Characterizing X–ray, UV, and optical variability in NGC 6814 using high–cadence Swift observations from a 2022 monitoring campaign

We present the first results of a high–cadence Swift monitoring campaign (3 − 4 visits per day for 75 days) of the Seyfert 1.5 galaxy NGC 6814 characterizing its variability throughout the X–ray and UV/optical wavebands. Structure function analysis reveals an X–ray power law ( 𝛼 = 0 . 5 + 0 . 2 − 0 . 1 ) that is significantly flatter than the one measured in the UV/optical bands ( ⟨ 𝛼 ⟩ ≈ 1 . 5), suggesting different physical mechanisms driving the observed variability in each emission region. The structure function break–time is consistent across the UV/optical bands ( ⟨ 𝜏 ⟩ ≈ 2 . 3 d), suggesting a very compact emission region in the disc. Correlated short time–scale variability measured through cross–correlation analysis finds a lag–wavelength spectrum that is inconsistent with a standard disc reprocessing scenario ( 𝜏 ∝ 𝜆 4 / 3 ) due to significant flattening in the optical wavebands. Flux–flux analysis finds an extremely blue AGN spectral component ( 𝐹 𝜈 ∝ 𝜆 − 0 . 85 ) that does not follow a standard accretion disc profile ( 𝐹 𝜈 ∝ 𝜆 − 1 / 3 ). While extreme outer disc truncation ( 𝑅 out = 202 ± 5 𝑟 𝑔 ) at a standard accretion rate ( (cid:164) 𝑚 Edd = 0 . 0255 ± 0 . 0006) may explain the shape of the AGN spectral component, the lag–wavelength spectrum requires more modest truncation ( 𝑅 out = 1 , 382 + 398 − 404 𝑟 𝑔 ) at an extreme accretion rate ( (cid:164) 𝑚 Edd = 1 . 3 + 2 . 1 − 0 . 9 ). No combination of parameters can simultaneously explain both results in a self–consistent way. Our results offer the first evidence of a non–standard accretion disc in NGC 6814.


INTRODUCTION
Active galactic nuclei (AGNs) host a central supermassive black hole (SMBH) that is actively accreting material from its surrounding environment, producing the most continuously luminous source of emission in the Universe.The extreme compactness of the SMBH and its environment, however, renders this region unresolved by current telescopes.Indirect methodologies, such as the analysis of multiwavelength flux variability, have therefore been developed to study the physical processes at work in this extreme region.
X-ray variability studies have revealed that close to the SMBH exists an extremely hot ( ∼ 10 9 K), optically thin ( < 1), compact ( ∼ 10   ; e.g.Alston et al. 2020;Gallo et al. 2021) X-ray corona.This corona illuminates an optically thick, geometrically thin disc of accreting material that surrounds the SMBH (Pringle & Rees 1972;Shakura & Sunyaev 1973;Novikov & Thorne 1973;Lynden-Bell & Pringle 1974), which itself produces thermal UV emission at the inner region and optical emission further out due to a radial temperature profile that follows  ∝  −3/4 (Shakura & Sunyaev 1973).The coronal and disc emission comprise the high-energy spectral energy distribution (SED) of AGNs, and act as a photoionising continuum for material at larger distances.
Optical variability studies have measured the size of the so-called broad line region (BLR) in AGNs, responsible for the production ★ E-mail: adam.gonzalez@smu.ca(AGG) of broad optical emission lines, through the reverberation mapping technique (Blandford & McKee 1982;Peterson 1993).By measuring time delays between correlated optical continuum and emission line variations, and assuming any measured time delay between the two is solely due to light travel time effects, the size of the BLR can be estimated as 10 3 − 10 5   (e.g.Gallo et al. 2023).Combining this result with the width of the corresponding emission line, a SMBH mass estimate can also be made, which has yielded an ever growing sample of reverberation-based SMBH mass estimates (e.g.Bentz & Katz 2015).
On smaller scales, time delays between the X-ray and UV/optical emission in AGNs have been used to estimate the size of the accretion disc.X-rays illuminating the accretion disc will be both reprocessed and emitted by the disc (i.e.so-called X-ray reflection; George & Fabian 1991) as well as absorbed by the disc, thereby heating it and producing thermal UV/optical emission.By correlating the X-ray variability with UV/optical continuum emission in various energy bands, corresponding time delays can be measured that can therefore probe both the location of the X-ray corona (e.g.Kammoun et al. 2021a) and the size of the accretion disc (e.g.McHardy et al. 2023).
NGC 6814 ( = 0.00522; Springob et al. 2005) is a Seyfert 1.5 (Véron-Cetty & Véron 2006) hosted by a nearly face-on grand-design spiral galaxy (see Figure 1 of Bentz et al. 2019)  1 .Its SMBH mass is estimated to be  BH = (1.85 ± 0.35) × 10 7  ⊙ based on H measurements (Bentz et al. 2009), though a range of  BH ≈ (0.55 − 6.1) × 10 7  ⊙ have also been reported using a variety of H and He emission lines (Bentz et al. 2010), with dynamical BLR modelling finding  BH = 2.6 +1.9 −0.9 × 10 6  ⊙ (Pancoast et al. 2014).Recently, we presented evidence of a partial X-ray eclipse in NGC 6814 that lasted ∼ 42 ks (Gallo et al. 2021).Combined with the duration of ingress and egress (both ∼ 13.5 ks), we were able to estimate the distance to the obscuring material as  ≈ 2700   and its density as   ≈ 10 10 cm −3 , consistent with a BLR origin (e.g.Rees et al. 1989).The X-ray corona was also estimated to have a diameter of ∼ 25   , consistent with X-ray reverberation measurements (e.g.Alston et al. 2020).New work by Pottie et al. (2023) has revealed that the obscuring medium is likely inhomogeneous, with denser clumps embedded in an ionised halo.
Here we present the results of a high-cadence (3 − 4 visits per day) multi-wavelength monitoring campaign of NGC 6814 with the Swift (Gehrels et al. 2004) observatory carried out from 28 August to 10 November 2022 (75 days).We outline our data processing steps in Section 2, with all data products and results of the various analyses performed presented in Section 3. We discuss our findings in Section 4 and conclude in Section 5.

