Gravitational-wave measurements of the mass and angular momentum of a black hole

A deformed black hole produced in a cataclysmic astrophysical event should undergo damped vibrations which emit gravitational radiation. By fitting the observed gravitational waveform h(t) to the waveform predicted for black-hole vibrations, it should be possible to deduce the hole’s mass M and dimensionless rotation parameter a=(c/G)(angular momentum)/$M^2$. This paper estimates the accuracy with which M and a can be determined by optimal signal processing of data from laser-interferometer gravitational-wave detectors. It is assumed that the detector noise has a white spectrum and has been made Gaussian by cross correlation of detectors at different sites. Assuming, also, that only the most slowly damped mode (which has spheroidal harmonic indices l=m=2) is significantly excited—as probably will be the case for a hole formed by the coalescence of a neutron-star binary or a black-hole binary—it is found that the one-sigma uncertainties in M and a are ΔM/M≃2.2${\rho }^{\mathrm{-}1}$(1-a${)}^{0.45}$, Δa≃5.9${\rho }^{\mathrm{-}1}$(1-a${)}^{1.06}$, where ρ≃$h_s$(π$S_h$${)}^{\mathrm{-}1/2}$ (1-a${)}^{\mathrm{-}0.22}$. Here ρ is the amplitude signal-to-noise ratio at the output of the optimal filter, $h_s$ is the wave’s amplitude at the beginning of the vibrations, f is the wave’s frequency (the angular frequency ω divided by 2π), and $S_h$ is the frequency-independent spectral density of the detectors’ noise. These formulas for ΔM and Δa are valid only for ρ≳10. Corrections to these approximate formulas are given in Table II.