The rmsflux relation in accreting objects: not a simple “volume control”
A response to Koen 2016
^{1} Anton Pannekoek Institute, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands
email: p.uttley@uva.nl
^{2} Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, UK
^{3} Department of Physics and Astronomy, University of Leicester, Leicester LE1 7RH, UK
Received: 10 November 2016
Accepted: 13 March 2017
The light curves of a diverse range of accreting objects show characteristic linear relationships between the shortterm rms amplitude of variability and the flux as measured on longer timescales. This behaviour is thought to be imprinted on the light curves by accretion rate fluctuations on different timescales, propagating and coupling together through the accretion flow. Recently, a simple mathematical interpretation has been proposed for the rmsflux relation, where shortterm variations are modulated by a single slower process. Here we show that this model was already considered and ruled out by another publication on the grounds that it did not produce the observed broad timescale dependence of the rmsflux relation and associated lognormal flux distribution. We demonstrate the problems with the model via mathematical arguments and a casestudy of Cyg X1 data compared with numerical simulations. We also highlight another conclusion of our original work, which is that a linear rmsflux relation is easy to produce using a variety of models with positively skewed flux distributions. Observing such a relation in a nonaccreting object (e.g. in solar flares) does not necessarily imply a phenomenological connection with the behaviour of accretion flows, unless the relation is seen over a similarly broad range of timescales.
Key words: accretion, accretion disks / methods: statistical
© ESO, 2017
1. Introduction
An apparently universal feature of the aperiodic “flickering” flux variability seen from accreting compact objects is that they show a linear relationship between the rms amplitude of shortterm variability and flux variations on longer timescales. This socalled “rmsflux relation” was first discovered by Uttley & McHardy (2001) in the Xray light curves of Xray binary systems (XRBs, both neutron star and black hole) as well as active galactic nuclei (AGN). Since then, rmsflux relations have been found to be ubiquitous in the Xray emission from black hole XRBs in different spectral states (Gleissner et al. 2004; Heil et al. 2012) and are also seen in the fast optical variability from hard state black hole XRBs (Gandhi 2009). Linear rmsflux relations are also present in the Xray variability from ultraluminous Xray sources (Heil & Vaughan 2010; HernándezGarcía et al. 2015) as well as the shorttimescale optical variability of a blazar (Edelson et al. 2013). Expanding its scope beyond the most compact accreting objects, linear rmsflux relations are found in the broadband noise variability seen from accreting white dwarfs (Scaringi et al. 2012; Van de Sande et al. 2015; Dobrotka & Ness 2015) and even in young stellar objects (YSOs, Scaringi et al. 2015).
A very important and often not wellappreciated property of the rmsflux relation, is that in cases where data are good enough that it can be studied on different timescales, it turns out that the linear rmsflux relation occurs on all measured timescales (Uttley et al. 2005, henceforth UMV05). In other words, not only does the rms measured in short (e.g. 1 s) segments track the flux on longer timescales (e.g. 10 s), but the rms variations produced on those longer timescales also track the flux variations on even longer timescales, for example, minutes (see UMV05 and a more extensive discussion in Vaughan & Uttley 2008). This result has the corollary that the variability process is inherently nonlinear and that, if the variability process is statistically stationary on long timescales and stochastic fluctuations on different timescales multiply together, the resulting (stationary) flux distribution should be lognormal. Such lognormal flux distributions are indeed observed from the data for Xray binaries, AGN, and accreting white dwarfs (Gaskell 2004; UMV05; Scaringi et al. 2012). All of these results tie in with the idea that the particular rmsflux relation seen in accreting systems is linked to the common feature: the accretion flow itself, with variations produced by turbulent fluctuations in massaccretion rate arising at different radii, which propagate through the flow so that variability is coupled together over and between a broad range of timescales (Lyubarskii 1997; King et al. 2004; Arévalo & Uttley 2006; Ingram & van der Klis 2013; Scaringi 2014; Cowperthwaite & Reynolds 2014; Hogg & Reynolds 2016). It is possible that linear rmsflux relations are produced by other types of process but these are not expected to show the relation across and between a broad range of timescales, as well as the corresponding lognormal flux distributions (see Appendix in UMV05 and Sect. 4 of this Letter).
Recently, Koen (2016, henceforth K16) proposed a simple model for the linear rmsflux relation seen from accreting objects. The K16 model invokes a simple scaling of a stationary statistical process meaning that the rms on the timescales dominated by that process scales linearly with the scaling factor, which itself is timevariable. However, only two components couple together. The scaling factor is coupled to the shortterm variability and there is no coupling of multiple variability components across a wider range of timescales, as suggested by UMV05 and consistent with observations as well as predicted by models for accretion variability. It is therefore important to check whether or not this simple prescription is really a valid description of the observed light curves. Here we show that it is not, and that this was, in fact, already discussed and ruled out by UMV05.
