Testing General Relativity with the Stellar-mass Black Hole in LMC X-1 Using the Continuum-fitting Method

Einstein's theory of general relativity was proposed over a century ago and has successfully passed a large number of observational tests, mainly in the weak field regime (Will 2014). Thanks to new observational facilities, nowadays tests of general relativity in the strong field regime are of broad interest and undergoing intense study. Astrophysical black holes are ideal laboratories for testing strong gravity. The spacetime geometry around these objects is thought to be well approximated by the Kerr solution (Kerr 1963; Carter 1971), as deviations induced by a nonvanishing electric charge or by the presence of accretion disks or nearby stars are usually very small (Bambi et al. 2009, 2014; Bambi 2018). On the other hand, macroscopic deviations from the Kerr geometry are predicted in models with exotic fields (e.g., Herdeiro & Radu 2014), in a number of modified theories of gravity (e.g., Ayzenberg & Yunes 2014), or as a result of large quantum gravity effects near the black hole event horizon (e.g., Dvali & Gomez 2013; Giddings & Psaltis 2018; Carballo-Rubio et al. 2020).

Black holes can be tested either with electromagnetic or gravitational-wave techniques, and the two approaches are complementary. Electromagnetic tests (Johannsen 2016; Bambi 2017; Krawczynski 2018), strictly speaking, are more suitable to probe the interplay between the gravity and the matter sectors (geodesic motion of particles and non-gravitational physics in presence of gravity). Gravitational-wave tests (Gair et al. 2013; Yunes & Siemens 2013; Yagi & Stein 2016) can probe Einstein's theory of general relativity in the dynamical regime. It may be possible that new physics only shows up in one of the two spectra (i.e., either the electromagnetic or the gravitational-wave spectrum) and not in the other one (Barausse & Sotiriou 2008; Bambi 2014; Will 2014; Li et al. 2020). When the two approaches can be compared to test the Kerr metric around black holes, they seem to provide similar constraints (Cardenas-Avendano et al. 2020).

The two leading electromagnetic techniques for measuring black-hole spins under the assumption of the Kerr metric are X-ray reflection spectroscopy (Brenneman & Reynolds 2006; Reynolds 2014), aka analysis of the iron line and the Compton hump, and the continuum-fitting method (Zhang et al. 1997; McClintock et al. 2014), or analysis of the thermal spectrum. Both techniques can be naturally extended to test the Kerr black-hole hypothesis (Schee & Stuchlík 2009; Bambi & Barausse 2011; Bambi 2012, 2013; Bambi et al. 2017). In the past few years, there has been significant work to test black holes using X-ray reflection spectroscopy and there are now a number of published results on observational constraints using XMM-Newton, Suzaku, and NuSTAR data (Cao et al. 2018; Abdikamalov et al. 2019; Tripathi et al. 2019a, 2019b; Zhang et al. 2019). In this paper, we start testing the Kerr metric using the continuum-fitting method. We employ our new model nkbb (Non-Kerr multi-temperature BlackBody spectrum; Zhou et al. 2019) and we analyze 17 Rossi X-ray Timing Explorer (RXTE) observations of the stellar-mass black hole in LMC X-1 to constrain possible deviations from the Kerr solution.

The continuum-fitting method is the analysis of the thermal spectrum of a geometrically thin and optically thick accretion disk around a black hole (Zhang et al. 1997). Geometrically thin accretion disks are normally described by the Novikov–Thorne model (Novikov & Thorne 1973; Page & Thorne 1974), which is the relativistic generalization of the Shakura–Sunyaev model (Shakura & Sunyaev 1973). The disk is assumed to be on the equatorial plane, perpendicular to the black-hole spin, and the inner edge of the disk is set at the innermost stable circular orbit (ISCO). From the conservation of mass, energy, and angular momentum, we infer the time-averaged radial structure of the disk. In the Kerr spacetime, the thermal spectrum of the disk turns out to depend only on five parameters: black-hole distance D, inclination angle of the disk with respect to the line of sight of the observer i, black-hole mass M, black-hole spin parameter a_*, and mass accretion rate $\dot{M}$. Since the spectrum is degenerate with respect to these five parameters, spin measurements require independent estimates of D, i, and M, and the fit can provide the spin parameter a_* and the mass accretion rate $\dot{M}$ (Zhang et al. 1997; McClintock et al. 2014).

