© The Authors 2026

1. Introduction

The fundamental components and geometry of active galactic nuclei (AGNs) are now well constrained (see, e.g., Urry & Padovani 1995; Krolik 1999; Padovani et al. 2017, for general reviews). In the standard framework, an AGN consists of a central supermassive black hole (SMBH) surrounded by an accretion disk with a compact hot corona above it, a broad emission-line region (BLR), and a dusty or molecular torus. Together, these components produce the characteristic broadband AGN spectrum, in which hard X-ray emission arises from the corona, UV/optical radiation is generated predominantly in the accretion disk, and near-infrared (NIR) originates in the dusty torus. In type 1 AGNs, all of these elements are directly visible, whereas in type 2 sources the torus partially obscures the nuclear regions due to the high inclination angle (i). In addition, a subset of AGNs exhibits powerful relativistic radio jets that further contribute to the observed emission.

Over the past decades, light-echo studies, collectively known as reverberation mapping (RM; Blandford & McKee 1982; Peterson 1993) have not only confirmed this structural picture but have also provided a powerful means of probing the innermost regions of AGNs in detail (see Cackett et al. 2021, for a recent review). By exploiting time variability, these techniques effectively use temporal resolution for spatial resolution, enabling us to map the structure and dynamics of regions that are otherwise unresolvable.

However, several aspects of the innermost AGN structure, particularly in the immediate vicinity of the central SMBH, remain poorly understood. In this work, we focus specifically on these central regions, with an emphasis on the location and geometry of the hot corona. To avoid complications introduced by obscuration or jet-dominated emission, we restrict our analysis to type 1 AGNs, in which the nuclear components are most clearly visible.

The physical nature and geometry of the hot corona have been debated since its earliest formulation (Haardt & Maraschi 1991, 1993). The initial plane-parallel configuration proposed in these models was unable to reproduce the extremely hard X-ray power-law spectra observed in some sources, prompting the development of more general geometrical prescriptions (e.g., Svensson 1984; Haardt et al. 1994). Subsequent models incorporated increasingly sophisticated treatments of Comptonization and pair-production cooling (e.g., Poutanen & Svensson 1996), significantly improving the physical realism of corona simulations. Nonetheless, the stationary spectral shape alone has proven insufficient to uniquely determine the true geometry of the corona, leaving key aspects of its structure unresolved. Consequently, most studies have adopted the lamppost model as a practical approximation, even though the literature highlights significant limitations and potential inconsistencies associated with this approach (Zoghbi et al. 2021; González-Buitrago et al. 2023).

New insights into AGN inner structures have emerged from spectral studies of disk irradiation by the corona, particularly those incorporating relativistic effects and modeling the formation of the relativistically smeared Fe Kα line (George & Fabian 1991; Campana & Stella 1995; Dovčiak et al. 2004). For such studies, a simplified description of the corona as a compact, point-like X-ray source located on the rotational axis – the so-called lamppost model – has generally provided an adequate first-order approximation (Matt et al. 1991; Martocchia & Matt 1996). Using this framework, numerous studies have derived corona heights and reflection signatures for a wide range of AGNs (George & Fabian 1991; Matt et al. 1991; Martocchia & Matt 1996; Reynolds & Begelman 1997; Miniutti & Fabian 2004; Dauser et al. 2013; Emmanoulopoulos et al. 2014; Kammoun et al. 2019; Ursini et al. 2020). However, in some cases the basic lamppost formulation required further refinement, either by modifying the disk structure (e.g., invoking an inner hot flow, Lohfink et al. 2013; Fabian et al. 2014; Serafinelli et al. 2023) or by allowing the corona itself to be dynamic (outflowing or inflowing, Beloborodov 1999). Such modifications help resolve discrepancies between the predicted and observed normalization of the reflected component relative to the incident continuum, an issue that can be naturally explained by Doppler boosting of radiation away from or toward the disk (Beloborodov 1999).

Building upon these studies, long X-ray monitoring campaigns enabled the measurement of both Fourier-resolved time delays and energy-dependent time delays (see Uttley et al. 2014, for an overview), a technique hereafter referred to as X-ray RM. The first hard lags were detected in Cygnus X-1 by Miyamoto et al. (1988) and subsequently in many other AGNs (e.g., Papadakis et al. 2001; McHardy et al. 2004), but they seemed to be related rather to spectral evolution within the Comptonizing medium (Chainakun et al. 2019; Zhang et al. 2023). Later, however, soft X-ray lags were discovered (Fabian et al. 2009), which are consistent with reprocessing in the inner accretion disk (Zoghbi et al. 2011). Importantly, these soft lags were found to scale approximately linearly with the black hole mass (M_BH) and can be interpreted in terms of the corona height (h) above the disk (e.g., De Marco et al. 2013). The measured lag amplitudes range from roughly 100 s to 500 s (Kara et al. 2016), depending on M_BH, formally corresponding to light-travel distances of less than ∼6r_g. Lags associated with the Fe Kα line were also measured, indicating slightly larger delays, typically in the range of 1−10 r_g (Zoghbi et al. 2013; Kara et al. 2013). Comparison of observational results with detailed modeling has, however, remained relatively uncommon. Caballero-García et al. (2020) successfully fitted both the lag and energy spectra of IRAS 13224−3809, demonstrating that consistent modeling is possible in at least some cases. In contrast, Caballero-García et al. (2018) report limitations in the simple lamppost geometry for three additional sources, and Zoghbi et al. (2021) likewise find substantial difficulties with the lamppost model in the two systems they analyzed. Specifically, MCG-5-23-16 shows no evidence of relativistic reverberation, while SWIFT J2127.4+5654 exhibited some signatures of such behavior, but only for parameter values inconsistent with the time-averaged spectrum. The delay of the Kα line was reproduced within a lamppost framework, but only over a restricted frequency range, and the inferred corona height was below 10 r_g (Epitropakis et al. 2016).

Most of these results were derived assuming a compact, lamppost-like corona, although extended corona models have occasionally been explored in this context (Hancock et al. 2023). These studies have demonstrated that both the height and radial extension of the corona can vary over time, as demonstrated through the long-term monitoring of 1H 0707−495 and IRAS 13224−3809.

Independent of these X-ray studies, the lamppost model has also been widely applied to explain UV–optical variability as the result of disk irradiation by a variable X-ray source (e.g., Rokaki et al. 1993). This framework quickly became a standard approach for modeling UV–optical inter-band time delays measured in photometric continuum RM campaigns (Sergeev et al. 2005; McHardy et al. 2014; Shappee et al. 2014; Fausnaugh et al. 2016; Mudd et al. 2018; Kara et al. 2021; Guo et al. 2022; Edelson et al. 2024; Mandal et al. 2025; Pozo Nuñez et al. 2025; Mandal et al. 2026), particularly after the inclusion of corona height as a key parameter (e.g., Cackett et al. 2007; Starkey et al. 2017; Kammoun et al. 2019, 2021b,a).

A natural question is whether the constraints on the corona height derived from X-ray RM studies and those obtained from UV–optical time delay fitting in photometric continuum RM are consistent for a given source. At present, there is no clear answer. Historically, these two lines of investigation, i.e., corona height measurements from X-ray RM and from UV–optical continuum RM have been conducted independently, often for different sources and using distinct models or software packages. Consequently, the methodologies employed in X-ray RM and UV–optical continuum RM differ substantially in their fitting procedures and analysis frameworks. To achieve a consistent physical interpretation, it is essential to revisit these approaches and analyze them using a unified methodology.

