Testing the no-hair property of black holes with x-ray observations of accretion disks

Accretion disks around black holes radiate a significant fraction of the rest mass of the accreting material in the form of thermal radiation from within a few gravitational radii of the black hole ($r\lesssim 20GM/c^2$). In addition, the accreting matter may also be illuminated by hard x rays from the surrounding plasma which adds fluorescent transition lines to the emission. This radiation is emitted by matter moving along geodesics in the metric; therefore the strong Doppler and gravitational redshifts observed in the emission encode information about the strong gravitational field around the black hole. In this paper the possibility of using the x-ray emission as a strong field test of general relativity is explored by calculating the spectra for both the transition line and thermal emission from a thin accretion disk in a series of parametrically deformed Kerr metrics. In addition the possibility of constraining a number of known black hole spacetimes in alternative theories of gravity is also considered.

Figure 1

The dependence of the Kerr iron line profile on various parameters. All spectra were normalized to the same peak flux. Unless otherwise stated in the figure all parameters were set to their fiducial values; $a=0.5$, $\iota =\pi /4$, $q=3$ and $r_{\mathrm{out}}=30$.

Figure 2

The flux as a function of radius, for various values of the spin parameter. Also shown is the power law, $(r/M{)}^{-3}$. All other parameters were set to their fiducial values, $\iota =\pi /4$, $M=M_{\odot }$, ${\dot{M}}_0={10}^{-12}{M}_{\odot }/\mathrm{yr}$ and $r_{\mathrm{out}}=30$.

Figure 3

The dependence of the Kerr black-body spectrum on various parameters. Unless indicated otherwise all parameters were set to their fiducial values, $a=0.5$, $\iota =\pi /4$, $M=M_{\odot }$, ${\dot{M}}_0={10}^{-12}{M}_{\odot }/\mathrm{yr}$ and $r_{\mathrm{out}}=30$.

Figure 4

A set of images showing the resolved appearance of an accretion disk around a Kerr black hole. The color indicates the redshift of the light from that portion of the disk; the top row shows the disk with no light bending and the bottom row shows the disk including light bending. The inclination angle is varied between the plot; from left to right it takes the values $\iota =\{\pi /10,2\pi /10,3\pi /10,4\pi /10\}$. The spin parameter and outer radius of the disk were fixed to $a=0.5$ and $r_{\text{out}}=10$.

Figure 5

A comparison of the likelihood surface predicted by the Fisher matrix [indicated by $1\sigma$ (black) and $2\sigma$ (red) contours] and the true posterior [shown as a density plot using samples produced from a Markov chain Monte Carlo (MCMC) algorithm]. All parameters were set to their fiducial values; $a=0.5$, $\iota =\pi /4$, $q=3$ and $r_{\mathrm{out}}=30$.

Figure 6

Each panel of this figure gives the error in the column parameter versus the value of the row parameter for fixed fiducial values of the other parameters; $a=1$, $\iota =\pi /4$, $q=3$ and $r_{\mathrm{out}}=30$.

Figure 7

A series of iron line spectra for accretion disks around KS black holes varying the value of $\omega$. All other parameters were set to fiducial values, $a=0.5$, $\iota =\pi /4$, $q=3$ and $r_{\mathrm{out}}=30$.

Figure 8

The $1\sigma$ and $2\sigma$ contours from a Fisher matrix analysis on the KS metric. It can be seen that introducing the extra degree of freedom in the KS deformation parameter has introduced a clear degeneracy with the spin parameter. From the bottom right-hand plot it can be seen that the data are almost equally consistent with any value of $\omega$ in the range $(1/2,\infty )$, so no bound may be placed in the deformation. All parameters were set to their fiducial values; $a=0.5$, $\iota =\pi /4$, $q=3$, $r_{\mathrm{out}}=30$ and $Y\equiv 1/\omega =0$.

Figure 9

The left-hand panel shows a pair of iron line spectra for accretion disks around a Kerr black hole and a quadratic in a spin CS black hole (CS2 metric) with a large deformation, $\zeta =1$. Also shown in the right-hand panel is the residual plot. All parameters were set to their fiducial values; $a=0.5$, $\iota =\pi /4$, $q=3$ and $r_{\mathrm{out}}=30$.

Figure 10

A series of iron line spectra for accretion disks around B2 bumpy black holes with varying values of the bump parameters. All other parameters were set to fiducial values; $a=0.5$, $\iota =\pi /4$, $q=3$, $r_{\mathrm{out}}=30$ and ${\gamma }_{i,j}=0$ unless otherwise indicated.

Figure 11

The bounds it is possible to place on the different ${\mathcal{B}}_N$ bump parameters for different values of the spin parameter, $a$. The tightest bounds can typically be placed for low values of $N$ (i.e. ${\mathcal{B}}_2$) and for the most highly spinning black holes. All other parameters were set to fiducial values; $\iota =\pi /4$, $q=3$, $r_{\mathrm{out}}=30$ and $\epsilon =0$.