nkbb follows the so-called bottom-up approach (Zhou et al. 2019). The thermal spectrum of the accretion disk is calculated in a parametric black-hole spacetime, where possible deviations from the Kerr geometry are quantified by introducing ad hoc deformation parameters. The spirit behind this approach is to perform a null-experiment and check whether astronomical data require that all deformation parameters vanish; that is, the Kerr metric is sufficient to model the data well. If a possible deviation from the Kerr solution is found, this approach is not suitable to measure the deviation from the Kerr geometry and it is necessary to use a different method. Among the many proposed parametric black-hole spacetimes in literature, here we use the Johannsen metric (Johannsen 2013).

In its simplest form, which is the version employed in our work, the Johannsen metric has only one deformation parameter, named α₁₃ (see Johannsen 2013 for the origin of this parameter). In Boyer–Lindquist coordinates, the line element reads (we use units in which G_N = c = 1)

$$\begin{eqnarray}\begin{array}{rcl}{{ds}}^{2} & = & -\displaystyle \frac{{\rm{\Sigma }}\left({\rm{\Sigma }}-2{Mr}\right)}{{A}^{2}}\,{{dt}}^{2}+\displaystyle \frac{{\rm{\Sigma }}}{{\rm{\Delta }}}\,{{dr}}^{2}+{\rm{\Sigma }}\,d{\theta }^{2}\\ & & +\,\displaystyle \frac{\left[{\left({r}^{2}+{a}^{2}\right)}^{2}{\left(1+\delta \right)}^{2}-{a}^{2}{\rm{\Delta }}{\sin }^{2}\theta \right]{\rm{\Sigma }}{\sin }^{2}\theta }{{A}^{2}}\,d{\phi }^{2}\\ & & -\,\displaystyle \frac{2a\left[2{Mr}+\delta \left({r}^{2}+{a}^{2}\right)\right]{\rm{\Sigma }}{\sin }^{2}\theta }{{A}^{2}}\,{dt}\,d\phi ,\end{array}\end{eqnarray} \tag{ 1 }$$

where $a={a}_{* }M$, ${\rm{\Sigma }}={r}^{2}+{a}^{2}{\cos }^{2}\theta$, ${\rm{\Delta }}={r}^{2}-2{Mr}+{a}^{2}$, and

$$\begin{eqnarray}&&A={\rm{\Sigma }}+\delta \left({r}^{2}+{a}^{2}\right),\quad \delta ={\alpha }_{13}{\left(\displaystyle \frac{M}{r}\right)}^{3}.\end{eqnarray} \tag{ 2 }$$

For α₁₃ = 0, we recover the Kerr metric. In order to have a regular exterior region, we have to impose $| {a}_{* }| \leqslant 1$ (as in the Kerr metric, for $| {a}_{* }| \gt 1$ there is no event horizon and the central singularity is naked) and the following restriction on α₁₃ (Tripathi et al. 2018)

$$\begin{eqnarray}&&{\alpha }_{13}\gt -\displaystyle \frac{1}{2}{\left(1+\sqrt{1-{a}_{* }^{2}}\right)}^{4}.\end{eqnarray} \tag{ 3 }$$

Note that α₁₃ enters the metric coefficients g_tt, g_tϕ, and g_ϕϕ. It thus affects the structure of the accretion disk, modifying the Keplerian gas motion and moving the ISCO radius. Qualitatively speaking, α₁₃ > 0 (< 0) increases (decreases) the strength of the gravitational force, so it moves the ISCO radius to higher (lower) values and this explains the strong correlation with the spin parameter in the plots that will be presented in the next sections.

LMC X-1 was discovered in 1969 as the first extragalactic X-ray binary (Mark et al. 1969; Price et al. 1971). The system consists of a stellar-mass black hole and an O-giant companion star. The distance of the source, the black-hole mass, and the inclination angle of the orbit, which are three key-quantities in the continuum-fitting method, have been estimated to be D = 48.10 ± 2.22 kpc, M = 10.91 ± 1.54 M_⊙, and i = 3638 ± 202, respectively (Gou et al. 2009; Orosz et al. 2009). LMC X-1 is characterized by a quite stable bolometric luminosity, which is about 16% of its Eddington luminosity L_Edd (Gou et al. 2009) and thus nicely meets the standard criterion required to use the continuum-fitting method: sources with an accretion luminosity in the range 5%–30% L_Edd (McClintock et al. 2014).