In the present work, we adopted a consistent modeling framework for both spectral and timing analyses across the X-ray and UV–optical spectral ranges. The model was originally developed by Dovčiak et al. (2004) for modeling X-ray spectra with full relativistic effects and includes an option for time-resolved spectral modeling. This framework was later extended to UV–optical time delay fitting by Kammoun et al. (2021a) and Dovčiak et al. (2022). Our goal is to use this uniform framework to test whether a single set of global parameters, such as the black hole mass, accretion rate, and corona height can simultaneously reproduce the observed properties inferred independently from X-ray RM and UV–optical continuum RM using both spectral and time-domain data. Here, we present the first results obtained for one of the most intensively studied sources, the Seyfert 1 galaxy NGC 5548. The paper is organized as follows. Section 2 describes the model employed to fit both the X-ray and UV–optical spectral and time-domain data. Section 3 provides details of the selected AGN and the observational data used in the analysis. The analysis and the resulting findings are presented in Sect. 4, followed by a discussion in Sect. 5. Finally, the main findings are summarized in Sect. 6.

2. Model

For our analysis, we employed the publicly available code KYNXiltr¹, described in Kammoun et al. (2023). This code builds upon KYNSED, the spectral model introduced by Dovčiak et al. (2022), with several components first presented in Kammoun et al. (2021a). The KYNXiltr framework has been used to model UV–optical inter-band delays in nine AGNs, including NGC 5548 (Kammoun et al. 2021b, 2023). Additionally, Langis et al. (2024) applied the same approach to fit the time-lag measurements of luminous AGNs and found that a corona height of roughly more than 40 r_g can reasonably account for the observed delays. As the code has undergone continuous development over the years, we summarize below the key assumptions underlying the current version relevant to our work.

The model fully incorporates general relativity (GR) effects and adopts a specific geometric configuration. It assumes a lamppost geometry for the hot corona, making the corona height an explicit free parameter. Disk irradiation is calculated under the assumption of a point-like source. However, the corona cannot be accurately described as point-like: its finite size plays a crucial role in shaping the hard X-ray spectrum. In particular, the size of the corona determines the fraction of soft photons from the disk that are Comptonized within it. This effect is explicitly accounted for in the code. The hard X-ray emission from the corona is characterized by its spectral slope and high-energy cutoff, while the low-energy cutoff is determined by the temperature of the seed photons originating from the accretion disk. Because the disk temperature varies with radius, the disk emission is radially integrated to obtain the overall spectral shape and, consequently, the effective seed-photon temperature for Comptonization. The corona is assumed to emit isotropically in its rest-frame.

The energy emitted by the hot corona constitutes a fraction of the total accretion energy budget. This energy is supplied by a Keplerian accretion disk of negligible geometrical thickness. Consequently, the global physical parameters of the model, such as M_BH, accretion rate (ṁ in units of Eddington ratio), black hole spin (a*), and the fractional division of accretion power between the corona and the disk (L_transf/L_disk), jointly determine the overall solution.

In principle, the model also allows for an alternative scenario in which the corona is powered by a mechanism independent of the accretion disk. Such power could be extracted from the black hole spin or arise from processes operating within the innermost stable circular orbit (ISCO). However, we do not consider this alternative configuration in the present analysis.

In the lamppost geometry, the compact X-ray source illuminates the accretion disk, where the incident radiation is partly reflected and partly absorbed and reemitted, thereby modifying the local disk temperature. The disk albedo varies with radius and is computed under the assumption of a constant number density in the disk atmosphere, while accounting for the appropriate local flux. The underlying atomic physics is taken from the XillverD tables of García et al. (2016), calculated for a representative density of n = 10¹⁵ cm⁻³. The ionization state of the disk is determined by the local (radial) dependence of the reprocessing efficiency on the incident flux, which naturally leads to an ionization parameter that varies with radius.

The code tracks photon trajectories, allowing it to follow not only spectral modifications but also the associated time delays. This framework enables the computation of the broadband spectral energy distribution (SED) using KYNSED as well as the time-dependent response of the disk to an impulsive flash from the corona using KYNXiltr. The latter provides the wavelength-dependent response function of the disk. Additionally, geometrical effects, such as the viewing angle of an observer, are fully incorporated into the calculation.

Lag-spectrum modeling is carried out using two complementary approaches: one designed for X-ray RM, where energy and frequency-resolved lags are measured, and another for modeling UV/optical inter-band time delays. This separation reflects the distinct observational strategies required in these two wavelength regimes. Nonetheless, a single physical model (packages inside KYNXiltr and KYNSED) is employed to ensure consistency and to minimize potential systematics arising from heterogeneous analysis methods.

In the X-ray band, we performed a Fourier analysis of the model transfer functions. This procedure provides Fourier-resolved time delays within selected energy intervals, as well as energy-resolved delays computed over specific frequency ranges. These quantities are directly comparable to those obtained from the observational data.

In the UV–optical bands, the mean arrival time of the signal at each wavelength was computed directly from the corresponding transfer function. Inter-band time delays were then obtained by measuring these mean arrival times relative to a chosen reference band. The resulting model-derived delays can be directly compared with the inter-band time delays inferred from the observational data.

3. Sample and data

We focused on identifying sources for which both X-ray Fourier- and energy-resolved delays and UV–optical inter-band time delays have been measured. However, such objects remain rare.

Kara et al. (2016) conducted X-ray RM of Seyfert 1 galaxies using archival observations from the XMM-Newton observatory that were publicly available up to January 1, 2015. Their final sample included 43 Seyfert galaxies exhibiting a wide range of flux levels, exposure times, and variability strengths. They reported both frequency- and energy-resolved lags, along with the X-ray SED for this sample. To explore possible connections between X-ray and longer-wavelength reverberation signatures, we crossmatched this sample with AGNs for which continuum UV–optical time-delay measurements are available from various photometric continuum RM campaigns in the literature. This comparison yielded three common sources: NGC 5548, NGC 4151, and Mrk 335.

Among these, NGC 5548 stands out as having the highest-quality data for both X-ray RM and UV–optical continuum delays. The latter were obtained by Fausnaugh et al. (2016), who carried out an extensive photometric monitoring campaign combining space-based HST observations from the HST/COS UV RM program with simultaneous ground-based observations from sixteen observatories. The optical data were obtained in multiple broadband filters, including Johnson/Cousins BVRI and Sloan Digital Sky Survey (SDSS) ugriz, covering the period from December 2013 to August 2014. We adopted the results of the X-ray analysis presented by Kara et al. (2016), including their source spectrum, frequency, and energy-resolved lag spectra (see the corresponding panel in their Fig. A1). These results are based on a long XMM-Newton light curve with a duration of 9.55 × 10⁴ s. We note that we did not perform an independent analysis of these data.

Furthermore, Kammoun et al. (2024) investigated the time-averaged and variable broadband X-ray/UV/optical SEDs of NGC 5548 using data from Swift, HST, and ground-based facilities. Their goal was to test whether the observed broadband spectral behavior could be explained by the X-ray illumination scenario, despite the relatively modest correlation observed between X-ray and longer-wavelength variations. For our analysis, we used their best-fit results obtained from individual spectral fits with the KYNSED model, which are summarized in Table 1. Additionally, we utilized the broadband SED provided by Mehdipour et al. (2015) in our analysis. Given its extensive multiwavelength coverage and well-characterized variability properties, NGC 5548 thus serves as an ideal laboratory for testing the consistency between SED fitting and lag-spectrum analysis when both are derived from a coherent, homogeneous approach.

4. Analysis and results

The number of free parameters in the adopted model is considerable, including, M_BH, ṁ, a*, h, L_transf/L_disk, photon index (Γ), extinction parameter determined by E(B − V)_host, among others. Consequently, rather than performing a blind search for parameter combinations capable of reproducing the broadband SED, the X-ray Fourier- and energy-resolved time delays, and the UV–optical inter-band delays, we adopted a multistep strategy.

