Wiggly tails: A gravitational wave signature of massive fields around black holes

Massive fields can exist in long-lived configurations around black holes. We examine how the gravitational wave signal of a perturbed black hole is affected by such “dirtiness” within linear theory. As a concrete example, we consider the gravitational radiation emitted by the infall of a massive scalar field into a Schwarzschild black hole. Whereas part of the scalar field is absorbed/scattered by the black hole and triggers gravitational wave emission, another part lingers in long-lived quasibound states. Solving numerically the Teukolsky master equation for gravitational perturbations coupled to the massive Klein-Gordon equation, we find a characteristic gravitational wave signal, composed by a quasinormal ringing followed by a late time tail. In contrast to “clean” black holes, however, the late time tail contains small amplitude wiggles with the frequency of the dominating quasibound state. Additionally, an observer dependent beating pattern may also be seen. These features were already observed in fully nonlinear studies; our analysis shows they are present at linear level, and, since it reduces to a $1+1$ dimensional numerical problem, allows for cleaner numerical data. Moreover, we discuss the power law of the tail and that it only becomes universal sufficiently far away from the dirty black hole. The wiggly tails, by constrast, are a generic feature that may be used as a smoking gun for the presence of massive fields around black holes, either as a linear cloud or as fully nonlinear hair.

Figure 1

GW signal ($\ell =2$) for $M\mu =0.48$ (observer at $500M$). Small amplitude wiggles can be seen in the late time behavior, in an otherwise apparently power law tail.

Figure 2

Scalar ($\ell =1$) and GW ($\ell =2$) signal for $M\mu =0.40$ (observer at $70M$). The dominant frequency of the scalar field is that of the first quasibound state overtone, with ${\omega }_1=0.3955-i2.2668\times {10}^{-4}$; the next leading quasibound state is the second overtone, with ${\omega }_2=0.3975-i1.0035\times {10}^{-4}$.

Figure 3

Late time behavior of the GW signal ($\ell =2$) for several values of $M\mu$ (observer at $70M$). The beating becomes increasingly more visible as the mass is decreased from $M\mu =0.48$ to $M\mu =0.4$. For even smaller values the beating becomes again suppressed (not shown).

Figure 4

Quadrupolar GW signal (observer at $100M$). We show a power law fit $t^a$ with $a=1.5$ for the late time tail. This power law fits well the three curves with highest mass, but it fails for the lowest mass.