The measurement of the spin parameter of the black hole in LMC X-1 using the continuum-fitting method was presented in Gou et al. (2009), and here we follow that study. There are 55 pointed observations of LMC X-1 with the Proportional Counter Array (PCA) on board RXTE (Swank 1999). To use the continuum-fitting method, it is desirable to choose thermal dominant spectral data, which are defined by three conditions (Remillard & McClintock 2006): (i) the flux of the thermal component accounts for more than 75% of the total 2–20 keV unabsorbed flux, (ii) the root mean square (rms) variability in the power density spectrum in the 0.1–10 Hz range is lower than 0.075, and (iii) quasi-periodic oscillations are absent or very weak. Imposing these three conditions, we only have three observations, which are named "gold spectra" in Gou et al. (2009) as they are supposed to be more suitable for the continuum-fitting method. Relaxing condition (i), we have 14 more observations, which are named "silver spectra" in Gou et al. (2009).⁶ Additionally, we require that the accretion luminosity is in the range 5%–30% in order to assure that the accretion disk is geometrically thin and the inner edge is at the ISCO (McClintock et al. 2014), but this is always satisfied in the RXTE observations of LMC X-1.

In our study, we analyzed 17 observations of the PCA. PCA was on board the RXTE, which was launched in 1995 and decommissioned in 2012. PCA was designed to study far-away faint sources in the 2–60 keV energy range and consisted of five proportional counter units (PCUs), which comprised of xenon layers to detect photons. We used the Heasoft version 6.25 to reduce the data and unprocessed data files were downloaded from the HEASARC website.

We used the pulse-height spectra of only PCU-2 because it is the best calibrated PCU and is the most operational one. "Standard 2" mode data were used for reduction. To derive the spectra, data from all the xenon layers were combined for PCU-2. Background spectra were obtained by the latest "faint source" background model provided by the RXTE team and using the ftool pcabackest. The final spectra were then obtained by subtracting the background spectra from the total spectra. The response files were constructed and combined for each layer using the ftool pcarsp. The data were corrected for calibration using the python script pcacorr (García et al. 2014). Finally, we added a systematic error of 0.1% to all the PCA energy channels.

We only used data in the energy range 3–20 keV. Below 3 keV, calibration effects become dominant (see, e.g., Jahoda et al. 2006) and, in particular, the pcacorr correction is not valid in this regime. Above 20 keV, the background becomes dominant and the calibration is uncertain.

We fit each of the 17 observations with the XSPEC model (Arnaud 1996) TBabs×(simpl × nkbb).

TBabs describes the Galactic absorption (Wilms et al. 2000). We use the abundances of Wilms et al. (2000) and we freeze the hydrogen column density to N_H = 4.6 × 10²¹ cm⁻² (Orosz et al. 2009); however, its exact value does not matter considering it is low and we are analyzing RXTE data that do not cover the low energies that are especially sensitive to low column densities. nkbb describes the thermal spectrum of the accretion disk (Zhou et al. 2019). The distance of the source D, the black-hole mass M, and the inclination angle of the orbit i are frozen to 48.10 kpc, 10.91 M_⊙, and 3638, respectively; at this stage, we ignore their uncertainties. The hardening factor is frozen to 1.55 for all observations, which is the value found in Gou et al. (2009). We checked that its impact is weak and we get very similar results even if we use 1.45 or 1.65. The hardening factor varies with luminosity, but LMC X-1 varies modestly around 16% of its Eddington limit,⁷ which justifies the use of a fixed value. The spin parameter a_* and the mass accretion rate are always free parameters to be determined by the fit. The deformation parameter α₁₃ is first set to zero (Kerr metric) in order to check if we can recover the results of Gou et al. (2009), and then it is left free in order to measure possible deviations from the Kerr spacetime. simpl converts a fraction f_SC of thermal photons into a power-law-like spectrum with photon index Γ to describe the radiation from the corona (Steiner et al. 2009), providing a superior description of the Comptonization at low energies as compared to a power law. Since the data do not permit us to determine Γ, we freeze it to 2.5, as in Gou et al. (2009). Employing a different value for Γ has a marginal impact on the estimate of the other parameters (Gou et al. 2009). In the end, the model has three free parameters (a_*, $\dot{M}$, and f_SC) when we assume the Kerr metric and four (even α₁₃) otherwise.