First, we fixed most of the global model parameters based on the broadband X-ray/UV/optical SED fitting of NGC 5548 presented by Kammoun et al. (2024), who employed the same physical model. Using these parameters, we computed the X-ray properties predicted by the SED fit and compared them directly with those inferred from the observed X-ray timing analysis. Next, we explored how the model-predicted X-ray characteristics depend on key parameters, identifying those which primarily govern the agreement between the model and the data. Finally, we inverted the procedure: starting from the parameters that best reproduce the X-ray timing properties, we examined how this parameter set translates into UV–optical behavior and assessed its consistency with the observed inter-band delays of the source.

4.1. Testing X-ray properties at the basis of the broadband SED fitting

We extracted several representative data points from the observed X-ray SED of NGC 5548 presented in Kara et al. (2016). These are shown as black points in Fig. 1, while the gray-shaded region marks the vertical extent of the original data. Since Kara et al. (2016) did not provide the underlying numerical values, we digitized the spectrum directly from their published figure. Using these extracted points, we calculated the reduced χ² per degree of freedom (χ_ν²) and the corresponding upper-tail χ² probability (p-value) to enable a rough comparison with our model fits. This approach provides only an approximate assessment; a more accurate and robust comparison would require rederiving the original data from Kara et al. (2016) using a consistent reduction and calibration procedure, which lies beyond the scope of the present work. Nevertheless, the constructed spectrum clearly shows two prominent features: a soft X-ray excess below approximately 1 keV and a broad Fe Kα emission line centered near 6.4 keV.

Fig. 1. Model fits overlaid on the observed X-ray SED for MBH = 7 × 107, ṁ = 0.05, i = 40°, and fcol = 1.7. Top: results for spin a *  = 0.0. Bottom: results for a *  = 0.998. Representative data points from the observed X-ray SED reported in Kara et al. (2016) are shown as black points, and the gray-shaded region indicates their vertical range. Model curves corresponding to different parameter sets are plotted in various colors, while the best-fitting model with the lowest reduced χν2 is shown in black.

Building on this, we calculated the model predictions as based on the optical best-fitting results. We adopted the same set of input parameters as in Kammoun et al. (2024), generating 15 different models corresponding to 15 segments of the UV−X-ray light-curve data studied independently by Kammoun et al. (2024). These segments were characterized by various combinations of the corona height h, photon index Γ, and energy fraction going to the hot corona and referred to as transfer ratio L_transf/L_disk, as they evolved with time (see Table 1 for reference). The M_BH was fixed at M_BH = 7 × 10⁷ M_⊙, with an accretion rate of ṁ = 0.05 and an inclination angle of i = 40°, consistent with the values adopted by Kammoun et al. (2024). We adopted a constant color-correction factor of f_col = 1.7 (Shimura & Takahara 1995; Zampieri et al. 2001; Hui et al. 2005; Kammoun et al. 2023, 2024) throughout our analysis.

We first show the resulting X-ray spectral models for different parameter sets in Fig. 1, with the colors representing distinct combinations of input parameters. The top and bottom panels correspond to spin values of a *  = 0 and a *  = 0.998, respectively. For each model, we calculated the reduced χ_ν² with p-value to evaluate the fit quality. The best-fitting model, corresponding to the lowest χ_ν², is highlighted with a solid black line.

For both the non-spinning (a *  = 0) and maximally spinning (a *  = 0.998) cases, we obtain largely consistent results, with a few exceptions reflected in the significant differences in the corresponding reduced χ_ν² values. In both cases the best fit is provided by parameter set S10, characterized by h = 20.5 r_g, Γ = 1.60, and L_transf/L_disk = 0.70.

However, for the parameter sets considered, the models systematically deviate from the observed spectrum, resulting in p-values close to zero. In particular, they fail to reproduce the soft X-ray excess, and the best-fitting model tends to overestimate the flux at higher energies, indicating that additional physical components or alternative parameter configurations may be needed to accurately describe the X-ray emission of NGC 5548. Moreover, the modeled Fe Kα line appears broader than observed. Although the X-ray and optical data are not from the same epoch, variability alone is unlikely to account for the discrepancies between the model and the observations. However, Kammoun et al. (2024) accounted for intrinsic X-ray absorption, including possible warm absorbers while fitting the broadband SED, whereas our model fitting did not incorporate any intrinsic absorption component. Kara et al. (2016) did not explicitly model the X-ray flux spectrum but rather presented the observed X-ray spectral data. Consequently, absorption effects were not accounted for in the X-ray spectral data used in our study. Therefore, inconsistencies already exist between the data and the parameter sets adopted in modeling the spectrum, which may contribute to the discrepancies observed in our comparisons.

To assess whether the soft X-ray excess and the potential X-ray obscuration feature around 0.9 keV influence the comparison between our model predictions and the observed X-ray spectrum, we performed an additional test in which the spectral fitting was restricted to the 1.5−10 keV energy range. The only notable change is a systematic reduction in the reduced ${\chi }_{\nu }^2$ values across all parameter sets, resulting in a better fit to the data. In the restricted-band analysis, parameter set S12 (S2) provides the best fit, yielding ${\chi }_{\nu }^2=0.6$ (0.2) for a *  = 0 (a *  = 0.998), whereas parameter set S10 was preferred when the full 0.3–10 keV range was considered.

Nevertheless, our primary objective is to reproduce the observational framework of Kara et al. (2016), which includes both energy- and frequency-resolved lag spectra derived from the entire 0.3−10 keV band. Restricting the analysis to the 1.5−10 keV range would therefore prevent a comprehensive comparison between the model predictions and the full set of observational constraints provided by the X-ray lag spectra. Consequently, we do not pursue the restricted-band analysis further and instead focus on the results obtained over the complete energy range.

Next, we employed the code to model the lag-energy and lag-frequency spectra of NGC 5548 obtained from X-ray RM. In the lag-energy spectrum, the time lags were computed over a specific frequency range ([0.4 − 3]×10⁻⁴) Hz between a narrow band of interest (0.3 − 1 keV) and a broad reference band covering the full 0.3 − 10 keV range. Similarly, the lag-frequency spectrum was derived by measuring the lags between the soft (0.3 − 1 keV) and hard (1 − 4 keV) energy bands. These energy ranges were chosen to maintain consistency with those adopted by Kara et al. (2016). During the fitting, we used the same set of input parameters described above.

The resulting model fits are shown in Fig. 2, with the top and bottom panels corresponding to the lag-energy and lag-frequency spectra, respectively. We find that the model fails to reproduce the observed data in both spectra when the input parameters from Table 1 are adopted. In the lag-energy spectrum, the model remains nearly constant across the entire energy range and does not capture the negative lags observed between ∼0.8 and 5 keV. For example, parameter set S8 yields the lowest χ_ν² = 2.0 corresponding to a p-value of 0.022. Similarly, in the lag-frequency spectrum, the model cannot reproduce the observed negative lags between ∼5 × 10⁻⁵ and 10⁻³ Hz, while the model predicts negative time lags at much lower frequencies, resulting in p-values less than 0.001.

Fig. 2. Top: model-fitted lag-energy spectra overlaid on the observed values (black points with error bars) from Kara et al. (2016) for MBH = 7 × 107, ṁ = 0.05, i = 40°, a *  = 0.0, and fcol = 1.7. Bottom: model-fitted lag-frequency spectra in different colors for the same set of input parameters, overlaid on the observed data shown as black points with error bars. The model curves for different parameter sets are shown in various colors, with the best-fitting model corresponding to the lowest reduced χν2, highlighted in black.

In summary, these results indicate that the current parameter sets are insufficient to fully capture the observed X-ray RM signatures of NGC 5548. This highlights the need for additional physical components, alternative parameter configurations, or a more complex treatment of the corona and disk structure to accurately model the lag-energy and lag-frequency behavior.

