 007 ( M i + 23) + 0 . 14 +0 . 02 -0 . 02 log 10 ( M BH / 10 9 M ⊙ ). They stressed that the DRW parameters could be constrained better by a longer baseline. Burke et al. (2021) measured the damping timescale of 67 AGNs, and obtained the relationship between τ and M BH , i.e., τ = 107 +11 -12 days ( M BH 10 8 M ⊙ ) 0 . 38 +0 . 05 -0 . 04 . They proposed that the AGN SMBH mass can be estimated using this relation. Stone et al. (2022) studied the optical variability of 190 quasars in SSDS Stripe 82 with 20-year photometric light curves, and obtained a weak wavelength dependence of τ ∝ λ 0 . 2 . Zhang et al. (2024) obtained τ for 34 blazars and 7 microquasars from the Fermi-Large Area Telescope and the XMM-Newton X-ray telescope, respectively. They obtained the relation between τ and M BH , i.e., τ = 120 +15 -18 days ( M BH 10 8 M ⊙ ) 0 . 57 +0 . 02 -0 . 02 .  \nThe CHAR model (Sun et al. 2020) can be used to predict the relationship between τ and AGN properties. Indeed, Zhou et al. (2024) investigated the damping timescale in the CHAR model and predicted a new relation between the theoretical/intrinsic damping timescale and the AGN properties, i.e.,  \nlog 10 ( τ/ days) =0 . 65 log 10 ( M BH /M ⊙ ) + 0 . 65 log 10 ˙ m +1 . 19 log 10 ( λ rest / ˚ A) -6 . 04 ,  \n(2)  \nwhere ˙ m represents the Eddington ratio (i.e., the ratio of the bolometric luminosity to the Eddington luminosity). Why are the relations obtained from theory and observation inconsistent? Is the discrepancy real or caused by some systematic bias?  \nThe length of a light curve (i.e., the baseline) must be much longer than the theoretical/intrinsic damping timescale when fitting the light curve with the DRW model. Otherwise, the damping timescale obtained from an insufficiently long light curve will be strongly underestimated (e.g., Kozlowski 2017; Suberlak et al. 2021; Hu et al. 2024). Zhou et al. (2024) emphasized that the measured damping timescale can still be significantly biased even if it is less than 10% of the baseline; note that this requirement is often adopted in observational studies; hence, these observational studies still obtained biased damping timescales. Instead, the correct criterion to ensure that the measured timescale is unbiased is that the intrinsic/expected (rather than measured) damping timescale is less than 10% of the baseline.  \nIn this paper, we constructed unbiased samples based on the relationships between the theoretical/intrinsic damping timescale and the AGN properties obtained from Kelly et al. (2009) (i.e., Equation (1)) and Zhou et al. (2024) (i.e., Equation (2)). We stress that all sources in our samples satisfy the criterion that the theoretical/intrinsic damping timescale (rather than the measured timescale) is less than 10% of the baseline. Hence, the measured damping timescales should be unbiased if the relationship of Kelly et al. (2009) or Zhou et al. (2024) is correct. Then, we can use the measured damping timescale to test the two works critically. This manuscript is formatted as follows. In Section 2, we describe the sample construction procedures. In Section 3, we introduce the data analysis. In Section 4, we present the results. In Section 5, we discuss the possible implications of our results. Conclusions are made in Section 6.", '2. OBSERVATIONS AND SAMPLE SELECTION': "In this work, the quasar optical light curves are obtained from the Zwicky Transient Facility 1 (ZTF; Bellm et al. 2019; Graham et al. 2019; Masci et al. 2019) and several references. Quasar physical parameters (e.g., black hole mass, luminosity, and redshift) are obtained from the second data release of the Swift BAT AGN Spectroscopic Survey 2 (BASS DR2; Koss et al. 2022a,b).  \n2.1. Physical parameters  \nThe initial sample is obtained from the BASS DR2 (Koss et al. 2022b) which provide the 858 hard-X-rayselected AGNs in the Swift BAT 70-month sample. This sample provides the key properties for each AGN, e.g., R.A. ( α J2000 ) and Declination. ( δ J2000 ); AGN type defined by optical spectroscopy, including Sy1 (broad H β emission line), Sy1.9 (narrow H β and broad H α emission lines), and Sy2 (narrow H β and H α emission lines). For most of Sy1 AGNs, the redshifts ( z ) are measured by [O iii ] λ 5007 or broad emission lines (e.g., broad Mg ii and C iv ; see Koss et al. 2022b). The redshifts of narrow-line sources are obtained by the emission-line [O iii ] λ 5007 (e.g., Koss et al. 2022b); the majority of black hole masses are calculated by either broad H β or stellar velocity dispersion measurements. The spectra used for the above measurements were observed by the Palomar Hale 5 m telescope Double Beam Spectrograph (DBSP) or the Very Large Telescope (VLT) X-shooter spectrograph (e.g., Koss et al. 2022a). The bolometric luminosity ( L bol ) is calculated by the 14-150 keV