OBSERVATIONS & DATA REDUCTION
Below we describe our processing steps for data collected with the Xray Telescope (XRT; Burrows et al. 2005) and the UltraViolet/Optical Telescope (UVOT; Roming et al. 2005) onboard Swift.
The XRT was operated in Photon Counting (PC) mode for the entire 2022 observing campaign.We processed the XRT data using the online XRT Product Builder tool (Evans et al. 2007(Evans et al. , 2009)), which is a pipe-line that produces data products that have been treated for pile-up, dead pixels, and vignetting.Using the per-snapshot binning scheme, we extracted the X-ray light curve in four energy bands of 0.3−10 (broad), 0.3−1 (soft), 1−4 (medium), and 4−10 (hard) keV to explore any possible energy-dependence of the forthcoming results.In the end, we found all of our results to be consistent across all four energy bands, and therefore present only the broad band X-ray results here to be concise (references to the X waveband throughout thus refer to 0.3 − 10 keV).We also extracted the time-averaged XRT spectrum for our campaign using the same tool, including all events with valid grades (i.e.0 − 12).
The UVOT was operated in imaging mode for the entire observing campaign, where for each pointing we used the UVW2, UVM2, UVW1, U, B, and V filters.We extracted the source and background fluxes from each image in the standard way as outlined in the UVOT Data Analysis guide 2 .We used the uvotsource task from the ftools package included in heasoft version 6.28 using a source region of radius 5 ′′ centred on the source position as available on the NASA/IPAC Extragalactic Database (NED) and a nearby offsource background region of 15 ′′ .To account for so-called UVOT dropouts, in which the source flux is observed to be significantly decreased compared to its neighbours due to bad pixels on the UVOT detector, we followed the steps outlined in the Appendix of Edelson et al. (2015).This iterative filtering procedure removes 2, 2, 9, 0, 1,  1.

Light curves
In Figure 1 (left panels) we present the light curves obtained during our 2022 observing campaign of NGC 6814.All wavebands exhibit large amplitude, rapid variability that decreases at longer wavelengths.A common long-term trend is apparent in all of the UV/optical data, while the long-term X-ray variability appears distinctly different by comparison, suggesting unrelated or disconnected physical processes responsible for the long-term variability in each wavelength regime.
To quantify the long-term variability, we compute the fractional variability ( var ) following Edelson et al. (2002) as: where  is the light curve count rate,  2 is its variance, and  2 err is the square of its standard error.Its corresponding uncertainty is then computed as: where  is the number of light curve data points over which  var is computed.Here, we compute the fractional variability over the entirety of each light curve.The results are given in Table 1, where a clear trend of decreased variability with increased wavelength is observed.
In order to isolate the short-term variations, which may be associated with the reprocessing of X-rays in the accretion disc, from long-term variations, which may be due to changes in the behaviour of the accretion flow itself, we de-trended the light curves using a Savitzky-Golay filter width of 21.5 d, which was determined by minimizing the W2 auto-correlation function variance (see Appendix A for details).Moving forward, any reference to de-trended data has had the aforementioned trend line subtracted from the unaltered data, as shown in Figure 1 (right panels).The forthcoming analysis is conducted on both the unaltered and de-trended light curves, which will be presented alongside one another throughout.While our interpretation is based solely on the de-trended results they are largely independent of the usage of unaltered or de-trended data.This procedure renders the V de-trended light curve indistinguishable from a constant count rate ( 2  = 1.03), indicating that it is dominated by long-term variations not due to the reprocessing of X-rays by the accretion disc.All other de-trended light curves are inconsistent with a constant ( 2  > 2).