The best-fit values of our 17 observations are reported in Table 1 for α₁₃ = 0 (left column) and for α₁₃ free (right column). The constraints on the spin parameter a_* and the deformation parameter α₁₃ for every observation are shown in Figure 1. The analysis is done with the standard XSPEC routines and the plots in Figure 1 are calculated with the steppar command in XSPEC.

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Table 1.  Best-fit Values for Observations 1−17 assuming the Kerr Spacetime (α₁₃ = 0) and without Such an Assumption (α₁₃ Free).

No.	UT	α13 = 0				α13 Free
		a*	$\dot{M}$	fSC	χ2/dof	a*	α13	$\dot{M}$	fSC	χ2/dof
1	1996 Jun 9	${0.936}_{-0.015}^{+0.014}$	${1.39}_{-0.08}^{+0.09}$	${0.073}_{-0.005}^{+0.005}$	38.08/42	${0.992}_{-0.30}^{+({\rm{P}})}$	${0.29}_{-2.2}^{+0.05}$	${1.34}_{-0.05}^{+0.15}$	${0.073}_{-0.005}^{+0.005}$	37.99/41
2	1996 Aug 1	${0.881}_{-0.018}^{+0.017}$	${1.86}_{-0.10}^{+0.10}$	${0.066}_{-0.004}^{+0.004}$	23.13/42	${0.998}_{-0.20}$	${0.59}_{-({\rm{P}})}^{+0.10}$	${1.91}_{-0.23}^{+0.03}$	${0.066}_{-0.004}^{+0.004}$	23.08/41
3	1997 Mar 9	${0.943}_{-0.010}^{+0.009}$	${1.47}_{-0.07}^{+0.07}$	${0.050}_{-0.004}^{+0.004}$	45.15/42	${0.998}_{-0.27}$	${0.29}_{-1.7}^{+0.01}$	${1.44}_{-0.10}^{+0.07}$	${0.050}_{-0.004}^{+0.004}$	45.02/41
4	1997 Mar 21	${0.948}_{-0.008}^{+0.007}$	${1.48}_{-0.05}^{+0.05}$	${0.046}_{-0.003}^{+0.003}$	34.48/42	${0.998}_{-0.28}$	${0.27}_{-2.0}^{+0.04}$	${1.45}_{-0.02}^{+0.22}$	${0.046}_{-0.003}^{+0.003}$	34.24/41
5	1997 Apr 16	${0.921}_{-0.010}^{+0.009}$	${1.64}_{-0.06}^{+0.06}$	${0.056}_{-0.003}^{+0.003}$	45.47/42	${0.998}_{-0.24}$	${0.39}_{-1.6}^{+0.01}$	${1.68}_{-0.16}^{+0.23}$	${0.056}_{-0.003}^{+0.003}$	45.07/41
6	1997 May 7	${0.938}_{-0.008}^{+0.008}$	${1.55}_{-0.05}^{+0.06}$	${0.053}_{-0.003}^{+0.003}$	37.71/42	${0.993}_{-0.48}^{+({\rm{P}})}$	${0.29}_{-2.2}^{+0.05}$	${1.50}_{-0.04}^{+0.12}$	${0.053}_{-0.003}^{+0.003}$	37.51/41
7	1997 May 28	${0.964}_{-0.015}^{+0.012}$	${1.31}_{-0.10}^{+0.11}$	${0.058}_{-0.007}^{+0.007}$	29.01/42	${0.995}_{-0.34}^{+({\rm{P}})}$	${0.2}_{-3.1}^{+0.2}$	${1.28}_{-0.18}^{+0.16}$	${0.058}_{-0.009}^{+0.007}$	28.92/41
8	1997 May 29	${0.938}_{-0.018}^{+0.016}$	${1.42}_{-0.10}^{+0.11}$	${0.049}_{-0.006}^{+0.006}$	29.83/42	${0.996}_{-0.40}^{+({\rm{P}})}$	${0.31}_{-4.4}^{+0.01}$	${1.38}_{-0.05}^{+0.05}$	${0.049}_{-0.005}^{+0.006}$	29.76/41