intrinsic luminosity with the bolometric correction of 8, and assuming the photon index Γ = 1.8. In this work, we focus on Seyfert 1 AGNs ('Type' = Sy1). There are 359 Seyfert 1 AGNs out of the 858 hard-X-ray-selected AGNs.", '2.2. Light curves': "The light curves of the 359 Seyfert 1 AGNs are obtained from ZTF. ZTF is a time-domain survey using the 48-inch Samuel Oschin Telescope of Palomar Observatory. The entire northern visible sky has been scanned at the optical band (including three custom filters, zg , zr , and zi ) by the ZTF camera every 2 days since 2018, and the camera has a 47 deg 2 field with 600 megapixels. The effective wavelengths of the zg , zr , and zi bands are 4753, 6369, and 7915 ˚ A, respectively (Rodrigo et al. 2012; Rodrigo & Solano 2020). The ZTF data are released every 2 months, and the baselines of current g and r band light curves are ∼ 1800 days. Such highquality ZTF light curves, including the long baseline and the high cadence, are very suitable for time-domain science, such as studying quasar optical variability. The zi band light curves are not utilized in this work because their baselines are not long enough for our research. In this work, we retrieved the zg and zr light curves for each source in our sample from ZTF DR21 via the IRSA service 3 , and the cross-matching radius is 1 '' . The observations are obtained from all CCDs. For each light  \ncurve, we removed observations with poor quality (i.e., 'catflags' > 0).", '2.3. Sample selection': 'As emphasized by several previous works (e.g., Kozlowski 2017; Suberlak et al. 2021; Kozlowski 2021; Hu et al. 2024; Zhou et al. 2024), the damping timescale is significantly biased if the light-curve baseline is less than ten times of the theoretical damping timescale. As we have stressed in Section 1, the commonly adopted criterion, which is that the baseline is at least ten times longer than the best-fitting damping timescale, cannot eliminate this bias (also see Zhou et al. 2024). Hence, in the following two sections, we aim to construct samples according to our new criterion, i.e., the theoretical damping timescale should be less than 10% of the baseline. As a result, the sample selection depends critically upon the damping timescale relation we aim to test in this work.', '2.3.1. Sample selection based on Kelly et al. (2009)': "In this section, we aim to construct a sample to test the damping timescale relation of Kelly et al. (2009), i.e., Equation 1. Given that the rest-frame baseline for ZTF light curves is about 1800 / (1 + z ) days ( where z represents the redshift), we expect that the measured damping timescale can be unbiased if the theoretical damping timescale is less than 180 / (1 + z ) days (i.e., less than 10% of the baseline). This requirement puts a strong constraint on M BH , luminosity, and redshift. The black hole mass of each source can be obtained from the BASS DR2 (see the description in Section 2.1). Sources without mass estimates ('logMBH' = 0) are removed. The monochromatic luminosity at 5100 ˚ A ( L 5100 ) of each source can be calculated as follows, i.e., L 5100 = L bol / 9 . 26 (e.g., Richards et al. 2006; Shen et al. 2011). For each source, we use M BH , L bol , and Equation 1 to calculate the theoretical damping timescale ( τ th , Kelly+2009 ). Only sources with τ th , Kelly+2009 < 180 / (1 + z ) days are selected. There are 142 sources in the zg band and 142 sources in the zr band of the 359 Seyfert 1 that merit this requirement and are selected for subsequent analysis, respectively.", '2.3.2. Sample selection based on Zhou et al. (2024)': "Zhou et al. (2024) reported the relationship between τ and AGN properties (i.e., M BH , ˙ m , and λ rest ), i.e., Equation (2). The dimensionless accretion ratio ˙ m = L bol / ((1+ k ) × L edd ), k = 1 3 represents the ratio between the magnetic fluctuations' power of the corona and the dissipation rate of the disk turbulent magnetic power (for details, see Sun et al. 2020), where L edd = 1 . 26 ×  \n10 38 × ( M BH /M ⊙ ) erg s -1 is the Eddington luminosity. Hence, Equation (2) can be expressed in the form of  \nlog 10 ( τ/ days) = 0 . 65 log 10 ( L bol / erg s -1 ) +1 . 19 log 10 ( λ rest / ˚ A) -30 . 9 . (3)  \nWe calculate the theoretical damping timescale ( τ th , Zhou+2024 ) of Zhou et al. (2024) for each of the 359 Seyfert 1 AGNs in BASS DR2. Then, we again only select sources with τ th , Zhou+2024 < 180 / (1+ z ) days. There are 70 and 38 sources of the 359 Seyfert 1 that meet our requirement for the zg band and zr band, respectively.", '2.4. Other sources obtained from the literature': "Several narrow-line Seyfert 1 galaxies' light curves have been analyzed in the literature. The Kepler light curve ( λ = 6600 ˚ A) of Zw 229-15 has been analyzed by Chen & Wang (2015). The black hole mass of Zw 229-15 is log 10 ( M BH /M ⊙ ) = 6 . 