Faint AGNs Favor Unexpectedly Long Inter-band Time Lags

Active Galactic Nucleus (AGN) continua at various UV/optical bands vary coherently, and the long-wavelength emission usually lags the short-wavelength emission with time delays of days. Such short time delays are unexpected in the static Shakura & Sunyaev disk (SSD; Shakura & Sunyaev, 1973) model as the relevant radial propagation timescale (i.e., the viscous timescale) is hundreds to thousands of years. Instead, the inter-band cross correlations and time lags ($\tau$) are often understood in the framework of X-ray reprocessing (e.g., Krolik et al., 1991). In this scenario, the central compact X-ray corona can illuminate the disk surface and the absorbed X-ray emission is thermalized and reprocessed as UV/optical emission. The inter-band time lags account for the differences in the light-travel timescales from the corona to the emission regions of various UV/optical wavelengths.

Recent high-cadence multi-band observations of several Seyfert 1 AGNs suggest that the measured inter-band time lags are longer than the predictions of the X-ray reprocessing of a static SSD by a factor of $2$ – $3$ (e.g., Fausnaugh et al., 2016; Cackett et al., 2018; McHardy et al., 2018; Edelson et al., 2019). This result is further supported by Pan-STARRS observations (Jiang et al., 2017), which might, however, suffer from significant selection bias (see Appendix A of Homayouni et al., 2019). Possible explanations involve alternative reprocessors, e.g., SSDs with powerful winds which have flatter disk temperature profiles (Li et al., 2019; Sun et al., 2019), SSDs with non-blackbody disk emission (Hall et al., 2018), inhomogeneous SSDs with global temperature fluctuations (Cai et al., 2020; Sun et al., 2020a), or non-disk UV/optical continuum emission from the more extended broad-line clouds (Cackett et al., 2018; Lawther et al., 2018; Sun et al., 2018a; Chelouche et al., 2019; Korista & Goad, 2019).

In contrast, very recent light-curve studies of distant survey quasars (which in general are more luminous than the targeted studies of nearby Seyfert 1 AGNs) from the Dark Energy Survey (DES; e.g., Flaugher et al., 2015)-Australian DES (OzDES; e.g., Lidman et al., 2020) reverberation mapping project (Yu et al., 2020), the Sloan Digital Sky Survey Reverberation Mapping (SDSS-RM) project (Homayouni et al., 2019) and the quasar PG 2308+098 (Kokubo, 2018) suggest that the measured inter-band time lags are actually consistent with (or only slightly larger than) the predictions of the X-ray reprocessing of a static SSD.

Microlensing observations of quasars find oversized disks (Morgan et al., 2018), which are actually inconsistent with the time-lag observations of quasars with similar luminosities but agree with the time lags of fainter nearby Seyfert 1 AGNs.

These apparently conflicting observational results might be symptoms of our poor understanding of AGN accretion physics. Meanwhile, Tie & Kochanek (2018) point out that the accretion-disk size induced microlensing time lags add significant systematic uncertainties to gravitational-lensing time delay cosmology. Hence, determining the real accretion-disk sizes are vital for our understanding of AGN central-engine physics and measuring cosmological parameters via gravitational lensing of distant quasars.

Here, we collect a large set of inter-band time-lag measurements from different previously published studies to demonstrate that the ratios of the measured time lags to the expectations of the SSD model anti-correlate with AGN luminosity ($L$). That is, faint AGNs tend to have larger-than-expected inter-band time lags. This result is unexpected in the X-ray reprocessing of a static SSD (hereafter the lamp-post SSD model) and might provide critical clues to the AGN disk-size problem.

We use the inter-band time lags reported in previous studies to estimate the $\lambda=2500\ \mathrm{\AA}$ time lags (hereafter $\tau_{\mathrm{obs}}$); the estimation procedures are presented in Appendix A. Our sample consists of the available local AGNs listed in Table 1 and more distant quasars from DES standard star fields (Yu et al., 2020, hereafter Y20), SDSS-RM (Homayouni et al., 2019, hereafter H19) and PG 2308+098 (Kokubo, 2018). We exclude the data from Jiang et al. (2017) and Mudd et al. (2018) as their relevant cadences are sparser than for H19 or Y20 and thus the time lags are likely biased to larger values.