9	1997 Jul 9	${0.917}_{-0.019}^{+0.017}$	${1.57}_{-0.10}^{+0.11}$	${0.046}_{-0.005}^{+0.005}$	28.33/42	${0.274}_{-0.075}^{+0.003}$	$-{5}^{+0.05}$	${1.014}_{-0.204}^{+0.005}$	${0.049}_{-0.005}^{+0.003}$	27.71/41
10	1997 Aug 20	${0.945}_{-0.008}^{+0.007}$	${1.57}_{-0.06}^{+0.06}$	${0.048}_{-0.003}^{+0.003}$	34.47/42	${0.998}_{-0.34}$	${0.29}_{-2.9}^{+0.05}$	${1.53}_{-0.03}^{+0.04}$	${0.048}_{-0.003}^{+0.003}$	34.36/41
11a	1997 Sep 12	${0.906}_{-0.011}^{+0.010}$	${1.76}_{-0.06}^{+0.06}$	${0.048}_{-0.003}^{+0.003}$	31.29/42	${0.2234}_{-0.0014}^{+0.0424}$	$-{5}^{+0.05}$	${1.189}_{-0.085}^{+0.003}$	${0.051}_{-0.003}^{+0.002}$	29.62/41
12	1997 Sep 19	${0.964}_{-0.010}^{+0.008}$	${1.31}_{-0.06}^{+0.07}$	${0.063}_{-0.004}^{+0.004}$	28.92/42	${0.994}_{-0.19}^{+({\rm{P}})}$	${0.18}_{-0.72}^{+0.02}$	${1.27}_{-0.13}^{+0.10}$	${0.063}_{-0.004}^{+0.004}$	28.77/41
13	1997 Dec 12	${0.925}_{-0.011}^{+0.010}$	${1.65}_{-0.06}^{+0.07}$	${0.042}_{-0.003}^{+0.003}$	26.01/42	${0.93}_{-0.82}^{+({\rm{P}})}$	${0.0}_{-({\rm{P}})}^{+0.3}$	${1.65}_{-0.10}^{+0.08}$	${0.042}_{-0.003}^{+0.003}$	26.01/41
14	1998 Mar 12	${0.966}_{-0.007}^{+0.007}$	${1.40}_{-0.04}^{+0.06}$	${0.054}_{-0.004}^{+0.004}$	30.37/42	${0.998}_{-0.20}$	${0.19}_{-0.44}^{+0.21}$	${1.39}_{-0.17}^{+0.05}$	${0.054}_{-0.004}^{+0.004}$	30.36/41
15	1998 May 6	${0.948}_{-0.009}^{+0.008}$	${1.43}_{-0.06}^{+0.06}$	${0.048}_{-0.003}^{+0.003}$	28.01/42	${0.998}_{-0.43}$	${0.27}_{-2.5}^{+0.05}$	${1.40}_{-0.04}^{+0.12}$	${0.048}_{-0.003}^{+0.003}$	27.90/41
16a	1998 Jul 20	${0.936}_{-0.009}^{+0.009}$	${1.52}_{-0.05}^{+0.06}$	${0.042}_{-0.003}^{+0.003}$	21.07/42	${0.55}_{-0.20}^{+0.28}$	$-{2.9}_{-({\rm{P}})}^{+3.2}$	${1.19}_{-0.36}^{+0.15}$	${0.044}_{-0.003}^{+0.003}$	20.50/41
17a	2004 Jan 7	${0.935}_{-0.022}^{+0.018}$	${1.45}_{-0.13}^{+0.13}$	${0.046}_{-0.005}^{+0.005}$	19.02/42	${0.54}_{-0.23}^{+0.26}$	$-{3.0}_{-({\rm{P}})}^{+3.3}$	${1.1}_{-0.4}^{+0.3}$	${0.048}_{-0.005}^{+0.005}$	18.96/41

Notes. The reported uncertainties correspond to the 90% confidence level for one relevant parameter (Δχ² = 2.71). Note that the maximum value of a_* allowed by the model is 0.998, while the minimum value of α₁₃ is −5. In several cases, the best fit is stuck at a_* = 0.998 or at α₁₃ = − 5 and therefore we do not report the upper/lower uncertainty. (P) means that the 90% confidence level reaches the maximum/minimum value of the parameter.