4.2. X-ray lag-energy and lag-frequency dependence on the global parameters

To investigate the source of the discrepancy, we relaxed the model constraints imposed by Kammoun et al. (2024) and listed in Table 1 and focused on identifying the key parameters that drive the predicted X-ray behavior. The model depends on several parameters, including M_BH, ṁ, h, i, L_transf/L_disk, and Γ. Among these, M_BH, h, and ṁ are expected to have the strongest influence on the model predictions. To assess their impact, we first examined each parameter individually, beginning with the corona height.

To study the dependence of the lag-energy spectrum on the corona height h, we generated model spectra for a range of h values from 5 r_g to 50 r_g, keeping all other parameters fixed at their previous values (e.g., M_BH = 7 × 10⁷ M_⊙, ṁ = 0.05, and i = 40°). We also fixed the photon index to Γ = 1.58, consistent with the observed value of Γ = 1.58 ± 0.02 reported for NGC 5548 by Ursini et al. (2015), and adopted L_transf/L_disk = 0.9. The resulting models are shown in the top panel of Fig. 3. As the corona height increases, the model-predicted lag amplitude near 1 keV becomes progressively larger and changes from negative to positive values, deviating further from the observed lag-energy spectrum in Fig. 2, upper panel. Even at the smallest height (h = 5 r_g), the model still fails to reproduce the observed negative lag near 1 keV.

Fig. 3. Top: Model-fitted lag-energy spectra for different corona heights h, assuming fixed parameters of MBH = 7 × 107 M⊙, ṁ = 0.05, i = 40°, Γ = 1.58, Ltransf/Ldisk = 0.9, and fcol = 1.7. Middle: model-fitted lag-energy spectra for different MBH, with ṁ = 0.05, a constant height of h = 5 rg, and other parameters fixed to the above values. Bottom: Same as the middle panel but for ṁ = 0.1. The observed data points are shown by black points with error bars in the middle and bottom panel.

However, it is worth noting that the corona heights reported in the literature for NGC 5548 are generally higher. For instance, Starkey et al. (2017) arbitrarily fixed the height at 6 r_g, while Kammoun et al. (2021b) obtain h ∼ 29.1 [23.2, 58.5] r_g from UV–optical lag-spectrum fitting assuming a spin of a *  = 0.0. Jaiswal et al. (2025) report an even larger value of h = 48.3 r_g from simultaneous UV–optical lag-spectrum and SED fitting. On the other hand, Gardner & Done (2017) used h = 10 r_g, although in that case it represented a ring of matter located further out rather than a classical lamppost corona.

Another way to increase the negative lag amplitude between ∼0.8 and 5 keV, as observed in the X-ray RM data, is by adjusting the black hole mass, M_BH. However, the plausible range for M_BH cannot be arbitrarily large. M_BH itself cannot change, but its determination highly depends on the BLR state. Multiple RM campaigns have revealed that, in many AGNs, the measured Hβ time delay can vary substantially between observing epochs, often reflecting different luminosity states. Such variations imply that BLR can undergo dynamic structural changes on timescales of just a few years (e.g., Kaastra et al. 2014; Pancoast et al. 2018). These changes could be driven by variations in the accretion rate (Cackett & Horne 2006; Elitzur et al. 2014), inhomogeneities in the BLR gas distribution (Wanders 1995), or the effects of radiation pressure (Chen et al. 2023). Nonetheless, the underlying cause of this variability remains uncertain, since the observed changes in luminosity are frequently small relative to the associated measurement errors.

NGC 5548 provides a notable example of such dynamic BLR behavior, being one of the best-studied AGNs with over twenty years of RM observations. For this source, the Hβ time delay varies from 2.3 days at L₅₁₀₀ ∼ 10^42.7 erg s⁻¹ (Denney et al. 2010) to 21.5 days at L₅₁₀₀ ∼ 10^43.4 erg s⁻¹ (Peterson et al. 2002, 2004), providing a clear illustration of the well-known BLR “breathing” phenomenon. Assuming a constant virial factor across campaigns, these variations correspond to RM-based M_BH estimates ranging from 0.7 to 9 × 10⁷ M_⊙, implying a mass uncertainty of roughly one dex for NGC 5548. Note that the virial factor used to convert the virial product (R_BLRv²/G) into M_BH depends on the geometry and kinematics of the BLR. Consequently, it can vary across different RM campaigns, introducing an additional uncertainty of more than 0.1 dex (Woo et al. 2015), when a constant virial factor is assumed.

Motivated by this observed variability, we performed model fitting for different black hole masses between 2 and 9 × 10⁷ M_⊙, keeping h = 5 r_g, ṁ = 0.05, i = 40°, Γ = 1.58, and L_transf/L_disk = 0.9. The results, shown in the middle panel of Fig. 3, reveal that decreasing the black hole mass enhances the negative lag amplitudes between ∼0.8 and 5 keV and provides a better match to the observed lag-energy spectrum (also see Sect. A and Fig. A.1 for further discussion). Consequently, the X-ray RM data appear to favor a smaller black hole mass, around M_BH ∼ 2 × 10⁷ M_⊙, yielding a reduced ${\chi }_{\nu }^2=1.1$ and a p-value of 0.362. In contrast, adopting a higher mass of 7 × 10⁷ M_⊙, as assumed by Kammoun et al. (2024) and consistent with the value reported by Horne et al. (2021), results in a poorer fit with χ_ν² = 2.1 and a p-value of 0.020.

We repeated the same experiments for different values of M_BH but with a higher accretion rate of ṁ = 0.1 and show the resulting model fits in the bottom panel of Fig. 3. Here, we obtain an even better model-fit for the lowest mass M_BH ∼ 2 × 10⁷ M_⊙ with χ_ν² = 1.04, and a p-value of 0.440. Overall, the model fits to the X-ray RM data consistently favor a lower black hole mass and higher accretion rate than those inferred from broadband SED fitting.

Building upon these results, we then examined the lag-frequency spectrum using the best-fit parameters inferred from the lag-energy modeling, namely, M_BH = 2 × 10⁷ M_⊙, h = 5 r_g, i = 40°, Γ = 1.58, and L_transf/L_disk = 0.9. To assess the effect of accretion rate on the model performance, we computed fits for both ṁ values of 0.05 and 0.1. The resulting fits, shown in the top panel of Fig. 4, provide a significantly improved agreement with the observed data compared to the earlier models shown in the bottom panel of Fig. 2. In particular, both fits reproduce the overall shape of the lag-frequency spectrum and match all data points within uncertainties, except for the large positive lag observed at the lowest frequency. Among the two cases, the model with ṁ = 0.1 yields a slightly better fit (χ_ν² = 3.9, p-value = 0.002) than that with ṁ = 0.05 (χ_ν² = 4.1, p-value = 0.001), again indicating that the X-ray RM data favor a smaller black hole mass accreting at a relatively higher rate.

Fig. 4. Top: model-fitted lag-frequency spectra with MBH = 2 × 107 M⊙, h = 5 rg, i = 40°, Γ = 1.58, Ltransf/Ldisk = 0.9, and fcol = 1.7 for ṁ = 0.05 (blue) and 0.1 (red). Bottom: Model fit on top of the observed X-ray SED for MBH = 2.7 × 107, h = 10 rg, i = 40°, ṁ = 0.1, Γ = 1.58, Ltransf/Ldisk = 0.9, and fcol = 1.7 shown by the red line. The values for MBH = 7 × 107, ṁ = 0.05, h = 3.9 rg, i = 40°, Γ = 1.71, Ltransf/Ldisk = 0.81, and fcol = 1.7 are shown by the black line. The power-law best-fit is shown as a dot-dashed blue line. Lower sub-panel: data to model ratio for each best-fit model in their respective colors, with a horizontal dashed line indicating a ratio of unity.

