Mode coupling of Schwarzschild perturbations: Ringdown frequencies

Within linearized perturbation theory, black holes decay to their final stationary state through the well-known spectrum of quasinormal modes. Here we numerically study whether nonlinearities change this picture. For that purpose we study the ringdown frequencies of gauge-invariant second-order gravitational perturbations induced by self-coupling of linearized perturbations of Schwarzschild black holes. We do so through high-accuracy simulations in the time domain of first and second-order Regge-Wheeler-Zerilli type equations, for a variety of initial data sets. We consider first-order even-parity ($\ell =2$, $m=\pm 2$) perturbations and odd-parity ($\ell =2$, $m=0$) ones, and all the multipoles that they generate through self-coupling. For all of them and all the initial data sets considered we find that—in contrast to previous predictions in the literature—the numerical decay frequencies of second-order perturbations are the same ones of linearized theory, and we explain the observed behavior. This would indicate, in particular, that when modeling or searching for ringdown gravitational waves, appropriately including the standard quasinormal modes already takes into account nonlinear effects.

Figure 1

Solution to the first-order Zerilli equation at observer location $r=51.8M$.

Figure 2

Numerical errors for different spatial resolutions using a fixed time step $\Delta t=0.01M$ (top), and for different time steps using a fixed spatial resolution of $N=60$ points per domain (bottom). Both panels show the differences between several resolutions and the most accurate one, which is $N=60$ for the top panel and $\Delta t_4=0.0005M$ for the bottom one. In both cases the observer is located at $r=51.8M$. We see exponential convergence and errors in the order of double precision round-off.

Figure 3

Numerical errors in the second-order Zerilli function ${}^{\{2\}}\Psi _4^4$ for different spatial resolutions and a fixed time step $\Delta t=0.001M$. The errors are to be interpreted as in the previous figures. For $N=30$ (which are the ones used in the rest of this paper) and higher, they are at the level of double precision round-off.

Figure 4

Absolute value of first- and second-order Zerilli functions extracted at $r=51.8M$.

Figure 5

Case A simulations: first-order ($\ell =2$, $m=2$) Zerilli function and second-order ($\ell =2$, $m=0$) one, along with the source term for the second-order master equation [Eq. (7)], as functions of time for different initial data profiles. The source plays a role only at very early times in exciting the second-order modes, afterward being much smaller than the first- and second-order solutions.

Figure 6

Case A simulations: snapshots of the first-order Zerilli function and second-order ($\ell =4$, $m=0$) one, along with the source term, for ingoing initial data. The generic behavior of the source is to rapidly decrease several orders of magnitude below the solutions themselves. (Note: in the first snapshot the second-order Zerilli function vanishes because it corresponds to $t=0$ and our setup of the physical problem.)

Figure 7

Residual from fitting a second-order Zerilli function to a complex frequency mode. The fitted frequency corresponds to the standard fundamental quasinormal one for that multipole index, and the residual does not appear to contain traces of twice that frequency (or any other).