It is now widely accepted that most massive galaxies contain a supermassive black hole (SMBH) at their center. The masses of these SMBHs correlate with properties of the galaxy itself, such as the mass of the bulge and stellar velocity dispersion (Magorrian et al., 1998;Gebhardt et al., 2000), evincing a link between the growth of the galaxy and the SMBH. A small fraction of galaxies are known as active galaxies and host Active Galactic Nuclei (AGNs), in which strong electromagnetic radiation is emitted across the entire spectrum from their central region, emission that is powered by the release of gravitational potential energy as material accretes onto the SMBH. This accretion can drive winds and outflows that influence the AGN host galaxy and explain the link between galaxy and SMBH properties (Silk & Rees, 1998;Fabian, 1999). This important process in galaxy evolution is often referred to as 'AGN feedback' (see Fabian, 2012, for a review). Understanding the process of accretion and the geometry and structure of gas in the inner regions of AGNs is therefore an important piece in understanding galaxy evolution. A simplified sketch of the inner regions and basic structure of an AGN is shown in Fig. 1, showing the accretion disk, X-ray corona, broad emission line regions, or BLR, and dusty torus (see Netzer, 2015, and references therein for a detailed review of AGNs).
The angular size of the vast majority of AGNs is too small to be spatially resolved with current techniques. For example, if we consider a relatively nearby AGN at a distance of 50 Mpc, with a black hole mass of M = 10 7 M , a size scale of 10,000 R G (similar to the size scale of the BLR; R G = GM/c 2 ) corresponds to an angular size of approximately 2 × 10 -5 arcsec. There are a few limited, and notable, cases where the AGN is near enough and the black hole massive enough that the angular size is within reach of some methods. For instance, the Event Horizon Telescope used interferometry at sub-mm wavelengths to image the black hole shadow in M87 (Event Horizon Telescope Collaboration et al., 2019) and the GRAVITY interferometer on the VLT has also been able to resolve the BLR in 3C 273 and IRAS 09149-6206 (Gravity Collaboration Figure 1: A cross-sectional schematic of an AGN, highlighting the main components and the four types of reverberation discussed in this review, X-ray reverberation (Section 2), optical/UV continuum reverberation (Section 3), broad line region reverberation (Section 4), and dust reverberation (Section 5) with general radial scales from the black hole indicated by labels. et al., 2018, 2020). Whereas size scales of ∼ 10 5 R G are currently inaccessible to all but a few AGN, in the future, X-ray interferometry may eventually make it possible to resolve scales of a few R G in the nearest Seyfert galaxies (Uttley et al., 2019). However, the vast majority of AGNs remain out of reach of current interferometric techniques, thus, other methods are needed to study the innermost regions of AGNs.
The focus of this review is reverberation mapping, a technique that swaps spatial resolution for time resolution, allowing black hole mass measurements and the determination of the size scales of emitting regions associated with Xrays, the UV/optical continuum, BLR, and dusty torus. Measuring the size and structure of the BLR was the first proposed application for reverberation mapping. Bahcall et al. (1972) recognized that time variability in the intensity of emission from a central source would affect the observed intensity of emission from photoionized gas in active galaxies and some novae. Building on this, Blandford & McKee (1982) developed a framework for inverting the observed time-dependence of broad emission-line variations to map out the structure of the BLR. In essence, time resolution can provide constraints on the emissivity and position of photoionized gas in a spatially-unresolved source, even when the source is located at a distance of millions to billions of parsecs.
The basic principle of reverberation mapping is to measure the time delay, or lag, between flux variations in an ionizing source, and the flux variations from the surrounding region that is being irradiated. Changes in the irradiating flux will drive changes in the reprocessed emission so light curves from the two regions should be strongly correlated. However, since the light from the reprocessed emission has to travel an extra distance to get to us (see Fig. 1), the variations in the reprocessed emission will arrive later than the variations in the ionizing flux. The lag, τ , will depend on the exact geometry of the system, but the average lag will be of the order R/c, where R is the typical radius of the emitting region.
The basic simplifying assumptions of reverberation mapping are that (a) the irradiating flux originates from a single central source, (b) that the light-travel time is the most important timescale, and that (c) the relationship between the observed reprocessed and ionizing fluxes is linear. If we describe the ionizing and reprocessed light curves, F r (t) and F i (t) respectively, as consisting of a constant plus variable component, i.e. F i (t) = Fi + ∆F i (t) and F r (t) = Fr + ∆F r (t), then the relationship between the variable components of each light curve (∆F i (t) and ∆F r (t)) can be described by: ∆F r (t) = τmax 0 Ψ(τ )∆F i (t -τ )dτ
(1)
where Ψ(τ ) is the transfer function (or impulse response function) which encodes all the information about the geometry of the reprocessing region. In other words, the reprocessed light curve is a blurred, delayed version of the ionizing light curve, where Ψ(τ ) can be thought of as the blurring kernel. This framework can be extended further to include velocity (or energy) dependence to the light curves and transfer function, allowing for a study of the kinematics of the reprocessing region also.
Reverberation mapping therefore requires measuring the flux variations of the AGN at different wavelengths. From this, we first aim to recover the lag between the light curves since this gives the responsivity-weighted radius of the reprocessing region. But, ultimately we would like to recover the transfer function in order to fully map out the geometry and dynamics of the region.
Methods employed to measure the lag depend on the nature of the observed light curve. For instance, optical light curves obtained from the ground are never continuous, with poor weather causing gaps. This necessitates a time domain approach to measuring the lag. In X-rays, however, the relevant timescales are short enough that continuous light curves can often be obtained, allowing for Fourier analysis techniques. We briefly describe both approaches below.