91 and the redshift is z = 0 . 0275 (Koss et al. 2022b). The Swift UVOT UVM 2 band ( λ = 2600 ˚ A) light curve of WPVS 007 has been analyzed by Li et al. (2019), whose M BH = 4 . 1 × 10 6 M ⊙ (Leighly et al. 2015), and z = 0 . 02882 (Grupe et al. 1995). The Swift UVOT UVW 2 band ( λ = 2120 ˚ A) light curve of Mrk 335 has been analyzed by Griffiths et al. (2021); the black hole mass and redshift of Mrk 335 are log 10 ( M BH /M ⊙ ) = 7 . 32 and z = 0 . 0259, respectively (Koss et al. 2022b). Burke et al. (2021) reported the characteristic optical variability timescale of 67 AGNs, and 6 sources (including NGC 4395, NGC 5548, DES J021822.51-043036.0, SDSS J025007.03+002525.3, SDSS J153425.58+040806.7, and SDSS J160531.85+174826.3) of 67 AGNs satisfy the condition τ th , Kelly+2009 / baseline < 10%, and τ th , Zhou+2024 / baseline < 10%. Five sources were added to our sample; NGC 4395 has been reduced since this target is included in our Swift BAT sample. Zhou et al. (2024) measured the damping timescale for 3 local AGNs (see Table 5 in Zhou et al. 2024), including NGC 4151, NGC 7469, and NGC 3516. These sources were added to our sample. We confirm that our main conclusions remain unchanged if these sources were excluded in our subsequent analysis.", '3. DATA ANALYSIS: LIGHT CURVE FITTING': "AGN light curves can be well fitted by the DRW model based on the taufit 4 code (Burke et al. 2021). This code is built upon the fast Gaussian process solver celerite (Foreman-Mackey et al. 2017). celerite fits the time series to the following specified kernel function using the Gaussian process regression  \nFigure 1. Fitting the zg band light curve (in the observed-frame) of Z 493-2 with the DRW model via the taufit code. Top panel: the solid red curve and shaded regions are the best-fitting DRW model and the 3 σ confidence intervals. The gray and black data points correspond to the observations that lie outside or inside the shaded regions, respectively. Bottom-left panel: the posterior distributions of the DRW parameters obtained in our new DRW fit (i.e., the DRW fit with the black data points in the top panel). The parameter σ DRW is the long-term variability amplitude, τ DRW is the damping timescale, and σ n is the excess white noise amplitude. The medians of the posterior distributions are adopted as the best-fitting parameters, and the 1 σ uncertainties are obtained by the 16-th and 84-th percentiles of the posterior distributions. The red shaded regions in the bottom-left panel correspond to where τ DRW is greater than 10% of the baseline. Bottom-right panel: the gray curve is the PSD of the data (the black data points in the top panel), and the DRW PSD (the red solid curve) and its 1 σ uncertainties (the orange shaded regions). The red-shaded regions represent the regions of frequency space not sampled by the light curve, including the regions with timescales shorter than the minimum cadence and the timescales longer than 20% of the baseline.  \n<!-- image -->  \n=  \n+0.04  \n<!-- image -->  \nlog10 ( DRW)  \n0.48  \n0.03  \nk ( t nm ) = 2 σ 2 DRW exp( -t nm /τ DRW ) + σ 2 n δ nm , (4)  \nwhere t nm = | t n -t m | represents the time interval between two measurements ( t n , t m ) of the light curve. σ DRW represents the long-term standard deviation of variability. τ DRW defines the damping timescale. σ n is the excess white noise amplitude, which is added to the DRW kernel to account for the possible underestimation of the photometric uncertainties. σ 2 n represents the variance of the noise component. δ nm is the Kro-  \nr delta function. The prior distribution of τ DRW was set to be a uniform distribution whose lower and upper bounds are the minimum cadence to the 10 times the baseline of each light curve, respectively. The bestfitting τ DRW is obtained by maximizing the posterior probability. The joint posterior probability distributions of the DRW parameters from celerite are obtained by the Markov Chain Monte Carlo (MCMC) code sampler emcee (Foreman-Mackey et al. 2013) implemented in Python. The median values of the posterior distributions are adopted as the best-fitting parameters, and  \nFigure 2. The redshift distributions of the samples in this work. Left panel: the selected sources with theoretical damping timescales, which are calculated from Equation 1 (Kelly et al. 2009), smaller than 10% of the light curve durations. The redshift ranges from 0.001 to 0.823, and the median redshift is 0.030 (the black dashed line). Right panel: the selected sources with theoretical damping timescales, which are calculated from Equation 2 (Zhou et al. 2024), smaller than 10% of the light curve durations. The redshift ranges from 0.001 to 0.823, and the median redshift is 0.016 (the black dashed line).  \n<!