2. Koen’s model for the rmsflux relation
Following the notation of K16, the model can be written as: (1)where g(t) is a “smooth trend” while z(t) is a more rapidly varying stationary stochastic process (K16 assume either a simple Gaussian independent and identically distributed variable, or a timeseries produced by an autoregressive process). Thus, since the variations of z(t) are assumed to be independent of those of g(t), both the mean and the rms of variations of X(t) on timescales dominated by z(t) will simply scale with the value of g(t), to produce a linear rmsflux relation. However, while the K16 model limits the number of light curve components which must be multiplied together, this is also the downfall of the model, such that it does not reflect the observed light curve behaviour in accreting systems in two important ways.
Firstly, the K16 model does not predict that the rmsflux relation applies over and between a broad range of timescales: the time series g(t) does not itself scale with another longertimescale time series, contrary to what is seen in observed light curves of accreting objects. In fact, the model of K16 was already explicitly considered and rejected for this reason by UMV05 (see Sect. 3.2 of that paper), where it is described as being analogous to a simple “volume control” on an amplifier or sound system. The rms amplitude of shorttimescale variations corresponds to the volume setting, which can itself be varied on longer timescales. However, to reproduce the data, these longer timescale variations would also need to follow a linear rmsflux relation (i.e. they must also be considered as a volume setting to be varied on even longer timescales), with the amplitudes of those even longer timescale variations modulated in the same way, and so on. This crucial ingredient is missing from the K16 model.
Secondly, the K16 model does not predict the observed lognormal flux distributions which (as shown in UMV05) arise from stationary time series when the rmsflux relation applies on all measured timescales. This is simply because log [X(t)] = log [g(t)] + log [z(t)], meaning that X(t) is only lognormally distributed if g(t) and z(t) are both also lognormally distributed, which would only be the case if they also followed linear rmsflux relations on all timescales (UMV05). We note here that the simulated light curves presented in K16 are in any case not stationary and bear little resemblance to the “broadband noise” or “flickering”type light curves of real accreting sources (with similar amplitudes of variability per decade range in timescale). It would be easy to adapt the K16 model so that the light curves are stationary, for example, by using an autoregressive timeseries model for g(t). Such an adaptation would still not produce the required lognormal flux distribution, however, unless both g(t) and z(t) were already lognormally distributed.
Finally, it is important to stress here that the fact that the rmsflux relation applies across a broad range of measured timescales and that the flux distributions are lognormal are not merely “ingredients” of the model of UMV05, they are observational results which support the interpretation that variations must multiply together over a broad range of timescales. Any model that does not replicate those basic observational results (including the model of K16) is falsified, even if it can explain other aspects of the light curves such as the shape of the power spectrum.
3. A case study^{1}
To demonstrate that the K16 model for the rmsflux relation does not adequately describe the other key properties of the data described in the last section (lognormal flux distribution and linear rmsflux on a broad range of timescales), even while it can reproduce the observed power spectrum, we now conduct a simple casestudy using the Cyg X1 December 1996 hard state Rossi Xray Timing Explorer (RXTE) observations that were also used in UMV05. We will use this real light curve to determine the shape of the power spectrum and then generate a light curve using the K16 model to match the shape of the power spectrum. We will then demonstrate that the K16 simulated data shows a flux distribution and rmsflux relations which bear little resemblance to those of the real data. Finally we will demonstrate that the model of UMV05, where the flux is forced to be lognormally distributed, can reproduce both the power spectrum and rmsflux properties of the real data.
For simplicity we obtained the “Standard 1” light curves, which contain the net countrate in 0.125 s intervals across all channels of the RXTE Proportional Counter Array (PCA). These light curves cover a broader energy range than the 2−13 keV band used in UMV05, but since the count rates are dominated by the softer energies, we do not expect this to make much difference to our results compared to UMV05 (and we confirm this to be the case). We chose the same relatively stationary section of the light curve used in UMV05, which includes a subset of the data from ObsIDs 10236010102, −020 and −021 with total exposure 38.5 ks. This is ~20 per cent longer than the light curve used in UMV05, due to more stringent goodtimeintervals used in that paper, which are relaxed now that the RXTE mission calibration has matured.
Fig. 1 Comparison of real Cyg X1 hard state data (blue circles) with simulated data showing the same power spectrum, made using the K16 model (green triangles) or the “exponentiated” model of UMV05 (open red squares). Top left: power spectra of the light curves, including the bestfitting Lorentzian decomposition for the real data (black lines, total model line in solid red). Top right: flux distributions (flux normalised by mean count rate), models (coloured lines) and data/model ratios for a fit to a lognormal model with constant offset constrained to be nonnegative. Bottom panels: the rmsflux relations of the light curves obtained for rms measured on two different timescale ranges, short (0.125−1 s, bottom left panel) and long (4−32 s, bottom right panel). Bestfitting linear (plus constant) relations are also shown as dotted lines. For the flux distributions and rmsflux relations, the K16 model is strongly inconsistent with the data, while the UMV05 model works reasonably well. 

