Living on a Flare: Relativistic Reflection in V404 Cyg Observed by NuSTAR during Its Summer 2015 Outburst

In Figure 5 (top panel), we show the average spectrum for the flares, extracted by selecting only periods where the count rate (per FPM) was >4000 $\mathrm{ct}\,{{\rm{s}}}^{-1}$. The total good exposure in the resulting spectrum is only ∼110–120 s. In contrast to the average spectrum, there is no visually apparent edge at ∼7 keV, indicating that the line-of-sight absorption is much weaker during these periods, and that we therefore have a cleaner view of the intrinsic spectrum. The flare spectrum is very hard, and there is still visible structure in the iron K band. In Figure 5 (bottom panel), we show the data/model residuals to a simple model consisting of a power-law continuum with a high-energy exponential cutoff, modified by a neutral absorption column, which is free to vary above a lower limit of 10²² cm⁻² (set by prior constraints on the Galactic column; see above). We use the TBABS absorption model, adopting the ISM abundances reported in Wilms et al. (2000) as our "solar" abundance set, and the cross-sections of Verner et al. (1996), as recommended. This model is fit to the 3–4, 8–10, and 50–79 keV energy ranges in order to minimize the influence of any reflected emission present in the spectrum. The photon index obtained is very hard, ${\rm{\Gamma }}\sim 1.5$, with a cutoff energy of ${E}_{\mathrm{cut}}\sim 160\,\mathrm{keV}$.

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A very strong Compton hump is visible around ∼20–30 keV, indicating a significant contribution from X-ray reprocessing by optically thick material. The iron emission is also rather strong, and, although there is a narrow core to the line profile, the majority of the line emission is broadened with a clear red-wing, a hallmark of relativistically broadened reflection from an accretion disk (referred to as a "diskline" profile; e.g., Fabian et al. 1989; Laor 1991). Modeling the 3–10 keV bandpass with the simple continuum model above (fixing the cutoff energy to its best-fit value, given the limited energy range being considered), and including both an unresolved Gaussian at 6.4 keV and a RELLINE component (Dauser et al. 2010) to account for the narrow core and the iron emission from the accretion disk, respectively, we find that the RELLINE component has an equivalent width of $\mathrm{EW}\sim 400\,\mathrm{eV}$, while the narrow core is much weaker, with $\mathrm{EW}\sim 25\,\mathrm{eV}$.

Isolating and modeling the reprocessed emission from the accretion disk is of significant importance becuase this provides information on both the spin of the black hole (e.g., Risaliti et al. 2013; Walton et al. 2013, 2014; Reynolds 2014) and the geometry/location of the illuminating X-ray source (e.g., Wilkins & Fabian 2012). This is of particular interest for the intense flares, since such X-ray flares are often associated with jet ejection (e.g., Corbel et al. 2002). However, constraining the disk reflection is not necessarily straightforward from the iron band alone. In order to aid in disentangling the contributions from reprocessing by the accretion disk and by more distant material to the spectrum, the main body of this work focuses on modeling the broadband evolution of V404 Cyg as a function of flux during the flaring phase of our NuSTAR observation.

4.2.1. Data Selection

One of the main complications for broadband modeling is the strong and variable absorption that is present throughout this observation. In order to mimimize this issue, based on the flare spectrum (Figure 5), we select only periods of similarly low absorption for our flux-resolved spectral analysis. In order to identify such periods, we define a narrow-band hardness ratio (hereafter ${R}_{\mathrm{edge}}$), with the softer band (6.5–7.0 keV) just below the sharp edge seen in the average spectrum, and the harder band (7.5–8.0 keV) just above, such that we can track the strength of the edge throughout the flaring period. With these narrow bands, a stronger absorption edge (and thus more absorption) would appear to have a softer spectrum (i.e., a lower hardness ratio). We show the behavior of ${R}_{\mathrm{edge}}$ in Figure 6, in the form of a similar HRI diagram to Figure 3. Note that we are forced to adopt a coarser temporal binning (100 s) in order for ${R}_{\mathrm{edge}}$ to be well constrained owing to the narrow energy bands used; hence the peak 3–79 keV count rates differ in this figure. Nevertheless, it is clear that in terms of ${R}_{\mathrm{edge}}$, the highest count rates (i.e., the strongest flares) show the hardest spectra, with ${R}_{\mathrm{edge}}\gtrsim 0.7$, consistent with the lack of absorption seen in Figure 5. Furthermore, though the majority of the observation shows a much stronger edge, there are other non-flare periods in which the absorption is similarly weak. These periods are spread randomly throughout the flaring portion of the observation, and span a broad range of flux.

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We therefore select only data with ${R}_{\mathrm{edge}}\geqslant 0.7$ for the lower flux intervals (i.e., <4000 $\mathrm{ct}\,{{\rm{s}}}^{-1}$), and then divide these periods into four flux bins: 100–500, 500–1000, 1000–2000, and 2000–4000 $\mathrm{ct}\,{{\rm{s}}}^{-1}$ (per FPM, using the count rates from the finer 10 s binning). The lower limit to the data considered is set to 100 $\mathrm{ct}\,{{\rm{s}}}^{-1}$ in order to avoid the low flux region in which the flux and the broadband hardness ratio are correlated (see Figure 3), because the source behavior is clearly distinct in this regime. We therefore have five flux bins in total (referred to as F1–5, in order of increasing flux), including the flare spectrum extracted from >4000 $\mathrm{ct}\,{{\rm{s}}}^{-1}$ in Section 4.1. Details of these flux bins are given in Table 1 and the extracted spectra are shown in Figure 7; the lack of strong, visible absorption edges in any of these spectra demonstrates the general success of our low-absorption selection procedure. As with the flare spectrum, despite the lack of strong absorption, the spectra from lower fluxes are also very hard. There are a couple of trends that can be seen from a visual inspection of these data. First, the relative contribution from the narrow core of the iron emission is stronger at lower fluxes. Second, the continuum above ∼10 keV shows a lot more spectral curvature at higher fluxes.

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Table 1.  Details of the Five Flux Bins Used in Our Flux-resolved Spectral Analysis of the Peruids of Low Absorption (Selected to Have ${R}_{\mathrm{edge}}$ $\geqslant$ 0.7)

Flux	Count	Good	Broad
Bin	Rate	Exposure	Fe K EW
	(ct s−1 FPM−1)	(FPMA/B; s)	(eV)
F1	100–500	1074/1105	${260}_{-60}^{+70}$
F2	500–1000	1067/1120	${350}_{-50}^{+60}$
F3	1000–2000	722/769	${390}_{-80}^{+40}$
F4	2000–4000	260/280	${460}_{-100}^{+60}$
F5	>4000	112/121	${440}_{-90}^{+50}$

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Before proceeding with our more detailed spectral analysis, we repeat our phenomenological modeling of the 3–10 keV bandpass performed above for the flare spectrum, and fit the data for each of these flux bins with a combination of a broad and narrow iron emission component. In order to minimize parameter degeneracies, given the limited bandpass utilized, we make the simplifying assumption that the profile of the broad iron emission is the same for all fluxes. With this simple modeling, though the individual uncertainties are relatively large, we find that the strength of the broad iron emission increases with increasing flux (see Table 1).

4.2.2. Basic Model Setup

4.2.3. Results

To begin with, we assume that the disk extends into the innermost stable circular orbit (ISCO) for all flux states and that the corona is not outflowing, and we allow the reflection fraction to vary as a free parameter (Model 1; a summary of all the models considered in our flux- and flare-resolved analyses is given in Table 2). This model provides a good fit to the global data set, with ${\chi }^{2}$ = 10599 for 10308 degrees of freedom (DoF). We observe several trends in the fits, which are presented in Table 3. Most notably, we find that the strength of the disk reflection increases with increasing flux (see Figure 8). This is a strong indicator that the (average) geometry of the innermost accretion flow evolves as a function of source flux. In addition to these variations, the ionization of the disk increases as the observed flux increases, as would broadly be expected for an increasing ionizing flux, and there are changes in the intrinsic continuum, with the high-energy cutoff decreasing in energy as the flux increases.