Commonly, measuring lags in the time domain uses the cross-correlation function. Essentially one shifts the light curves in time with respect to each other and measures the degree of correlation at each lag. With non-continuous light curves, the question becomes how to deal with the gaps. Often linear interpolation is used to fill between the gaps (Gaskell & Sparke, 1986;White & Peterson, 1994;Peterson et al., 2004), or alternatively interpolation can be avoided by using the discrete correlation function (Edelson & Krolik, 1988;White & Peterson, 1994). One advantage of the cross-correlation approaches are that they are simple, making no assumptions, and rely on the data alone.
However, when the data are not well-sampled they do not perform well. More recently, Markov Chain Monte Carlo (MCMC) methods that assume the underlying AGN variability can be described by a damped random walk or by Gaussian processes, have been developed to fit the light curves and determine the lag (Zu et al., 2011(Zu et al., , 2016;;Starkey et al., 2016;Yu et al., 2020a). The advantage of these methods is that with some assumptions about the variability characteristics and transfer function shape they can provide lag estimates with less well-sampled data than traditional cross correlation methods, with the disadvantage being the dependence on the validity of the assumptions made.
The Fourier analysis techniques employed in X-ray reverberation analysis is described in detail in Uttley et al. (2014). Briefly, the cross-spectrum of the light curves in two bands is used to determine the lag at each Fourier frequency, and is calculated as follows. If the light curve in one energy range is s(t) and its corresponding Fourier transform is S(f ), and in the other band the light curve and Fourier transform are h(t) and H(f ), the cross-spectrum is calculated via
where S * (f ) is the complex conjugate of S(f ). The phase of the cross-spectrum, φ(f ), gives the phase lag at each Fourier frequency. This can be converted to a time lag, τ (f ) = φ(f )/2πf . In practice the lags are averaged over a range of Fourier-frequencies, and/or a number of light curve segments.
This technique therefore gives the lag as a function of Fourier frequency, and so can separate lags from different processes occurring on different timescales. Not all X-ray telescopes provide long, continuous light curves where the approach can be directly applied, and so there is a need to also be able to determine Fourier-resolved lags from data containing gaps. Several methods have been developed using maximum likelihood fitting and MCMC approaches (Zoghbi et al., 2013;Wilkins, 2019).
Reverberation mapping was first used to measure the sizes of the BLR by measuring lags between the UV or optical continuum and broad emission lines such as C IV and Hβ. But, the principle has been extended to study the dusty torus, the accretion disk (UV/optical continuum) and the X-ray corona. In this review we will outline the application of reverberation mapping to each of these different size-scales and structures around the SMBH (Fig. 1). The scale of interest is measured in terms of the gravitation radius, R G , and the equivalent light-crossing time is given by t c = R G /c (see Table 1 for the light-crossing times). First, in Section 2 we discuss the innermost X-ray-emitting part of the accretion flow at just a few to tens of R G . Next (Section 3), we'll move outwards to discuss the UV and optical emitting continuum and accretion disk at scales of 100 to 10000 R G . Moving further out still, in Section 4 we will discuss the broad emission line region at scales of thousands to tens of thousands of R G .
Finally, Section 5 discusses the inner edge of the dust-emitting region at scales of tens of thousands to hundreds of thousands of R G and beyond.
As gas funnels in towards the SMBH at the center of a galaxy, collisions and angular momentum conservation cause the formation of an optically thick, geometrically thin accretion disk around the black hole, classically described by Shakura & Sunyaev (1973). This disk is mildly ionized and produces weak magnetic fields that are responsible for the outward transfer of angular momentum, allowing for material to fall towards the black hole. In luminous, actively accreting AGN, material flows inwards through the accretion disk up to the innermost stable circular orbit (ISCO), beyond which the gas plunges towards the black hole on a ballistic trajectory 1 . The location of the ISCO is dictated by the spin of the black hole, as General Relativistic frame dragging effects will support more orbits closer to the black hole as the spin increases. X-ray reverberation aims to exploit this fact, and measure the inner edge of the accretion disk in luminous, radiatively efficient AGN, in order to measure a fundamental black hole parameter, its spin.
The spin of the black hole also has important astrophysical implications, as the overall radiative efficiency of the accretion process is largely determined by 1 The situation is quite different in lower accretion rate AGN, where the density of the flow is so low that very little accretion energy is radiated away, thus heating up the inner flow, and effectively truncating the inner edge of the accretion disk at radii larger than the ISCO (Narayan et al., 1996).
the black hole spin (6% for non-rotating black holes vs. 42% for maximally spinning black holes). Moreover, the spin is a by-product of the underlying growth mechanism of the SMBH, as gas that accretes on to the black hole via a prograde accretion disk will eventually spin up the black hole (Volonteri et al., 2003). Randomly-oriented black hole-black hole mergers, on the other hand, will tend to spin-down the black hole. Therefore, if we can measure a distribution of black hole spins across the universe, we can understand the relative importance of the accretion process versus mergers in the growth of SMBHs (Berti & Volonteri, 2008).