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As we have already noted, the K16 model assumes a deterministic long timescale driving timeseries g(t). Such a timeseries model, with large longterm variability trends and a correspondingly steep power spectrum, does not produce power spectra resembling those of real accreting sources, which in the hard state of black hole XRBs resemble a sum of broad Lorentzian components (Pottschmidt et al. 2003). Therefore, for the K16 model, we instead assume a more plausible driving signal, where the lowestfrequency Lorentzian in the broadband PSD is assumed to drive the variations of the sum of the higherfrequency Lorentzians. To construct the model we first fit the powerspectrum of the real data, with three broad Lorentzian components, using the model prescription of Pottschmidt et al. (2003). We allow all parameters (Lorentzian peak frequency, integrated fractional rms and the frequency coherence q) to vary except q for the highestfrequency Lorentzian, which we fix to q = 0.3 since this component lies at the edge of the measured frequency range and is otherwise degenerate in its parameters. The resulting fit is reasonable given the simplicity of the model and good data quality (χ^{2} = 81.3 for 67 d.o.f.) and the data and bestfitting model (including the decomposition in terms of separate Lorentzians) is shown in Fig. 1 (top left panel).
We next confirmed one of the key results of UMV05 by fitting a threeparameter lognormal model to the flux distribution of the real light curve (flux is normalised by the mean count rate). The observed distribution, bestfitting model (see UMV05 for the model description) and data/model ratio are shown in Fig. 1 (top right panel). The lognormal model provides a good fit to the data over most of the measured flux range, although the fit is poorer in the tails of the distribution (resulting in χ^{2} = 86.6 for 52 d.o.f., similar to the quality obtained by UMV05). The bestfitting parameters are shape parameter σ = 0.36, location parameter τ = 0.16 and scale parameter ln(μ) = 0.79. We do not quote errors on these values since the fit is not formally acceptable, but we note that they are consistent with the parameters obtained by us for the fit in UMV05. The deviations from the model may result from a number of subtle aspects of the data, for example, the fact that the lognormal distribution is slightly distorted by Poisson deviations in the observed count rate, or that the light curve is not strictly stationary, or perhaps due to real physical effects not accounted for by such a simple mathematical model. Nevertheless, we will see that the lognormal fit is a much better match to the data than to a simulated light curve based on the K16 model.
To simulate the K16 version of the light curve, we used the approach of Timmer & Koenig (1995) to simulate zeromean Gaussian fluxdistributed light curves (with the same length as the original data) for each of the three Lorentzian components (using the Lorentzian parameters obtained from the fit to the real data), which we call (in order of frequency) l_{1}(t), l_{2}(t), and l_{3}(t). We then construct the K16 model light curve as follows: (2)that is, l_{1}(t) subsumes the role of the slowly varying timeseries g(t) (although in this case, it is Gaussian distributed and stationary; the latter being required to fit the power spectrum), while (l_{2}(t) + l_{3}(t)) takes the role of the shortterm Gaussiandistributed component z(t). Finally, we scaled the light curve to the observed mean count rate and included Poisson noise. The power spectrum of this K16 simulated light curve is also included in Fig. 1 (top left panel), which shows that it is a close match to the power spectrum of the real data.
We next fitted the lognormal model to the flux distribution of the K16 light curve. If we apply no constraints to the model parameters, we can return a reasonable fit, but only with negative values of the location parameter, which when combined with a large scaleparameter can produce distributions which are close to normally distributed, rather than pure, lognormal. However, this situation would correspond to unphysical negative constant offsets in the flux distribution, as opposed to the physically plausible situation of a positive offset as seen in the real data, which could be explained by the presence of a constantflux component. Therefore we restrict the offset τ to be >0, which makes the fit to the lognormal model much worse than seen in the real data (as can be seen Fig. 1, top right panel).
We also compared the rmsflux relations of the light curves obtained for two different timescales. The shortterm rmsflux relation was obtained by binning the variance obtained in 1 s segments according to flux (then subtracting the expected Poisson noise contribution and taking the squareroot to get the rms). The longterm rmsflux relation was obtained by first binning up the 0.125 s resolution light curves to 4 s resolution, and then measuring the rmsflux relation in the same way as for the shortterm relation, except using 32 s segments. Thus the shortterm relation is driven by the subsecond rms of the l_{2} and l_{3} Lorentzians responding to modulation by the l_{1} Lorentzian, while the longterm rms is dominated more by the l_{1} Lorentzian, which is not modulated by any additional timescales.