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Table 2.  A Summary of the Lamp-post Reflection Models Applied during Our Flux- and Flare-resolved Analyses, Presented in Sections 4.2 and 4.3, Respectively

Model	Data Set	Source Emission	Notes
1	Flux-resolved	Lamp-post only	${R}_{\mathrm{disk}}$ a free parameter, ${r}_{\mathrm{in}}$ fixed at the ISCO
2	Flux-resolved	Lamp-post only	${R}_{\mathrm{disk}}$ calculated self-consistently, ${r}_{\mathrm{in}}$ free to vary, h constant
3	Flux-resolved	Lamp-post only	${R}_{\mathrm{disk}}$ calculated self-consistently, ${r}_{\mathrm{in}}$ fixed at the ISCO, h free to vary
4	Flux-resolved	Lamp-post + disk	${R}_{\mathrm{disk}}$ calculated self-consistently, ${r}_{\mathrm{in}}$ free to vary, h constant
5	Flare-resolved	Lamp-post only	${R}_{\mathrm{disk}}$ calculated self-consistently, ${r}_{\mathrm{in}}$ fixed at the ISCO
6	Flare-resolved	Lamp-post + disk	${R}_{\mathrm{disk}}$ calculated self-consistently, ${r}_{\mathrm{in}}$ fixed at the ISCO
6i	Flare-resolved	Lamp-post + disk	Same as model 6, but i limited to ≥45°

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Table 3.  Results for the Free Parameters in the Basic Lamp-post Reflection Model (Model 1) Constructed for the Spectral Evolution As a Function of Flux

Model Component	Parameter		Global	Flux Level
				F1	F2	F3	F4	F5
TBABS${}_{\mathrm{src}}$	${N}_{{\rm{H}}}$	(1021 cm−2)	⋯	${9.1}_{-1.7}^{+2.2}$	${9.1}_{-2.1}^{+1.6}$	$\lt 0.7$	$\lt 1.6$	$\lt 0.4$
RELXILLLP	Γ	⋯	⋯	1.43 ± 0.02	1.49 ± 0.03	${1.42}_{-0.01}^{+0.02}$	${1.41}_{-0.01}^{+0.02}$	${1.40}_{-0.02}^{+0.01}$
	${E}_{\mathrm{cut}}$	(keV)	⋯	$\gt 840$ a	${610}_{-210}^{+340}$	${240}_{-20}^{+10}$	${150}_{-10}^{+20}$	120 ± 10
	a*	⋯	$\gt -0.1$	⋯	⋯	⋯	⋯	⋯
	i	(°)	27 ± 2	⋯	⋯	⋯	⋯	⋯
	h	$({r}_{{\rm{H}}})$	⋯	${6.0}_{-2.0}^{+7.0}$	${4.7}_{-1.0}^{+4.0}$	${3.9}_{-1.1}^{+3.0}$	${3.7}_{-0.9}^{+2.8}$	${3.2}_{-1.1}^{+2.5}$
	$\mathrm{log}\xi$	log ($\mathrm{erg}\,\mathrm{cm}\,{{\rm{s}}}^{-1}$)	⋯	${3.01}_{-0.01}^{+0.02}$	3.02 ± 0.01	${3.09}_{-0.02}^{+0.01}$	${3.15}_{-0.02}^{+0.05}$	${3.47}_{-0.04}^{+0.05}$
	${A}_{\mathrm{Fe}}$	(solar)	${1.9}_{-0.1}^{+0.3}$	⋯	⋯	⋯	⋯	⋯
	${R}_{\mathrm{disk}}$	⋯	⋯	1.1 ± 0.2	${1.5}_{-0.2}^{+0.3}$	${1.7}_{-0.2}^{+0.4}$	${2.0}_{-0.2}^{+0.6}$	${3.0}_{-0.5}^{+0.8}$
	Norm	⋯	⋯	${0.15}_{-0.02}^{+0.03}$	${0.23}_{-0.03}^{+0.05}$	${0.33}_{-0.09}^{+0.08}$	${0.47}_{-0.06}^{+0.09}$	${0.65}_{-0.15}^{+0.34}$
XSTAR ${}_{\mathrm{abs}}$	$\mathrm{log}\xi$	log ($\mathrm{erg}\,\mathrm{cm}\,{{\rm{s}}}^{-1}$)	${4.6}_{-0.3}^{+0.8}$	⋯	⋯	⋯	⋯	⋯
	${N}_{{\rm{H}}}$	(1021 cm−2)	⋯	${3.7}_{-2.3}^{+5.0}$	${4.2}_{-1.5}^{+4.8}$	$\lt 3.5$	${3.4}_{-1.3}^{+4.0}$	${3.0}_{-1.1}^{+2.7}$
XILLVER	ib	(°)	$\lt 11$	⋯	⋯	⋯	⋯	⋯
	${A}_{\mathrm{Fe}}$ b	(solar)	$\lt 0.91$	⋯	⋯	⋯	⋯	⋯
	Norm	$({10}^{-2})$	⋯	${1.2}_{-0.8}^{+1.1}$	${8.6}_{-1.1}^{+0.7}$	${12.7}_{-0.7}^{+0.8}$	${18.7}_{-1.5}^{+1.6}$	${33.0}_{-3.0}^{+2.7}$
XSTAR ${}_{\mathrm{emis}}$	$\mathrm{log}\xi$	log ($\mathrm{erg}\,\mathrm{cm}\,{{\rm{s}}}^{-1}$)	⋯	$\lt 1.7$	⋯	⋯	⋯	⋯
	Norm	(104)	⋯	1.3 ± 0.3	⋯	⋯	⋯	⋯
${\chi }^{2}$/DoF			10599/10308
${F}_{3-79}$ c	(10−8 $\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$)		⋯	3.78 ± 0.02	8.85 ± 0.03	16.06 ± 0.06	28.0 ± 0.1	54.6 ± 0.3

Notes.

^a ${E}_{\mathrm{cut}}$ is constrained to be ≤1000 keV following García et al. (2015). ^bThese act as dummy "shape" parameters to allow this component the flexibility to deviate from the simple slab approximation adopted in the XILLVER model (see the main text), and in turn allow for our lack of knowledge with regards to the geometry of the distant reprocessor. ^cAverage flux in the 3–79 keV bandpass.

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The black hole spin is not well constrained with this model (though the majority of negative spins are excluded: ${a}^{* }\gt -0.1$). However, during the flares, the disk reflection is very strong, (${R}_{\mathrm{disk}}\sim 3$). Caution over the exact value is necessary here, because the strength of the reflection obtained is dependent to some extent on the form of the high-energy curvature included in the input continuum model (a simple exponential cutoff in this work), and there is also some degeneracy between the ${R}_{\mathrm{disk}}$ and ${E}_{\mathrm{cut}}$ parameters. However, taking the result at face value, this would imply a scenario in which strong gravitational light bending enhances the disk reflection (e.g., Miniutti & Fabian 2004). In turn, this would imply that V404 Cyg hosts a rapidly rotating black hole (e.g., Dauser et al. 2014; Parker et al. 2014). Although we are using an idealized lamp-post geometry in this work, as long as the disk is thin, then this is the case regardless of the precise geometry of the X-ray source, because the disk must extend close to the black hole in order to subtend a sufficiently large solid angle to produce the high reflection fraction; the validity of the thin disk assumption (which is currently implicit in the RELXILL models) for these flares is discussed further in Section 5.2. Potential evidence for strong reflection during bright flares has also been seen from INTEGRAL observations of this outburst (Natalucci et al. 2015; Roques et al. 2015). We stress, though, that despite any degeneracy between these parameters, the variations in both ${E}_{\mathrm{cut}}$ and ${R}_{\mathrm{disk}}$ are significant; if we try to force one of these two parameters to be the same for each of the flux states and only allow the other to vary, the fits are significantly worse (${\rm{\Delta }}{\chi }^{2}\gt 80$ for four fewer free parameters). While the absolute values themselves are somewhat model dependent, the trend of increasing ${R}_{\mathrm{disk}}$ with increasing flux appears to be robust to such issues.

For the disk reflection fraction to vary in such a manner, the solid angle subtended by the disk as seen by the X-ray source must decrease as the observed flux decreases. A few potential scenarios could produce such behavior: (1) the disk itself could evolve (e.g., truncate) such that it genuinely covers a smaller solid angle at lower fluxes; (2) the corona could evolve and vary its location/size, such that the degree of gravitational light bending is reduced; (3) the corona could alternately vary its velocity, such that the beaming away from the disk is increased. While some combination of these three effects is of course possible, and probably even likely should the flares be related to jet ejection events, from a practical standpoint their individual effects on the observed reflection emission are rather similar (Dauser et al. 2013; Fabian et al. 2014). Therefore, in order to investigate the potential geometric evolution without introducing further parameter degeneracies, we also modify our basic model to consider two limiting scenarios representing the first two of these three possibilities (Models 2 and 3, respectively), now making use of the fact that given a combination of black hole spin, inner disk radius, and X-ray source height, RELXILLLP can self-consistently compute the expected ${R}_{\mathrm{disk}}$ from the lamp-post geometry.¹⁵ First, we assume that the corona remains static and the disk progressively truncates as the flux decreases, and second, we assume that the disk remains static and the corona progressively moves away from the disk (note that this could relate to either a physical movement of the corona, or a vertical expansion of the electron cloud).