Structure function
Due to the gappy, unevenly sampled light curves that comprise our data set, we are unable to implement Fourier-transform based techniques, such as the power density spectrum, without binning the data and therefore losing information on the shortest time-scales.To explore the distribution of power in our light curves as a function of time-scale () we instead compute the structure function (SF) according to the method of Collier & Peterson (2001) as: where  () is the light curve count rate at time , and the summation is performed over those data pairs  () that satisfy  =   −   .We then subtract the noise variance (2 2 noise ) and normalize by the signal variance (2 2 signal ) such that the SF converges to 1 with increasing time-scale.Within some range of time-scales where the variability is correlated the SF will take on a power law shape ( ∝   ) before flattening after some characteristic break-time.Similarly to the power density spectrum, the slope () and break-time ( break ) of the SF can inform us about the nature of the physical mechanism driving the observed variability (e.g.Gallo et al. 2018).
The SFs of the unaltered and de-trended light curves are shown in Figure 2. We have fit the de-trended SFs using a simple broken power law model 3 such that  ( <  break ) =   and  ( ≥  break ) =   break .To estimate parameters and their credible intervals we implement a Markov Chain Monte Carlo (MCMC) sampler via the emcee (Foreman-Mackey et al. 2013) python package.In our MCMC runs we used a Gaussian likelihood function with uniform priors for each parameter, ran 30 walkers across the parameter space, burned the first 1, 000 iterations, and computed relevant parameter values (50 th percentile, i.e. median) and credible intervals (16 th and 84 th percentiles) from the posterior distributions formed from a total of 10, 000 iterations per walker.We note that only the portion of the SF 3 We note that we do not include an additional free-to-vary constant in our model as we have already subtracted the noise variance component.0.17 for which  < 5.1 d was fit, as above this value the de-trended SFs begin to oscillate around  () = 1.Furthermore, at  ≈ 5.1 d the unaltered SFs begin to deviate significantly from the nearly identical trend (albeit shifted in normalization) observed in the de-trended SFs.
The median model curves are plotted alongside the data in Figure 2, with parameter values and credible intervals listed in Table 2.We find that the UV/optical SF slopes and break-times exhibit no significant wavelength dependence and are broadly consistent with their respective weighted means of ⟨⟩ ≈ 1.46 and ⟨ break ⟩ ≈ 2.30 d.These results suggest a common emission region in the disc is responsible for the observed variability across all UV/optical wavebands.The X-ray data, however, exhibit a significantly flatter SF slope than the mean UV/optical results (at the 6.3 level) with a comparable SF break-time (within ∼ 1).While the similar X-ray and UV/optical break-times may suggest that the variability observed in both is produced in a similar emission region, the significant difference in SF slopes between the two wavelength regimes suggests a different physical process drives the observed variability in each.

Interpolated cross-correlation function
To explore the correlated variations between light curves in different wavebands, we computed the interpolated cross-correlation function4 (ICCF; Gaskell & Sparke 1986;Gaskell & Peterson 1987;White & Peterson 1994).This procedure computes the crosscorrelation function (CCF) between two unevenly sampled light curves,  () and  (), by linearly interpolating between the data points in each curve.The ICCF between two light curves shifted by a time-lag () is computed as: where  denotes the standard deviation of the corresponding light curve count rate.In practice, our ICCFs were computed following the modifications by Edelson et al. (2019): (i) we did not compute the ICCF for pairs outside of the time series, (ii) all quantities in Equation 4 were computed locally (i.e. over only the subset in each time series that overlaps for a given time-lag) rather than globally, and (iii) all ICCFs were computed using 'two-way' interpolation by averaging over the individual results obtained by first interpolating In this work we used the W2 light curve as the reference band, computing all ICCFs and thus all time-lags relative to it.We interpolated the light curves using a sampling rate a factor of 10 smaller than the median cadence (see Table 1).
To evaluate the significance of any peaks observed in the ICCFs, we simulated  = 1, 000 realizations of the W2 light curve using the light curve generation method of Emmanoulopoulos et al. (2013) by estimating the slope of the power density spectrum () to be  W2 ≈  W2 + 1 = 2.4 and using the observed flux distribution as the input probability density function.We computed the ICCF for each realization, producing a distribution of noisy ICCFs.We then computed the 1 st and 99 th percentiles from the ensemble of noisy ICCFs at each time-lag, and only considered ICCF peaks from the data to be a significant result if they laid below/above these thresholds.
Each ICCF was found to contain a single significant positive peak near  ≈ 0 d.We therefore used the subset of each ICCF > 99 th percentile from the corresponding simulations and computed the centroid ( cent ) as: . (5) To estimate the credible interval on each centroid measurement, we implemented the flux randomization/random subset selection (FR/RSS) method of Peterson et al. (1998), in which we performed  = 1, 000 iterations and measured the centroid each time.From this distribution of centroid measurements we then computed the 16 th and 84 th percentiles to estimate the 68 per cent credible interval for the observed centroid measurements.The ICCFs and centroid distributions are shown in Figure 3, with centroid measurements and credible intervals given in Table 3.We note that the use of unaltered versus de-trended light curves does not significantly change our results; de-trended centroid estimates produce credible intervals ∼ 10 per cent smaller than the unaltered ones.As is visually evident in Figure 1, all of the de-trended UV/optical light curves are found to be strongly correlated with the W2 data (i.e.max [ICCF ()] ≳ 0.8), with only the V band data exhibiting a significantly weaker correlation strength (max [ICCF ()] ≈ 0.5) and broader centroid distribution, likely due to its lower observed rapid variability.The X-rays exhibit a moderately strong correlation with the W2 data (max [ICCF ()] ≈ 0.7), and while the peak of the X-ray ICCF indicates a modest lead with respect to the W2 data of  peak ≈ 0.2 d, due to the asymmetry of the peak its centroid is measured as a lead of  cent ≈ 0.4 d.
An alternative approach to estimating inter-band continuum lags is to model some driving light curve as a damped random walk which then becomes lagged, smoothed, and scaled in the various response light curves.This approach is implemented in the javelin code (Zu et al. 2013), which we use here to compare with our ICCF centroid measurements by using our unaltered light curves 5 .Our inter-band lag measurements with javelin were computed using its built-in MCMC sampler with 100 walkers over 100, 000 iterations after burning the first 100, 000 iterations.Parameter values and 68 per cent credible intervals were estimated by taking the median as well as 16 th and 84 th percentiles of the corresponding posterior distribution.Our results are given in Table 3 where we find that, apart from producing credible intervals ∼ 55 per cent smaller than the de-trended ICCF ones, the javelin results are completely consistent with our ICCF results.We note, however, that we are unable to place any constraint on the X-ray lag with javelin as it is extremely sensitive to the top-hat width of the response function implemented in the code, which we do not know a priori.
For a standard accretion disc where  ∝  −3/4 the lag-wavelength spectrum is expected to follow  ∝  4/3 as  ∝  −1 and  = /.Our results, however, exhibit a completely different behaviour as we find that the optical bands all exhibit a nearly constant lag of ∼ 0.3 days relative to W2.Furthermore, while the U band lag may appear to follow the standard prediction, it is likely that the measured lag relative to W2 is enhanced due to significant contamination from the H emission line originating in the BLR, and is thus unlikely to be entirely due to disc reprocessing.We note that the optical flattening of the lag-wavelength spectrum was also observed by Troyer et al. (2016) using a 2012 Swift campaign.However, the low-cadence of that campaign (75 observations over 3 months) resulted in much larger uncertainties on the measured inter-band lags (a factor of ∼ 10 larger than those presented here) and thus no conclusive evidence of a flattened lag-wavelength spectrum.