The best probe of the ISCO in AGN is in the X-ray band. X-rays are ubiquitous in AGN systems and originate within a few gravitational radii of the black hole. The accretion disk is hot enough to emit thermal radiation that peaks in the Extreme Ultraviolet band (see schematic, Fig. 1). Some of those UV photons scatter off of mildly relativistic electrons in an region close to the black hole that is known as the corona (Haardt & Maraschi, 1991). This scattering boosts the UV photons to X-ray energies via inverse Compton scattering. Some X-ray photons reach our telescopes directly, and some photons irradiate the accretion disk and are reprocessed (e.g. via photoelectric absorption and fluorescence, thermalization, Compton scattering). Reprocessed X-ray emission from the disk is generically referred to as a 'reflection' component in the X-ray spectrum. The most prominent reflection features are the iron Kα emission line at 6.4 keV and the Compton scattering 'hump' peaking at ∼ 20 keV (e.g., Risaliti et al. 2013;Marinucci et al. 2014;Walton et al. 2014;Parker et al. 2014 and see Fig. 3-left.) We observe the iron K emission line and other reflection features to be broadened due to Doppler motion in the accretion disk, and also by the gravitational redshift from the strong potential well of the black hole (Fabian et al., 1989). If the black hole is spinning rapidly and the ISCO is small, then a large gravitational redshift is observed. For the iron Kα line, a rapidly rotating black hole results in a strong red wing of the line that can extend down to 3-4 keV in the most extreme cases. This simple idea has been used to measure the spins of ∼ 50 nearby AGN (Reynolds, 2019).
While measuring spin from broadened reflection features is a simple idea, astronomers have been plagued by systematic uncertainties that have cast doubt on black hole spin parameter estimations. For instance, from the spectrum alone, it is not always clear what emission is from the primary coronal continuum and what is relativistically broadened reflection. Moreover, we do not have strong observational constraints on the geometry of the X-ray corona, which effects the emissivity (irradiation profile) of the accretion disk and therefore can affect one's ability to uniquely identify the inner edge of the accretion disk (e.g. Dauser et al. 2014, Wilkins & Fabian 2011).
Fortunately, in recent years we have made a breakthrough in our understanding of the X-ray emission from close to black holes with the discovery of X-ray reverberation. Implicit in the model described above is the idea that there should be a light travel time delay between the primary emission from the corona and the reflected emission off of the accretion disk. Reverberation time delays were first discovered in 2009 using the XMM-Newton Observatory (Fabian et al. 2009, Zoghbi et al. 2010and Uttley et al. 2014 for an early review). These short light travel lags generally confirm the picture that the X-rays are reprocessing off the inner accretion disk within a few gravitational radii. The compact size of the X-ray emitting region (e.g. Reis & Miller 2013) is independently confirmed through microlensing (e.g., Chartas et al., 2009Chartas et al., , 2012) ) and X-ray eclipses (e.g. Gallo et al. 2021).
X-ray variability from black holes is generally characterized as a red noise process, with variability from different processes contributing to the observed variability. It is for this reason that the technique used in the X-ray band is to measure Fourier-frequency-resolved time delays (see above for a brief overview emission line. The frequency-resolved approach allowed for zeroing in on just the shortest timescale variability, where the reflection dominated band was observed to lag behind the continuum by roughly 30 seconds, or 2 R g /c for a 10 6 M black hole (see Fig. 2). In the years following this discovery, high-frequency reverberation lags were observed both in low-energy X-rays (e.g. Zoghbi et al. 2011;Cackett et al. 2013;Alston et al. 2014;De Marco et al. 2013), but also in the iron K emission line (Fig. 3-right and see Zoghbi et al. 2012;Kara et al. 2013bKara et al. , 2016;;Vincentelli et al. 2020) and more tentatively, the Compton reflection hump above 10 keV (Zoghbi et al. 2014;Kara et al. 2015; thanks to the launch of the hard X-ray NuSTAR Observatory, and the development of techniques to deal with gaps in light curves; Zoghbi et al. 2013). However, follow-up hard X-ray studies with much longer exposures suggest little statistical deviations from log-linear lags (Zoghbi et al., 2021). We point readers to Uttley et al.
(2014) for a review of many of these early observational discoveries.
To date, X-ray reverberation lags have been measured in roughly 2 dozen AGN (De Marco et al., 2013;Kara et al., 2016). In these sources, the lag amplitudes scale with the masses of the black holes, with some significant scatter in the relation (Fig. 4-left). The spread is likely due to in part to uncertainties in the black hole mass, but also likely due to intrinsic variance in the geometry of the corona or inner accretion disk. Such effects are, for instance, important in IRAS 13224-3809 (Kara et al. 2013a, and confirmed with a much more extensive dataset in Alston et al. 2020). In this highly variable AGN, as the intrinsic luminosity of the corona increases, the reverberation lags become longer . By modeling the reverberation lags with General Relativistic ray-tracing simulations of a compact corona irradiating a thin accretion disk (see Section 2.2), Alston et al. (2020) showed that the height of the corona increases with increasing luminosity. This finding illustrates the ability of X-ray reverberation not just to make a picture of the black hole inner accretion flow, but a movie. Moreover, this work demonstrates that if the geometry of the inner accretion flow can be well constrained, X-ray reverberation lags can place constraints on fundamental black hole parameters, namely mass and spin.
Reverberation with General Relativistic Ray-Tracing Simulations X-ray reverberation near the black hole requires treatment of general relativistic effects. The basic approach is to model the response of the accretion disk (including relativistic dynamical distortions) to a sudden impulse of photons from the X-ray corona (usually assumed, for computational reasons, to be a point source above the accretion disk, positioned on the spin axis of the black hole; Campana & Stella 1995;Reynolds et al. 1999;Wilkins & Fabian 2013).
The first attempt to compare these impulse response functions to observations of lag-frequency spectra was presented in Wilkins & Fabian (2013), and later Cackett et al. (2014) fitted the iron K lag with a grid of impulse response functions in order to put constraints on, e.g. disk inclination and coronal height . This was extended to a larger sample fitting the lag-frequency spectra in Emmanoulopoulos et al. (2014). The next step was to calculate the energy-dependent lags across the entire reflection spectrum (Chainakun & Young, 2015;Caballero-García et al., 2018). This model was used to fit the lagfrequency spectra in IRAS 13224-3809, discussed in Section 2.1 (Alston et al., 2020).