Finally, to assess the consistency of the parameters inferred from the lag-energy and lag-frequency fits, we tested whether the same parameter set can also reproduce the observed X-ray SED. The KYNSED model was parameterized with both M_BH and ṁ, which primarily determine the overall normalization of the spectrum. Lowering M_BH requires a higher accretion rate to maintain a comparable total energy flux. This adjustment, however, also introduces changes in the spectral shape. For example, at high Eddington ratios or for small M_BH, the soft X-ray excess becomes clearly visible below 1 keV, making the combined spectrum appear harder. By contrast, variations in L_transf/L_disk have a much weaker impact on the spectral shape and therefore cannot fully compensate for the changes induced by the other two parameters (see Appendix in Dovčiak et al. 2022). We thus refitted the X-ray SED using parameter values close to those obtained from the timing analyses. As shown by the solid red line in the bottom panel of Fig. 4, we find that the model provides a better fit to the observed SED, with improved χ_ν² values for M_BH = 2.7 × 10⁷ M_⊙, h = 10 r_g, ṁ = 0.1, i = 40°, Γ = 1.58, and L_transf/L_disk = 0.9. However, the p-value remains close to zero. For comparison, we also fitted the observed X-ray SED with a simple power-law model, shown by the dot-dashed blue line. The fit yields an energy dependence of ∝E^2 − Γ, corresponding to a photon index of Γ = 1.58, which is in excellent agreement with the value reported by Ursini et al. (2015) and adopted in our model fits. This consistency across the lag-energy, lag-frequency, and X-ray SED analyses suggests a coherent physical scenario in which NGC 5548 is better described by a relatively smaller black hole mass accreting at a comparatively higher rate.

4.3. UV–optical lag-spectrum fitting

In this section, we discuss modeling the UV–optical lag spectrum of NGC 5548 using the physically motivated X-ray reflection code KYNXiltr (Kammoun et al. 2021a, 2023), implemented within the same modeling framework used for the X-ray RM data. Our goal is to test whether the physical parameters inferred from X-ray RM analyses can reproduce the observed inter-band lags in the UV–optical regime. To this end, we fitted the observed lag spectrum using the same input parameters derived from our X-ray RM results, as described in the previous sections. Specifically, we adopted a black hole mass of M_BH = 2.7 × 10⁷ M_⊙, i = 40°, ṁ = 0.1, Γ = 1.58, and f_col = 1.7. The corona height, h, was treated as a free parameter, while two different ratios, L_transf/L_disk = 0.5 and 0.9, were considered.

The resulting model fits for the two spin configurations, a *  = 0.0 and a *  = 0.998, are shown in Fig. 5 as dashed purple and teal lines, respectively. Although these fits employ physically consistent parameters, the model significantly underpredicts the observed UV–optical inter-band delays, resulting in reduced χ_ν² values of 4.3 and 8.2, with p-values less than 0.001 for the nonrotating and maximally rotating cases, respectively. Moreover, the best-fit corona heights in both scenarios exceed 90 r_g, far larger than the typical values of h ∼ 5–10 r_g inferred from X-ray RM and X-ray SED analyses. This pronounced mismatch indicates that the parameter set derived exclusively from the X-ray RM data modeling is insufficient to reproduce the UV–optical lag spectrum of NGC 5548.

Fig. 5. UV–optical lag spectrum of NGC 5548. The black circular points with error bars show the observed rest-frame inter-band time delays reported by Fausnaugh et al. (2016). The median best-fit KYNXiltr-model curves for spins a *  = 0.0 and a *  = 0.998 are shown as solid and dashed lines. The fit ranges corresponding to the two adopted values of Ltransf/Ldisk (0.5 and 0.9) are indicated by the shaded regions. The model fits for the different combinations of input parameters are shown in different colors. The vertical and horizontal dotted lines indicate the rest-frame reference wavelength and zero rest-frame lag, respectively.

To further investigate this discrepancy, we next repeated the fits using a comparable black hole mass of M_BH = 2 × 10⁷ M_⊙, this time allowing both the accretion rate, ṁ and h to vary while keeping all other parameters same as before. The resulting fits, shown in Fig. 5 as dashed red and brown lines for a *  = 0.0 and a *  = 0.998, respectively, reproduce the observed inter-band delays more successfully, with improved χ_ν² values of 1.8 (p-value = 0.026) and 3.5 (p-value < 0.001). However, these fits required a considerably higher accretion rate, ṁ ∼ 0.499, along with an increased corona height exceeding 90 r_g. Despite these high values, the close match between the model and observed lag spectrum indicates that the UV–optical delays can, in principle, be explained using a black hole mass consistent with X-ray RM results, provided that both the accretion rate and corona height are significantly enhanced. However, NGC 5548 is unlikely to exhibit such a high accretion rate of 0.499 if the smaller black hole mass inferred from X-ray RM is adopted.

Finally, we tested whether the higher M_BH inferred from broadband SED modeling can reconcile these discrepancies. We therefore fitted the observed UV–optical lag spectrum using M_BH = 7 × 10⁷ M_⊙, as in the time-averaged and variable X-ray/UV/optical SEDs fitting of NGC 5548 by Kammoun et al. (2024). The other parameters (i, Γ, L_transf/L_disk, and f_col) were kept the same, while ṁ and h were allowed to vary freely. With these inputs, the model once again reproduces the observed inter-band delays well, as represented by the solid black and blue lines for a *  = 0.0, and 0.998, respectively, achieving a χ_ν² value of 1.5 with p-values of 0.086 and 0.090, respectively. Note that Kammoun et al. (2024) previously performed lag-spectrum fitting assuming M_BH = 7 × 10⁷ M_⊙. We repeated the same analysis to ensure full internal consistency within our framework. From this fitting, the best-fit accretion rates are ṁ ∼ 0.05 for a *  = 0.0 and ṁ ∼ 0.1 for a *  = 0.998, consistent with those inferred from broadband SED fitting by Kammoun et al. (2024), and X-ray SED fitting only in this work, respectively. However, the corresponding corona heights remain relatively large, ranging from ∼23 to 60 r_g, which are still higher than the values typically inferred from X-ray RM analyses, yet consistent with those reported by Kammoun et al. (2021b) and Jaiswal et al. (2025). The results of the UV–optical lag-spectrum model fits obtained using different values of M_BH are summarized in Table 2.

Overall, these results reveal a systematic discrepancy between the M_BH, corona heights, and accretion rates derived from X-ray RM and those obtained from UV–optical lag-spectrum modeling. Specifically, reproducing the UV–optical lag spectrum requires a significantly higher and more extended corona, as well as a higher accretion rate, compared to those inferred from the X-ray domain, assuming a consistent M_BH across the two wavelength domains. In contrast, when the model fitting is performed separately within each spectral regime, i.e., X-ray (lag energy, lag frequency, and X-ray SEDs) or UV–optical (broadband SED in Kammoun et al. (2024) and UV–optical lag spectrum), the derived parameters remain internally consistent within each respective domain. This divergence highlights the challenge of simultaneously reconciling the X-ray and UV–optical timing properties of NGC 5548 within a single, unified reprocessing framework.

5. Discussion

We modeled the Fourier-resolved lag-energy and lag-frequency spectra, together with the X-ray SED inferred from X-ray RM and the UV–optical lag spectrum from photometric continuum RM in a self-consistent manner using a unified set of codes within a single modeling framework. However, this approach did not produce mutually consistent results.

In particular, the Fourier-resolved X-ray delays generally imply a relatively small M_BH. However, for such a low black hole mass, the accretion rate and corona height inferred from the Fourier-resolved X-ray lag spectra become incompatible with those derived from the UV–optical continuum RM lag spectrum in NGC 5548. The problem with the black hole mass is coupled with the problem of accretion rate.