Extracting the variable spectral component
The fractional variability results in Table 1 decrease rapidly at longer wavelengths, likely due to the presence of a constant component that dominates the redder wavebands, such as emission from the host galaxy.To disentangle the variable AGN component from the invariable host galaxy emission, we implement a flux-flux analysis (e.g.McHardy et al. 2018).
We first extract the flux density light curve (  (, )) in each UVOT filter and de-redden them according to a Galactic extinction of  ( − ) = 0.207 (Willingale et al. 2013) using the Cardelli et al. (1989) extinction law with   = 3.1.We filtered the light curves for observations that used every UVOT filter (total of 217 points) and fit them using  2 minimization with a linear trend line as:  4. No obvious deviations from a linear trend are present in the data ( 2  = 1.26), thus suggesting no significant changes in the spectral shape during the 6 We tested allowing each point in  ( ) to be free-to-vary, finding no significant differences in interpretation of the final results.4).The expected accretion disc spectrum of   ∝  −1/3 is plotted alongside the RMS and AGN spectra, scaled by the V band flux of each, as the dashed dark grey curves.The Sc spiral template is shown as the solid light grey curve.
campaign.The spectrum is dominated by host galaxy emission in the B and V bands, thus resulting in the rapid decrease in variability in those bands relative to the others.We fit the host galaxy emission using the spiral galaxy templates available in the SWIRE template library7 (Polletta et al. 2007), finding the Sc spiral template to provide the best description of the data, consistent with the host galaxy morphological classification of NGC 6814 as an SAB(rs)bc galaxy by de Vaucouleurs et al. (1991).We note that the excess emission measured in the W1 band relative to the Sc spiral template may be due to the 'small blue bump' feature consisting of blended Fe II and Balmer continuum emission (see, e.g., Figure 5 of Mehdipour et al. 2015).
The intrinsic AGN variability spectrum does not follow a typical accretion disc spectrum, for which   ∝  −1/3 , but instead appears extremely blue such that   ∝  −0.86 .While we do not consider the effects of extinction in the source frame here, we note that its effect would be to make the intrinsic AGN variability spectrum even bluer and thus more extreme.

