Gravitational lensing by phantom black holes

In some models dark energy is described by phantom scalar fields (scalar fields with the “wrong” sign of the kinetic term in the Lagrangian). In the current paper we study the effect of phantom scalar field and/or phantom electromagnetic field on gravitational lensing by black holes in the strong deflection regime. The black-hole solutions that we have studied have been obtained in the frame of the Einstein-(anti-)Maxwell-(anti)dilaton theory. The numerical analysis shows a considerable effect of the phantom scalar and electromagnetic fields on the angular position, brightness, and separation of the relativistic images.

Figure 1

The photon sphere $x_{ps}$ (red), the event horizon $x_{+}$ (blue), and the inner horizon $x_{-}$ (black) of the EMD and $\mathrm{E}\overline{\mathrm{M}}\mathrm{D}$ black holes for three values of $\gamma$: $\gamma =-0.5$ (dotted-dashed), $\gamma =0$ (dashed), and $\gamma =0.5$ (solid). $x_{-}<x_{+}<x_{ps}$.

Figure 2

The photon sphere $x_{ps}$ (red), the event horizon $x_{+}$ (blue), and the inner horizon $x_{-}$ (black) of the $\mathrm{EM}\overline{\mathrm{D}}$ and $\mathrm{E}\overline{\mathrm{MD}}$ black holes for three values of $\gamma$: $\gamma =-0.5$ (dotted-dashed), $\gamma =0$ (dashed), and $\gamma =0.5$ (solid). $x_{-}<x_{+}<x_{ps}$.

Figure 3

The EMD lens parameters $a$, $b$, and $u_{ps}$ for the following values of $Q/M$: $Q/M=0$ (black), $Q/M=0.25$ (red), $Q/M=0.5$ (orange), $Q/M=0.75$ (green), $Q/M=1$ (blue), $Q/M=1.25$ (purple), $Q/M=\sqrt{2}$ (brown). Steeper curves correspond to higher values of $Q/M$.

Figure 4

The observables ${\theta }_1^{\mathrm{pro}}$, $r_m$, and $s_1^{\mathrm{pro}}$ for the EMD black hole. The values of $Q/M$ are the same as on Fig. 3.

Figure 5

The $\mathrm{E}\overline{\mathrm{M}}\mathrm{D}$ lens parameters $a$, $b$, and $u_{ps}$ for the following values of $Q/M$: $Q/M=0$ (black), $Q/M=0.25$ (red), $Q/M=0.5$ (orange), $Q/M=0.75$ (green), $Q/M=1$ (blue), $Q/M=1.25$ (purple), $Q/M=1.5$ (brown). Steeper curves correspond to higher values of $Q/M$.

Figure 6

The observables ${\theta }_1^{\mathrm{pro}}$, $r_m$, and $s_1^{\mathrm{pro}}$ for the $\mathrm{E}\overline{\mathrm{M}}\mathrm{D}$ black hole. The values of $Q/M$ are the same as on Fig. 5.

Figure 7

The $\mathrm{EM}\overline{\mathrm{D}}$ lens parameters $a$, $b$, and $u_{ps}$ for the following values of $Q/M$: $Q/M=0$ (black), $Q/M=0.25$ (red), $Q/M=0.5$ (orange), $Q/M=0.75$ (green), $Q/M=1$ (blue). Steeper curves correspond to higher values of $Q/M$.

Figure 8

The observables ${\theta }_1^{\mathrm{pro}}$, $r_m$, and $s_1^{\mathrm{pro}}$ for the $\mathrm{EM}\overline{\mathrm{D}}$ black hole. The values of $Q/M$ are the same as on Fig. 7.

Figure 9

The $\mathrm{E}\overline{\mathrm{MD}}$ lens parameters $a$, $b$, and $u_{ps}$ for the following values of $Q/M$: $Q/M=0$ (black), $Q/M=0.25$ (red), $Q/M=0.5$ (orange), $Q/M=0.75$ (green), $Q/M=1$ (blue), $Q/M=1.25$ (purple), $Q/M=1.5$ (brown). Steeper curves correspond to higher values of $Q/M$.

Figure 10

The observables ${\theta }_1^{\mathrm{pro}}$, $r_m$, and $s_1^{\mathrm{pro}}$ for the $\mathrm{E}\overline{\mathrm{MD}}$ black hole. The values of $Q/M$ are the same as on Fig. 9.

Figure 11

The photon sphere for $Q/M=0$ corresponding to the Schwarzschild black hole (grey) and $Q/M=0.8$ for the other four cases—EMD (thick), $\mathrm{E}\overline{\mathrm{M}}\mathrm{D}$ (dotted-dashed), $\mathrm{EM}\overline{\mathrm{D}}$ (dashed), $\mathrm{E}\overline{\mathrm{MD}}$ (dotted).

Figure 12

The lens parameters $a$ and $b$ for $Q/M=0$ corresponding to the Schwarzschild black hole (grey) and $Q/M=0.8$ for the other four cases—EMD (thick), $\mathrm{E}\overline{\mathrm{M}}\mathrm{D}$ (dotted-dashed), $\mathrm{EM}\overline{\mathrm{D}}$ (dashed), $\mathrm{E}\overline{\mathrm{MD}}$ (dotted).

Figure 13

The impact parameter $u_{ps}$ and the angular position of the first relativistic image for prograde photons ${\theta }_1^{\mathrm{pro}}$ for $Q/M=0$ corresponding to the Schwarzschild black hole (grey) and $Q/M=0.8$ for the other four cases—EMD (thick), $\mathrm{E}\overline{\mathrm{M}}\mathrm{D}$ (dotted-dashed), $\mathrm{EM}\overline{\mathrm{D}}$ (dashed), $\mathrm{E}\overline{\mathrm{MD}}$ (dotted).

Figure 14

The flux ratio $r_m$ and the angular separation between the first and second relativistic image for prograde photons $s_1^{\mathrm{pro}}$ for $Q/M=0$ corresponding to the Schwarzschild black hole (grey) and $Q/M=0.8$ for the other four cases—EMD (thick), $\mathrm{E}\overline{\mathrm{M}}\mathrm{D}$ (dotted-dashed), $\mathrm{EM}\overline{\mathrm{D}}$ (dashed), $\mathrm{E}\overline{\mathrm{MD}}$ (dotted).