In addition to general relativistic corrections, the observed X-ray lags are complex because they not only show reverberation lags, but also contain lags associated with the continuum variability (referred to here as 'continuum lags').
These lags dominate on long timescales, and are thought to be associated with the propagation of mass accretion rate fluctuations through the accretion flow (Lyubarskii, 1997;Arévalo & Uttley, 2006). Recently, Ingram et al. (2019) developed a general relativistic ray-tracing model that also includes the continuum lag. In this model, the energy spectrum from the coronal continuum is modelled as a powerlaw that fluctuates in photon index and normalization at a few per cent level, which causes time lags between soft and hard continuum photons.
This can model both the long timescale continuum lags and high frequency reverberation signature (Mastroserio et al., 2018;Wang et al., 2021).
Beyond a simple lamppost corona irradiating a razor thin accretion disk, there have been recent efforts to account for more complex geometries of the corona and disk, and indeed, the data seem to suggest that the time lags would be better modelled with a corona extended over tens of gravitational radii (Mas-
X-rays are not only a probe of the inner accretion flow, but also the hard X-ray continuum is absorbed, scattered, and reprocessed off circumnuclear gas flows beyond the inner disk, including from disk winds, the broad line region and dusty torus. This has led to an alternative explanation for X-ray lags as due to scattering of X-rays passing through an absorbing medium (e.g., Miller et al., 2010), for instance a disk wind (Mizumoto et al., 2019). But, these models require that most of the X-ray emitting gas is along our line of site (Zoghbi et al., 2011) and do not explain the difference in lag-energy properties at high and low frequency (Kara et al., 2013c). However, there is mounting evidence that absorption can add to the complexity in observed lags, at least in some systems. Time-dependent changes in the absorber properties can lead to changes in observed lags (Silva et al., 2016), which has been suggested to explain lags in several objects which do not simply look like disk reverberation and which show absorption in their spectra (Kara et al., 2015;Zoghbi et al., 2019).
The most prominent feature from distant reflection is the narrow iron K emission line, that is commonly seen in AGN spectra. This line has typically been associated with the torus, though recent Chandra/HETG studies of the line profile suggest Doppler broadening effects that would suggest that the lines originate in the outer accretion disk, or inner broad line region (Miller et al., 2018). This was supported by the correlation of the narrow iron line and continuum flux on short timescales (Zoghbi et al., 2019), with limits on the narrow iron K lag suggesting it originates in a region within the optical BLR. Future telescopes with X-ray micro-calorimeters, like XRISM and Athena will put important spectral constraints that can confirm these results, and allow for measurement of X-ray reverberation lags to probe larger-scale gas flows around the AGN, complimenting longer wavelength reverberation that probes the outer disk and broad line region.
While this review focuses on AGN reverberation, we would be remiss not to mention an important development in X-ray reverberation studies in the last few years, the discovery of X-ray reverberation in black hole X-ray bina- the second brightest X-ray source in the sky (after only Sco X-1). NICER observations of the luminous hard state of MAXI J1820+070 showed the timescale of the reverberation lags shortened by an order of magnitude over a period of weeks, while the shape of the broadened iron K emission line remained remarkably constant, suggesting that the change in lags before the state transition were due to a decrease in the spatial extent of the corona, rather than a change in the inner edge of the accretion disk (Kara et al., 2019). This decrease in spatial extent of the corona tracks the decrease in radio luminosity, supporting earlier theoretical and observational work suggesting the corona is related to the base of the relativistic jet (Markoff et al., 2005;Fender et al., 1999;Homan et al., 2005).
UV/optical continuum reverberation mapping looks for lags between continuum light curves measured at different wavelengths. The accretion disk reprocessing scenario posits that a central source (presumably emitting in the X-rays or far-UV) irradiates the accretion disk and drives the UV and optical variability, with the hot inner part of the accretion disk seeing the irradiating light before the cool outer part of the disk (see e.g., Collier et al., 1999;Cackett et al., 2007, for detailed descriptions). This therefore predicts that the UV and optical continuum light curves should be well-correlated, with the UV leading the optical (as is observed, see Fig. 5), and the lag dependent on the temperature profile of the disk. For a standard optically thick and geometrically thin (Shakura & Sunyaev, 1973) accretion disk, the temperature profile follows bands, with the correlation peaking at a lag of around 0.9 days for UVW1 and 4 days for i.
Data from Fausnaugh et al. (2016).
, where Ṁ is the mass accretion rate. Since the lag gives a measure of the radius (τ = R/c), and for blackbody radiation we have
where ṀE = L bol /L Edd is the Eddington fraction. One can generalize this further, as a disk with T (R) ∝ R -b leading to wavelength-dependent lags of
Early BLR reverberation campaigns showed that the UV and optical continua were well-correlated with inter-band lags of less than a few days (Stirpe et al., 1994;Wanders et al., 1997). Observed lags showed evidence for the expected wavelength-dependence, though at relatively low significance (Collier et al., 1998(Collier et al., , 2001)). Multi-year optical photometric monitoring of 14 AGNs by Sergeev et al. (2005) showed both that lags increase with wavelength and that more luminous objects exhibited longer lags, as expected in the disk re-processing scenario. Again, however, few of the lags were significant at >3σ.
Observationally, it is a challenge to measure these lags with traditionally scheduled ground-based telescopes, since temporal sampling of < 24 hours is required over a baseline of weeks to months.
The field has changed in the last decade for two reasons. Firstly, The Neil Gehrels Swift Observatory (hereafter Swift; Burrows et al., 2005) has made it possible to perform high cadence (often multiple observations per day), high signal-to-noise observations in the X-ray, UV and optical. Secondly, the introduction of robotic observatories dotted across the globe has made it easier to perform high cadence optical monitoring with significantly fewer losses to poor weather, while also avoiding many of the logistical issues involved in running long-term monitoring campaigns.
