The usefulness of the PSD is highly dependent on the S/N and level of variability in a source. Therefore it can only be employed on the brightest and most variable objects and is the reason we only expect at most 22 AGN to have well measured PSDs allowing for extensive analysis.
However, this does not mean there are no other tools available to study the variability properties of AGN. One of the most simple and widely used ones is known as “excess variance“ first proposed by \cite{Nandra_1997}. It is a measure of the extra variance above that caused by Poisson variations and can be calculated using the following equation. \[\label{eq:xs_var} \sigma^2_{\rm XS} = \frac{S^2 - \overline{\sigma^2_{\rm err}}}{\mu^2}\] \(S^2\) is the standard sample variance of the light curve, \(\overline{\sigma^2_{\rm err}}\) is the average error in the count rate, and \(\mu\) is the mean count rate. \(\sigma^2_{\rm XS}\) is also the integral of the PSD over the sampled timescales of the light curve and can be used for lower flux and variable sources since it works in the time rather than frequency domain.
\citet{Soldi_2014} used a maximum likelihood estimate of \(\sigma^2_{\rm XS}\) \citep{Almaini_2000} to study the variablilty properties of a much larger sample of the BAT AGN than ours in \citet{Shimizu_2013}. However, just as in our PSD analysis, the lack of short timescales able to be probed (\citet{Soldi_2014} had to use time bins of 1 month) limited their results to only weak or nonexistent relationships between the variability and intrinsic properties of the AGN. With NuSTAR lightcurves, we will be much closer and much more affected by the changes of the break frequency as shown in Figure \ref{fig:sims}. If the ultra-hard X-ray break frequency is changing as function of \(M_{\rm BH}\) and/or \(L_{\rm Bol}\) we should detect it not only through our PSD analysis, but also in an analysis of \(\sigma^2_{\rm XS}\) over a large sample of AGN.
The NuSTAR archive provides just such a large sample. A search of the archive returns 114 BAT AGN with at least one observation with \(>10\) ks on-source exposure with most having \(\sim20\) ks on-source exposure as part of one of NuSTAR’s legacy surveys. These archival observations will allow us to significantly study the general variability properties of AGN at high energies and determine if and how they are related to the AGN physical properties such as \(M_{\rm BH}\), \(L_{\rm Bol}\), \(N_{\rm H}\), \(\Gamma\) (power law spectral index), etc.
A further advantage of the archival NuSTAR observations is the broadband energy range from 3–79 keV. We can then generate light curves in specific energy ranges and characterize the variability as function of energy for both our PSD and excess variance analysis.
A test of the energy dependence would give insight into the physical processes that are controlling the emission in each energy range. At low energies, emission is thought to be dominated by a power-law continuum and is also affected by absorption. At higher energies, absorption is negligible, but reflection becomes important which is considered to be fairly constant. At the highest energies, changes in the cutoff energy will greatly affect the variability.
Both \citet{Shimizu_2013} and \citet{Soldi_2014} found that the variability strength over the broad 2–10 keV and 14–150 keV energy ranges are consistent with being equal with even a hint it is stronger at higher energies. This would argue against a constant reflection dominated spectrum in the 14–150 keV range. However these comparisons were based on extrapolating the 2–10 keV PSD to the low frequencies covered by Swift/BAT which could introduce large uncertainties. Further, the comparisons are over different epochs. With NuSTAR, we can construct PSDs as well as calculate \(\sigma^2_{\rm XS}\) over the same timescales for different energy ranges that cover the same points in time.
Finally, as a bonus, NuSTAR also provides high fidelity spectra along with the light curves. This will allow us to make detailed comparisons between the spectral and temporal characteristics for a large sample of AGN. Using XSPEC we will be able to fit complex models to determine the relative strengths of absorption, power-law continuum, and reflection to the emission and see how this affects the flux variability. If the main source of variability is a changing power law continuum (in both normalization and spectral index) \citet[e.g.][]{Parker_2014}, then we would expect continuum dominated AGN to exhibit larger levels of variability while reflection dominated AGN to show lower levels of variability. Within a Comptonization model, confirmation of this source of variability would indicate an increase of seed photons from the accretion disk that illuminates the corona increases cooling of the electrons and causes the spectrum to steepen. This is seen in the “softer-when-brighter” behavior of many X-ray spectra of AGN \cite{Markowitz_2004, Caballero_Garcia_2012}.
An exciting aspect of the study of variability is the prospect that it can be used to measure black hole masses. \citet{Kelly_2013} explains that if the PSD break frequency and the amplitude of the high frequency section of the PSD are correlated with \(M_{\rm BH}\), then a simple variability analysis will provide an estimate of the black hole mass with an uncertainty that is comparable to using reverberation mapping and the width of the broad emission lines. The possible advantage of using X-ray variability is the relative low observational cost and the ability to use it to measure black hole masses for Seyfert 2’s where obscuration hinders the effectiveness of reverberation mapping and hides the broad emission lines. In this study, we will thoroughly test the dependence of both the PSD parameters and \(\sigma^2_{\rm XS}\) on \(M_{\rm BH}\). If the dependence is strong this would greatly increase the value of the NuSTAR archive as well as provide an easy method for future large scale surveys of AGN to measure black hole mass, especially at high redshift.