-- image -->  \n16-th and 84-th percentiles of the posterior distributions are adopted as the 1 σ uncertainties for the bestfitting parameters. We compare the Akaike Information Criterion (AIC; Akaike 1998) of this fit (AIC best ) with the AICs of two alternative fits. The first alternative fit (AIC low ) is that we set τ DRW to be an extremely small value (0 . 02 days, i.e., about one-hundredth of the typical ZTF cadence); the second (AIC up ) is that we set τ DRW to be an extremely large value (18000 days, i.e., about 100 times of the typical ZTF baseline). The AIC of each fit is calculated as -ln(Likelihood) + 2 N , where N = 3 is the number of parameters. Statistically speaking, if the AIC difference between two models is larger than 10, the model with a larger AIC value has relatively little support (e.g., Burnham et al. 2011). Hence, we propose that the damping timescale is well constrained only if ∆AIC up = AIC up -AIC best > 10 and ∆AIC low = AIC low -AIC best > 100 (the larger ∆AIC corresponds to the better constraint of the damping timescale).  \nIn this section, the light curve(in the observed-frame) of each source mentioned in Section 2 is fitted by the above method. A DRW fit example of Z 493-2 is shown in Fig. 1. The top panel shows the original light curve of Z 493-2, and the predicted light curve from the DRW model obtained by the best-fitting parameters. Note that, we removed the observations outside the 3 σ of the predicted light curve (the gray points in the top panel), and then repeated the fitting process until all observations were included within 3 σ of the predicted light curve obtained by the DRW model. The bottom left panel shows the best-fitting DRW parameters (i.e.,  \nσ DRW , τ DRW , and σ n ) obtained by the taufit . The bottom right panel shows the PSD of the original light curve (the gray curve), and the PSD of the light curve obtained from the posterior distribution of the celerite fit (the red curve). The model PSD is shown in the orange region (with 1 σ uncertainty).  \nIn summary, our sample selection criteria are summarized as follows.  \n- · The theoretical damping timescale obtained from Kelly et al. (2009) (i.e., τ th , Kelly+2009 calculated by Equation 1) or Zhou et al. (2024) (i.e., τ th , Zhou+2024 calculated by Equation 3) should be smaller than the 10% baseline in the rest frame, i.e., τ th , Kelly+2009 < 10% × baseline / (1 + z ), or τ th , Zhou+2024 < 10% × baseline / (1 + z );\n- · Following Burke et al. (2021), the best-fitting damping timescale obtained by the DRW model fitting ( τ obs ) is larger than 1.5 times cadence, i.e., τ obs > 1 . 5 × cadence;\n- · Following Burke et al. (2021), the source should have the high signal-to-noise ratio, i.e., σ 2 DRW > σ 2 n +d y 2 , where d y is the median magnitude err of each light curve;\n- · As described in Section 3, in order to ensure that the damping timescales of the sources in our sample are indeed constrained by the data, we require that ∆AIC low > 100, and ∆AIC up > 10.  \nThe key parameters of each source in our sample include baseline, cadence (the median cadence of each  \nFigure 3. The relationship between the damping timescales (log 10 ( τ obs ); rest-frame values) obtained from the DRW fittings and the theoretical predictions (log 10 ( τ th , Kelly+2009 )) of Kelly et al. (2009) (i.e., Equation 1). The cyan and orange solid dots indicate the damping timescales obtained by the zg and zr band light curves, respectively. The blue and purple dots represent the damping timescales obtained from Burke et al. (2021) and Zhou et al. (2024), respectively. The dark green star, diamond, and square indicate the damping timescales of Mrk 335, Zw 229-15, and WPVS 007, respectively. For more information on these sources, see Table A1. The gray point (ESO 424-12 zr band) was not considered in our analysis because of its unusually small measured damping timescale (the DRW fitting result of ESO 424-12 zr band light curve is shown in Figu. B1). The vertical solid gray line represents 10% of the baseline of the ZTF light curve. The solid black line indicates a one-to-one line, and the black dashed line indicates the uncertainties of 0.2 dex. The reduced χ 2 between log 10 ( τ th , Kelly+2009 ) and log 10 ( τ obs ) is 3.86, and the Kendall's correlation coefficient is ρ = 0.15 ( p -value = 1 . 68 × 10 -3 ). Most of the data points are located above the one-to-one line. That is, log 10 ( τ th , Kelly+2009 ) significantly underestimates the observed damping timescales.  \n<!