In the truncation scenario (which we call Model 2), we therefore allow the inner radius of the disk (${r}_{\mathrm{in}}$) to vary, though given the results above, we assume that during the flares the disk does reach the ISCO, and we link h across all flux states. For the lower flux states, ${r}_{\mathrm{in}}$ is computed in units of ${r}_{\mathrm{ISCO}}$, so that we can ensure that ${r}_{\mathrm{in}}\geqslant {r}_{\mathrm{ISCO}}$, as simulations find that emission from within the ISCO is negligible (e.g., Reynolds & Fabian 2008; Shafee et al. 2008). In the dynamic corona scenario (Model 3), we assume that the disk reaches the ISCO for all fluxes, and instead vary h. The results from these two scenarios are presented in Table 4; we focus only on the key RELXILLLP parameters because the parameters for the other components generally remain similar to Model 1.

Table 4.  Results for the High-spin Solutions Obtained with the Lamp-post Reflection Models Constructed to Investigate Potential Geometric Evolution Scenarios As a Function of Flux (Models 2 and 3)

Model Component	Parameter		Global	Flux Level
				F1	F2	F3	F4	F5
Model 2: truncating disk, static corona
RELXILLLP	Γ	⋯	⋯	${1.41}_{-0.03}^{+0.02}$	${1.44}_{-0.03}^{+0.02}$	1.40 ± 0.01	1.37 ± 0.02	1.37 ± 0.01
	${E}_{\mathrm{cut}}$	(keV)	⋯	$\gt 540$ a	330 ± 60	190 ± 10	${125}_{-6}^{+8}$	${91}_{-3}^{+4}$
	a*	⋯	$\gt 0.95$	⋯	⋯	⋯	⋯	⋯
	i	(°)	36 ± 1	⋯	⋯	⋯	⋯	⋯
	h	$({r}_{{\rm{H}}})$	${2.3}_{-0.1}^{+0.4}$	⋯	⋯	⋯	⋯	⋯
	${A}_{\mathrm{Fe}}$	(solar)	3.0 ± 0.1	⋯	⋯	⋯	⋯	⋯
	${r}_{\mathrm{in}}$	(${r}_{\mathrm{ISCO}}$)	⋯	${2.5}_{-0.3}^{+0.4}$	${2.3}_{-0.2}^{+0.1}$	2.0 ± 0.1	${1.7}_{-0.1}^{+0.2}$	1 (fixed)
	${R}_{\mathrm{disk}}$ b	⋯	⋯	1.3	1.5	1.7	1.9	3.0
	Norm	⋯	⋯	${0.61}_{-0.10}^{+0.11}$	${0.80}_{-0.16}^{+0.18}$	1.07 ± 0.15	${1.53}_{-0.31}^{+0.24}$	${2.61}_{-0.51}^{+0.19}$
${\chi }^{2}$/DoF			10656/10313
Model 3: stable disk, dynamic corona
RELXILLLP	Γ	⋯	⋯	${1.36}_{-0.01}^{+0.03}$	1.41 ± 0.01	${1.37}_{-0.01}^{+0.02}$	1.36 ± 0.01	1.38 ± 0.01
	${E}_{\mathrm{cut}}$	(keV)	⋯	${540}_{-50}^{+80}$	280 ± 20	180 ± 10	${123}_{-3}^{+5}$	94 ± 3
	a*	⋯	$\gt 0.88$	⋯	⋯	⋯	⋯	⋯
	i	(°)	${28}_{-2}^{+1}$	⋯	⋯	⋯	⋯	⋯
	h	$({r}_{{\rm{H}}})$	⋯	${5.2}_{-0.4}^{+1.5}$	5.6 ± 0.3	${4.6}_{-0.2}^{+0.8}$	${4.1}_{-0.2}^{+0.3}$	4.4 ± 0.2
	${A}_{\mathrm{Fe}}$	(solar)	3.03 ± 0.05	⋯	⋯	⋯	⋯	⋯
	${R}_{\mathrm{disk}}$ b	⋯	⋯	1.7	1.6	1.8	2.0	1.9
	Norm	⋯	⋯	0.18 ± 0.01	0.22 ± 0.01	${0.38}_{-0.01}^{+0.05}$	${0.63}_{-0.04}^{+0.09}$	${1.02}_{-0.07}^{+0.12}$
${\chi }^{2}$/DoF			10678/10313

Notes.

^a ${E}_{\mathrm{cut}}$ is constrained to be ≤1000 keV following García et al. (2015). ^bFor these models, ${R}_{\mathrm{disk}}$ is calculated self-consistently in the lamp-post geometry from a*, h, and ${r}_{\mathrm{in}}$. Because it is not a free parameter, errors are not estimated.

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Both of these scenarios provide reasonable fits to the data (${\chi }^{2}$/DoF = 10656/10313 and 10675/10312 for Models 2 and 3, respectively), though the truncation scenario formally provides the better fit, and both are worse fits than Model 1 (in which ${R}_{\mathrm{disk}}$ is a free parameter), owing to the additional physical constraints imposed. With these additional constraints, both scenarios require the spin to be at least moderate (${a}^{* }\gtrsim 0.6$), but above this value the ${\chi }^{2}$ landscape becomes complex. Both scenarios show distinct minima that provide similarly good fits (different by ${\rm{\Delta }}{\chi }^{2}\lt 5$ in both cases) at a high spin value and at a more moderate-spin value (see Figure 9). For the truncating disk scenario (Model 2) the high spin solution (${a}^{* }\sim 0.97$) is marginally preferred over the lower spin solution (${a}^{* }\sim 0.82$), while for the dynamic corona scenario (Model 3) the lower spin solution (${a}^{* }\sim 0.65$) is marginally preferred over the high spin solution (${a}^{* }\sim 0.97$), perhaps indicating that even when allowing the height of the X-ray source to vary the data still want an evolution in the inner radius of the disk at some level. In both of these scenarios, we present the results for the high spin solution (which in the latter case gives a fit of ${\chi }^{2}$/DoF = 10678/10312) because our subsequent modeling of the individual flares strongly suggests that the black hole in V404 Cyg is rapidly rotating (see Section 4.3). However, for completeness, we also present the parameter constraints for the lower-spin solutions in the Appendix; where the solutions are not the global best fit, errors are calculated as ${\rm{\Delta }}{\chi }^{2}=2.71$ around the local ${\chi }^{2}$ minimum. Separating out the solutions in this manner also allows the evolution required in ${r}_{\mathrm{in}}$ and h to be more clearly seen because both of these parameters are scaled by the spin in our model implementation. As expected, we see that either the inner radius of the accretion disk moves outward (Model 2), or the source height moves upward (Model 3), as the flux decreases. In Model 2, the inner disk radius evolves from the ISCO (assumed) out to ∼2.5 ${r}_{\mathrm{ISCO}}$, and in Model 3 the source height evolves from ∼4 to ∼6 ${r}_{{\rm{H}}}$.

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Finally, although the X-ray emission in the NuSTAR bandpass is clearly dominated by a hard, high-energy continuum and reprocessed emission, we also test for the presence of any thermal emission from an accretion disk. As the self-consistent evolutionary scenario that formally provides the best fit, we focus on the truncating disk scenario, and modify our model for the intrinsic emission from V404 Cyg to include a multi-color blackbody accretion disk (Model 4), using the DISKBB model (Mitsuda et al. 1984). In the XSPEC model outlined in Section 4.2.2, we thus update the source term to be TBABS_src × (DISKBB + RELXILLLP). ${R}_{\mathrm{disk}}$ is still calculated self-consistently in the lamp-post geometry from a*, h, and ${r}_{\mathrm{in}}$. During the fitting process for this model, we found that the DISKBB component only makes a significant contribution during the highest flux state (F5), and so fixed its normalization to zero for F1–4.