^aThis marks the three "golden spectra," which meet all the conditions for thermal dominant spectral data.

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Roughly speaking, the continuum-fitting method measures the position of the inner edge of the disk, which is set at the ISCO in the Novikov–Thorne model and only depends on the spin parameter in the Kerr spacetime. This allows us to measure the black-hole spin when we assume the Kerr metric. Relaxing the Kerr hypothesis, the situation is more complicated. As we can see from Figure 1, there is a strong correlation between a_* and α₁₃. Now the ISCO is set by a_* and α₁₃ (see Kong et al. 2014 for more details), so it is difficult to constrain the values of a_* and α₁₃.

We combine all observations together following the standard approach of averaging the χ² at each grid-point in the $({a}_{* },{\alpha }_{13})$ plane. The measurement of a_* and α₁₃ is (90% confidence level for one relevant parameter)

$$\begin{eqnarray}&&{a}_{* }={0.998}_{-0.44},\quad {\alpha }_{13}={0.32}_{-3.1}^{+0.04}.\end{eqnarray} \tag{ 4 }$$

Figure 2 shows the constraints on the spin parameter a_* and the deformation parameter α₁₃ at the 68%, 90%, and 99% confidence level for two relevant parameters (Δχ² = 2.30, 4.61, and 9.21, respectively).

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In the Kerr spacetime, the uncertainty on the final measurement of the black-hole spin parameter is dominated by the observational uncertainties on D, M, and i. In order to evaluate the impact of the observational uncertainties of these three parameters on our measurements of a_* and α₁₃, we proceed as in Gou et al. (2009). We generate 2000 parameter sets for D, M, and i with Monte Carlo simulations assuming that the uncertainties in these parameters are normally and independently distributed. For every set, we find the best fit of the combined observations. We thus calculate the median values of the best-fit values of the spin parameter a_* and of the deformation parameter α₁₃. Our result is shown in Figure 3. Contrary to the Kerr case, in which the uncertainties of D, M, and i provide the main contribution on the final uncertainty on a_*, in the non-Kerr case their impact is small and eventually subdominant with respect to the statistical uncertainty shown in Equation (4). This is because the degeneracy between a_* and α₁₃ plays an important role in the final measurement. Neglecting the subdominant contributions from the uncertainties of D, M, and i, our final measurement of a_* and α₁₃ is thus given in Equation (4).

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In this paper, we have presented the first attempt to use the continuum-fitting method with real data and test the Kerr nature of an astrophysical black hole. We have employed the new model nkbb to fit 17 RXTE observations of the X-ray binary LMC X-1 and constrain the deformation parameter α₁₃ of the Johannsen metric. In the past couple of years, similar tests of the Kerr metric using XMM-Newton, Suzaku, and NuSTAR data were done by analyzing the reflection spectrum, never with the thermal spectrum.

We find that it is quite challenging to constrain a_* and α₁₃ at the same time. This point is clear if we compare the spin measurement reported in Gou et al. (2009) assuming the Kerr metric, ${a}_{* }={0.92}_{-0.07}^{+0.05}$, and our spin measurement with free α₁₃, a_* > 0.56. When α₁₃ is free, the constraint on a_* is much weaker. Moreover, in the traditional continuum-fitting method for the Kerr metric, the main sources of uncertainty in the final spin measurement are the uncertainties on the black-hole mass, distance, and inclination angle of the disk (Kulkarni et al. 2011; McClintock et al. 2014), three quantities that must be determined from independent measurements, often with optical observations. When we test the Kerr metric, the uncertainties on these three parameters seem to be subdominant. Even assuming they are known without uncertainty, the intrinsic degeneracy between the black-hole spin and the deformation parameter does not permit precise measurements of a_* and α₁₃.