As a second step, we use the estimated black hole mass ($M_{\mathrm{BH}}$) and AGN bolometric luminosity ($L$) to calculate the expected $\lambda=2500\ \mathrm{\AA}$ static SSD lag (hereafter $\tau_{\mathrm{SSD}}$; see Eq. 7 of Fausnaugh et al., 2018),

where $f_{\mathrm{Edd}}$ is the ratio of $L$ to the Eddington luminosity. Note that this formula is valid only if the radiative efficiency is $\eta=0.1$; $L$ is estimated from the continuum luminosities at rest-frame $1350\ \mathrm{\AA}$, $3000\ \mathrm{\AA}$, or $5100\ \mathrm{\AA}$ (depending on redshift) with the bolometric corrections recommended by Richards et al. (2006). The factor $X$ is chosen to be $5.04$; that is, $\tau_{\mathrm{SSD}}$ corresponds to the average time lag of the variable flux (Tie & Kochanek, 2018) and is $1.5$ times larger than the flux-weighted light-travel time lag adopted in some previous studies (e.g., Fausnaugh et al., 2016). In the third step, the ratio ($\tau_{\mathrm{diff}}$) of the measured $\tau_{\mathrm{obs}}$ to $\tau_{\mathrm{SSD}}$ is obtained.

We find that $\tau_{\mathrm{diff}}$ anti-correlates with $L$ (see Figure 1), with a Spearman correlation coefficient $\rho_{\mathrm{spearman,obs}}=-0.78$ (the corresponding $p$-value is $3\times 10^{-4}$).¹¹1We use the local-AGN measurements and the median values of the sources from H19 and Y20 to calculate the Spearman coefficient unless otherwise specified. This is because, comparing with local AGN measurements, the results of H19 and Y20 have larger uncertainties; thus, we decide to use their median values. Nevertheless, we also calculate the corresponding Spearman coefficient by treating the local AGNs and the quasars from H19 and Y20 equally and find that the coefficient and the $p$-value are $-0.29$ and $3\times 10^{-3}$, respectively; that is, the anti-correlation still exists. We also confirm that $\tau_{\mathrm{diff}}$ anti-correlates with $M_{\mathrm{BH}}$ (the Spearman coefficient and the $p$-value are $-0.69$ and $4\times 10^{-3}$, respectively); however, the partial correlation between $\tau_{\mathrm{diff}}$ and $M_{\mathrm{BH}}$ is insignificant once $L$ is controlled (the Spearman partial-correlation coefficient, which is calculating via the package “ppcor” in R, is $-0.05$). Our results also suggest an apparent disagreement between the microlensing disk-size measurements and the inter-band time-lag observations: the former focus on luminous AGNs but find over-sized disks and the latter suggest that such AGNs have consistent-with-expectation time lags.

To test whether the lamp-post SSD model can explain our results or not, we perform the following experiment. First, we use the measured $M_{\mathrm{BH}}$ and $L_{\mathrm{bol}}$ to calculate the static SSD effective temperature ($T_{\mathrm{eff}}$) profile (we again adopt $\eta=0.1$). Second, we add the additional surface heating due to the X-ray illumination (which is modeled by a damped random-walk process) to obtain the fluctuations of $T_{\mathrm{eff}}$ and the corresponding mock UV/optical light curves (by integrating the SSD blackbody emission over the whole disk). Note that, the adopted UV/optical bands are the same as real observations. Third, the mock UV/optical inter-band time lags are estimated by utilizing Javelin to fit the mock light curves. Fourth, we calculate the corresponding mock time lags at $2500\ \mathrm{\AA}$ (hereafter $\tau_{\mathrm{SSD(sim)}}$) following the same recipes aforementioned. We repeat this experiment $200$ times to account for statistical fluctuations.