This model provides a good fit to the data, with ${\chi }^{2}$/DoF = 10581/10311 (i.e., an improvement of ${\rm{\Delta }}{\chi }^{2}=75$ for two additional free parameters over Model 2). We do not tabulate the parameter values because the vast majority have not varied significantly from the values presented for Model 2 in Table 4, but a few key parameters are worth highlighting individually. The best-fit disk temperature for the average flare spectrum is ${0.41}_{-0.07}^{+0.10}$ keV, such that this component only contributes close to the lower boundary of the NuSTAR bandpass. However, this temperature is similar to values reported from X-ray observatories with coverage extending to lower energies throughout this outburst (e.g., Radhika et al. 2016, Rahoui et al. 2017). The inclusion of this additional continuum component at the lower end of the NuSTAR bandpass allows the high energy power-law continuum to take on a harder photon index (${\rm{\Gamma }}=1.32\pm 0.01$), and subsequently a lower energy cutoff (${E}_{\mathrm{cut}}=75\pm 4$ keV), such that this primary continuum emission exhibits stronger curvature in the NuSTAR band. In turn, this allows a slightly lower reflection fraction (${R}_{\mathrm{disk}}=2.5$), with the source height increasing slightly to $h={2.5}_{-0.1}^{+0.5}$ ${r}_{{\rm{H}}}$. The black hole spin remains high, with ${a}^{* }\gt 0.82$ (and noteably the ${\chi }^{2}$ contour only displays a single solution; see Figure 9). The data/model ratios for the five flux states are shown in Figure 10 for this model, and the best-fit model along with the relative contributions of the various emission components are shown in Figure 11 for the highest flux state (F5).

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It is worth noting that all of these flux-resolved models have returned inclinations for the inner disk of ∼30°. This inclination would mark a large difference between the inclination of the inner disk and the orbital plane, which the latest optical studies during quiescence have estimated to be ${i}_{\mathrm{orb}}\sim 65$° (Khargharia et al. 2010), with literature estimates covering a range from 50° to 75° (Shahbaz et al. 1994, 1996; Sanwal et al. 1996). While evidence of misalignment between the inner and outer regions of the disk has been seen in other sources, e.g., Cygnus X-1 (Tomsick et al. 2014; Walton et al. 2016), a difference this large would likely be unphysical (e.g., Fragos et al. 2010; Nealon et al. 2015). We will return to this issue in the following section.

We also investigate a number of the individual flares, focusing on the six that reach or exceed ∼10,000 $\mathrm{ct}\,{{\rm{s}}}^{-1}$ (labeled in Figure 2). Following the reduction procedure used in Section 4.1, we extracted NuSTAR spectra for each of these six flares individually. These spectra are shown in Figure 12. While they are all reasonably similar, as suggested by their similar broadband hardness ratios (Figure 3), there are also obvious differences between them, so there is still some clear averaging of different states in our flux-resolved analysis. For example, the first flare has a harder spectrum at lower energies than the subsequent flares, and the third flare has a softer spectrum than the rest over the NuSTAR bandpass.

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We performed a joint fit of each of these flare spectra with the lamp-post model discussed in Sections 4.2.2 and 4.2.3 (excluding the photoionized emission component, which makes no contribution to the flux-resolved fits at high fluxes). As with our flux-resolved analysis, we link the black hole spin, iron abundance, accretion disk inclination, and ionization state of the photoionized absorption across all of the flares. Additionally, for the distant reprocessor, we fix the shape parameters (iron abundance, slab inclination) to the values found in the flux-resolved work. We also assume that the disk extends to the ISCO and again compute the reflection fraction self-consistently assuming a lamp-post geometry. Finally, given the results presented in Section 4.2.3, we fit the lamp-post model both with and without an accretion disk contribution, again using the DISKBB model. While fitting the model with the DISKBB component, we found the disk temperatures to be consistent among all of the flares, and so linked this parameter across the data sets for simplicity. The results obtained with both these models (Models 5 and 6, respectively) are presented in Table 5.

Table 5.  Results Obtained for the Free Parameters in the Lamp-post Reflection Models Constructed for the Joint Fits to the Individual Flare Spectra (Models 5 and 6)

Model Component	Parameter		Global	Flare
				1	2	3	4	5	6
Model 5: lamp-post only
TBABS${}_{\mathrm{src}}$	${N}_{{\rm{H}}}$	(1022 cm−2)	⋯	${3.3}_{-0.8}^{+0.6}$	$\lt 0.2$	$\lt 0.3$	$\lt 0.2$	$\lt 0.1$	$\lt 0.1$
RELXILLLP	Γ	⋯	⋯	${1.22}_{-0.12}^{+0.06}$	${1.44}_{-0.03}^{+0.01}$	${1.63}_{-0.12}^{+0.08}$	${1.52}_{-0.04}^{+0.03}$	${1.28}_{-0.03}^{+0.02}$	${1.58}_{-0.03}^{+0.01}$
	${E}_{\mathrm{cut}}$	(keV)	⋯	${50}_{-10}^{+6}$	210 ± 30	47 ± 4	${94}_{-15}^{+12}$	${60}_{-6}^{+4}$	${140}_{-10}^{+20}$
	a*	⋯	$\gt 0.99$	⋯	⋯	⋯	⋯	⋯	⋯
	i	(°)	42 ± 2	⋯	⋯	⋯	⋯	⋯	⋯
	h	$({r}_{{\rm{H}}})$	⋯	$\lt 2.2$	$\lt 2.4$	${4.0}_{-1.0}^{+3.5}$	${2.7}_{-0.3}^{+1.6}$	$\lt 2.1$	${8.0}_{-0.9}^{+3.3}$
	${A}_{\mathrm{Fe}}$	(solar)	${2.9}_{-0.6}^{+0.3}$	⋯	⋯	⋯	⋯	⋯	⋯
	$\mathrm{log}\xi$	log ($\mathrm{erg}\,\mathrm{cm}\,{{\rm{s}}}^{-1}$)	⋯	3.2 ± 0.1	${3.5}_{-0.1}^{+0.2}$	${3.1}_{-0.1}^{+0.2}$	${3.4}_{-0.1}^{+0.2}$	${3.36}_{-0.05}^{+0.07}$	2.2 ± 0.1
	${R}_{\mathrm{disk}}$ a	⋯	⋯	5.3	4.4	2.3	3.4	5.3	1.5
	Norm	⋯	⋯	${4.2}_{-0.5}^{+0.3}$	${4.4}_{-0.7}^{+0.9}$	${1.1}_{-0.4}^{+0.3}$	${2.1}_{-1.1}^{+0.5}$	${3.9}_{-1.2}^{+0.3}$	1.3 ± 0.2
XSTAR${}_{\mathrm{abs}}$	$\mathrm{log}\xi$	log ($\mathrm{erg}\,\mathrm{cm}\,{{\rm{s}}}^{-1}$)	${5.3}_{-0.3}^{+0.4}$	⋯	⋯	⋯	⋯	⋯	⋯
	${N}_{{\rm{H}}}$	(1021 cm−2)	⋯	${43}_{-10}^{+12}$	$\lt 1$	$\lt 3$	$\lt 7$	$\lt 3$	$\lt 12$
XILLVER	Norm	⋯	⋯	0.22 ± 0.04	${0.42}_{-0.08}^{+0.09}$	0.25 ± 0.06	${0.34}_{-0.07}^{+0.04}$	0.45 ± 0.05	0.40 ± 0.04
${\chi }^{2}$/DoF			6039/5859
Model 6: lamp-post with disk emission
TBABS${}_{\mathrm{src}}$	${N}_{{\rm{H}}}$	(1022 cm−2)	⋯	${5.1}_{-1.1}^{+1.0}$	${2.3}_{-0.6}^{+0.9}$	${3.6}_{-1.3}^{+1.4}$	${2.2}_{-1.3}^{+1.5}$	${3.3}_{-1.0}^{+1.1}$	${3.8}_{-0.9}^{+0.8}$
DISKBB	${T}_{\mathrm{in}}$	(keV)	0.49 ± 0.04	⋯	⋯	⋯	⋯	⋯	⋯
	Norm	(105)	⋯	${1.5}_{-0.7}^{+1.3}$	${1.9}_{-1.1}^{+1.5}$	${3.4}_{-1.5}^{+2.9}$	${2.2}_{-1.1}^{+1.4}$	${2.7}_{-1.2}^{+2.2}$	${2.8}_{-1.2}^{+1.9}$
RELXILLLP	Γ	⋯	⋯	$\lt 1.04$ b	${1.27}_{-0.03}^{+0.04}$	${1.29}_{-0.08}^{+0.12}$	${1.33}_{-0.06}^{+0.07}$	$\lt 1.12$ b	${1.22}_{-0.04}^{+0.06}$
	${E}_{\mathrm{cut}}$	(keV)	⋯	${40}_{-2}^{+3}$	${113}_{-28}^{+16}$	${32}_{-4}^{+5}$	${60}_{-9}^{+12}$	${42}_{-3}^{+5}$	${68}_{-6}^{+8}$
	a*	⋯	$\gt 0.98$	⋯	⋯	⋯	⋯	⋯	⋯
	i	(°)	${52}_{-3}^{+2}$	⋯	⋯	⋯	⋯	⋯	⋯
	h	$({r}_{{\rm{H}}})$	⋯	$\lt 2.3$	$\lt 3.8$	$\lt 3.6$	$\lt 3.5$	$\lt 2.4$	${7.5}_{-1.9}^{+8.7}$
	$\mathrm{log}{\xi }_{\mathrm{disk}}$	log ($\mathrm{erg}\,\mathrm{cm}\,{{\rm{s}}}^{-1}$)	⋯	3.5 ± 0.1	3.8 ± 0.1	3.4 ± 0.1	3.6 ± 0.1	3.6 ± 0.1	3.5 ± 0.1
	${A}_{\mathrm{Fe}}$	(solar)	${5.0}_{-0.4}^{+0.7}$	⋯	⋯	⋯	⋯	⋯	⋯
	${R}_{\mathrm{disk}}$ a	⋯	⋯	5.3	3.9	4.1	4.1	5.3	1.6
	Norm	⋯	⋯	${3.5}_{-0.8}^{+0.2}$	${3.1}_{-1.7}^{+2.0}$	${1.9}_{-0.9}^{+1.0}$	${2.7}_{-1.3}^{+1.4}$	${3.5}_{-1.2}^{+0.2}$	${1.0}_{-0.2}^{+0.4}$
XSTAR${}_{\mathrm{abs}}$	$\mathrm{log}\xi$	log ($\mathrm{erg}\,\mathrm{cm}\,{{\rm{s}}}^{-1}$)	5.7 ± 0.2	⋯	⋯	⋯	⋯	⋯	⋯
	${N}_{{\rm{H}}}$	(1021 cm−2)	⋯	${88}_{-3}^{+4}$	$\lt 6$	$\lt 16$	$\lt 30$	$\lt 23$	${19}_{-11}^{+17}$
XILLVER	Norm	⋯	⋯	0.13 ± 0.05	${0.22}_{-0.08}^{+0.09}$	0.13 ± 0.07	0.19 ± 0.09	0.29 ± 0.06	${0.26}_{-0.03}^{+0.06}$
${\chi }^{2}$/DoF			5906/5852
${F}_{3-79}$ c	(10−8 $\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$)		⋯	50.7 ± 0.6	62.4 ± 0.6	34.4 ± 0.5	50.6 ± 0.6	57.0 ± 0.6	60.0 ± 0.4