-- image -->  \nlight curve), τ obs (i.e., obtained by the DRW fit), σ DRW , σ n , L bol , M BH , λ , and z (see Tables A1 and A2). The theoretical damping timescale ( τ th , Kelly+2009 and τ th , Zhou+2024 ) of our sample can be calculated by Equation 1 and Equation 3. Based on the damping timescale predicted by Kelly et al. (2009) (i.e., Equation 1) and the above criteria, 112 sources (including 187 light curves) were included in our final sample (hereafter sample K) to test the Kelly et al. (2009) relation. More details of these sources are listed in Table A1. Meanwhile, 49 sources (including 68 light curves) were included in our final sample (hereafter sample Z) based on the damping timescale predicted by Zhou et al. (2024); i.e., Equation 3 and the above conditions. More details of these sources  \nFigure 4. The relationship between the damping timescales (log 10 ( τ obs ); rest-frame values) obtained from DRW fitting and the theoretical predictions (log 10 ( τ th , Zhou+2024 )) of Zhou et al. (2024) (i.e., Equation 3). The meaning of the data points and lines in this figure is the same as in Fig. 3. The triangles represent the damping timescales of the high and low states of zg and zr of the CL AGN NGC 5273, respectively. For more information on these sources, see Table A2. The reduced χ 2 between log 10 ( τ th , Zhou+2024 ) and log 10 ( τ obs ) is 1.81, and the Kendall's correlation coefficient is ρ = 0.41 ( p -value = 9 . 23 × 10 -7 ). The observations and the theoretical predictions of Zhou et al. (2024) are roughly consistent.  \n<!-- image -->  \nare listed in Table A2. The redshift distribution of these sources ranges from 0.001 to 0.823 (see Fig. 2), with medians of 0.030 and 0.016, respectively. All sources in the sample are low redshift AGNs.", '4.1. Observed damping timescales vs. theoretical predictions': "The relationship between the damping timescales (log 10 ( τ obs )) obtained from the DRW fitting and the theoretical predictions (log 10 ( τ th , Kelly+2009 )) of Kelly et al. (2009) (i.e., Equation 1) is shown in Fig. 3. There are 187 data points: 93 light curves obtained from the zg band (i.e., the cyan solid dots), 83 light curves obtained from the zr band (i.e., the orange solid dots), 5 data points obtained Burke et al. (2021) (i.e., the blue solid dots), 3 data points obtained from Zhou et al. (2024) (i.e., the purple solid dots), and 3 green dots (Mrk 335, Zw 229-15, and WPVS 007) are obtained from the literature (i.e., Griffiths et al. 2021; Chen & Wang 2015; Li et al. 2019). We found that there are unusually dense observations for the zr band light curve of ESO 424-12, which leads to the damping timescale of this light curve being unusually small, i.e., about 1 day (the  \n<!-- image -->  \nFigure 5. Left panel: the same as Fig. 3 but for sample C, which consists of sources that satisfy the criteria according to Kelly et al. (2009) (i.e., Equation 1) and Zhou et al. (2024) (i.e., Equation 3). Right panel: the same as Fig. 4 but for sample C. Again, observations are more consistent with the theoretical predictions of Zhou et al. (2024) than those of Kelly et al. (2009).  \n<!-- image -->  \nDRW fitting result of ESO 424-12 zr band light curve is shown in Fig. B1). We reject this light curve from our sample. Note that our conclusion remains unchanged if this target is included. The vertical solid gray line represents 10% of the baseline of the ZTF light curve (i.e., ZTFbaseline / 10 days). It is important to emphasize that all sources in our sample satisfy the theoretically predicted damping timescale being less than 10% of its baseline in the rest frame. As shown in Fig. 3, the vast majority of data points (184 of 187) lie to the left of the vertical dashed gray line. There are three data points located to the right of the vertical line because their light curves are not obtained from the ZTF. Their light curves have baselines much larger than 1800 days. We calculated the reduced χ 2 between log 10 ( τ th , Kelly+2009 ) and log 10 ( τ obs ) to be 3.86 and found that the Kendall's correlation coefficient to be ρ = 0 . 15 with a p -value of 1 . 68 × 10 -3 . The reduced χ 2 is χ 2 reduced = 1 K -n ∑ K i =1 ( (log 10 ( τ obs ,i ) -log 10 ( τ th , Kelly+2009 ,i )) 2 /σ 2 i ) = 3 . 86, where K is the total number of data points, n = 0 is the number of free parameters in Equation 1, log 10 ( τ obs ,i ) is the value at the i -th observed damping timescale, log 10 ( τ th , Kelly+2009 ,i ) is the value at the i -th predicted damping timescale, and σ i is the error associated with the i -th observed damping timescale. The large reduced χ 2 and small correlation coefficient suggest that the measured timescales cannot be well described by Equation 1 (Kelly et al. 2009). Indeed, most of the points are located above the one-to-one line. That is, log 10 ( τ th , Kelly+2009 ) significantly underestimates the observed damping timescales.  \nSimilarly, the relationship between the damping timescales (log 10 ( τ obs )) obtained from the DRW fitting  \nand the theoretical predictions (log 10 ( τ th , Zhou+2024 )) of Zhou et al. (2024) (i.e., Equation 3) is shown in Fig. 4. There are 68 data points: 35 are obtained from the zg band (i.e., the cyan solid dots), 22 are obtained from the zr band (i.e., the orange solid dots), 5 data points are obtained from Burke et al. (2021) (i.e., the blue solid dots), 4 data points are obtained from Zhou et al. (2024) (i.e., the purple solid dots), and 2 green dots (Mrk 335 and WPVS 007) are obtained from the literature (i.e., Griffiths et al. 2021; Li et al. 2019). NGC 5273 is a CL AGN, and four light curves of the high and low states of this source are studied separately. We again removed ESO 424-12 from our sample. As shown in Fig. 4, the vast majority of data points (65 of 68) lie to the left of the vertical dashed gray line. There are three data points located to the right of the vertical line because their light curves are not obtained from the ZTF. We calculate the reduced χ 2 between log 10 ( τ th , Zhou2024 ) and log 10 ( τ obs ) to be 1.81 and the Kendall's correlation coefficient to be ρ = 0 . 41 with a p -value of 9 . 23 × 10 -7 . This indicates that the measured timescales can be well described by the relation of Zhou et al. (2024). It can be noticed that over half of the data points (38 of 68) lie within 0.2 dex of the uncertainty of the one-to-one line. That is, the observations and the theoretical predictions of Zhou et al. (2024) are roughly consistent.  \nWe also construct a new sample (hereafter sample C) by selecting sources whose theoretical predictions according to Kelly et al. (2009) (i.e., Equation 1) and Zhou et al. (2024) (i.e., Equation 3) satisfy our criteria (see Section 3); that is, sample C is an intersection of the sample in Fig. 3 and the sample in Fig. 4. There are 38 sources (including 54 light curves) in sample C. The  \nFigure 6. The probability distributions of the parameters a and b in the fitting relation log 10 ( τ obs ) = a log 10 ( L bol ) + 1 . 19 log 10 ( λ rest ) + b . Bottom-left panel: the joint probability distributions of a and b . Top-left panel: the distribution of a . The median value of the distribution is adopted as the best-fitting a , and the gray region is the 1 σ uncertainties of a . The vertical solid orange line indicates the same parameter predicted by Zhou et al. (2024), which is 0 . 65. Bottomright panel: the distribution of b . The median value of the distribution is adopted as the best-fitting b , and the gray region is the 1 σ uncertainties of b . The vertical dashed orange line represents the same parameter predicted by Zhou et al. (2024), which is -30 . 9. Our best-fitting parameters and the theoretical predictions of Zhou et al. (2024) are statistically consistent.  \n<!-- image -->  \nrelationship between the theoretical/intrinsic damping timescales (log 10 ( τ obs )) obtained from the DRW fittings and the theoretical predictions (log 10 ( τ th )) of sample C are shown in Fig. 5. The reduced χ 2 between log 10 ( τ th ) of Equation 1 and log 10 ( τ obs ) is 5 . 92 (the left panel of Fig. 5), and the Kendall's correlation coefficient is ρ = 0 . 25 with a p -value of 8 . 96 × 10 -3 . There are less than half of the data points (22 of 54) lie within 0.2 dex of the uncertainty of the one-to-one line, and most of the points are located above the one-to-one line. Meanwhile, the reduced χ 2 between log 10 ( τ th ) of Equation 3 and log 10 ( τ obs ) is 1 . 79 (the right panel of Fig. 5), and the Kendall's correlation coefficient is ρ = 0 . 41 with a p -value of 1 . 69 × 10 -5 . More than half of the data points (32 of 54) lie within 0.2 dex of the uncertainty of the one-to-one line. In summary, the measured damp-  \nes can be better described by the relation of Zhou et al. (2024) rather than Kelly et al. (2009).", '4.2. The dependencies of τ obs upon L bol and λ rest': 'As described in Section 1, some previous works reported the established empirical relations between the rest frame damping timescale and several physical parameters. The empirical relations between τ obs and L bol , and λ rest are also constructed in this section. The adopted sample is the same as in Fig. 4 because the observations are more consistent with the theoretical predictions of Zhou et al. (2024) than that of Kelly et al. (2009). The fitted equation is  \nlog 10 ( τ obs days ) = a log 10 ( L bol erg s -1 ) +1 . 19 log 10 ( λ rest ˚ A ) + b. (5)  \nThe best-fitting parameters are obtained by minimizing the χ 2 statistic. The χ 2 statistic is χ 2 = ∑ (( τ obs , i -τ model , i ) 2 / (( τ err , obs , i ) 2 + ( a × δ log 10 L bol , i ) 2 )), where τ err , obs , i is the 1 σ uncertainty of τ obs , i , and δ log 10 L bol , i represents the typical 0.2 dex uncertainty of log 10 L bol for each source. We use bootstrap with replacements to obtain the statistical distributions of a and b . The median values of the bootstrap distributions are adopted as the best-fitting parameters, and the 1 σ uncertainties are taken as 16-th to 84-th percentiles of the distributions. The results are shown in Fig. 6, with the best-fitting parameters with 1 σ uncertainties being a = 0 . 