Swift can rapidly slew to targets, which allows it to observe many targets per day, each with relatively short (1 ks) visits. This is perfect for continuum reverberation mapping since Swift can even observe the same target multiple times per day. Moreover, it has both an X-ray and a UV/optical telescope (UVOT), allowing for X-ray spectroscopy and UV/optical photometry in 6 filters from 1928 Å (UVW2) to 5468 Å (V ). The first uses of Swift for continuum reverberation demonstrated its potential to measure lags to much higher precision than had previously been achieved, and also demonstrated that the lags were longer than expected from the standard thin disk model (Shappee et al., 2014;McHardy et al., 2014). A breakthrough was achieved with monitoring of than doubled the number of visits with all 6 UVOT filters in either of the previous two campaigns, achieving a sampling rate of better than once per 0.5 day (Edelson et al., 2015;Fausnaugh et al., 2016). The main results of that campaign -that the wavelength-dependent lags approximately follow λ 4/3 , that (Fausnaugh et al., 2016), NGC 4593 (Cackett et al., 2018), and Mrk 142 (Cackett et al., 2020). Data from Swift (black circles),
HST (red squares) and ground-based observatories (blue triangles). The solid lines show the best-fitting τ ∝ λ 4/3 relation (excluding the X-rays and u/U bands), while the dashed lines show the predicted lags based on reasonable estimates of the black hole mass and accretion rate (eqn. 12 in Fausnaugh et al., 2016)). The difference between the observed (solid line)
and predicted (dashed line) relations is the 'accretion disk size problem'. The lags in the u/U bands (around 3500 Å) show a clear excess, and in NGC 4593 the lag spectrum reveals a broad excess around the Balmer jump. Finally, note also how there is no consistent relationship between the X-rays and the best-fitting λ 4/3 relation.
the accretion disk is bigger than expected by a factor if ∼3, that the lag in the U band (3465 Å) is in excess of the λ 4/3 relationship (Edelson et al., 2015;Fausnaugh et al., 2016), and that the X-ray light curve is not consistent with being the driving light curve (Gardner & Done, 2017;Starkey et al., 2017) are common to most of the Swift continuum reverberation campaigns that have followed (Edelson et al., 2017(Edelson et al., , 2019;;Cackett et al., 2018Cackett et al., , 2020;;McHardy et al., 2018;Hernández Santisteban et al., 2020). Fig. 6 shows some examples of these recent results.
A different approach to the intensive high-cadence monitoring of small numbers of objects is to look at a larger sample of objects but at a lower cadence.
This approach has been undertaken utilizing large surveys. While generally individual objects will not yield high-fidelity lag-wavelength relations, the pop-ulation of objects can be studied as a whole. This powerful approach is being taken by PanSTARRS (Jiang et al., 2017), DES (Mudd et al., 2018;Yu et al., 2020b) and SDSS (Homayouni et al., 2019)
The observed wavelength-dependent lags are usually fit with a simple model of the form τ = τ 0 (λ/λ 0 ) β -1 , where λ 0 is the reference band wavelength (usually chosen to be the best-sampled and highest S/N band), τ 0 is the normalization, and the value of β is 4/3 for a standard thin disk. Generally it is found that β ∼ 4/3, though the best-fitting results from NGC 5548 prefer a slightly shallower value close to 1 (Fausnaugh et al., 2016). τ 0 can be predicted based on the mass and mass accretion rate of the target (see equation 12 in Fausnaugh et al., 2016). When the predicted τ 0 is compared to the best-fitting value, it has frequently been found to be 2 -3 times larger than expected. This has been seen both in the intensive disk reverberation studies (Shappee et al., 2014;McHardy et al., 2014;Edelson et al., 2015Edelson et al., , 2017Edelson et al., , 2019;;Fausnaugh et al., 2016Fausnaugh et al., , 2018;;Cackett et al., 2018Cackett et al., , 2020;;Pozo Nuñez et al., 2019) and also from surveys (Jiang et al., 2017;Mudd et al., 2018), but is not always the case (Edelson et al., 2019;Hernández Santisteban et al., 2020;Homayouni et al., 2019;Yu et al., 2020b). Lags that are longer than expected indicate that the radius for a given temperature is larger than expected, or that some of the assumptions are wrong, such as the mass accretion rate estimate or the choice of factor to convert from wavelength to temperature. This is often referred to as the accretion disk size problem (see the comparison of predicted and measured relations in Fig. 6). Interestingly, independent results using gravitational microlensing of quasars indicates that the optical emitting region in those objects is similarly larger than expected (e.g. Morgan et al., 2010).
Several explanations for the disk size problem involve disks being more complex than the simple Shakura & Sunyaev (1973) picture. For instance, to explain gravitational microlensing results Dexter & Agol (2011) suggest that local fluctuations lead to an inhomogeneous, time-dependent disk structure and can reproduce the larger than expected disk sizes. Alternatively, the assumption that the disk emits as a blackbody may also be incorrect. Hall et al. (2018) suggest that the larger disk sizes seen by continuum reverberation can be explained by accretion disks with a sufficiently low-density scattering atmosphere, which leads to a substantially different temperature profile. Such a scattering atmosphere gives longer lags, a flatter wavelength dependence, and can match the observed NGC 5548 lags. Alternative disk models also predict different temperature profiles (Mummery & Balbus, 2020). More sophisticated disk reprocessing models using general relativistic ray-tracing give systematically longer lags than the simple analytic relation frequently used, and are able to reproduce the lags with reasonable mass and mass accretion rate estimates, potentially solving the accretion disk size problem (Kammoun et al., 2019(Kammoun et al., , 2021a,b),b). Interestingly, the implied height of the X-ray source is typically larger than seen from X-ray reverberation, which may suggest that the corona is vertically extended (see Section 2.1).
