Phenomenological inclusion of alternative dispersion relations to the Teukolsky equation and its application to bounding the graviton mass with gravitational-wave measurements

Existing constraints on the graviton mass from gravitational-wave detections rely on the phase difference developed between different frequencies during the propagation. Effects on the quasinormal mode frequencies of the black-hole ringdown due to the graviton mass are often ignored. While perturbation theories of black holes have been well developed in the context of general relativity, this is not the case for modified gravity theories. We propose a phenomenological modification to the Teukolsky equation of perturbed black holes to include the dispersion relation due to a gravitational field of nonzero mass. Solving this modified Teukolsky equation by logarithmic perturbation theory, we compute the shift of the quasinormal mode frequencies due to the presence of a graviton mass. This hypothetical shift can be used to constrain the graviton mass with ringdown signals, either standalone or in conjunction with the phase difference accumulated due to the wave propagation. We estimate that constraints on the graviton mass of $m_g\le 10^{-15}eV$ can be put with a detection of the ringdown signal alone by second generation gravitational-wave detectors.

Figure 1

Quasinormal mode frequencies of 022 (top panel) and 033 (bottom panel) overtone of a spinless black hole of mass $100M_{\odot }$ with $m_g=0,2,4,\dots ,20{\mathrm{s}}^{-1}$. The point of the smallest ${\omega }_{nlm}^{\mathrm{Re}}$ and the largest ${\omega }_{nlm}^{\mathrm{Im}}$ corresponds to $m_g=0$. Both frequencies and lifetimes are enhanced by mass of gravitational field.

Figure 2

The plus mode time domain ringdown waveforms emitted by a black hole of $100M_{\odot }$ at 400 Mpc for $\log {\lambda }_g=6.0$ (blue), 6.5 (green), and $\log {\lambda }_g=\infty$ ($m_g=0$) (red). As $\log \lambda$ approaches 7, waveforms of $m_g>0$ and $m_g=0$ overlap almost completely. Major overtones of $nl|m|=022$, 122, 033, 133, 044, 055, 021, 032, and 034 are included in the plots.

Figure 3

(Top panel) The posterior of $\log {\lambda }_g$ obtained from the ringdown signal by a black hole of $50M_{\odot }$ (blue), $100M_{\odot }$ (red), $200M_{\odot }$ (green), and $290M_{\odot }$ (black) at 400 Mpc. The posteriors of different black hole masses are in step-function shape. Beyond $\log {\lambda }_g\sim 6–8$, the ringdown waveforms of $m_g>0$ are almost indistinguishable from $m_g=0$ to the sampler. (Bottom panel) The 90% confidence interval of $m_g$ as a function of the final black hole mass $M_f$. The credible interval increases with the final mass of black holes, despite some fluctuation due to noise. Increase of signal-to-noise ratio with mass leads to a better constraint by a black hole of higher mass.

Figure 4

The real (top panel) and imaginary part (bottom panel) of $⟨u|V^{(1)}|u⟩/⟨u|u⟩$ as functions of the upper limit ($R_{+}$) of the inner products for $m_g=1$ and $a=0.7$. $u$ is the analytic solution for the 022-quasinormal mode for Kerr black holes. $u$ involves summation of hypergeometric function from $n=-N$ to $n=+N$. For $N=20$, 30, and 40, the real part of $⟨u|V^{(1)}|u⟩/⟨u|u⟩$ trends to 1 as $R_{+}$ is extended to $R_{+}\sim r_{+}+50\frac{\epsilon \kappa }{\omega }$. This serves as a numerical proof of Eq. (5).