72 +0 . 18 -0 . 13 and b = -34 . 2 +5 . 7 -7 . 9 (i.e., the histograms and gray shaded areas in the upper left and lower right panels). The solid orange lines indicate the parameters (i.e., a Zhou+2024 = 0 . 65, b Zhou+2024 = -30 . 9) reported by Zhou et al. (2024). The best-fitting parameters obtained in this work and those of Zhou et al. (2024) are statistically consistent. Note that the parameters of the wavelength term are fixed to 1 . 19 according to Zhou et al. (2024). This is because most of the sources in our sample are from the ZTF and the wavelengths in the zg and the zr bands are close, i.e., ( λ r -λ g ) / ( λ r + λ g ) = 0 . 15. Hence, the current sample data are unable to robustly constrain the dependence of the damping timescale upon wavelength.', '5. DISCUSSIONS': 'The observed damping timescales ( τ obs ) in this work are more consistent with the theoretical predictions of Zhou et al. (2024) than that of Kelly et al. (2009). Compared with previous observational studies, we carefully select sources to ensure that the damping timescales are not biased. In order to obtain the unbiased damping  \ntimescales, the theoretical/intrinsic damping timescale rather than the measured one should be smaller than the 10% baseline. We construct two well-defined samples in this work, and each source in our samples follows the condition that the theoretical damping timescale (i.e., τ th , Kelly+2009 or τ th , Zhou+2024 ) is smaller than the 10% baseline in the rest frame, i.e., τ obs in this work are unbiased. Our measured damping timescales are inconsistent with the relationship of Kelly et al. (2009) (as shown in Section 4.1, and Fig. 3), possibly because this relationship is biased. In addition, Burke et al. (2021) also established a similar relationship between the damping timescale and the black hole mass. Zhou et al. (2024) has shown that the sample of Burke et al. (2021) is biased.  \nWe obtained the dependencies of τ obs upon L bol and λ rest by our well-defined sample, i.e., log 10 ( τ obs ) = 0 . 72 +0 . 18 -0 . 13 log 10 ( L bol ) + 1 . 19 log 10 ( λ rest ) -34 . 2 +5 . 7 -7 . 9 (as shown in Section 4.2, Fig. 4 and 6). There is a positive correlation between τ obs and L bol , and the coefficient ( a = 0 . 72 +0 . 18 -0 . 13 ) of L bol term is close to that of Zhou et al. (2024) ( a Zhou+2024 = 0 . 65), and the parameter b = -34 . 2 +5 . 7 -7 . 9 is also close to that of Zhou et al. (2024) ( b Zhou+2024 = -30 . 9). Hence, this suggests that the origin of AGN UV/optical variations can be described by the CHAR model; that is, the UV/optical variations of AGN are caused by the thermal fluctuations triggered by turbulence in the accretion disk, whose characteristic timescale depends mostly upon the mass accretion rate (or luminosity). In addition, as shown in Fig. 4, the CLAGN NGC 5273 has different damping timescales in the low and high states. Therefore, our results suggest that the damping timescale might be used to measure the luminosity rather than the black hole mass of quasars.  \nThe unavoidable magnetohydrodynamic (MHD) turbulence plays an important role in the SMBH accretion process as it is responsible for removing the angular momentum of the accreted gas (e.g., Balbus & Hawley 1991; Balbus 2003). The MHD turbulence in the accretion disk can drive significant variability in the UV/optical emission of the accretion disks. Zhou et al. (2024) demonstrated that in the MHD turbulence scenario, the UV/optical luminosity should vary on the thermal timescale, and the damping timescale is the average of thermal timescales at different radii of the accretion disk. Dwarf galaxies exhibit flux variations on short timescales because of their low luminosities. That is, the damping timescales of dwarf galaxies can be well constrained even for short-duration light curves. Hence, dwarf galaxies are a promising class of targets that can be used to study the MHD turbulence-driven thermal fluctuations in accretion disks.', '6. CONCLUSION': 'We have constructed unbiased samples in which each source in our sample satisfies the criterion that the intrinsic/expected damping timescale is smaller than 10% of the baseline. We have obtained the best-fit parameters between the damping timescales ( τ obs ) and the bolometric luminosity ( L bol ) and the wavelength in the rest frame ( λ rest ), i.e., log 10 ( τ obs ) = 0 . 72 +0 . 18 -0 . 13 log 10 ( L bol ) + 1 . 19 log 10 ( λ rest ) -34 . 2 +5 . 7 -7 . 