The problem of testing the Kerr metric with electromagnetic data without an independent measurement of the black-hole spin parameter is well known in literature (see, e.g., Krawczynski 2012; Johannsen & Psaltis 2013; Kong et al. 2014; Hoormann et al. 2016). We usually meet this issue when the deformation parameter affects the metric coefficients g_tt, g_tϕ, and/or g_ϕϕ, which can have a strong impact on the location of the ISCO radius and, in turn, on the shape of the spectrum. This is the case of the deformation parameter α₁₃ of the Johannsen metric and the simple shape of the thermal spectrum of the disk, which is simply a multi-temperature blackbody spectrum, cannot break the parameter degeneracy. Note that the correlation between the measurements of a_* and α₁₃ is usually quite similar among different electromagnetic techniques, since all of them are mainly sensitive to the exact location of the inner edge of the disk, while other relativistic effects more specific of the particular spectral component have often a weaker impact.

A comparison between the constraints in Figure 2 and those found from the analysis of the reflection spectrum of the disk of other sources in previous studies is not straightforward because these measurements are quite sensitive to the specific source and the quality of the data, so it may be dangerous to generalize the results found from our analysis of 17 RXTE observations of LMC X-1. In general, a correlation between the estimates of a_* and α₁₃ is common even when we analyze the reflection spectrum. However, such a degeneracy can be broken when the inner edge of the disk is very close to the compact object (Tripathi et al. 2018, 2019b; Zhang et al. 2019). For example, Tripathi et al. (2019a) analyzed simultaneous XMM-Newton and NuSTAR observations of MCG–6–30–15 obtaining ${a}_{* }={0.976}_{-0.013}^{+0.007}$ and ${\alpha }_{13}={0.00}_{-0.20}^{+0.07}$ (90% confidence level for one relevant parameter). Even if there is a correlation between these measurements of a_* and α₁₃, see Figure 6 in Tripathi et al. (2019a), we can get quite stringent constraints on both parameters. This is not the case with the analysis of the thermal component presented in this paper. When we assume the Kerr metric, observations 7, 12, and 14 of LMC X-1 give quite high spin values (see left column of Table 1), comparable to the XMM-Newton and NuSTAR observations of MCG–6–30–15. However, when we leave α₁₃ free it is not easy to constrain a_* and α₁₃ at the same time. While the limited energy resolution of RXTE with respect to XMM-Newton may have some effect, the key-point is in the difference between the reflection spectrum and the thermal one. The former is characterized by many features, notably, but not only, the iron Kα complex around 6–7 keV. Such features help to break the parameter degeneracy, even if the reflection spectrum has several parameters to fit. The thermal spectrum, on the contrary, has quite a simple shape and there is an intrinsic degeneracy among the model parameters. This is true even when we assume the Kerr metric: it is possible to measure the black-hole spin only when we have independent estimates of the black-hole mass, distance, and inclination angle of the disk. If we want to use the continuum-fitting method to test the Kerr metric and we add a deformation parameter, the problem of degeneracy between a_* and α₁₃ should not be a surprise.

While the analysis of the reflection spectrum of an accreting black hole is likely a more powerful technique for getting stringent constraints on possible deviations from the Kerr geometry, we can expect that the combination of the two methods to test the same source can provide more reliable and stronger constraints. In general, the possibility of a combined analysis is not automatic, because the two methods require, respectively, a strong reflection component and a strong thermal component. Some sources do not have data suitable for both techniques. Moreover, the continuum-fitting method requires independent estimates of the black-hole mass, distance, and inclination angle of the disk, while most sources do not have reliable measurements of these three quantities. We plan to present the combined constraints from the analysis of the reflection and thermal components of the same source in a forthcoming paper.

This work was supported by the Innovation Program of the Shanghai Municipal Education Commission, grant No. 2019-01-07-00-07-E00035, and the National Natural Science Foundation of China (NSFC), grant No. 11973019. V.G. is supported through the Margarete von Wrangell fellowship by the ESF and the Ministry of Science, Research and the Arts Baden-Württemberg. A.T., C.B., V.G., H.L., and J.F.S. are members of the International Team 458 at the International Space Science Institute (ISSI), Bern, Switzerland, and acknowledge support from ISSI during the meetings in Bern.