Notes. For model 6, the high-spin solution is given.

^aFor these models, ${R}_{\mathrm{disk}}$ is calculated self-consistently in the lamp-post geometry from a* and h. Because it is not a free parameter, errors are not estimated. ^bThe RELXILLLP model is only calculated for ${\rm{\Gamma }}\geqslant 1$. ^cAverage flux in the 3–79 keV bandpass.

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The pure lamp-post model (Model 5) fits the data well, with ${\chi }^{2}$/DoF = 6039/5859. The spin is constrained to be very high, ${a}^{* }\gt 0.99$ (see Figure 9), and there is a slight increase in the inclination inferred for the inner disk; while the flux-resolved analysis typically found $i\sim 30$°, here we find $i\sim 40$°. We find that the first flare shows stronger absorption than the subsequent flares, both in terms of the neutral and the ionized absorption components. The former results in the harder spectrum seen from this flare at lower energies, and there is a clear absorption line from ionized iron at ∼6.7 keV produced by the latter, similar to that reported by King et al. (2015), which is not seen in any of the subsequent flares. As with the flux-resolved analysis, we find that during the flares the height inferred for the X-ray source is very close to the black hole, always within ∼10 ${r}_{{\rm{G}}}$.

The model including the disk emission (Model 6) again provides a substantial improvement over the basic lamp-post model, resulting in an outstanding fit to the data (${\chi }^{2}$/DoF =5906/5852, i.e., an improvement of ${\rm{\Delta }}{\chi }^{2}=133$ for seven additional free parameters). We show the data/model ratios for the individual flares with this model in Figure 13. The disk temperature is again similar to that reported by lower energy missions, ${T}_{\mathrm{in}}\sim 0.5\,\mathrm{keV}$, and as before we see that the inclusion of this emission allows the high-energy continuum to take on a harder form, subsequently resulting in lower-energy cutoffs. The neutral absorption inferred also increases to compensate for this additional low-energy continuum emission.

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With regards to the black hole spin, we again find a situation in which two solutions exist that provide statistically equivalent fits (separated by ${\rm{\Delta }}{\chi }^{2}\lt 1;$ see Figure 9): one at high spin (${a}^{* }\gt 0.98$), which is marginally preferred, and another broad, local minimum at a more moderate spin (${a}^{* }\sim 0.5$). In this case, the dual solutions are related to a significant degeneracy between the spin and the disk inclination, resulting from the combination of the additional continuum component, and the lower total S/N utilized in these fits (these data represent ∼80% of the exposure from which the F5 spectrum considered in the previous section is extracted).

For the best-fit, high spin solution we find that the inclination has further increased to $i\sim 52$°, which is similar to the estimates for the orbital inclination of the system (${i}_{\mathrm{orb}}\sim 50^\circ \mbox{--}75^\circ$; e.g., Shahbaz et al. 1994; Khargharia et al. 2010). In contrast, for the more moderate-spin solution, we find that the associated inclination is $\lt 20$°. This would imply an even more extreme disk warp than the flux-resolved analysis, which we deem unphysical. This degeneracy between the spin and the inclination is distinct from the traditional sense of a parameter degeneracy, in which two parameters are correlated such that any value of one can be made acceptable by adjusting the other; rather, there are two solutions that are acceptable in distinct areas of parameter space. We therefore present the results from the high-spin solution in Table 5, though again the parameter constraints for the lower-spin solution are presented in the Appendix, and recalculate the confidence contour for the black hole spin with the inclination constrained to be $i\geqslant 45$° for this model (which we refer to as Model 6i; see Figure 9) in order to ensure a reasonable agreement between the inner and outer disk. This constraint strongly requires a rapidly rotating black hole. We also assess the degree to which the assumed geometry is driving the spin constraint in this scenario by relaxing the requirement that ${R}_{\mathrm{disk}}$ is set self-consistently and allowing this to vary as a free parameter for each of the six flares (but keeping the $i\geqslant 45$° constraint). Although the constraint on the spin is naturally looser, we still find that ${a}^{* }\gt 0.7$ and the constraints on ${R}_{\mathrm{disk}}$ are all consistent with the values presented in Table 5. If we exclude unphysically large disk warps, a rapidly rotating black hole is still required regardless of any additional geometric constraints.

The range of heights inferred for the X-ray source remains similar to the pure lamp-post case. However, one issue of note with this model is that the iron abundance has increased to ${A}_{\mathrm{Fe}}/\mathrm{solar}\sim 5$ (for both solutions), in order to compensate for the harder irradiating continuum and reproduce the observed line flux. All our previous models had typically found ${A}_{\mathrm{Fe}}/\mathrm{solar}\sim 2\mbox{--}3$, which is similar to the iron abundance of ${A}_{\mathrm{Fe}}/\mathrm{solar}\sim 2$ found for the companion star by González-Hernández et al. (2011). While this is certainly not always the case (e.g., El-Batal et al. 2016), similarly high iron abundances have also been reported for a few other Galactic BHBs observed by NuSTAR when using the XILLVER based family of reflection models (e.g., Fuerst et al. 2016; Parker et al. 2016; Walton et al. 2016). The abundance inferred may be dependent on the reflection code utilized; Walton et al. (2016) note that for the Galactic binary Cygnus X-1, the iron abundances obtained with the XILLVER family of reflection models are generally a factor of ∼2 larger than those obtained with the REFLIONX (Ross & Fabian 2005) family of models (see also Miller et al. 2015). Should the iron abundance here be systematically overpredicted by a similar factor, this would bring the abundance derived back down to ${A}_{\mathrm{Fe}}/\mathrm{solar}\sim 2.5$, which would again be similar to that reported by González-Hernández et al. (2011). However, we stress that the key results obtained here do not strongly depend on this issue. If we fix the iron abundance to ${A}_{\mathrm{Fe}}/\mathrm{solar}=2$ in Model 6, the fit worsens slightly (but is still excellent, ${\chi }^{2}$/DoF = 5933/5853). The spin is strongly constrained to be very high (${a}^{* }\gt 0.997$), and the requirement for small source heights further tightens. The most noteable change is that the best-fit inclination further increases to $i\sim 60^\circ$, which is still in good agreement with the range estimated for the orbital plane.