9 . Our results are roughly consistent with those of Zhou et al. (2024) which is based on the CHAR model (Sun et al. 2020). This agreement supports the idea that the observed UV/optical variability is due to thermal fluctuations in SMBH accretion disks. It is important to note that these samples are unable to robustly test the dependence of the damping timescale upon wavelength. In the future, sources with a larger wavelength range are needed to better test our conclusion and put strong constraints on the SMBH accretion physics.', '7. ACKNOWLEDGMENTS': 'G.W.R. would like to thank Dr. Q.Z. Yu for helpful discussions. We acknowledge support from the National Key R&D Program of China (No. 2023YFA1607903). G.W.R., S.Y.Z., and M.Y.S. acknowledge support from the National Natural Science Foundation of China (NSFC-12322303) and the Natural Science Foundation of Fujian Province of China (No. 2022J06002). Y.Q.X. acknowledges support from the National Natural Science Foundation of China (NSFC-12025303).  \nBased on observations obtained with the Samuel Oschin Telescope 48 inch and the 60 inch Telescope at the Palomar Observatory as part of the Zwicky Transient Facility project (IRSA 2022). Z.T.F. is supported by the National Science Foundation under grants No. AST-1440341 and AST-2034437 and a collaboration including current partners Caltech, IPAC, the Weizmann Institute for Science, the Oskar Klein Center at Stockholm University, the University of Maryland, Deutsches Elektronen-Synchrotron and Humboldt University, the TANGO Consortium of Taiwan, the University of Wisconsin at Milwaukee, Trinity College Dublin, Lawrence Livermore National Laboratories, IN2P3, University of Warwick, Ruhr University Bochum, Northwestern University, and former partners the University of Washington, Los Alamos National Laboratories, and Lawrence Berkeley National Laboratories. Operations are conducted by COO, IPAC, and UW.  \n(Hunter 2007), Numpy (Harris et al. 2020), Scipy (Virtanen et al. 2020), taufit (Burke et al. 2021).', 'A. RESULTS OF DRW FITTING TO THE LIGHT CURVE SAMPLES IN THIS WORK': 'Table A1 . Sample with τ th , Kelly+2009 / baseline < 10%  \nContinued on next page  \nTable A1 . Sample with τ th , Kelly+2009 / baseline < 10%  \nContinued on next page  \nTable A1 . Sample with τ th , Kelly+2009 / baseline < 10%  \nContinued on next page  \nTable A1 . Sample with τ th , Kelly+2009 / baseline < 10%  \nContinued on next page  \nTable A1 . Sample with τ th , Kelly+2009 / baseline < 10%  \nColumn notes : Col. (1): Source name (sources with * represent objects that are not selected from the Swift BAT sample); Col. (2): The redshift; Col. (3): The black hole mass; Col. (4): The bolometric luminosity; Col. (5): Band Col. (6): The rest-frame wavelength; Col. (7): The rest-frame best-fitting damping timescale; Col. (8): The theoretical damping timescale; Col. (9): The baseline in the rest-frame; Col. (10): The τ th , Kelly+2009 -to-baseline ratio. The redshift, black hole mass, and the bolometric luminosity are adopt from BASS DR2 (Koss et al. 2022a,b). The samples with the same source name and the different λ rest correspond to the different band light curves of the same source. In comparison with Table A2, log 10 ( τ obs ) for the same light curve may be slightly different because log 10 ( τ obs ) in Tables A1 and A2 were obtained from the MCMC code. Similarly, the baseline for the same light curve may be slightly different due to the slightly different outlier rejections in each DRW MCMC fit.  \nTable A2 . Sample with τ th , Zhou+2024 / baseline < 10%  \nContinued on next page  \nTable A2 . Sample with τ th , Zhou+2024 / baseline < 10%  \nContinued on next page  \nTable A2 . Sample with τ th , Zhou+2024 / baseline < 10%  \nColumn notes : Col. (1): Source name (sources with * represent objects that are not selected from the Swift BAT sample); Col. (2): The redshift; Col. (3): The black hole mass; Col. (4): The bolometric luminosity; Col. (5): Band Col. (6): The rest-frame wavelength; Col. (7): The rest-frame best-fitting damping timescale; Col. (8): The theoretical damping timescale; Col. (9): The baseline in the rest-frame; Col. (10): The τ th , Kelly+2009 -to-baseline ratio. The redshift, black hole mass, and the bolometric luminosity are adopted from BASS DR2 (Koss et al. 2022a,b). The samples with the same source name and the different λ rest correspond to the different band light curves of the same source. In comparison with Table A1, log 10 ( τ obs ) for the same light curve may be slightly different because log 10 ( τ obs ) in Tables A1 and A2 were obtained from the MCMC code. Similarly, the baseline for the same light curve may be slightly different due to the slightly different outlier rejections in each DRW MCMC fit.  \nFigure B1. Fitting the zr band light curve (in the observed-frame) of ESO 424-12 with the DRW model via the taufit code. The meaning of the data points and lines in this figure are the same as in Fig. 1. 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