Estimating the parameters of nonspinning binary black holes using ground-based gravitational-wave detectors: Statistical errors

We assess the statistical errors in estimating the parameters of nonspinning black hole binaries using ground-based gravitational-wave detectors. While past assessments were based on partial information provided by only the inspiral and/or ring-down pieces of the coalescence signal, the recent progress in analytical and numerical relativity enables us to make more accurate projections using complete inspiral-merger-ring-down waveforms. We employ the Fisher information-matrix formalism to estimate how accurately the source parameters will be measurable using a single interferometric detector as well as a network of interferometers. Those estimates are further vetted by full-fledged Monte Carlo simulations. We find that the parameter accuracies of the complete waveform are, in general, significantly better than those of just the inspiral waveform in the case of binaries with total mass $M\gtrsim 20M_{\odot }$. In particular, for the case of the Advanced LIGO detector, parameter estimation is the most accurate in the $M=100–200M_{\odot }$ range. For an $M=100M_{\odot }$ system, the errors in measuring the total mass and the symmetric mass-ratio are reduced by an order of magnitude or more compared to inspiral waveforms. Furthermore, for binaries located at a fixed luminosity distance $d_L$, and observed with the Advanced LIGO-Advanced Virgo network, the sky-position error is expected to vary widely across the sky: For $M=100M_{\odot }$ systems at $d_L=1\mathrm{Gpc}$, this variation ranges mostly from about a hundredth of a square degree to about a square degree, with an average value of nearly a tenth of a square degree. This is more than 40 times better than the average sky-position accuracy of inspiral waveforms at this mass range. For the mass parameters as well as the sky position, this improvement in accuracy is due partly to the increased signal-to-noise ratio and partly to the information about these parameters harnessed through the post-inspiral phases of the waveform. The error in estimating $d_L$ is dominated by the error in measuring the wave’s polarization and is roughly 43% for low-mass ($M\sim 20M_{\odot }$) binaries and about 23% for high-mass ($M\sim 100M_{\odot }$) binaries located at $d_L=1\mathrm{Gpc}$.

Figure 1

Noise amplitude spectrum ($\sqrt{S_h(f)}$) of different detectors considered in this paper.

Figure 2

Errors in estimating the total mass $M$ (top left), symmetric mass ratio $\eta$ (top middle), chirp mass $M_c$ (top right), time of arrival $t_0$ (bottom left), and effective distance $d_{\mathrm{eff}}$ (bottom right) in the case of the Advanced LIGO noise spectrum, plotted against the total mass of the binary. The errors of $M$, $\eta$, $M_c$, and $d_{\mathrm{eff}}$ are in percentages and the errors of $t_0$ are in seconds. The value of the symmetric mass ratio $\eta$ is shown in the legends. The solid lines correspond to a search using complete BBH templates and the dashed lines correspond to a search using 3.5 PN-accurate post-Newtonian templates in the SPA, truncated at the Schwarzschild ISCO. The binary is placed optimally oriented at an effective distance of 1 Gpc.

Figure 3

Same as in Fig. 2 except that the binary is placed at an effective distance of 100 Mpc and the noise PSD corresponds to that of Initial LIGO.

Figure 4

Same as in Fig. 2 except that the binary is placed at an effective distance of 100 Mpc and the noise PSD corresponds to that of Enhanced LIGO.

Figure 5

The overlap function (i.e., the ambiguity function maximized over $t_0$ and ${\phi }_0$) between waveforms constructed at different points in the parameter space. The horizontal axis reports the total mass $M$ of the binary while the vertical axis reports its symmetric mass ratio $\eta$. Each panel shows the overlap of different waveforms with one “target waveform.” The total mass of the target waveform is chosen to be $M=20M_{\odot }$, $100M_{\odot }$, $200M_{\odot }$, $400M_{\odot }$, respectively, for the four columns starting from the left. The symmetric mass ratio of the target waveforms is chosen to be $\eta =0.25$, 0.222, 0.16, in the top, middle, and bottom rows, respectively.

Figure 6

Errors in estimating the total mass $M$ (top left), symmetric mass ratio $\eta$ (top middle), chirp mass $M_c$ (top right), time of arrival $t_0$ (bottom left), and effective distance $d_{\mathrm{eff}}$ (bottom right) in the case of the Advanced LIGO noise spectrum, plotted against the total mass of the binary. The errors of $M$, $\eta$, $M_c$, and $d_{\mathrm{eff}}$ are in percentage and the errors of $t_0$ are in seconds. Symmetric mass ratio $\eta$ is shown in the legends. The solid lines correspond to a search using complete BBH templates and the dashed lines correspond to 3.5 PN-accurate post-Newtonian templates in the SPA, truncated at the Schwarzschild ISCO. The errors correspond to a fixed SNR of 10.

Figure 7

Same as Fig. 6 except that the noise PSD corresponds to that of Initial LIGO.

Figure 8

Same as Fig. 6 except that the noise PSD corresponds to that of Enhanced LIGO.