Different geometries have also been considered, for instance, Gardner & Done (2017) suggest the inner disk puffs up to prevent hard X-rays directly irradiating the UV/optical disk and propose that the lags may be associated with the timescale for the outer disk vertical structure to vary. Another class of models explains AGN UV/optical variability and disk lags as due to a different process (not the light-crossing time). In the corona-heated accretion-disk reprocessing model (Sun et al., 2020a,b) the X-ray-emitting corona and the accretion disk are coupled via the magnetic field. Magnetic turbulence in the corona causes X-ray variability and temperature fluctuations in the disk, with the temperature fluctuations delayed with respect to the X-ray variability as the magnetohydrodynamic waves propagate outwards at the Alfvén velocity. This produces longer lags than the simple disk reverberation model.
Other considerations are lags not associated with the disk. For instance, the addition of winds can lengthen the lags (Sun et al., 2019). The contributions from broad emission lines that fall within photometric filter bands are not enough to significantly skew the lags (Fausnaugh et al., 2016), however, as we discuss in detail next, continuum emission from the BLR gas may have an important influence.
A common feature of continuum lags is that the U band (3465 Å) lag is consistently in excess of an extrapolation of the trend through the rest of the UV/optical, by a factor of around 2 (e.g., Edelson et al., 2019). This can be explained if emission from the 'diffuse continuum' in the BLR is significant. The 'diffuse continuum' (thermal free-bound and free-free, plus scattering) emanates from the same clouds that emit the broad emission lines (Korista & Goad, 2001, 2019;Lawther et al., 2018;Netzer, 2020). It is expected to contribute substantially to the continuum at all wavelengths, but particularly around the and suggest that lags originate from photoionized material above the outer disk, rather than from disk reprocessing closer in.
In addition to the signature of the diffuse BLR in the lags, this component should also show up prominently in the variable (rms) spectrum. Recent studies (e.g. McHardy et al., 2018;Cackett et al., 2020;Hernández Santisteban et al., 2020) that perform a flux-flux analysis to calculate the variable spectrum show that the u/U bands is also enhanced there (e.g. approximately 15% in Mrk 142), but overall the variable spectrum follows a simple power-law. Future work that combines lag and spectral analysis together is needed to properly separate out the contributions from the disk and diffuse BLR.
The Swift continuum reverberation campaigns have also shown that the relationship between the X-ray and UV/optical variability is complex. The peak correlation coefficient between the X-ray and UVW2 light curves is consistently much weaker than that between UVW2 and the other UV/optical light curves (Edelson et al., 2019). Moreover, the X-ray lag is often seen to be offset with respect to the best-fitting UV/optical relation (see Fig. 6). It can even be longer than expected from a straightforward lamp-post scenario (Noda et al., 2016).
Even when the X-ray lag is consistent, analysis shows that in some cases, e.g.
NGC 5548, the X-ray light curve does not look like the light curve needed to drive the variability at longer wavelengths (Starkey et al., 2017;Gardner & Done, 2017). Other AGNs show no correlation between X-ray and UV/optical variability at all (Buisson et al., 2018;Morales et al., 2019). This lack of a clear connection between the X-ray and UV/optical is puzzling and a problem for the simple disk reprocessing picture, and therefore may indicate more complex geometries or absorption (Gardner & Done, 2017), or that different variability processes are taking place on different timescales (Pahari et al., 2020). In the future, it will be important to connect the implied geometry from X-ray reverberation to the UV/optical continuum geometry.
On-going and upcoming projects involving continuum reverberation promise to further pursue these issues and shed light on the UV/optical continuumemitting region in AGNs.
The first reverberation mapping experiments for AGNs focused on trying to measure the BLR response to variations in the continuum flux. However, these early investigations were hampered by erroneously large expectations for the size of the BLR. While numerous studies of NGC 4151 indicated that the BLR size in that AGN was smaller than 30 light days in radius (Cherepashchuk & Lyutyi, 1973;Antonucci & Cohen, 1983;Bochkarev, 1984), the low-luminosity and high ionization parameter of NGC 4151 provided a plausible way out of the difficulties introduced by this small size. It was not until Peterson et al. (1985) carried out repeated observations of the luminous Seyfert Akn 120 over a period of 4 years and clearly demonstrated that the predicted BLR size from photoionization equilibrium arguments was a factor of ∼ 10 too large, that an unavoidable deficiency in our understanding was revealed. Additional studies of other AGNs confirmed these results (e.g., Perez et al. 1989;Maoz et al. 1990).
Thus it was not until the early 1990s that the first accurate measurements of broad-line reverberation were obtained. The breakthrough was facilitated largely by the efforts of the International AGN Watch consortium, a large-scale coordinated endeavor involving simultaneous UV and optical photometric and spectroscopic monitoring, from space and from the groundfoot_0 . The initial results, summarized by Peterson (1993), showed decisively that the broad lines respond rapidly to continuum variations, with high ionization lines like C IV and He II responding first and then lower ionization lines such as Hβ responding later, indicating ionization stratification in the BLR (see Fig. 8 for a recent example).
Additionally, gross radial motions that would indicate infall or outflow as the dominant motions in the BLR, were ruled out (Korista et al., 1995), demonstrating that rotation dominates the bulk motions of BLR gas in the local Seyferts that were targeted for study. Around the same time, the shortcoming in early photoionization models was determined to be the use of a single representative cloud for the BLR, whereas a "locally optimally emitting cloud (LOC)" model (Baldwin et al., 1995) more successfully reproduced early reverberation results, including the BLR size and ionization stratification. Unlike single-cloud models, the LOC model included a wide range of gas conditions and found that BLR emission preferentially arises from the locations where the gas conditions are optimal.