DISCUSSION
In the previous sections, we have presented an analysis of the temporal properties of NGC 6814 in the X-ray, UV, and optical wavebands from our 2022 Swift campaign consisting of ∼ 250 observations over 75 days.We have found that the long-term X-ray variability is distinctly different than the long-term UV/optical variability, as evidenced by its unique long-term trend (Figure 1) and comparatively flat SF slope (Figure 2 and Table 2).Short-term variations in the X-ray band were found to lead the UV (W2) ones by ∼ 0.4 d (Figure 3 and Table 3).The UV and optical wavebands all exhibit similar long-term trends (Figure 1) and SF properties (Figure 2 and Table 2), suggesting a common origin.Interestingly, the inter-band lags do not follow the  ∝  4/3 predicted relation of a standard disc reprocessing scenario (Table 3), nor does the intrinsic AGN variability spectrum follow the   ∝  −1/3 predicted relation of a standard accretion disc (Figure 4 and Table 4).
To probe the origin of these peculiar results, we use the kynxiltr (Kammoun et al. 2023, submitted) and kynsed8 (Dovčiak et al. 2022) codes to self-consistently compute the inter-band lags and AGN SED, respectively, for a selection of physical parameter combinations.Both codes self-consistently compute the X-ray power law, reflected X-ray, as well as thermal disc emission spectra and response functions based on the following set of physical parameter inputs: black hole mass  BH , black hole spin  * , line-of-sight inclination  where  = 0 is face-on and  = 90 is edge-on, mass accretion rate  Edd , colour-temperature correction factor  col , fraction of total accretion luminosity powering the corona  transf , outer disc radius  out , Fe abundance  Fe , X-ray coronal height ℎ, X-ray photon index Γ, source redshift , and model normalization 1/ 2 L where  L is the luminosity distance to the source.We note that here we have assumed the following standard values in all forthcoming computations: an inner disc radius of  in =  ISCO where  ISCO is the radius of the innermost stable circular orbit, a constant density throughout the disc of  e = 10 15 cm −3 , and a high energy cut-off for the X-ray power law of  cut = 300 keV.Below we describe our choice of parameter values, which are fixed to the listed values unless otherwise stated.
Based on the X-ray spectral fits of Gallo et al. (2021), we assume  * = 0.998,  Fe = 4, and Γ = 2.While those authors found  = 65 • to best fit the 2016 XMM-Newton spectrum of NGC 6814, we find that the time-averaged XRT spectrum of our Swift monitoring campaign is best fit with  = 45 • , and therefore use the latter value here.Gallo et al. (2021) also estimated the size of the extended X-ray corona to be ∼ 25   in diameter.Here, we approximate such an extended coronal geometry under the lamp-post coronal geometry used by the kynxiltr and kynsed codes by assuming ℎ = 10   , which is comparable to a lamp-post placed along the spin axis of the black hole at nearly the maximum radial extent of the extended coronal geometry.The SMBH mass in NGC 6814 has been measured as  BH = (1.85 ± 0.35) × 10 7  ⊙ based on its H emission line properties (Bentz et al. 2009).While significantly lower mass estimates exist (e.g.Pancoast et al. 2014), we find that such low SMBH masses consistently produce disc spectra that significantly over-predict the observed soft X-ray flux which cannot be reconciled even with extreme X-ray absorption, and thus we do not consider them further here.We assume  col = 1.7, consistent with the colour-temperature correction factor of black hole X-ray binary accretion discs (e.g.Shimura & Takahara 1995), and note that while the precise value used does affect the other estimated disc parameters, the interpretation of our results does not change (see Appendix B for details).Lastly, we use  = 0.00522 (Springob et al. 2005) and  L = 21.65 Mpc (Bentz et al. 2019).
Therefore, we have left  out and  Edd as variable parameters to explore.Of all the parameters explored by Kammoun et al. (2021b), only the reduction of outer disc radius was found to produce significant flattening of the lag-wavelength spectrum in a way comparable to our lag measurements in the B and V bands.We take  out = 10, 000   as a fiducial value for this parameter to compare with other values sampled in our fits.The observed accretion rate ( Edd ) may be estimated by estimating the bolometric luminosity ( bol ) as  bol ≈  X  2−10 keV = 3.1 × 10 43 ergs s −1 , where  X = 15.5 is the bolometric correction factor estimated based on Equation 3 of Duras et al. (2020) and  2−10 keV ≈ 2 × 10 42 ergs s −1 is the unabsorbed X-ray luminosity of the time-averaged XRT spectrum, yielding a sub-Eddington observed accretion rate of  Edd =  bol / Edd ≈ 0.0133.We use this observed accretion rate as a reasonable approximation of the intrinsic accretion rate (i.e.within a factor of a few), and therefore take  Edd = 0.0133 as a fiducial value for this parameter to compare with other values sampled in our fits.
We computed the inter-band lags and AGN SED with kynxiltr and kynsed, respectively, by fixing all aforementioned physical parameters to the given values while testing various combinations of  out and  Edd .In practice, we fit the observed lag-wavelength spectrum by computing the predicted inter-band lags in the XRT (0.3 − 10 keV) and UVOT (as listed in Table 1 of Edelson et al. 2015) band-passes by stepping through  out /  ∈ [55, 5, 000] by Δ log  out /  = 0.01 and through  Edd ∈ [0.003, 30] by Δ log (  Edd ) = 0.01.We note that  transf was fixed for each  Edd to ensure that the model predicted X-ray flux matched the observed one.For each ( out ,  Edd ) pair, we evaluated the goodness of fit of the predicted lag values compared to the measured ones using the  2 statistic.The U band was excluded in all model evaluations due to possible contamination from the H emission line and/or diffuse continuum emission (DCE; e.g.Korista & Goad 2019) originating in the BLR.While some physical scenarios, such as reprocessing between the X-ray and UV/optical emitting regions (e.g.McHardy et al. 2018), may produce X-ray lags (relative to the UV/optical) significantly different than those predicted by a purely disc repro- Each panel in the top row displays the results from individually fitting the lag-wavelength spectrum (red) and SED (blue) with kynxiltr and kynsed, respectively, as well as a fiducial (grey) curve based on appropriate parameter values for comparison.We include the relevant colour-coded fit statistics in each panel of the top row.For the lag fits, we have computed fit statistics by excluding the X-ray and U band results as well as by including all of the data.For the SED fits, the fit statistics for the UV/optical ( 2 ) and X-ray () data are shown separately, near the corresponding data.The bottom three rows display the colour-coded best-fit model residuals corresponding to the fiducial, SED, and lag fits from top to bottom.
cessing scenario, we found that exclusion of the X-ray lag value during model evaluation resulted in an extremely poor constraint on the accretion rate; therefore we included the X-ray lag measurement in all of our model evaluations.The intrinsic AGN variability spectrum 9 and optimally binned 10 (Kaastra & Bleeker 2016) timeaveraged XRT spectrum were jointly fit with kynsed using xspec version 12.12.1 (Arnaud 1996) by computing the  2 statistic for the intrinsic AGN variability spectrum and the Cash statistic (i.e.; Cash 1979) for the XRT spectrum to evaluate the model likelihood.Here,  transf was left free-to-vary in order to fit the observed X-ray flux.We note that the 0.3 − 2 keV band exhibits evidence of significant neutral absorption in the source frame.This was fit using partcov*zphabs in xspec, which we found to have a column density of  H ≈ 10 22 cm −2 and covering fraction of  cov ≈ 0.4, which were fixed throughout all of our fits.Galactic absorption was fit using phabs with  H = 1.53 × 10 21 cm −2 (Willingale et al. 2013).Since the intrinsic AGN variability spectrum was already de-reddened for Galactic extinction (see Section 3.4), we do not need to account for it in our xspec model.We recorded the parameters that best reproduced the inter-band lags and AGN SED individually, and computed 9 The data were converted into PHA and RSP files readable by xspec using the ftflx2xsp ftools task. 10The data were binned using the grouptype=opt flag of the ftgrouppha ftools task.