The 3–79 keV fluxes observed from these spectra are also given in Table 5. However, the average count rates during the periods from which these spectra are extracted are obviously significantly lower than the peaks of the flares. Assuming a similar spectral form, scaling these fluxes up to the peak incident count rates observed during these flares—as determined from light curves with 1s time bins—corresponds to peak 3–79 keV fluxes ranging from 0.8–2.0$\,\times \,{10}^{-6}$ $\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$. For a 10 ${M}_{\odot }$ black hole at a distance of 2.4 kpc, these fluxes equate to 3–79 keV luminosities of ∼0.4–1.0 ${L}_{{\rm{E}}}$ (where ${L}_{{\rm{E}}}=1.4\times {10}^{39}$ $\mathrm{erg}\,{{\rm{s}}}^{-1}$ is the Eddington luminosity). The bolometric fluxes observed from the DISKBB component in these spectra, which assumes a thin disk as described by Shakura & Sunyaev (1973), equate to disk luminosities of ∼0.1 ${L}_{{\rm{E}}}$ (assuming the disk is viewed close to $i\sim 60$°). Temperatures of ${T}_{\mathrm{in}}\sim 0.5\,\mathrm{keV}$ are not unreasonable for such luminosities (e.g., Gierliński & Done 2004; Reynolds & Miller 2013). Assuming these fluxes also scale up during the peaks of the flares, the peak disk fluxes would equate to luminosities of ∼0.3–0.5 ${L}_{{\rm{E}}}$.

As the final component of our analysis in this work, we track the evolution of the spectrum across one of the major flares considered in Section 4.3. We focus on Flare 4 (see Figure 2) because it is followed by a relatively long, uninterrupted period of low absorption (as determined by our analysis in Section 4.2.1). As such, we should have a relatively clean view of the flare and its subsequent decline. In order to track the evolution of the spectrum, we split the data into bins with 40 s durations, and extracted spectra from each, again following the method outlined in Section 2. While significant variability obviously occurs on shorter timescales (e.g., P. Gandhi et al. 2017, in preparation), a 40 s duration was found to offer a good balance between retaining good time resolution and the need for reasonable S/N in the individual spectra. We start immediately prior to the flare, and continue until the point that the observed count rate (as averaged over 40 s) starts to rise again after the decline of the flare, resulting in 14 time-resolved spectra (per FPM) in total (hereafter T1–14). These spectra are shown in Figure 14.

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There are too many data sets to undertake a joint analysis of all the data, so we fit the data from each of the time bins individually, using the same lamp-post-based model utilized in our joint analysis of the major flares observed (Section 4.3). Specifically, we use the model that includes the thermal disk emission (Model 6). However, the average good exposure time per FPM is only ∼11 s per bin (being higher for lower flux bins and vice versa, owing to the instrumental deadtime; Harrison et al. 2013), so the S/N per time bin is relatively low. We therefore limit ourselves to considering only a few key free parameters when fitting each of these data sets. Because there is no evidence for ionized iron absorption during this flare (only an upper limit is obtained on the column for this component during flare 4, see Table 5), we exclude the XSTAR absorption component from our analysis in this section. Furthermore, we fix all of the remaining global parameters (black hole spin, disk inclination, iron abundance, and disk temperature) to the best-fit values presented for Model 6 in Table 5. We also fix the ionization parameter to the value obtained in our flux-resolved analysis (see Table 3), based on the average count rate in that time bin, thus ensuring that the ionization increases as the flux increases. Finally, we are not able to simultaneously constrain both the inner radius of the disk and the height of the X-ray source, so we initially fix the latter at the best fit obtained for this flare in our flare-resolved analysis (h = 2.5 ${r}_{{\rm{H}}}$). The free parameters allowed to vary for each of the time-resolved data sets are therefore the (source intrinsic) neutral absorption column, the photon index and high-energy cutoff of the power-law continuum, the inner radius of the disk, and the normalizations of the various emission components. As before, the reflection fraction ${R}_{\mathrm{disk}}$ is calculated self-consistently from the spin, source height, and inner radius of the disk in the lamp-post geometry, which helps us to constrain ${r}_{\mathrm{in}}$ in these fits.

The results for a number of the key parameters, as well as a zoom-in on the light curve of this flare, are shown in Figure 15, which shows a characteristic fast rise, exponential decay profile. Aside from the first time bin, the absorption stays relatively low and stable throughout, as expected. Prior to the flare, the observed spectrum is relatively soft (in comparison to the spectra shown in Figures 7 and 12). Then, as the source flares, the spectrum hardens significantly (reaching ${\rm{\Gamma }}={1.14}_{-0.08}^{+0.04}$), and during the decline it softens again before gradually becoming harder as the source fades. We see a significant difference in the average cutoff energy before and after the flare. Finally, we also see a significant difference in the inner radius of the disk, the key geometry parameter in this analysis, across the evolution of the flare, being close to the ISCO prior to and during the rise of the flare, before moving out to ∼10 ${r}_{\mathrm{ISCO}}$ during the subsequent decline. The data are well modeled, with an average ${\chi }^{2}$/DoF of 1.02 (for an average of 302 DoF). As a sanity check, assuming a disk structure of ${h}_{{\rm{D}}}/{r}_{{\rm{D}}}\sim 0.2$ (where ${h}_{{\rm{D}}}$ is the scale height of the disk at a given radius ${r}_{{\rm{D}}};$ see Section 5.2), a standard viscosity parameter of $\alpha \sim 0.1$, and that the dynamical timescale is set by the Keplerian orbital timescale, we estimate that the viscous timescale for the disk should be ∼0.01 s at a radius of 10 ${r}_{{\rm{G}}}$ (for our best-fit spin, ${r}_{\mathrm{ISCO}}$ ∼ ${r}_{{\rm{G}}}$) for V404 Cyg. Significant evolution of the inner disk is therefore certainly possible over the timescales probed here.

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We also consider two additional iterations of this analysis. First, as with our flux-resolved analysis, we also consider the case in which h varies and ${r}_{\mathrm{in}}$ stays constant, fixing the latter to the ISCO throughout. Equivalent results are obtained, with the only difference being that h increases as the flare evolves instead of ${r}_{\mathrm{in}}$, starting at ∼2 ${r}_{{\rm{H}}}$ before jumping to ∼20 ${r}_{{\rm{H}}}$ in the decline of the flare. The fit statistics are very similar to the scenario in which ${r}_{\mathrm{in}}$ varies and h is constant. At least one of h or ${r}_{\mathrm{in}}$ must therefore increase across the flare; in reality, the two may evolve together. Second, we relax our assumption with regards to the ionization of the disk. While this would be expected to increase with increasing luminosity for a constant density, with the inner regions of the disk evolving its density may also vary. We therefore re-fit the data with the ionization as a further free parameter. While this increases the uncertainties on the other parameters, the same qualitative evolution is still seen, with the main difference being that the point at which ${r}_{\mathrm{in}}$ moves outward occurs later in time. Broadly speaking, the ionization of the disk does still appear to increase with increasing flux.

Finally, we note that V404 Cyg is known to exhibit a strong dust halo, which can produce emission that can potentially mimic an accretion disk component, particularly when the source is faint (Beardmore et al. 2016; Heinz et al. 2016; Motta et al. 2016; Vasilopoulos & Petropoulou 2016). However, during this work, we are largely focusing on periods when the source was very bright. Furthermore, in the analysis presented here, we find that the normalization of the DISKBB component included in the model varies across Flare 4 along with the overall flux. This is too fast for the response from dusty interstellar clouds, and so we cannot be mistaking a dust contribution for the accretion disk in this work.

