An Alternative Approach to Measuring Reverberation Lags in Active Galactic Nuclei

Abstract

Motivated by recent progress in the statistical modeling of quasar variability, we develop a new approach to measuring emission-line reverberation lags to estimate the size of broad-line regions (BLRs) in active galactic nuclei. Assuming that all emission-line light curves are scaled, smoothed, and displaced versions of the continuum, this alternative approach fits the light curves directly using a damped random walk model and aligns them to recover the time lag and its statistical confidence limits. We introduce the mathematical formalism of this approach and demonstrate its ability to cope with some of the problems for traditional methods, such as irregular sampling, correlated errors, and seasonal gaps. We redetermine the lags for 87 emission lines in 31 quasars and reassess the BLR size-luminosity relationship using 60 Hβ lags. We confirm the general results from the traditional crosscorrelation methods, with a few exceptions. Our method, however, also supports a broad range of extensions. In particular, it can simultaneously fit multiple lines and continuum light curves which improves the lag estimate for the lines and provides estimates of the error correlations between them. Determining these correlations is of particular importance for interpreting emission-line velocity-delay maps. We can also include parameters for luminosity-dependent lags or line responses. We use this to detect the scaling of the BLR size with continuum luminosity in NGC 5548.

X-ray reverberation around accreting black holes

“…Peterson 1993;Peterson et al 2004;Bentz et al 2009;Denney et al 2010). More recently, stochastic modelling and fitting of light curves has been applied to more accurately model the optical reverberation data (Zu et al 2011;Pancoast et al 2011). The situation for X-ray time-series analysis is different however: here the focus to date has been on Fourier analysis techniques (with just a few exceptions, e.g.…”

Size and disk-like shape of the broad-line region of ESO 399-IG20

“…An alternative method to estimate the time delay, called Stochastic Process Estimation for AGN Reverberation (SPEAR), has been worked out recently by Zu et al (2011). Through the modeling of the AGN light curves as a damped random walk (DRW; Zu et al 2011, 2013 andreference therein), this method appears to be consistent with previous crosscorrelation techniques (e.g.…”

A degeneracy in DRW modelling of AGN light curves

“…To simulate a light curve with N points, first, the (N × N ) covariance matrix of the signal Press et al 1992), where L is the upper triangular matrix. Next, the light curve is obtained from y = Lr, where r is a vector of Gaussian deviations with the variance of unity (e.g., Zu et al 2011). Finally, we add the photometric noise ni(yi) dependent on the magnitude yi, yi = yi + G(ni(yi)), that is drawn from a Gaussian (G) distribution of the true SDSS photometric noise (Ivezić et al 2007).…”

“…1. Subsequently, the light curves are modelled with DRW [see appendix in Koz lowski et al 2010 for fast (only O(N ) operations for a light curve with N points) modelling with DRW, also Kelly et al 2009;MacLeod et al 2010;Zu et al 2011Zu et al , 2013, i.e., with fixed β = 1.…”

“…The damped random walk (DRW) model is an increasingly successful method of quantifying the variability of active galactic nuclei (AGNs; Kelly et al 2009;Koz lowski et al 2010;MacLeod et al 2010;Zu et al 2011Zu et al , 2013. Kelly et al (2009) introduce DRW as an underlying stochastic process leading to AGN variability, also known as the continuous-time first order autoregressive process [CAR(1)] or Ornstein-Uhlenbeck process (Uhlenbeck & Ornstein 1930).…”