Formation of Supermassive Black Holes through Fragmentation of Torodial Supermassive Stars

We investigate new paths to supermassive black hole formation by considering the general relativistic evolution of a differentially rotating polytrope with a toroidal shape. We find that this polytrope is unstable to nonaxisymmetric modes, which leads to a fragmentation into self-gravitating, collapsing components. In the case of one such fragment, we apply a simplified adaptive mesh refinement technique to follow the evolution to the formation of an apparent horizon centered on the fragment. This is the first study of the onset of nonaxisymmetric dynamical instabilities of supermassive stars in full general relativity.

Figure 1

Time evolution of the $L^2$ norm of the Hamiltonian constraint for different resolutions. The time is normalized to the dynamical time scale $t_{\mathrm{D}}=r_{\mathrm{e}}\sqrt{r_{\mathrm{e}}/M}$.

Figure 2

Time evolution of the mode amplitudes in the standard resolution ($65\times 65\times 33$ points per patch). The amplitudes are obtained from a Fourier decomposition of the density profile on the equatorial plane circle at $\varpi =0.25r_{\mathrm{e}}$, the initial radius of highest density.

Figure 3

Time evolution of the equatorial plane density using the perturbation parameters ${\lambda }_m={\delta }_{1m}$. Shown are isocontours of the logarithm of the rest-mass density. The four snapshots extend to $0.37r_{\mathrm{e}}$ and are taken at $t/t_{\mathrm{D}}=0$, 6.43, 7.14, and 7.45, respectively. They show the formation and collapse of the fragment produced by the $m=1$ instability. The last slice contains an AH demarked by the thick white line. Note that the shades of gray used for illustration are adapted to the current maximal density at each time, and that darker shades denote higher densities.

Figure 4

Time evolution of the equatorial plane density using the perturbation parameters ${\lambda }_m={\delta }_{2m}$. The snapshot times are the same as in Fig. 3. In this case, two fragments are forming. Constraint violations have forced us to terminate the simulation before AHs could be located, (see main text).

Figure 5

Example of a gravitational wave signal. The extracted signal is normalized to a ${10}^6{M}_{\odot }$ source at a distance of 1 Gpc, and an observer inclination of $\theta =0$. Note that $t_{\mathrm{D}}\approx 182\mathrm{s}$ in this case.