A second discrepancy concerns the inferred height of the corona. X-ray RM studies typically favor a compact geometry, with heights on the order of ∼10 r_g, as commonly reported in X-ray analyses (e.g., De Marco et al. 2013). In contrast, optical studies often imply significantly larger values, reaching up to an order of magnitude higher. While the results of Kammoun et al. (2023) do not always require such large heights, they nevertheless span a broad range – from ∼10 r_g to ∼100 r_g, depending on the choice of other model parameters. Most notably, M_BH, and ṁ should remain consistent across both X-ray and optical datasets, and the fact that they do not poses a more fundamental challenge to achieving a fully self-consistent model.

Therefore, the disagreement in the inferred black hole mass, accretion rate, and corona height requires careful consideration. Several potential sources for this discrepancy include:

• data quality,

• underlying assumptions of the X-ray RM modeling, and

• underlying assumptions of the UV–optical RM modeling.

We discuss the potential sources of these discrepancies below. In this context, the present work should be regarded as a pilot study aimed at assessing the consistency of the model with the observed RM features in both the X-ray and UV–optical domains. Importantly, the lack of agreement between the model predictions and the observations is, in itself, a significant result, as it highlights the need for a more comprehensive and systematic investigation, which we intend to pursue in future work.

5.1. Data quality

The lag-energy and lag-frequency spectra considered in our analysis were simply taken from Kara et al. (2016), who computed them using data from a single X-ray observation. Their study relied on an XMM-Newton observation with an exposure of nearly 100 ks, which is longer than typical exposures and thus offers relatively high statistical quality. Throughout the exposure, the source maintained an average count rate of about five counts per second. However, despite the long exposure and reasonable count rate, the reported normalized excess variance is low, F_var = 0.039. This implies that variability on timescales shorter than ∼150 s, corresponding to a frequency of 6.6 × 10⁻³ Hz, is likely dominated by Poisson noise, although this value should be regarded only as a rough estimate. A more rigorous and comprehensive analysis of the available data is clearly required. We intend to pursue this in detail; however, the volume of data is substantial, encompassing not only observations from XMM-Newton but also datasets from multiple additional instruments.

For example, a more robust determination of the usable frequency range requires examining the coherence function, as illustrated for example in Fig. 1 of Epitropakis et al. (2016). Such tests, however, were not performed by Kara et al. (2016), leaving the reliability of the high-frequency lag measurements uncertain. Therefore, a more robust and careful analysis of X-ray data spanning a longer temporal baseline, potentially incorporating monitoring data from Swift is required. This is particularly important for NGC 5548, which hosts a relatively massive black hole and thus does not exhibit strong variability on timescales shorter than a day. We leave this detailed investigation to future work.

The optical data are of very high quality. Kammoun et al. (2024) independently analyzed several datasets and successfully modeled the SEDs. Moreover, the high-quality UV–optical lag spectrum constructed by Fausnaugh et al. (2016) was well reproduced using the same modeling approach by Kammoun et al. (2021b).

5.2. Assumptions in X-ray RM modeling

The model we employed incorporates several advanced features, including GR effects and a self-consistent treatment of hard X-ray reprocessing by the accretion disk. However, it does not capture all relevant physical processes. In particular, the irradiation was modeled using a lamppost geometry, while the hard X-ray emission was assumed to originate from a spherical hot corona.

A key parameter in this context is the compactness parameter, a dimensionless combination of the corona radius (when approximated as a sphere) and its luminosity (Guilbert et al. 1983; Svensson 1984). The interaction of soft photons with hot electrons additionally depends on the electron temperature (for a thermal electron distribution) and the optical depth, as pair creation can act as an important cooling mechanism, preventing the corona from becoming too compact (Bisnovatyi-Kogan et al. 1971; Svensson & Zdziarski 1994). Spectral analysis enables the evaluation of some coronal plasma parameters (e.g., Tortosa et al. 2018), and relatively recent observational studies suggest that corona sizes are on the order of 3–10 r_g (Fabian et al. 2015), highlighting the significant role of pair creation.

Thus, the point-like lamppost approximation represents a simplifying assumption. In the model, the radius of the hot corona was explicitly calculated and is typically of the order of half of the corona height; however, the disk irradiation was still treated as originating from a point-like source. Spectral data alone place only weak constraints on the detailed geometry of the corona, since the reflected emission from the underlying accretion disk is relatively insensitive to the exact spatial extent and shape of the irradiating region. Robust constraints on the corona geometry therefore require extremely high-quality data (e.g., Szanecki et al. 2020; Feng et al. 2025; Nekrasov et al. 2025). Consequently, the adoption of the lamppost approximation is not expected to significantly affect the results presented here, if the hot corona indeed has a spherical geometry.

However, this assumption may not hold in all cases. If the corona is, in fact, part of a (not very strong) jet (Henri & Pelletier 1991) with standing or propagating shocks, we can envision two distinct regions of relatively high dissipation. In such a scenario, a vertically extended corona could help resolve the apparent discrepancy in inferred coronal heights, since X-ray and UV–optical observations probe different reprocessing regions. Specifically, X-ray reflection arises within a few gravitational radii, where the lower portion of the corona is likely to dominate, whereas UV–optical emission is produced at distances of tens to hundreds of gravitational radii, where the upper regions of the corona may play a more significant role.

An additional challenge lies in incorporating variations in the hard X-ray spectral slope into the model in a fully self-consistent manner. Although this could, in principle, be achieved by adopting a time-dependent spectral slope directly from the observations, such an approach requires reliable estimates of the hard X-ray spectral slope obtained through a dedicated reanalysis of the X-ray data. Since no such analysis was performed in the present work, this aspect was not been implemented here; however, we plan to address it in future studies.

The role of the warm absorber is also not negligible. Turner & Miller (2009) highlighted a significant contribution of absorption (warm absorber) in addition to reflection, potentially affecting its properties. By contrast, Panagiotou & Walter (2019) concluded that in unobscured sources the reflection primarily originates from the disk. However, NGC 5548 does show traces of variable obscuration. Moreover, the mechanism responsible for hard X-ray positive lags may be coupled to soft negative flux variations, further complicating the model. If this absorption originates from a warm absorber along the line of sight with a density of ∼10⁹ cm⁻³, the corresponding recombination timescale on the order of one hour could further influence the observed variability characteristics. Furthermore, the variable absorber may also not cover the entire source, as for example envisioned in Fig. 14 of Wildy et al. (2021), which would introduce additional modifications to the measured time delays.

5.3. Assumptions in UV–optical RM modeling

A key assumption in the present study is that the contribution from the BLR is neglected. However, there is growing evidence that the BLR contributes to the UV–optical bands not only through strong emission lines but also via a reprocessed continuum (e.g., Korista & Goad 2001; Netzer 2022; Jaiswal et al. 2023).

If this additional BLR component is taken into account, the total observed delay would include both the intrinsic disk response and an extra contribution from the BLR. As a result, the disk’s intrinsic lag could be shorter than currently inferred, which in turn may lead to a lower estimate of the corona height from model fits. At present, the model employed in this work does not incorporate such a BLR contribution, and therefore this effect cannot yet be explored within our framework.

