Testing the space-time geometry around black hole candidates with the analysis of the broad $\mathbf{K}\alpha$ iron line

Astrophysical black hole candidates are thought to be the Kerr black holes predicted by general relativity, but there is not yet clear evidence that the geometry of the space-time around these objects is really described by the Kerr metric. In order to confirm the Kerr black hole hypothesis, we have to observe strong gravity features and check that they are in agreement with the ones predicted by general relativity. In this paper, I study the broad $\mathrm{K}\alpha$ iron line, which is often seen in the x-ray spectrum of both stellar-mass and supermassive black hole candidates and whose shape is supposed to be strongly affected by the space-time geometry. As found in previous studies in the literature, there is a strong correlation between the spin parameter and the deformation parameter; that is, the line emitted around a Kerr black hole with a certain spin can be very similar to the one coming from the space-time around a non-Kerr object with a quite different spin. Despite that, the analysis of the broad $\mathrm{K}\alpha$ iron line is potentially more powerful than the continuum-fitting method, as it can put an interesting bound on possible deviations from the Kerr geometry independently of the value of the spin parameter and without additional measurements.

Figure 1

Broad $\mathrm{K}\alpha$ iron line in the Kerr background for different values of the parameters of the model. Top left panel: effect of the spin parameter $a_{*}$ for a viewing angle $i=45°$, intensity profile with index $\alpha =-3$, and emissivity region $r_{\mathrm{out}}-r_{\mathrm{in}}=100M$. Top right panel: effect of the viewing angle for a spin parameter $a_{*}=0.7$, intensity profile with index $\alpha =-3$, and emissivity region $r_{\mathrm{out}}-r_{\mathrm{in}}=100M$. Bottom left panel: effect of the intensity profile index $\alpha$, for a spin parameter $a_{*}=0.7$, viewing angle $i=45°$, and emissivity region $r_{\mathrm{out}}-r_{\mathrm{in}}=100M$. Bottom right panel: effect of the emissivity region for a spin parameter $a_{*}=0.7$, viewing angle $i=45°$, and intensity profile with index $\alpha =-3$. See the text for details.

Figure 2

Broad $\mathrm{K}\alpha$ iron line in the JP background for different values of the deformation parameter ${ϵ}_3$. Left panel: spin parameter $a_{*}=0.4$. Right panel: spin parameter $a_{*}=0.7$. In both the panels, the viewing angle is $i=45°$, the intensity profile index is $\alpha =-3$, and the emissivity region is $r_{\mathrm{out}}-r_{\mathrm{in}}=100M$. See the text for details.

Figure 3

${\chi }_{\mathrm{red}}^2$ from the comparison of the broad $\mathrm{K}\alpha$ iron line produced in a Kerr space-time with spin parameter ${\tilde{a}}_{*}=0.70$ and the one in a JP space-time with spin parameter $a_{*}$ and deformation parameter ${ϵ}_3$. Top left panel: $\tilde{i}=i=45°$. Top right panel: $\tilde{i}=45°$ and $i$ free. Central left panel: $\tilde{i}=i=15°$. Central right panel: $\tilde{i}=15°$ and $i$ free. Bottom left panel: $\tilde{i}=i=75°$. Bottom right panel: $\tilde{i}=75°$ and $i$ free. The other parameters of the model are $\tilde{\alpha }=\alpha =-3$ and ${\tilde{r}}_{\mathrm{out}}-{\tilde{r}}_{\mathrm{in}}=r_{\mathrm{out}}-r_{\mathrm{in}}=100M$. ${\chi }_{\mathrm{red}}^2=0$ at $a_{*}=0.70$ and ${ϵ}_3=0.0$. See the text for details.

Figure 4

${\chi }_{\mathrm{red}}^2$ from the comparison of the broad $\mathrm{K}\alpha$ iron line produced in a Kerr space-time with spin parameter $\tilde{a}$ and the one in a JP space-time with spin parameter $a_{*}$ and deformation parameter ${ϵ}_3$. Top left panel: ${\tilde{a}}_{*}=0.20$ and $\tilde{i}=i=45°$. Top right panel: ${\tilde{a}}_{*}=0.20$, $\tilde{i}=45°$, and $i$ free. Bottom left panel: ${\tilde{a}}_{*}=0.92$ and $\tilde{i}=i=45°$. Bottom right panel: ${\tilde{a}}_{*}=0.92$, $\tilde{i}=45°$, and $i$ free. ${\chi }_{\mathrm{red}}^2=0$ at $a_{*}=0.20$ and ${ϵ}_3=0.0$ in the top panels, and at $a_{*}=0.92$ and ${ϵ}_3=0.0$ in the bottom panels. See the text for details.

