Gravitational signature of Schwarzschild black holes in dynamical Chern-Simons gravity

Dynamical Chern-Simons gravity is an extension of general relativity in which the gravitational field is coupled to a scalar field through a parity-violating Chern-Simons term. In this framework, we study perturbations of spherically symmetric black hole spacetimes, assuming that the background scalar field vanishes. Our results suggest that these spacetimes are stable, and small perturbations die away as a ringdown. However, in contrast to standard general relativity, the gravitational waveforms are also driven by the scalar field. Thus, the gravitational oscillation modes of black holes carry imprints of the coupling to the scalar field. This is a smoking gun for Chern-Simons theory and could be tested with gravitational-wave detectors, such as LIGO or LISA. For negative values of the coupling constant, ghosts are known to arise, and we explicitly verify their appearance numerically. Our results are validated using both time evolution and frequency domain methods.

Figure 1

${\omega }_{\mathrm{no}}$ as a function of $\ell$ for different values of $M^4\beta \le 0.4$. The dots indicate data from our numerical methods, continuous lines indicate the fit (4.2).

Figure 2

Time profiles for the $|\Psi |$ (upper panel) and $|\Theta |$ (lower panel) components with $\ell =2$ and $M^4\beta =5\times {10}^{-3}$, 0.25, 1.

Figure 3

Real (upper panel) and imaginary (lower panel) parts of the fundamental QNMs as functions of $\beta$ for $\ell =2$.

Figure 4

Time profile for the $\Psi$ component from the time-evolution approach (solid line) for $M^4\beta =0.3$ and $\ell =2$, compared with a combination of oscillatory and nonoscillatory modes (dashed line).

Figure 5

Time evolution of $|\Psi |$ and $|\Theta |$, for $M^4\beta =100$ and $\ell =2$.

Figure 6

Time evolution of $|\Psi |$ and $|\Theta |$ for several values of $\ell$. Straight lines indicate power-law decay.

Figure 7

Dependence of the gravitational-wave signal on the relative amplitude of the initial Gaussian profile, for $\beta =100$ and $\ell =2$.

Figure 8

Evolution in time of $|\Psi |$ and $|\Theta |$ for $\ell =2$ and negative values of $M^4\beta$.