Hence, to investigate the discrepancy between the M_BH and ṁ inferred from the X-ray RM and UV–optical RM modeling, together with their respective SED fits in NGC 5548, we applied an additional independent methodology. Specifically, we adopted the simultaneous UV–optical lag spectrum and broadband SED fitting approach described by Jaiswal et al. (2025). This framework incorporates a standard accretion disk, an inner hot flow, a warm corona covering part of the inner disk, lamppost irradiation of the outer disk, and a radiation-pressure regulated BLR based on the Failed Radiatively Accelerated Dusty Outflow (FRADO) model (Czerny & Hryniewicz 2011; Naddaf et al. 2021; Naddaf & Czerny 2022), which accounts for BLR contamination (e.g., Korista & Goad 2001; Lawther et al. 2018; Netzer 2022). However, this model does not include GR effects or the albedo contribution from the disk, and it instead assumes complete thermalization of the incident X-ray flux. As a result of these simplifications, the model is not well suited for describing X-ray reverberation phenomena.

Our goal is to assess which M_BH estimate – either that derived from X-ray SED and X-ray RM analysis or that obtained from UV–optical lag spectrum and SED fitting – is more consistent when confronted with an independent modeling strategy. Such a large discrepancy in both M_BH and ṁ is physically unexpected.

The top panel of Fig. 6 shows the model-recovered lag spectra for two trial black hole masses, M_BH = 2 × 10⁷ M_⊙ and M_BH = 7 × 10⁷ M_⊙, overlaid on the observed time-delay measurements. As part of the same procedure, we simultaneously fitted the UV–optical SED of NGC 5548 from Mehdipour et al. (2015). The bottom panel of Fig. 6 presents these SED fits, where the best-fit models for the two masses are shown as dashed red and solid red curves, respectively. The black squares mark the selected continuum points extracted from the observed spectrum (black line), and all contributing model components are displayed to illustrate how the total SED was constructed. In this fitting, the corona height and accretion rate were allowed to vary freely. The corresponding best-fit parameters are reported in Table 2.

Fig. 6. UV–optical lag spectrum and SED of NGC 5548. Top: observed rest-frame inter-band time delays, shown by the black circular points with error bars. The inter-band time delays recovered from the accretion disk combined with the BLR reprocessing model for two MBH values are shown by the red and blue circular points. The vertical and horizontal dotted lines indicate the rest-frame reference wavelength and zero rest-frame lag, respectively. Bottom: UV–optical spectrum, where the solid black line represents the observed data and the black squares mark the selected continuum points. The individual model components are shown in different colors, while the total best-fit models are indicated with red lines for MBH = 2 × 107 M⊙ and MBH = 7 × 107 M⊙, shown as dashed and solid lines, respectively.

We find that the model provides a better joint fit to both the UV–optical lag spectrum and the SED for M_BH = 7 × 10⁷ M_⊙, resulting in χ_ν² = 1.7 (p-value = 0.038) for the lag spectrum and χ_ν² = 0.84 for the SED, compared to χ_ν² = 2.1 (p-value = 0.007) and 1.97, respectively, for M_BH = 2 × 10⁷ M_⊙. These results further indicate that NGC 5548 favors a larger black hole mass of M_BH = 7 × 10⁷ M_⊙, together with a smaller accretion rate of ṁ ∼ 0.014 and a lamppost height of h = 18.6 r_g. It is worth noting that Jaiswal et al. (2025) fitted the same lag spectrum and SED assuming M_BH = 5 × 10⁷ M_⊙. They obtained broadly similar results, with ṁ ∼ 0.017 and a larger corona height of h = 48.3 r_g. Importantly, these estimates are broadly consistent with the constraints derived from the KYNXiltr modeling of the UV–optical lag spectrum and the broadband SED, once the dispersion associated with different black hole spin configurations is taken into account. Consequently, explaining the low M_BH and high ṁ solution inferred from the X-ray SED and Fourier-resolved X-ray RM analysis appears highly unlikely.

Notably, the model of Kammoun et al. (2024) incorporated GR effects but excluded contributions from the BLR, whereas the model of Jaiswal et al. (2025) included the BLR but did not account for GR effects. Despite these differences, both studies favored larger black hole masses and corona heights, together with relatively lower accretion rates, which successfully reproduce the corresponding SED shapes and UV–optical time delays across multiple wavelength bands. Therefore, further work on X-ray RM is clearly warranted.

5.4. Future prospects

The potential limitations affecting the X-ray results used in the present study prevent us from drawing firm conclusions that the observed X-ray−UV/optical discrepancy necessarily reflects differences in the underlying assumptions of the reverberation models. At the same time, the assumptions adopted in our modeling of the Fourier-resolved lags may themselves contribute to the apparent X-ray−UV/optical inconsistency.

The essence of the X-ray RM model is the assumption that the X-ray source emits a flux with a fixed spectral shape, a power law with photon index Γ that does not vary with time. This isotropic emission is partially observed directly by the observer and partially illuminates the accretion disk. The subsequent reprocessing involves selective absorption and reemission, which depend on the ionization state of the disk, calculated self-consistently from the disk surface density and the incident radiation. Consequently, a spectral evolution arises entirely from the reprocessing of radiation by progressively more distant disk regions. The disk’s internal dissipation is assumed to be constant; hence, its variability is solely a response to irradiation. Moreover, the model does not account for hard X-ray lags and assumes the absence of a warm absorber along the line of sight that could respond to changes in the lamp luminosity.

Each of these assumptions can potentially influence the results. The corona is compact. Thus, any internal changes can occur rapidly and may be coupled to variations in the soft photon flux from the innermost regions of the flow. Modeling such changes is beyond the scope of this study, given the limited understanding of the variability mechanisms in this complex region.

6. Summary

We modeled the X-ray SED, the Fourier-resolved lag-energy and lag-frequency spectra from X-ray RM as well as the inter-band lag spectrum from UV–optical continuum RM of NGC 5548 within a unified and physically motivated reflection-based framework. Despite adopting a consistent modeling strategy across all wavelength regimes, we are unable to identify a single set of physical parameters that can simultaneously reproduce both the spectral and timing properties of the source.

The Fourier-resolved X-ray RM data preferentially indicate a relatively small black hole mass, M_BH ∼ 2 − 3 × 10⁷ M_⊙, together with a higher accretion rate and a compact corona. This parameter combination also yields an improved description of the X-ray SED. In contrast, the UV–optical continuum RM data and broadband SED consistently favor a larger black hole mass, M_BH ∼ 5 − 7 × 10⁷ M_⊙, lower accretion rates, and a more spatially extended irradiating region. While discrepancies in the inferred corona height can be partly attributed to the simplified lamppost geometry and to the fact that X-ray and UV–optical reverberation probe markedly different spatial scales, the inconsistency in the inferred M_BH and ṁ represents a more fundamental challenge to achieving a fully self-consistent model.

In addition, we identify several potential limitations in the X-ray RM data used in Kara et al. (2016). The lag-energy and lag-frequency spectra were derived from a single XMM-Newton observation characterized by modest intrinsic variability and uncertain coherence at high frequencies. Together, these factors may bias the inferred Fourier-resolved lags and, consequently, the physical parameters derived from X-ray RM modeling. Moreover, the lack of a consistent treatment of intrinsic X-ray absorption and warm absorbers in the construction of the X-ray SED data may contribute to the significant deviations observed in the model fits. A reanalysis based on the full archival X-ray dataset, incorporating coherence-based frequency selection and a consistent treatment of absorption, is therefore essential to obtain more robust and reliable constraints.

We conclude that the most critical next step is to fully exploit the available X-ray data for this source by analyzing a larger dataset and making use of its full potential. At the same time, future progress will also require extending the current modeling framework by incorporating additional physical ingredients that are presently missing.

Acknowledgments

We thank the referee for comments and suggestions. We also thank Piotr T. Życki for helpful discussions. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. [951549]). The Czech-Polish Mobility program of the two Academies of Sciences, titled “Appearance and dynamics of accretion onto black holes”, is greatly appreciated.