The key to observational success measuring lags involved satisfying the following constraints in the planning and execution of reverberation mapping pro-grams:
• a program length at least 3 times the longest expected time delay (Horne et al., 2004);
• a temporal cadence fine enough to resolve the shortest expected time delay;
• high signal-to-noise ( 50) in the continuum and significantly higher in the emission lines, so that variability amplitudes of a few percent could be clearly detected;
• flux calibration with a precision of 2% or better across all the spectra obtained throughout the program (Peterson et al., 2004);
• and a dash of good luck, since AGN variability is stochastic and unpredictable, though longer program lengths increased the chances of detecting variability (e.g., MacLeod et al. 2016).
With a better understanding of the observational constraints necessary for a successful reverberation mapping program, the number of accurate broad line reverberation measurements began to rapidly accrue through the efforts of several groups. This also led to the development of new tools for improving the time delay measurements. Thus a consistent reanalysis of all previous reverberation measurements for UV and optical broad lines was carried out by Peterson et al. (2004), with the same tools and methods applied in all cases, so that results from different groups could be compared in a more straightforward manner. The various analysis methods that had been employed by different groups were also examined and compared, leading to a set of best practices for the field and laying the groundwork for continued and improved success of BLR reverberation mapping experiments. With a solid footing for broad-line lag measurements, RM goals have thus expanded to tackle new questions and focus on AGNs beyond the brightest local Seyferts, and the observational constraints needed to achieve these goals have continued to evolve as well.
With the availability of accurate broad-line time delay measurements, it became clear that reverberation mapping could be used to constrain the mass of the central black hole in AGNs (Peterson & Wandel, 1999, 2000). With the measurement of a time delay τ and line width V for a broad emission line:
where c is the speed of light, G is the gravitational constant, and f is an orderunity scaling factor that accounts for the (generally unknown) geometry and kinematics of the broad line region gas. Notably, when different emission lines are probed in the same AGN, the high ionization lines have short time delays and are found to be broader in width than the low ionization lines, generally agreeing with the expectations for virial motion and leading to consistent M BH determinations (e.g., Peterson & Wandel 2000;Kollatschny et al. 2001).
As one of the recommended best practices, Peterson et al. (2004) demonstrated that it was critical to measure the line width in the variable part of the emission line (see Fig. 8), so that the line-of-sight velocity was directly related to the reverberating gas, when determining M BH . Furthermore, focusing on the variable emission excludes narrow lines and other nonvariable emission that would otherwise bias the line width measurement, a point that is especially important for lines like C IV that have multiple components (cf. Denney 2012).
The population average value of f has been constrained through comparison of the M BH -σ relationship (Gültekin et al., 2009;Kormendy & Ho, 2013;McConnell & Ma, 2013) for AGNs and for quiescent galaxies. Under the assumption that all the galaxies are drawn from the same parent sample, the multiplicative scaling factor that is needed to bring the intercept of the AGN relationship into agreement with that of the quiescent galaxies gives f . When the second moment of the emission line is used to represent the velocity width (another best practice recommended by Peterson et al. 2004), values of f have ranged from 2.8 (Graham et al., 2011) to 5.5 (Onken et al., 2004), with most values settling around 4 -5 (e.g., Park et al. 2012;Grier et al. 2013; Batiste If inclination angle is assumed to be the largest contributor to the value of f , then f ≈ 4 -5 implies that the average AGN in the reverberation sample is viewed at an inclination angle of 25 -30 • , which also seems reasonable based on our current understanding of the AGN unification model (e.g., Urry & Padovani 1995). However, individual AGNs may have inclination angles that deviate from the population average, and so reverberation masses for individual objects that make use of f have an additional factor of 2 -3 uncertainty.
Another important product of reverberation mapping results is the scaling of the typical BLR size with the luminosity of the central AGN, the R BLR -L AGN relationship (Koratkar & Gaskell, 1991;Kaspi et al., 2000Kaspi et al., , 2005)), which was expected based on photoionization arguments and searched for with the earliest reverberation measurements. When the host-galaxy starlight contamination is properly removed from the AGN luminosity, the relationship for local Seyferts has the form R BLR ∝ L 1/2 (Bentz et al. 2006b(Bentz et al. , 2009(Bentz et al. , 2013)). Microlensing of the BLR in lensed quasars, where the magnification amplitude is dependent on the size of the emission region, provides an independent way to probe the R BLR -L AGN relationship, and the results agree well (Guerras et al., 2013). Additionally, the Pa α BLR size in the quasar 3C 273 was resolved in the near-infrared with the GRAVITY instrument on VLT, and the size compares well with those derived from reverberation mapping of the Balmer lines (Gravity Collaboration et al., 2018), although reverberation mapping determines responsivity-weighted radii while interferometry instead measures flux-weighted radii.
The R BLR -L AGN relationship has been used to estimate M BH for large samples of broad-lined AGNs with only a single spectrum per target (e.g., Shen et al. 2011). Rather than investing months or years to measure the BLR size for an AGN of interest, the R BLR -L AGN relationship allows R to be predicted from the AGN luminosity, which may then be combined with the broad line width and an adopted f factor to estimate M BH (e.g., Vestergaard & Peterson 2006).
The majority of BLR sizes thus far have been measured for the Hβ emitting region, so at this time, the most well-constrained R BLR -L AGN relationship is for Hβ. Efforts to bootstrap these results for the use of Mg and other emission lines become available (Kaspi et al., 2007;Hoormann et al., 2019;Grier et al., 2019). However, efforts to widen the population of AGNs that are studied by reverberation mapping have included objects expected to have higher mass accretion rates than those found among local Seyferts, as well as objects at higher redshift.