Figure 5

${\chi }_{\mathrm{red}}^2$ from the comparison of the broad $\mathrm{K}\alpha$ iron line produced in a Kerr space-time with spin parameter ${\tilde{a}}_{*}=0.98$ and the one in a JP space-time with spin parameter $a_{*}$ and deformation parameter ${ϵ}_3$. Top left panel: $\tilde{i}=i=45°$. Top right panel: $\tilde{i}=45°$ and $i$ free. Central left panel: $\tilde{i}=i=15°$. Central right panel: $\tilde{i}=15°$ and $i$ free. Bottom left panel: $\tilde{i}=i=75°$. Bottom right panel: $\tilde{i}=75°$ and $i$ free. The other parameters of the model are $\tilde{\alpha }=\alpha =-3$ and ${\tilde{r}}_{\mathrm{out}}-{\tilde{r}}_{\mathrm{in}}=r_{\mathrm{out}}-r_{\mathrm{in}}=100M$. ${\chi }_{\mathrm{red}}^2=0$ at $a_{*}=0.98$ and ${ϵ}_3=0.0$. See the text for details.

Figure 6

As in Fig. 3 for $\tilde{i}=i=45°$, $\tilde{\alpha }=-3$, and $\alpha$ free.

Figure 7

Left panel: ${\chi }_{\mathrm{red}}^2$ from the comparison of the thermal spectrum of a thin accretion disk around a Kerr BH with spin parameter ${\tilde{a}}_{*}=0.70$ and the one in a JP space-time with spin parameter $a_{*}$ and deformation parameter ${ϵ}_3$. Right panel: ${\chi }_{\mathrm{red},\mathrm{tot}}^2$ from the combination of the analysis of the broad $\mathrm{K}\alpha$ iron line and the continuum-fitting method. The parameters of the model are mass $M=10M_{\odot }$, mass accretion rate $\dot{M}=2·{10}^{18}\mathrm{g}{\mathrm{s}}^{-1}$, distance $D=10\mathrm{kpc}$, and viewing angle $i=45°$. ${\chi }_{\mathrm{red}}^2=0$ at $a_{*}=0.70$ and ${ϵ}_3=0.0$. See the text for details.

Figure 8

Left panel: ${\chi }_{\mathrm{red}}^2$ from the comparison of the thermal spectrum of a thin accretion disk around a Kerr BH with spin parameter ${\tilde{a}}_{*}=0.98$ and the one in a JP space-time with spin parameter $a_{*}$ and deformation parameter ${ϵ}_3$. Right panel: ${\chi }_{\mathrm{red},\mathrm{tot}}^2$ from the combination of the analysis of the broad $\mathrm{K}\alpha$ iron line and the continuum-fitting method. The parameters of the model are mass $M=10M_{\odot }$, mass accretion rate $\dot{M}=2·{10}^{18}\mathrm{g}{\mathrm{s}}^{-1}$, distance $D=10\mathrm{kpc}$, and viewing angle $i=45°$. ${\chi }_{\mathrm{red}}^2=0$ at $a_{*}=0.98$ and ${ϵ}_3=0.0$. See the text for details.

Figure 9

Top left panel: ${\chi }_{\mathrm{red}}^2$ from the comparison of the broad $\mathrm{K}\alpha$ iron line produced in a JP space-time with spin parameter ${\tilde{a}}_{*}=0.20$ and ${\widetilde{ϵ}}_3=7.0$ and the one in a JP space-time with spin parameter $a_{*}$ and deformation parameter ${ϵ}_3$. $\tilde{i}=i=45°$. Top right panel: as in the left panel, for $\tilde{i}=45°$ and $i$ free. Bottom left panel: ${\chi }_{\mathrm{red}}^2$ from the comparison of the thermal spectrum of a thin accretion disk around a JP BH with spin parameter ${\tilde{a}}_{*}=0.20$ and ${\widetilde{ϵ}}_3=7.0$ and the one in a JP space-time with spin parameter $a_{*}$ and deformation parameter ${ϵ}_3$. The parameters of the model are mass $M=10M_{\odot }$, mass accretion rate $\dot{M}=2·{10}^{18}\mathrm{g}{\mathrm{s}}^{-1}$, distance $D=10\mathrm{kpc}$, and viewing angle $i=45°$. Bottom right panel: ${\chi }_{\mathrm{red},\mathrm{tot}}^2$ from the combination of the analysis of the broad $\mathrm{K}\alpha$ iron line and the continuum-fitting method. ${\chi }_{\mathrm{red}}^2=0$ at $a_{*}=0.20$ and ${ϵ}_3=7.0$. See the text for details.