The rmsflux relations are shown in the bottom panels of Fig. 1. As expected, both the real data and the K16 model light curves show close to linear shortterm rmsflux relations, although the slope for the K16 simulation is significantly flatter than in the real data, which is likely due to the contribution of the l_{1} Lorentzian to the subsecond variability (since it adds an rms component which does not depend on the flux). The longterm rmsflux relation of the K16 model light curve is close to being constant, rather than linear, which is as expected since this component does not have any rmsflux relation built in due to variations on even longer timescales. In contrast, the real data show a clear linear rmsflux relation on both long and short timescales.
For comparison, we also simulated a light curve using the “exponentiation” approach of UMV05, which is designed explicitly to produce the observed lognormal flux distribution seen in the data and hence (as shown in UMV05) also produce the linear rmsflux relations seen on all timescales. The model works simply by taking the exponential of a Gaussiandistributed time series such that the resulting flux distribution is lognormal. In the case of our multipleLorentzian power spectrum, the operation simply involves taking the exponential of the sum of (zeromean) simulated Lorentzian light curves generated using the Timmer & Koenig (1995) approach, that is: (3)The primes denote that we followed the approach of UMV05 in reducing the amplitudes of the input Lorentzians used to make the original Gaussiandistributed light curves, to account for the effect of exponentiating the combined light curve (since this transformation increases the rms amplitude). Besides ensuring that the final fractional rms matches that of the data, we also choose the lognormal location parameter τ = 0.16, to match that of the real data. We apply the same analysis as for the real data and K16 simulation and the results are included in all the plots in Fig. 1.
The exponentiated light curve shows a similar power spectrum to the real data and has a lognormal flux distribution by construction which, as expected, looks very similar to that of the real data. The rmsflux relations are also similar to those of the real data on both long and short timescales. There are small differences but these are expected given that the exponentiation approach is very simplified, while real data shows more complex behaviour (e.g. in higherorder statistics such as the bicoherence, UMV05). In conclusion, our simulated exponentiated light curve shows a flux distribution and rmsflux relations, which appear more similar to the real data than the K16 model light curve does. Thus, our simulations demonstrate the arguments made in the previous section.
4. Some final remarks
It is important to note that a linear rmsflux relation by itself is not evidence for the same kind of variability process that produces lognormal flux distributions and linear rmsflux relations across a broad range of timescales. As noted in UMV05 (we refer to Appendix D of that paper), linear rmsflux relations can result generally from light curves with positively skewed flux distributions, provided that the rms, which is plotted versus flux, corresponds to a large fraction of the total variability amplitude (i.e. the rms of segments is relatively large compared to the amplitude of variations which produce the variations in mean flux of the segments). This situation can occur in time series that do not have very broad power spectra, unlike those seen in accreting objects where a broad range of frequencies contribute similar amounts to the total variability amplitude. This property of time series with positively skewed flux distributions and relatively narrowband power spectra can explain the presence of linear rmsflux relations in models distinct from the propagating fluctuations model, such as the “thundercloud model” of Merloni & Fabian (2001). This effect can also explain the presence of linear rmsflux relations in diverse data such as light curves of solar flares (Zhang 2007), where the rms variability amplitude in 1 s intervals is of a similar order to the highly skewed and nonlinear flux variation on longer timescales. The presence of linear rmsflux relations should therefore not be taken as an indication of similar variability processes to those seen in accreting systems, unless the effects of the flux distribution and powerspectral shape are also accounted for.
In the spirit of reproducible research, the data and the Python Jupyter Notebook used for this analysis is available at https://github.com/philuttley/uttleymchardyvaughan17
Acknowledgments
We would like to thank the anonymous referee for valuable comments. This research has made use of data provided by the High Energy Astrophysics Science Archive Research Center (HEASARC), which is a service of the Astrophysics Science Division at NASA/GSFC and the High Energy Astrophysics Division of the Smithsonian Astrophysical Observatory.
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All Figures
Fig. 1 Comparison of real Cyg X1 hard state data (blue circles) with simulated data showing the same power spectrum, made using the K16 model (green triangles) or the “exponentiated” model of UMV05 (open red squares). Top left: power spectra of the light curves, including the bestfitting Lorentzian decomposition for the real data (black lines, total model line in solid red). Top right: flux distributions (flux normalised by mean count rate), models (coloured lines) and data/model ratios for a fit to a lognormal model with constant offset constrained to be nonnegative. Bottom panels: the rmsflux relations of the light curves obtained for rms measured on two different timescale ranges, short (0.125−1 s, bottom left panel) and long (4−32 s, bottom right panel). Bestfitting linear (plus constant) relations are also shown as dotted lines. For the flux distributions and rmsflux relations, the K16 model is strongly inconsistent with the data, while the UMV05 model works reasonably well. 

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In the text 