Several studies show that many AGNs, including some of the highest accretion rate objects, fall below the usual R BLR -L AGN relation (see Fig. 9; Du et al. 2015Du et al. , 2016Du et al. , 2018;;Grier et al. 2017b;Fonseca Alvarez et al. 2020.) for large numbers of AGNs that span the full range of observed properties is more complicated than previously assumed.
Further advancements in the quality of reverberation datasets (Bentz et al., 2009;Grier et al., 2012;Fausnaugh et al., 2017;De Rosa et al., 2018) have finally begun to allow the initial goal of reverberation mapping -mapping out the geometry and kinematics of BLR gas -to be achieved for a handful of objects (Pancoast et al., 2014;Grier et al., 2017a;Williams et al., 2018).
While the details vary from object to object, there are similarities as well: the 2020), although a disk-like rather than shell-like geometry is preferred for C IV and Lyα. There is also evidence in the reverberation response for the presence of an azimuthal structure orbiting on the far side of the C IV and Lyα BLR during the monitoring period, as has been seen in several double-peaked AGNs (Gezari et al., 2007;Lewis et al., 2010;Schimoia et al., 2015Schimoia et al., , 2017).
An unexpected finding during the AGN STORM campaign was a period where all the emission lines, high-ionization absorption lines and longer wavelength continuum variations decoupled from the UV continuum variability (the so-called "BLR holiday"; Goad et al., 2016Goad et al., , 2019) ) for approximately 60-70 days -a breakdown of one of the fundamental assumptions of reverberation mapping.
It may be that a change in the ionizing SED incident on the BLR led to the anomalous line behavior (Mathur et al., 2017;Goad et al., 2019), likely caused by a disk wind (Dehghanian et al., 2020). This complication led to unexpected difficulties in the forward modeling and the velocity-delay map inversion analyses, however it also led to intense scrutiny of all the available information within this rich data set. Thus, the AGN STORM study of NGC 5548 was also the first reverberation experiment to explore absorption line reverberation (Kriss et al., 2019;Dehghanian et al., 2019), which provides a separate probe of the ionizing continuum and the physical conditions of the surrounding photoionized gas.
Near-IR emission in AGNs at about 2 -10µm is thought to be produced by thermal radiation from hot dust (e.g., Landt et al., 2011, 2019, andreferences therein). Since dust only survives at temperatures below the sublimation temperature (in the range 1300 -2000 K, depending on the type of dust grains), this determines the inner radius of the dusty torus. The dust will be illuminated by, and absorb radiation from, the inner X-ray and UV/optical emitting regions, and so variability at these shorter wavelengths will drive variability in the reradiated dust emission. The blackbody emission from this hot dust should peak close to the K band, meaning that the inner edge of the dusty torus can be measured through reverberation mapping in the near-IR. The larger size scale of the dusty torus compared to the disk and BLR leads to longer lags and slower and weaker variability, affording lower cadence monitoring but requiring longer campaigns.
Early efforts showed the dusty torus to be ∼light-year in Fairall 9 (Clavel et al., 1989), with dust reverberation of 10 AGNs establishing that the dust lags follow a radius-luminosity relation following approximately R ∝ L 1/2 , as expected for dust sublimation (Oknyanskij & Horne, 2001), although the typical radius of the inner wall of the torus is a factor of 2-3 smaller than was initially expected for a population of dust grains with properties similar to the ISM (e.g., Kishimoto et al. 2007). Long baseline interferometry in the near-infrared has confirmed these small sizes (see Burtscher et al. 2016 for a summary), which are now understood to be the result of a population of large graphite grains making up the inner torus wall, as they are the grains that are most resistant to sublimation (García-González et al., 2017;Hönig & Kishimoto, 2017).
In the last 20 years a number of dedicated dust reverberation campaigns have substantially increased the sample of lags (e.g., Suganuma et al., 2006;Koshida et al., 2014;Minezaki et al., 2019), as have surveys with repeated imaging that search for transient phenomena (Lyu et al., 2019;Yang et al., 2020), firmly establishing the dust R -L relation over 4 orders of magnitude in luminosity. On average the dust lags are ∼4 times longer than Hβ lags (e.g., Koshida et al. 2014;Netzer 2015; see Fig. 10 for a comparison of dust and BLR radii), demonstrating that the bulk of the BLR is enclosed within the dusty torus, as expected from AGN unification arguments.
The observable effects of central AGNs are expected to extend far beyond the dust torus. More than 20 years of spectrophotometric monitoring of NGC 5548 enabled the first and only measurement of narrow emission line reverberation thus far (Peterson et al., 2013), constraining the [O III] λλ 4959, 5007 emitting region to 1-3 pc in radius, with a light crossing time of approximately a decade.
The timescales required to probe similar and larger structures are generally band interferometric radii (Kishimoto et al., 2011;Weigelt et al., 2012), green dots at dust radii from SED fitting (Mor & Netzer, 2012) and blue error bars indicate BLR reverberation radii from Bentz et al. (2009). Dust radii are approximately a factor of 4 larger than the BLR radii.
prohibitive when compared to the lengths of human lifetimes let alone careers.
But just as light echoes from long-dead supernovae may still be found and studied within the Milky Way and its closest companions (e.g., Rest et al., 2008b,a), there are indications that distant echoes from AGNs may still be found and studied today, as has been suggested for Hanny's Voorwerp (Lintott et al., 2009).
The development of reverberation mapping has allowed AGN studies to go
EMC led Sections 1, 3 and 6. EK led Section 2. MCB led Section 4. EMC and MB co-led Section 5. All authors reviewed and edited the article.
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