The median ratios (and their $1\sigma$ uncertainties) of $\tau_{\mathrm{SSD(sim)}}$ to $\tau_{\mathrm{SSD}}$ (i.e., Eq. 1) are shown in the left panel of Figure 2, which suggests that $\tau_{\mathrm{SSD(sim)}}$ is on average less than $\tau_{\mathrm{SSD}}$; similar results have been reported by Chan et al. (2020) who therefore propose that the accretion-disk sizes estimated via Javelin are underestimates by $\sim 30\%$ (or $0.15$ dex). However, such a bias cannot account for our results. Indeed, the anti-correlation in the left panel of Figure 2 is statistically insignificant (the Spearman coefficient and the $p$-value are $-0.38$ and $0.14$, respectively) and much weaker than the observed one. Moreover, a clear anti-correlation between the ratios of $\tau_{\mathrm{obs}}$ to $\tau_{\mathrm{SSD(sim)}}$ and $L$ holds (see the right panel of Figure 2; the Spearman coefficient and the $p$-value are $-0.77$ and $5\times 10^{-4}$, respectively).

Our results are unexpected in the lamp-post SSD model. Some alternative solutions are proposed to account for the AGN disk-size problem. Below, we discuss the physical implications of our results for these models.

Our results in Figure 1 suggest that microlensing observations of quasars (which often find larger-than-expected disk sizes) are, in fact, inconsistent with the time-lag observations of quasars with similar luminosities. Note that the inter-band time lags of a gravitationally lensed AGN 0957+561 were obtained by Gil-Merino et al. (2012). By interpreting the time lags as the variability-weighted ones (i.e., Eq. 1 with $X=5.04$), we use their UV/optical measurements and the methodology aforementioned to obtain the corresponding $r$-band half-light radius (i.e., $\log(r_{\mathrm{half}}(r)/\mathrm{cm})=15.82$). Recently, Cornachione et al. (2020) obtained its microlensing $\log(r_{\mathrm{half}}(r)/\mathrm{cm})=16.66^{+0.37}_{-0.62}$. The difference of the two results is $0.84$ dex albeit with a large uncertainty. Hence, any X-ray reprocessing models that actually enlarge the effective disk sizes might not explain this discrepancy.

In our CHAR model, the actual disk sizes are similar to the SSD model. In addition, the median time lag of our CHAR model is considerably smaller than the SSD model (see the left panel of Figure 3). Disk winds are probably common in luminous AGNs (e.g., Laor & Brandt, 2002; Ganguly et al., 2007; Laor & Davis, 2014) and can make the actual disk sizes larger than the no-wind SSD model (e.g., Sun et al., 2019). If we also consider the CHAR model with winds, the predicted median time lag approaches the SSD model and is more consistent with observations. At the same time, the windy CHAR model has a larger half-light radius and can account for the microlensing observations (e.g., Morgan et al., 2018; Li et al., 2019). In summary, the possible discrepancy between the inter-band time lags and microlensing observations might be a unique probe of winds and corona-disk magnetic coupling. Future inter-band time-lag measurements of gravitationally lensed quasars can verify this idea.

We have collected the inter-band time lags for a large AGN sample. We find tentative evidence that the ratio of the observed to SSD time lags anti-correlates with $L$; this anti-correlation is unexpected in the lamp-post SSD (with or without winds and/or disk-atmosphere scattering) model. Our result indicates that the inter-band time lags are not solely determined by the spatial sizes of the emission regions but contain important information regarding the disk inner thermal fluctuations or the BLR structure. While both the BLR and CHAR models can explain our results, they have entirely different predictions. For the BLR model, we would expect that the time-lag ratios depend upon the BLR components in the variable spectra. For the CHAR model, we suggest that the time lags increase with the variability timescales. Future time-domain surveys such as the Rubin Observatory’s Legacy Survey of Space and Time (e.g., Ivezić et al., 2019) can measure inter-band time lags for a large number of AGNs with various properties, thereby starting a new era in testing MHD accretion-disk theory or BLR physics.