Estimating parameters of binary black holes from gravitational-wave observations of their inspiral, merger, and ringdown

We characterize the expected statistical errors with which the parameters of black hole binaries can be measured from gravitational-wave (GW) observations of their inspiral, merger, and ringdown by a network of second-generation ground-based GW observatories. We simulate a population of black hole binaries with uniform distribution of component masses in the interval $(3,80)M_{\odot }$, distributed uniformly in comoving volume, with isotropic orientations. From signals producing signal-to-noise ratio $\ge 5$ in at least two detectors, we estimate the posterior distributions of the binary parameters using the Bayesian parameter estimation code LALInference. The GW signals will be redshifted due to the cosmological expansion, and we measure only the “redshifted” masses. By assuming a cosmology, it is possible to estimate the gravitational masses by inferring the redshift from the measured posterior of the luminosity distance. We find that the measurement of the gravitational masses will be, in general, dominated by the error in measuring the luminosity distance. In spite of this, the component masses of more than 50% of the population can be measured with accuracy better than $\sim 25\%$ using the Advanced LIGO-Virgo network. Additionally, the mass of the final black hole can be measured with median accuracy $\sim 18\%$. Spin of the final black hole can be measured with median accuracy $\sim 5\%(17\%)$ for binaries with nonspinning (aligned-spin) black holes. Additional detectors in Japan and India significantly improve the accuracy of sky localization, and moderately improve the estimation of luminosity distance, and hence, that of all mass parameters. We discuss the implication of these results on the observational evidence of intermediate-mass black holes and the estimation of cosmological parameters using GW observations.

Figure 1

Median values of the $1\sigma$ errors in estimating various parameters of nonspinning binary black holes using the three-detector (Advanced LIGO-Virgo) network as a function of the (redshifted) total mass of the binary. The error in sky location is in square degrees (sq. deg.), while the rest are fractional errors. The error in estimating the physical mass parameters (left plot) are dominated by the error in estimating the luminosity distance. Hence, they are estimated with poorer accuracy as compared to their redshifted counterparts. For the case of the redshifted mass parameters (middle plot), the chirp mass ${\mathcal{M}}^z$ is the best estimated mass parameter at low masses ($M^z\lesssim 150M_{\odot }$), while at high masses ($M^z\gtrsim 200M_{\odot }$), the total mass $M^z$ and the final mass $M_f^z$ are better estimated.

Figure 2

The top plot shows the horizon distance (distance to which optimally located and oriented binaries can be observed with an optimal signal-to-noise ratio of 8) of the Advanced LIGO detector towards nonspinning, equal-mass binaries as a function of the (redshifted) total mass of the binary. The bottom plot shows the expected distribution of SNRs from the population of binary black holes that cross the detection criterion.

Figure 3

The left plot shows the distribution of the rest-frame component masses $m_{1,2}$ of the simulated population of nonspinning binaries (light colored histograms) and that of the subpopulation that crossed the detection threshold for the three-detector network (dark colored histograms). The two middle plots show the distributions of the redshifted component masses $m_{1,2}^z$ and the total mass $M^z$. Note that, even stellar-mass ($m_{1,2}<80M_{\odot }$) black holes can appear to have intermediate masses ($m_{1,2}^z>100M_{\odot }$) due to the degeneracy between mass and cosmological redshift. The right plot shows the distributions of the mass ratio $q\equiv m_2/m_1$. Note that comparable-mass $(q\gg 0)$ binaries are detected preferentially due their larger intrinsic luminosity.

Figure 4

The top panel shows the distribution of the $1\sigma$ errors in estimating the “redshifted” mass parameters (redshifted component masses $m_{1,2}^z$, chirp mass ${\mathcal{M}}^z$, total mass $M^z$, and the final mass $M_f^z$) from the simulated population of coalescing binary black holes. In each subplot, the three distributions correspond to nonspinning binary black holes detected by the 3-detector network and the 5-detector network that we consider, and the population of binary black holes with nonprecessing spins detected by the 3-detector network. The vertical lines show the median of the distribution. The bottom panels show the distribution of the errors in estimating the “physical” parameters. It can be seen that the error in measuring the “physical” mass parameters $\mathcal{M}$, $M$, and $M_f$ are significantly larger than that in measuring the “redshifted” parameters, and are dominated by the error in measuring the luminosity distance. The error in the component masses are dominated by that in measuring the mass ratio of the system, and hence, are not very different in the top and bottom panels. Note that the mass ratio $\eta$, the final spin $a_f/M_f$, the luminosity distance $d_L$, and the sky location $\mathrm{\Omega }$ are independent of the cosmological redshift.

Figure 5

Typical posteriors obtained using LALInference. The solid red lines indicate the injection values. The dashed magenta, cyan, and yellow contours indicate the 68.3%, 95.5%, and 99.7% confidence regions, respectively. The left panel shows the correlation between the component masses. For large masses, the total mass is roughly the most precisely estimated quantity. The posterior samples are thus correlated along the $M^z=\text{constant}$ line. The right panel shows the correlation between luminosity distance and inclination. In this case, $d_L/(1+{\cos }^2\iota )$ is the precisely estimated quantity. Because of this degeneracy, neither of the two physical quantities are well-estimated. These typical results are for an injection with a signal-to-noise ratio of 13 recovered with a 3-detector network.