Vortex String Formation in Black Hole Superradiance of a Dark Photon with the Higgs Mechanism

Black hole superradiance, which only relies on gravitational interactions, can provide a powerful probe of the existence of ultralight bosons that are weakly coupled to ordinary matter. However, as a boson cloud grows through superradiance, nonlinear effects from interactions with itself or other fields may become important. As a representative example of this, we use nonlinear evolutions to study black hole superradiance of a vector boson that attains a mass, via a coupling to a complex scalar, through the Higgs mechanism. For the cases considered, we find that the superradiant instability can lead to a transient period where the scalar field reaches its symmetry restoration value, leading to the formation of closed vortex strings, the temporary disruption of the exponential growth of the cloud, and an explosive outburst of energy. After the cloud loses sufficient mass, the superradiant growth resumes, and the cycle repeats. Thus, the black hole will be spun down but, potentially, at a much lower rate compared to when nonlinear effects are unimportant and with the liberated energy going primarily into bosonic radiation instead of gravitational waves.

Figure 1

The displacement of the minimum scalar field value from the VEV (solid curves) and the maximum vector magnitude (dashed curves) as a function of time for cases with $\alpha =0.4$ and $\lambda /g^2=25$. The blue and green curves have the same parameters as the black curves but have initial values for the fields that are, respectively, $\approx 2$ and $3\times$ smaller. At lower field values $\min (|\mathrm{\Phi }|/v)\approx 1-\max (A^2)/(2A_{\mathrm{c}}^2)$.

Figure 2

The three pairs of panels on the left show snapshots of the scalar complex phase $\arg (\mathrm{\Phi })$ in the equatorial plane of the black hole (top), and the scalar magnitude $|\mathrm{\Phi }|$ in a meridional slice intersecting the black hole and a string at that time, for subsequent times (the second and third are, respectively, $7M$ and $24M$ after the first) following vortex formation in a case with $\alpha =0.4$ and $\lambda /g^2=25$. The two rightmost panels show 3D contours of $|\mathrm{\Phi }|\approx 0.08v$ in green, indicating the extent of the strings, as well as the complex phase in the equatorial plane. The top rightmost panel is the same case and same time as the third column, but shows that the nearby pairs of vortices are actually part of the same small loops. (Note that $|\mathrm{\Phi }|$ is small, but does not go to zero on the black hole horizon, here.) The bottom rightmost panel is from the case with $\alpha =0.3$. For all cases, the scale can be judged by the size of the black hole horizon (black region), which has proper circumference $4\pi M$.

Figure 3

Top: Energy (solid lines) and angular momentum (dashed lines) as a function of time for $\alpha =0.4$ and $\lambda /g^2=25$. We show the energy and angular momentum contribution from the scalar field, including the interaction term, $E_{\mathrm{\Phi }}$/$J_{\mathrm{\Phi }}$, and the vector field (not including terms involving $\mathrm{\Phi }$) $E_A$/$J_A$ separately. Bottom: The flux of energy extracted (${\dot{E}}_{\mathrm{BH}}$) or absorbed ($-{\dot{E}}_{\mathrm{BH}}$) by the black hole compared to the radiation luminosity for the same case. Here, $t_{\max }$ is the time when the total energy is maximum.

Figure 4

Total energy as a function of time for cases with $\alpha =0.3$ and 0.4 and various values of $\lambda /g^2$. Here, $t_{\max }$ is the time when the total energy is maximum.