Figure 9

The curves labeled “IniLIGO” and “EnhLIGO” report the SNR produced by binaries located at an effective distance of 100 Mpc at Initial LIGO and Enhanced LIGO, respectively, as a function of the total mass. The curves labeled “AdLIGO” and “AdVirgo” report the same produced by binaries located at 1 Gpc at Advanced LIGO and Advanced Virgo. The solid lines correspond to complete waveforms and the dashed lines correspond to PN waveforms.

Figure 10

Scatter plot of parameters estimated from ${10}^4$ Monte Carlo simulations. The horizontal axis reports the total mass and the vertical axis reports the symmetric mass ratio. The left panel correspond to the injection with parameters $M=20M_{\odot }$ and $\eta =0.2222$, and the right panel correspond the injection with parameters $M=200M_{\odot }$ and $\eta =0.2222$. The injections correspond to an SNR of 20. Also overlaid in the left panel is a cartoon of the four different initial simplexes chosen for the maximization algorithm. The true values of the parameters are marked by a cross. Note that the eigendirections are different in the two plots.

Figure 11

Distribution of the estimated parameters from Monte Carlo simulations and expected probability distributions from Fisher-matrix calculation. The true parameters are $M=20M_{\odot }$, $\eta =0.16$.

Figure 12

The data points show the errors computed from Monte Carlo simulations in the case of Advanced LIGO noise PSD. The horizontal axis reports the total mass while the legends report the symmetric mass ratio. The errors are computed for a fixed SNR of 20. The dashed lines correspond to the same errors computed using Fisher-matrix formalism.

Figure 13

Errors computed from Monte Carlo simulations (crosses, dots, and diamonds) plotted against SNR. The horizontal axes report the SNR of the injections and the legends report the total mass in units of $M_{\odot }$. The top, middle, and bottom panels correspond to mass ratios $\eta =0.25$, 0.2222, and 0.16, respectively. The error bounds expected from the Fisher-matrix calculation are indicated by dashed lines.

Figure 14

The network SNR of a signal, corresponding to the complete waveform, from an equal-mass binary with $M=100M_{\odot }$ located at $d_L=1\mathrm{Gpc}$, plotted as a function of its sky position. The network here is the three-detector AdvLIGO-AdvVirgo network, such that the two 4 km-arm-length LIGO detectors in Hanford and Livingston have AdvLIGO noise PSDs and the Virgo detector in Cascina has AdvVirgo noise PSD. Above, $\theta$ and $\varphi$ are the polar and azimuthal angles specifying the location of the source in the sky in the geographic coordinate system.

Figure 15

The top plot shows the sky-position error ${log}_{10}$ [$\Delta \Omega$ (in square degrees)] and the bottom plot shows the fractional error in the luminosity distance ${log}_{10}$ [$\Delta d_L/d_L$ (in %)] as functions of the sky position of a BBH source. The source studied here is the same equal-mass binary considered in Fig. 14, and, $\theta$ and $\varphi$ are the polar and azimuthal angles specifying the location of the source in the sky in the geographic coordinate system. Note how the effect of the varying network sensitivity, as seen in the SNR plot in Fig. 14, is imprinted in the two error plots. Additionally, the error plots display a full “sine-wave” pattern, which comprises a set of sky positions for which the geometric independence of the LIGO-Virgo detectors is the weakest. Extraction of the signal’s polarization is affected the most at these locations. That in turn hurts the distance measurement accuracy. The same locations do not necessarily hurt the determination of the sky position, which is mostly driven by the measurement accuracy of the times of arrival of the signal at the three sites.

Figure 16

All-sky distribution of errors in estimating the solid angle $\Omega$ in the case of the AdvLIGO-AdvVirgo network. The left plots show the probability density and the right plots show the cumulative distribution. The top panels correspond to an equal-mass binary with $M=20M_{\odot }$, and the bottom panels to one with $M=100M_{\odot }$. In each plot the thick (red) traces correspond to the errors estimated using the complete waveforms while the thin (black) traces correspond to those estimated using restricted 3.5 PN waveforms. All the errors are computed for a network SNR of 10 for the respective waveforms.

Figure 17

All-sky distribution of errors in estimating $d_L$ in the case of the AdvLIGO-AdvVirgo network. The top plots show the probability density and the bottom plots show the cumulative distribution. The top panels correspond to an equal-mass binary with $M=20M_{\odot }$, and the bottom panels to one with $M=100M_{\odot }$. In each plot the thick (red) traces correspond to the errors estimated using the complete waveforms while the thin (black) traces correspond to those estimated using restricted 3.5 PN waveforms. All the errors are computed for a network SNR of 10 for the respective waveforms.

Figure 18

Same as Fig. 16 except that the binary is now placed at a fixed luminosity distance of 1 Gpc. Notice the strong similarity between the plots in the top panel above and those in the top panel of Fig. 16. This is because in the plots of the top panel above the average SNR is relatively close to 10. The plots in the bottom rows of the two figures are more disparate: The average SNR above is several [few] times better than the fixed SNR in Fig. 16 for the complete [inspiral-only] waveforms.

Figure 19

Same as Fig. 17 except that the binary is placed at a fixed luminosity distance of 1 Gpc. By comparing the above figure with Fig. 17, it is manifest that nearly all the improvement in the luminosity-distance measurement accuracy, when including the post-inspiral phases, arises due to the increased SNR.