the predicted corresponding curves, as well as computed a pair of fiducial curves based on  out = 10, 000   and  Edd = 0.0133.Error bars on forthcoming parameter values represent corresponding 68 per cent parameter credible intervals.
Our results are shown in Figure 5.We find that the measured lagwavelength spectrum is best fit by a moderately truncated outer disc radius of  out = 1, 382 +398 −404   at an accretion rate of  Edd = 1.3 +2.1 −0.9 .This model self-consistently explains the X-ray lag, requiring no extended reprocessor scenario, and exhibits only a minor deviation in the U band (1.4), suggesting only a marginal contribution from DCE.The corresponding predicted AGN SED, however, overpredicts the observed one by a factor of ∼ 10 across the UV/optical bands, and even extends well into the soft X-ray band where it overpredicts the observed 0.3 − 1 keV flux by a factor of ∼ 20.In contrast, we find that the AGN SED is best fit by an extremely truncated outer disc radius of  out = 202 ± 5   at a modest accretion rate of  Edd = 0.0255 ± 0.0006 that is comparable to our fiducial estimate.The corresponding predicted lag-wavelength spectrum, however, significantly under-predicts the measured one in the W1, U, and B bands, suggesting a significant contribution from DCE, and cannot self-consistently explain the X-ray lag, suggesting the presence of an extended X-ray reprocessor, providing a statistically unacceptable fit even when ignoring the X-ray and U band lags.Our fiducial model produces a statistically acceptable fit to the lag-wavelength spectrum if we consider that the X-ray and U band lags have origins other than due to disc reprocessing, however, the corresponding predicted disc spectrum provides a statistically unacceptable fit to the intrinsic AGN variability spectrum, thereby strongly disfavouring a standard disc scenario.It is therefore plausible that a standard disc illuminated by a lamp-post X-ray corona cannot self-consistently explain the observed properties of NGC 6814 given the fact that none of the above results adequately fit both the inter-band lags and AGN SED simultaneously.
One possible mechanism through which the outer disc radius may become truncated is via dust formation in the disc once the temperature falls below the dust sublimation temperature.Baskin & Laor (2018) showed that the minimum radius for which this occurs is: where  opt,45 =   at 4861 Å in units of 10 45 ergs s −1 .For  out = 202   and  Edd = 0.0255 we find that  opt ≈ 9.6 × 10 41 ergs s −1 yielding  dust ≈ 628   , while for  out = 1, 382   and  Edd = 1.3 we find that  opt ≈ 1.8 × 10 43 ergs s −1 yielding  dust ≈ 2, 700   .
Both of these estimates place the minimum dust sublimation radius far outside the corresponding outer disc truncation radius, thus it seems unlikely that dust formation in the disc is responsible for truncation of the outer disc here.
Outer disc truncation may also be achieved through disc selfgravity.According to Equation 15 of Laor & Netzer (1989), the self-gravity radius of a Shakura & Sunyaev (1973) accretion disc occurs at: where  9 =  BH /10 9  ⊙ and   is the -disc viscosity parameter.
Assuming   ≈ 0.02 (e.g.Mishra et al. 2016), for  Edd = 0.0255 we find  sg ≈ 428   while instead for  Edd = 1.3 we find  sg ≈ 2, 458   .Both of these estimates are significantly larger than the corresponding outer disc truncation radius, thus it seems that selfgravity is also unlikely to be responsible for truncation of the outer disc here.
The measured SF break-times of the de-trended UV/optical light curves were all found to be consistent with ⟨ break ⟩ ≈ 2.30 d.We may determine the most plausible origin in the disc of this characteristic time-scale as due to light-crossing ( lc = /), dynamical ( dyn = √︁  3 /  BH ), thermal ( th ≈  dyn /  ), or viscous ( vis ≈  th / 2 H , where  H = / ≈ 0.01 is the disc aspect ratio) variability timescales.Solving each relation for the radius that corresponds to  = 2.30 d finds  lc ≈ 2180   ,  dyn ≈ 168   ,  th ≈ 12.4   , and  vis ≪ 1   .We may immediately rule out both a light-crossing origin, as it exceeds both of our estimated disc sizes, and a viscous origin, as it is unphysically small (i.e.within the event horizon).
Regarding a dynamical origin, we note that the corresponding radius ( dyn ≈ 168   ) lies well within  out = 1, 382   for the  Edd = 1.3 disc scenario, and is nearly coincident with  out = 202   for the  Edd = 0.0255 disc scenario.In the latter scenario, the long-term variability of the UV/optical emission is driven by dynamical variations originating very close to the outer edge of the accretion disc.This may be plausible in the context of a bowllike disc geometry with a steep rim (e.g.Starkey et al. 2023), in which the outer regions of the disc are elevated and tilted toward our line-of-sight and therefore dominate the observed area of the disc.Moreover, these outer regions of the disc will then dominate the measured UV/optical inter-band lags such that they are significantly larger than predicted by a thin, flat disc geometry (see Figure 5 of Starkey et al. 2023), perhaps in a manner comparable to those measured here.For accretion discs that are misaligned with respect to the SMBH spin axis, torques due to Lense-Thirring precession can effectively warp (for low degrees of misalignment), or even tear apart (for high degrees of misalignment), the disc (e.g.Nixon et al. 2012;Dyda & Reynolds 2020), a phenomenon that may be even more prevalent in the case of binary SMBHs (Nixon et al. 2013), thereby revealing more of the outer disc region than in a thin, flat disc geometry.Therefore, whether intrinsically bowl-like or warped / torn to appear as such, non-standard disc geometries may offer a self-consistent description of the inter-band lags, AGN SED, and characterisitc variability time-scales observed in NGC 6814.This will be explored in more detail in a future work.
Regarding a thermal origin, we note that the corresponding radius ( th = 12.4   ) is exactly coincident with the outer edge of the extended X-ray corona measured by (Gallo et al. 2021), where a diameter of ∼ 25   was measured yielding  cor ≈ 12.5   .If this extended corona exists between the SMBH event horizon and inner disc radius such that  in =  cor (similar to the geometry invoked by, e.g., Done et al. 2012;Kubota & Done 2018), it is possible that the UV/optical SFs are measuring thermal variations at the inner edge of the accretion disc.Furthermore, we note that the measured Xray SF break-time ( break = 1.7 +0.5 −0.4 d) is broadly consistent with the mean UV/optical break-time (⟨ break ⟩ ≈ 2.30 d), possibly suggesting an emission region of similar size (i.e. at or within the inner disc radius).The difference between the measured X-ray and UV/optical SF slopes may then be the result of the different physical processes responsible for the production of the coronal and disc emission, respectively.
Our findings here make it abundantly clear then that NGC 6814 exhibits complex multi-wavelength variability.The extreme discrepancy found when comparing model predictions by fitting the inter-band lag spectrum and AGN SED individually as well as the essentially wavelength-independent characteristic variability timescales across all observed band-passes may, however, be indicative of a non-standard accretion disc geometry such as a bowl-like or warped/torn disc.NGC 6814 is a prime target for multi-wavelength follow-up across a wider waveband to more fully investigate the peculiar and interesting properties presented here.