We suggest that the strong flares observed by NuSTAR mark transient jet ejection events, with the jet becoming the source of the X-rays illuminating the disk (hence our use of the lamp-post geometry throughout our reflection modeling). There are a variety of lines of evidence from the X-ray band alone that support this claim. In other accreting black holes, strong X-ray flares are known to be associated with such events. The BHB XTE J1550-564 is particularly notable in this respect. During its 1998 outburst, an approximately Eddington level flare was observed by RXTE, which triggered the onset of super-luminal radio ejecta, and some time later ejecta were resolved from the central point source by Chandra (Corbel et al. 2002; Kaaret et al. 2003; Tomsick et al. 2003; Steiner & McClintock 2012). Such behavior has also been seen from the BHB H 1743−322 (Corbel et al. 2005; McClintock et al. 2009; Steiner et al. 2012), and the X-ray flux seems to be elevated in the hours prior to many of the radio ejections from the BHB GRS 1915+105 (e.g., Punsly & Rodriguez 2013; Punsly et al. 2016). In addition, some X-ray flares in active galaxies also appear to be associated with jet ejection events (e.g., the recent flares observed from Mrk 335 and M81; Wilkins et al. 2015; King et al. 2016). The fact that the intrinsic spectrum observed immediately prior to flare 4 in our time-resolved analysis is inferred to be quite soft (${\rm{\Gamma }}\sim 2.3$) is of potential importance here, because this is very similar to the "steep power-law state" identified by Remillard & McClintock (2006 also referred to as the "Very High State" in other works). For most LMXBs in outburst, transient jets are launched as they flare up to approximately Eddington during the transition from the hard state to the soft state, which occurs via the steep power-law state (e.g., Corbel et al. 2004; Fender et al. 2004; Steiner et al. 2011; Narayan & McClintock 2012).

In addition to this broader precedent, the nature of the X-ray spectrum observed during these flares also supports a jet scenario. Even after accounting for the reprocessed emission, the primary X-ray continuum is found to be extremely hard, despite the high flux; on average, we see ${\rm{\Gamma }}\sim 1.4$, and from our time-resolved analysis of flare 4, we see that the continuum even reaches ${\rm{\Gamma }}\sim 1.1$. This is not the spectrum that would be expected from an accretion flow radiating at approximately Eddington, which should be dominated by emission from a multi-color blackbody accretion disk, modified slightly by the effects of photon advection (e.g., Middleton et al. 2012, 2013; Straub et al. 2013). In addition, spectra this hard (particularly in the ${\rm{\Gamma }}\sim 1.1$ case) are difficult to produce via Compton scattering of thermal disk photons in a standard accretion disk corona. Strong illumination of the corona by the disk should cool the electrons and produce a softer spectrum. The hard X-ray source would therefore be required to be extremely photon starved (e.g., Fabian et al. 1988; Haardt & Maraschi 1993), in which case only a very small fraction of the disk emission would be scattered into the hard X-ray continuum, or some other process must serve to counteract the cooling of the electrons.

This may point to a magnetic origin for the flares, which would also support a transient jet scenario (e.g., Dexter et al. 2014). Furthermore, if we assume that immediately prior to this flare the high-energy continuum is produced by thermal Comptonization, following a similar calculation to Merloni & Fabian (2001) and taking h to be representative of the size-scale of the corona, we find that there is not enough thermal energy stored in the corona to power the flare by many orders of magnitude, which would also support a magnetic origin. While the spectrum during the peak is also likely too hard for direct synchrotron emission from a jet, which would be expected to give ${\rm{\Gamma }}\sim 1.7$ in the X-ray band, but synchrotron-self-Compton emission (e.g., Markoff et al. 2005) may be able to produce a high-energy continuum this hard.

The increase of roughly an order of magnitude in ${E}_{\mathrm{cut}}$ observed across flare 4, from ∼50 to ∼500 keV, would also appear to indicate that significant energy is being injected into the X-ray emitting electron population during this event, because ${E}_{\mathrm{cut}}$ is a proxy for the electron temperature ${T}_{{\rm{e}}}$. INTEGRAL  may have seen a similar evolution in the cutoff across one of the bright flares observed during its coverage of this outburst (Natalucci et al. 2015). If the height of the source does increase across this flare, the change in gravitational redshift experienced by the primary emission could contribute at least in part to the difference seen in ${E}_{\mathrm{cut}}$, since this correction is not yet incorporated into the RELXILLLP model (Niedźwiecki et al. 2016). However, in the most extreme scenario, where ${r}_{\mathrm{in}}$ remains constant while h varies (evolving from ∼2 to ∼20 ${r}_{{\rm{G}}}$), the movement of the source height should only result in a factor of ∼2 change in the observed cutoff energy (assuming no intrinsic variation). This is clearly insufficient to explain the difference observed, and so we conclude that the intrinsic cutoff energy does indeed increase across the flare.

Assuming the power-law emission is produced by Compton scattering at least during the times both prior to and after the main flare, this implies either an increase in the characteristic electron temperature if the particle distribution remains thermal, or perhaps a transition to a more power-law-like (non-thermal) distribution that extends up to significantly higher energies. With the spectral coverage of NuSTAR stopping at 79 keV, it can be difficult to distinguish between these two scenarios for sufficiently high electron temperatures in the thermal case, because the high-energy cutoff is shifted out of the NuSTAR bandpass, resulting in the observation of a power-law spectrum with little or no curvature. In turn, this results in a run-away effect in terms of the measured ${E}_{\mathrm{cut}}$, owing to the fact that the cutoff power-law model is constantly curving at all energies, while a thermal Comptonization continuum is more power-law-like until it rolls over with a sharper cutoff (see the discussion in Fürst et al. 2016), potentially explaining the fact that ${E}_{\mathrm{cut}}$ is often consistent with the maximum value currently permitted by the RELXILL models after the flare. Nevertheless, the evolution in ${E}_{\mathrm{cut}}$ observed here provides a good match to the jet model described in Markoff et al. (2005), in which electrons are accelerated into a power-law distribution within a region of ∼10–100 ${r}_{{\rm{G}}}$ above the jet's point of origin. Should the particle distribution instead be thermal both before and after the peak of the flare, assuming that the size of the corona increases across the flare (i.e., it expands as either ${r}_{\mathrm{in}}$ or h increase), then the evolution would be similar to that expected for a corona being kept close to its catastrophic pair production limit (see Fabian et al. 2015, and references therein).

Finally, the geometric results from our reflection modeling are also likely consistent with a jet scenario. We see evidence for either the disk truncating or the height of the X-ray source increasing, both on average as the source flux decreases, and also across one of the major flares individually. Although we cannot constrain the evolution of the inner disk radius and the source height simultaneously, as variations in the two produce similar results for the observed reflection spectrum (e.g., Fabian et al. 2014, hence our treatment of these two possibilities in isolation), as noted previously it is quite possible that both of these quantities evolve. Indeed, if we repeat the time-resolved analysis of flare 4 presented in Section 4.4, forcing this to be the case, linking the two with a simple linear relation and assuming that both evolve simultaneously just for illustration ([h/${r}_{{\rm{H}}}$] = 2[${r}_{\mathrm{in}}$/${r}_{\mathrm{ISCO}}$]), we again find the same qualitative evolution seen in Figure 15, and the fits are as good as the scenarios in which only one of ${r}_{\mathrm{in}}$ and h is allowed to vary. In this scenario, the magnitude of the changes in ${r}_{\mathrm{in}}$ and h are both reduced in comparison to the de-coupled scenarios discussed in Section 4.4, with ${r}_{\mathrm{in}}$ evolving from 1–5 ${r}_{\mathrm{ISCO}}$, and h evolving from 2–10 ${r}_{{\rm{H}}}$. We note that, should the ejecta have reached a significant outflow velocity, the reflection fraction would be reduced for a given combination of h and ${r}_{\mathrm{in}}$ (e.g., Beloborodov 1999) resulting in these quantities potentially being overestimated during the times after the flare, but again the same qualitative evolution should be seen. Furthermore, acceleration up to significant outflow velocities may not be expected so close to the black hole (see Section 5.1.2).

In a flare associated with transient jet ejection, obviously if the jet is the source of illumination, then one naturally expects the height of the source to increase across the flare. However, such ejection events may also be associated with an evacuation of the inner disk, as the same instability that results in the ejection also results in catastrophic accretion of the innermost portion of the disk (Szuszkiewicz & Miller 1998; Meier 2001). Chen et al. (1995) suggest that thin disk solutions should become unstable above luminosities of ∼0.3 ${L}_{{\rm{E}}}$, similar to the peak disk fluxes inferred here. Evidence for such behavior might be seen, for example, in GRS 1915+105, where radio ejections are also preceded by dips in X-ray intensity in some of the oscillatory states exhibited by this source, during which the inner radius of the accretion disk is inferred to increase (e.g., Pooley & Fender 1997; Mirabel et al. 1998; Klein-Wolt et al. 2002), though Rodriguez et al. (2008) suggest that the ejections might actually be associated with the post-dip flares observed in those cycles. Similar behavior may also have been seen in the radio galaxies 3C 120 (Marscher et al. 2002; Chatterjee et al. 2009; Lohfink et al. 2013), and 3C 111 (Chatterjee et al. 2011), where radio ejections appear to be preceeded by X-ray dips. Therefore, both an increasing source height and a truncation of the inner accretion disk may be expected for transient ejection events, consistent with the evolution seen in our analysis.