References

Appendix A: Fourier-resolved lag dependence on black hole mass in the model

The lag-energy spectrum was computed in a fixed Fourier frequency band of (0.4 − 3)×10⁻⁴ Hz (Kara et al. 2016), and this same band was adopted throughout our analysis. Because larger black holes have longer characteristic timescales, their variability power shifts to lower Fourier frequencies, whereas lower-mass AGNs exhibit variability at higher frequencies. Consequently, a fixed Fourier frequency band samples different regions of the transfer function for different M_BH.

For low M_BH, the characteristic frequencies are high, thus the chosen frequency band may fall below the break frequency, where propagation lags dominate or partially overlap the reverberation regime. In this case, the band probes relatively slow variability for that mass, resulting in larger measured lags. For high M_BH, however, the same frequency band may lie in the high-frequency tail of the transfer function, where the response is weak or even beyond the coherence limit; the band then probes variability faster than physically relevant for that mass, producing smaller lags (see Fig. 3). Thus, the Fourier-resolved lag amplitude decreases systematically with increasing M_BH when the frequency band is held fixed.

To illustrate this, we computed lag-energy spectra for a lower black hole mass, M_BH = 2 × 10⁷ M_⊙, and for a higher mass, M_BH = 7 × 10⁷ M_⊙, using both the original Fourier band of (0.4 − 3)×10⁻⁴ Hz and frequency bands scaled upward and downward by the mass ratio of 3.5. The corresponding mass-scaled frequency ranges are (4.65 − 7.25)×10⁻⁴ Hz and (0.10 − 1.40)×10⁻⁴ Hz, respectively. As shown earlier, comparing the two masses within the same fixed original Fourier band (solid lines) results in larger lag amplitudes for the lower-mass black hole. However, when the Fourier band is shifted upward for the lower-mass model, or downward for the higher-mass model, the resulting lag-energy spectra (dot-dashed black and dot-dashed red lines, respectively) align closely with those of the higher- or lower-mass models computed using the original frequency band (solid red and black lines) as shown in Fig. A.1.

Fig. A.1. Model-fitted lag-energy spectra for different MBH, computed with ṁ = 0.05, a fixed height of h = 5 rg, and all other parameters held constant except the Fourier frequency band. The solid black line shows the model for MBH = 2 × 107 M⊙, using the (0.4 − 3)×10−4 Hz band. The solid red line shows the same frequency band applied to MBH = 7 × 107 M⊙. The dot-dashed black and dot-dashed red lines show the models for MBH = 2 × 107 M⊙ and MBH = 7 × 107 M⊙, respectively, computed using the mass-scaled frequency bands of (4.65 − 7.25)×10−4 Hz and (0.10 − 1.40)×10−4 Hz.

In summary, the apparent decrease in Fourier-resolved lag amplitude with increasing M_BH arises naturally when a fixed frequency band is used, because the same band probes progressively faster and less responsive regions of the transfer function for higher-mass black holes. Adjusting the Fourier frequency band in proportion to M_BH restores consistency between the modeled lag-energy spectra, demonstrating that the mass dependence of the observed lags in the lag-energy spectrum is primarily a consequence of the characteristic timescale scaling with black hole mass rather than intrinsic changes in the reverberation response. However, since our simulations were performed using the same approach as in the data analysis of Kara et al. (2016), the discrepancy between the model predictions and the observed data persists.

All Tables

All Figures

Fig. 1. Model fits overlaid on the observed X-ray SED for MBH = 7 × 107, ṁ = 0.05, i = 40°, and fcol = 1.7. Top: results for spin a *  = 0.0. Bottom: results for a *  = 0.998. Representative data points from the observed X-ray SED reported in Kara et al. (2016) are shown as black points, and the gray-shaded region indicates their vertical range. Model curves corresponding to different parameter sets are plotted in various colors, while the best-fitting model with the lowest reduced χν2 is shown in black.

In the text

Fig. 2. Top: model-fitted lag-energy spectra overlaid on the observed values (black points with error bars) from Kara et al. (2016) for MBH = 7 × 107, ṁ = 0.05, i = 40°, a *  = 0.0, and fcol = 1.7. Bottom: model-fitted lag-frequency spectra in different colors for the same set of input parameters, overlaid on the observed data shown as black points with error bars. The model curves for different parameter sets are shown in various colors, with the best-fitting model corresponding to the lowest reduced χν2, highlighted in black.

In the text

Fig. 3. Top: Model-fitted lag-energy spectra for different corona heights h, assuming fixed parameters of MBH = 7 × 107 M⊙, ṁ = 0.05, i = 40°, Γ = 1.58, Ltransf/Ldisk = 0.9, and fcol = 1.7. Middle: model-fitted lag-energy spectra for different MBH, with ṁ = 0.05, a constant height of h = 5 rg, and other parameters fixed to the above values. Bottom: Same as the middle panel but for ṁ = 0.1. The observed data points are shown by black points with error bars in the middle and bottom panel.

In the text

Fig. 4. Top: model-fitted lag-frequency spectra with MBH = 2 × 107 M⊙, h = 5 rg, i = 40°, Γ = 1.58, Ltransf/Ldisk = 0.9, and fcol = 1.7 for ṁ = 0.05 (blue) and 0.1 (red). Bottom: Model fit on top of the observed X-ray SED for MBH = 2.7 × 107, h = 10 rg, i = 40°, ṁ = 0.1, Γ = 1.58, Ltransf/Ldisk = 0.9, and fcol = 1.7 shown by the red line. The values for MBH = 7 × 107, ṁ = 0.05, h = 3.9 rg, i = 40°, Γ = 1.71, Ltransf/Ldisk = 0.81, and fcol = 1.7 are shown by the black line. The power-law best-fit is shown as a dot-dashed blue line. Lower sub-panel: data to model ratio for each best-fit model in their respective colors, with a horizontal dashed line indicating a ratio of unity.

In the text

Fig. 5. UV–optical lag spectrum of NGC 5548. The black circular points with error bars show the observed rest-frame inter-band time delays reported by Fausnaugh et al. (2016). The median best-fit KYNXiltr-model curves for spins a *  = 0.0 and a *  = 0.998 are shown as solid and dashed lines. The fit ranges corresponding to the two adopted values of Ltransf/Ldisk (0.5 and 0.9) are indicated by the shaded regions. The model fits for the different combinations of input parameters are shown in different colors. The vertical and horizontal dotted lines indicate the rest-frame reference wavelength and zero rest-frame lag, respectively.

In the text

Fig. 6. UV–optical lag spectrum and SED of NGC 5548. Top: observed rest-frame inter-band time delays, shown by the black circular points with error bars. The inter-band time delays recovered from the accretion disk combined with the BLR reprocessing model for two MBH values are shown by the red and blue circular points. The vertical and horizontal dotted lines indicate the rest-frame reference wavelength and zero rest-frame lag, respectively. Bottom: UV–optical spectrum, where the solid black line represents the observed data and the black squares mark the selected continuum points. The individual model components are shown in different colors, while the total best-fit models are indicated with red lines for MBH = 2 × 107 M⊙ and MBH = 7 × 107 M⊙, shown as dashed and solid lines, respectively.

In the text

Fig. A.1. Model-fitted lag-energy spectra for different MBH, computed with ṁ = 0.05, a fixed height of h = 5 rg, and all other parameters held constant except the Fourier frequency band. The solid black line shows the model for MBH = 2 × 107 M⊙, using the (0.4 − 3)×10−4 Hz band. The solid red line shows the same frequency band applied to MBH = 7 × 107 M⊙. The dot-dashed black and dot-dashed red lines show the models for MBH = 2 × 107 M⊙ and MBH = 7 × 107 M⊙, respectively, computed using the mass-scaled frequency bands of (4.65 − 7.25)×10−4 Hz and (0.10 − 1.40)×10−4 Hz.

In the text