CONCLUSIONS
We have presented a thorough timing analysis of the X-ray, UV, and optical emission of NGC 6814 from our 2022 Swift monitoring campaign comprised of ∼ 250 observations over 75 days.
The long-term X-ray variability displays significantly different properties than the long-term UV/optical variability as seen in the smoothed long-term light curve trends of each waveband.
A SF analysis reveals that the UV/optical variability is likely driven by a common physical mechanism, while the X-ray variability likely driven by a separate mechanism.We find that the similar SF breaktimes observed across all wavebands are consistent with either dynamical variations originating near the outer edge of a highly truncated outer disc or thermal variations originating at the inner edge of a truncated inner disc.
A cross-correlation analysis evaluating inter-band lags finds that short-term variations in the X-ray band lead the UV ones by ∼ 0.4 d while the optical wavebands exhibit a constant lag of ∼ 0.3 d relative the UV.These measurements do not follow the standard disc X-ray reprocessing prediction (i.e. ∝  4/3 ), unless an extreme accretion rate (  Edd = 1.3 +2.1 −0.9 ) and moderately truncated outer disc radius ( out = 1, 382 +398 −404   ) are invoked.A flux-flux analysis reveals a constant, red component (i.e. the invariable host galaxy emission) as well as a variable, extremely blue (  ∝  −0.9 ) component (i.e. the intrinsic AGN variable emission).The variable component does not follow the standard disc spectrum (i.e.  ∝  −0.9 ), unless an extremely truncated out disc radius ( out = 202 ± 5   ) and modest accretion rate (  Edd = 0.0255 ± 0.0006) are invoked.
Our results suggest the possibility of a non-standard disc geometry in NGC 6814 that will be explored in a future work.These results also make a strong case for future studies including multi-wavelength follow-up across a wider combined band-pass to more thoroughly investigate the origin of the various measured peculiarities.1994; the details of our ICCF methodology are presented in Section 3.3).We then smoothed the data using a Savitzky-Golay filter for a range of filter widths /d ∈ [3,65] in steps of Δ (/d) = 0.5, approximately.We note, however, that since the Savitzky-Golay filter actually smooths over a specific number of data points, and given the gappy, uneven sampling of the data which results in irregularly changing time widths for the moving filter with a set number of data points, we give the nearest approximate value for all related filter widths, which are within ±0.25 d of their true values.For each filter width, we computed the W2 ACF and evaluated its standard deviation in the regions outside of the central peak at  = 0 d (i.e. for || greater than the first zero crossing points of the ACF peak).Minimizing this quantity minimizes the long-term variations present in the light curve not due to the thermal reprocessing of X-rays in the accretion disc that act to broaden the W2 ACF, and therefore all ICCFs that use it as a reference band, thereby enabling a more accurate and precise evaluation of the ICCF centroids.
The results of this procedure are shown in Figure A1, where we find an optimal Savitzky-Golay filter width of 21.5 d.We note that while the non-peak variance can be minimized further by using the narrowest filter widths tested, such smoothing also removes shortterm variations that we are interested in evaluating as the smoothed trend essentially replicates the data.