5.1.1. Radio Monitoring

A natural prediction of the jet scenario is that radio emission should be observed. Throughout this recent outburst, V404 Cyg was frequently monitored by the Arcminute Microkelvin Imager–Large Array (hereafter AMI; Zwart et al. 2008), a compact array of eight dishes operating in the 13-18 GHz frequency range. The full AMI campaign on V404 Cyg will be presented in R. Fender et al. (2017, in preparation; see also Mooley et al. 2015); here we focus on the coverage that is simultaneous with our NuSTAR observation. Flagging and calibration of the data were performed with the AMI REDUCE software (Perrott et al. 2013). The calibrated data were then imported into CASA and flux densities of V404 Cyg were extracted by vector averaging over all baselines; the absolute flux calibration uncertainty is ∼5%.

A comparison of the NuSTAR and AMI light curves is shown in Figure 16. Unfortunately, owing to occultation by the Earth, the majority of the flaring period observed by NuSTAR does not have simultaneous AMI coverage which, in combination with the frequent Earth-occultations experienced by NuSTAR, prevents any detailed analysis attempting to search for radio responses to specific X-ray flares. However, there is AMI coverage right at the beginning of this period, and toward the end of the NuSTAR observation. These short periods of overlap do clearly show that radio activity commences as the flaring phase of the NuSTAR observation begins, which then appears to persist throughout. The coincidence of this radio activity further supports our suggestion that the major flares seen by NuSTAR represent jet ejection events.

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5.1.2. Transient Jet Launching

Through our investigation of the inner accretion geometry, we are also able to place constraints on the spin of the black hole in V404 Cyg. Our initial modeling of the flux-resolved spectra provided some indication that the black hole spin is high, owing to the strong disk reflection inferred (${R}_{\mathrm{disk}}\sim 3$). This requires strong gravitational light bending, which in turn requires a high black hole spin, such that the disk can extend very close to the black hole and subtend a large solid angle as seen by the illuminating X-ray source (Miniutti & Fabian 2004; Dauser et al. 2013). Evidence for strong gravitational light bending has previously been observed in a wide variety of active galactic nuclei (e.g., Zoghbi et al. 2008; Fabian et al. 2012; Parker et al. 2014; Reis et al. 2014; Reynolds et al. 2014b; Chiang et al. 2015; Lanzuisi et al. 2016), but also in other Galactic BHBs (e.g., Rossi et al. 2005; Reis et al. 2013). Furthermore, as noted previously, potential evidence for strong reflection has also been seen during flares seen by the INTEGRAL coverage of this outburst from V404 Cyg (Natalucci et al. 2015; Roques et al. 2015).

A high spin is supported by our flux-resolved analysis with a self-consistent lamp-post geometry. There is some complexity in the results obtained for the two scenarios considered with the pure lamp-post reflection model (varying the inner radius of the disk while holding the height of the X-ray source constant, and vice versa; Models 2 and 3, respectively), with similarly good fits obtained with high and more moderate-spin solutions in both cases. However, we obtain a significant improvement in the global fit with the inclusion of a contribution from thermal disk emission at the highest fluxes in addition to the lamp-post component (Model 4); this is our best-fit model for the flux-resolved data. In this case, a high spin is unambiguously preferred: ${a}^{* }\gt 0.82$ (see Figure 9, top panel).

In addition, a high spin is also supported by our flare-resolved analysis, focusing on the peaks of the six most extreme flares observed. While this analysis utilizes much less total exposure than our flux-resolved analysis, it has the advantage of relying on much less averaging of different spectra (see Figure 12). The pure lamp-post model strongly requires a high spin (Model 5), but we again see a significant improvement in the fit with the inclusion of a thermal disk component (Model 6); this is our best-fit model for the flare-resolved data. In this case, we see a strong degeneracy between the black hole spin and the inclination of the inner accretion disk, resulting in high- and moderate-spin solutions again providing similarly good fits. The best-fit inclination, ${i}_{\mathrm{disk}}\sim 52$°, which corresponds to the high-spin solution, is in good agreement with the range inferred for the orbital plane of the binary system, ${i}_{\mathrm{orb}}\sim 50^\circ \mbox{--}75^\circ$ (e.g., Shahbaz et al. 1994; Khargharia et al. 2010). If we require the inclination to be in this range (Model 6i), then the spin is again strongly required to be high: ${a}^{* }\gt 0.98$ (see Figure 9, bottom panel). Taking a more conservative 99% confidence level, the spin constraint expands to ${a}^{* }\gt 0.92$. Given the lower degree of time-averaging of different spectral "states" in the data analyzed¹⁶ and the good agreement with the orbital inclination, we consider this to be the most robust spin constraint derived from any of our models.

The quantitative constraints on the black hole spin discussed here are the statistical parameter constraints obtained through our spectral modeling. There are additional systematic errors associated with the assumptions inherent to the models used here that are likely significant but difficult to robustly quantify. One issue common to any attempt to constrain black hole spin is the assumption that the accretion disk truncates quickly at the ISCO, and that no significant emission should be observed from within this radius. Numerical simulations suggest that, for thin disks, this is a reasonable assumption (e.g., Reynolds & Fabian 2008; Shafee et al. 2008), and that any additional uncertainty should be small (approximately a few percent), particularly for rapidly rotating black holes.

For the particular case of V404 Cyg considered here, given the extreme luminosities reached during the flares it is worth considering whether the assumption of a thin disk is reasonable. Standard accretion theory predicts that as the accretion flow becomes more luminous, its scale height should start to increase as vertical support from radiation pressure becomes more prominent (e.g., Shakura & Sunyaev 1973). Indeed, some thickness to the disk may be required in order for the disk to be able to anchor the magnetic fields required for jet ejections (e.g., Meier 2001; Tchekhovskoy & McKinney 2012). This is potentially important for both the issue of how quickly the disk truncates at the ISCO, as thicker disks are more able to exhibit emission that "spills over" the ISCO slightly (e.g., Reynolds & Fabian 2008), and also for the self-consistent lamp-post reflection models, which calculate the expected reflection contribution assuming a thin disk geometry (Dauser et al. 2013; García et al. 2014).

In Section 4.3, we estimated the peak disk luminosities to be ${L}_{\mathrm{disk}}\sim 0.3\mbox{--}0.5$ ${L}_{{\rm{E}}}$. Typically, the high-energy power-law emission from Galactic BHBs is assumed to arise from Compton up-scattering of disk photons, and so the intrinsic disk luminosities would have to be further corrected for the flux lost into the power-law component (e.g., Steiner et al. 2009). This may well be the case at times outside of the flare peaks. However, as noted above, during the flares the power-law emission is likely too hard to originate via Compton scattering of disk photons. If we are correct about the magnetic/jet ejection nature of these flares, then we should be able to take the peak disk fluxes at roughly face value. Therefore, we take ${L}_{\mathrm{disk}}$/${L}_{{\rm{E}}}$ ≲ 0.5. For the calculations of the expected disk structure presented in McClintock et al. (2006), this would correspond to a maximum scale height of ${h}_{{\rm{D}}}/{r}_{{\rm{D}}}\lesssim 0.2$, or equivalently a half-opening angle for the inner disk of ≲10°. This is unlikely to be large enough that our assumption of a thin disk would lead to large errors. Even if we are incorrect and the high-energy continuum does arise through up-scattering of disk photons, since photon number (rather than flux) is conserved, the peak intrinsic disk fluxes would only have been ∼20% larger, even accounting for the hard X-ray flux bent away from the observer in our strong light-bending scenario. Indeed, Straub et al. (2011) find that the X-ray spectrum of LMC X-3 is still fairly well described by a thin disk model up to luminosities of ${L}_{\mathrm{disk}}\sim 0.6$ ${L}_{{\rm{E}}}$. Furthermore, while the flare peaks are extreme, the majority of the good exposure obtained naturally covers lower fluxes, during which the thin disk approximation should be even more reliable in terms of the reflection modeling. We therefore expect that, while there may be some mild deviation from the thin disk approximation during the peaks of the flares that could serve to relax the constraints on the spin slightly, this is unlikely to result in major errors, and our conclusion that V404 Cyg hosts a rapidly rotating black hole is likely robust to such issues.