Figure 1 .
Figure 1.Left: Light curves for the 2022 observing campaign of NGC 6814.UVOT dropouts have been removed (see text for details).Solid curves in each panel represent the Savitzky-Golay 21.5 d smoothed trend (see Appendix A for details).Right: The de-trended (data minus smoothed trend in left panels) light curves.This procedure renders the V de-trended light curve indistinguishable from a constant count rate ( 2  = 1.03), indicating that it is dominated by long-term variations not due to the reprocessing of X-rays by the accretion disc.All other de-trended light curves are inconsistent with a constant ( 2  > 2).

Figure 2 .
Figure 2. Structure functions for each unaltered (grey) and de-trended (coloured) light curve.The horizontal dashed grey lines represent  (  ) = 1.Solid curves represent the median broken power law model plotted over the range of data points used to perform the fits.The V band data, once de-trended, are invariant and thus are not fit, though we show them for comparison.

Table 2 .
Structure function normalization ( ), slope (), and break-time ( break ) median values from the MCMC broken power law fits to the 2022 de-trended light curves, as well as reduced  2 fit statistic ( 2  ).Parameter uncertainties represent the 68 per cent credible interval.A ( * ) indicates a structure function that was not fit.

Figure 3 .
Figure 3. Interpolated cross-correlation functions for the unaltered (dotted) and de-trended (solid) light curves in each filter relative to the W2 light curve.Negative  values indicate a given band leads the W2 data; positive values indicate a lag.Colour-coded histograms represent the centroid distributions obtained from the  = 1, 000 FR/RSS evaluations of the de-trended ICCFs.Shaded horizontal regions are bound by the 1 st and 99 th percentiles of the  = 1, 000 light curve simulations (see text for details).The dashed lines are the zero-point for each axis.
where  () is a dimensionless driving light curve with mean of 0 and standard deviation of 1,  () is a wavelength-dependent shift factor, and  () is a wavelength-dependent scale factor representing the RMS spectrum.Here,  () is simply derived by computing the mean of the normalized (subtracting the mean, dividing by the standard deviation) UVOT light curves 6 .To isolate the intrinsic AGN variable component, we evaluate the best-fit line in each waveband at max [ ()] and min [ ()], and take the difference,  ().To isolate the host galaxy component, we solve for  () such that   (W2, ) = 0, then extrapolate the best-fit line of each other waveband to this value to estimate the minimum host galaxy contribution in each,  ().The flux-flux plots and best-fit lines are shown in Figure 4 (top panel), with parameters and corresponding computed values shown in Figure 4 (bottom panel) and given in Table

Figure 4 .
Figure 4. Top: Flux-flux plots for all de-reddened and filtered UVOT flux density light curves.Colour-coded solid lines represent the best-fit lines   (,  ) =  () +  ()  ( ), with fit parameters given in Table 4.The vertical grey dashed line represents the W2 zero crossing point, which is used to estimate the minimum host galaxy contribution in each other band.Bottom: Measured total,  (), and RMS,  (), spectra alongside the computed AGN,  (), and host galaxy,  (), spectra based on the best-fit lines to the flux-flux plots in the top panel (all values given in Table4).The expected accretion disc spectrum of   ∝  −1/3 is plotted alongside the RMS and AGN spectra, scaled by the V band flux of each, as the dashed dark grey curves.The Sc spiral template is shown as the solid light grey curve.

Figure 5 .
Figure5.The lag-wavelength spectrum (top left) and AGN SED (top right; observed as dotted curves, intrinsic as solid curves) are shown as the black data points.Each panel in the top row displays the results from individually fitting the lag-wavelength spectrum (red) and SED (blue) with kynxiltr and kynsed, respectively, as well as a fiducial (grey) curve based on appropriate parameter values for comparison.We include the relevant colour-coded fit statistics in each panel of the top row.For the lag fits, we have computed fit statistics by excluding the X-ray and U band results as well as by including all of the data.For the SED fits, the fit statistics for the UV/optical ( 2 ) and X-ray () data are shown separately, near the corresponding data.The bottom three rows display the colour-coded best-fit model residuals corresponding to the fiducial, SED, and lag fits from top to bottom.

Figure A1 .
Figure A1.Top: The optimal filter width determination method (see text for details).The 21.5 d filter width is shown as the red dot, which minimizes the standard deviation of the ACF outside the central peak at  = 0 d.Bottom: The unaltered (grey) and de-trended (coloured) W2 ACFs, using a Savitzky-Golay filter width of 21.5 d for the latter.The dashed lines are the zero-point for each axis, and the horizontal dotted line is ACF = 1.

Table 3 .
Interpolated cross-correlation function centroids ( cent ) relative to W2. Negative values indicate a given band leads the W2 data; positive values indicate a lag.Parameter errors represent 68 per cent credible intervals and are shown to the precision of the javelin results for more direct comparison.A ( * ) indicates that there was no reliable centroid measurement.() and leaving  () gappy, then by interpolating  () leaving  () gappy.

Table 4 .
Flux-flux fit parameters according to Equation 6 and computed values to isolate the AGN () and host galaxy () components (see text for details), in units of mJy.The fit to all UVOT light curves yields  2  = 1.26.