Parameter estimation for compact binary coalescence signals with the first generation gravitational-wave detector network

Compact binary systems with neutron stars or black holes are one of the most promising sources for ground-based gravitational-wave detectors. Gravitational radiation encodes rich information about source physics; thus parameter estimation and model selection are crucial analysis steps for any detection candidate events. Detailed models of the anticipated waveforms enable inference on several parameters, such as component masses, spins, sky location and distance, that are essential for new astrophysical studies of these sources. However, accurate measurements of these parameters and discrimination of models describing the underlying physics are complicated by artifacts in the data, uncertainties in the waveform models and in the calibration of the detectors. Here we report such measurements on a selection of simulated signals added either in hardware or software to the data collected by the two LIGO instruments and the Virgo detector during their most recent joint science run, including a “blind injection” where the signal was not initially revealed to the collaboration. We exemplify the ability to extract information about the source physics on signals that cover the neutron-star and black-hole binary parameter space over the component mass range $1{\mathrm{M}}_{\odot }–25{\mathrm{M}}_{\odot }$ and the full range of spin parameters. The cases reported in this study provide a snapshot of the status of parameter estimation in preparation for the operation of advanced detectors.

Figure 1

(Left) Posterior probability distributions for the chirp mass $\mathcal{M}$ of the nonspinning BBH hardware injection (Sec. 3a1) for the seven signal models considered. The injected value is marked with a vertical red line. (Right) Overlay of 90% probability regions for the joint posterior distribution on the component masses $m_1$, $m_2$ of the binary. The true value is marked by the blue star. Models which allow for nonzero spins find wider PDFs for the coupled mass parameters.

Figure 2

Joint posterior probability regions for the location and inclination angle of the nonspinning BBH hardware injection (Sec. 3a1) for the seven signal models considered. (Left) The binary is constrained to two neighboring regions of the sky. (Right) The distance and inclination, like the sky location, are estimated with a similar accuracy in models that include or exclude spins.

Figure 3

Posterior probability distributions for the dimensionless spin magnitude of the heavier (left) and lighter (right) components of the binary from the nonspinning BBH hardware injection (Sec. 3a1), as inferred in the model ST (Table 1), full-spin STPN. The injection was made with $a_1=a_2=0$.

Figure 4

(Left) Posterior probability distributions for the chirp mass $\mathcal{M}$ of the nonspinning BNS hardware injection (Sec. 3a2) for the seven signal models considered. The injected value is marked with a vertical red line. (Right) Overlay of 90% probability regions for the joint posterior distribution on the component masses $m_1$, $m_2$ of the binary. The true value is marked by the blue star.

Figure 5

Joint posterior probability regions for the location and inclination angle of the nonspinning BNS hardware injection (Sec. 3a2) for the seven signal models considered. (Left) The binary can be constrained to two regions of the sky representing the reflection of the source location through the plane of the detectors. (Right) The distance and inclination, like the sky location, are estimated with a similar accuracy in models that include or exclude spins. The data were such that the injected distance and inclination angle are far outside the 90% credible intervals.

Figure 6

(Left) Posterior probability distributions for the chirp mass $\mathcal{M}$ of the spinning BBH software injection (Sec. 3b1) for the seven signal models considered. The injected value is marked with a vertical red line. (Right) Overlay of 90% probability regions for the joint posterior distribution on the component masses $m_1$, $m_2$ of the binary.

Figure 7

Joint posterior probability regions for the location and inclination angle of the spinning BBH software injection (Sec. 3b1). (Left) The binary’s true location lies just outside of the 90% credible interval. (Right) The degeneracy in distance and inclination prevents either parameter from being accurately constrained individually.

Figure 8

Posterior probability distributions for the dimensionless spin magnitude of the heavier (left) and lighter (right) components of the binary from the spinning BBH software injection (Sec. 3b1), as inferred in the model ST (Table 1), full-spin STPN; the true values are shown with vertical red lines.

Figure 9

(Left) Posterior probability distributions for the chirp mass $\mathcal{M}$ of the spinning NSBH software injection (Sec. 3b2) for the seven signal models considered. The injected value is marked with a vertical red line. (Right) Overlay of 90% probability regions for the joint posterior distribution on the component masses $m_1$, $m_2$ of the binary.

Figure 10

Joint posterior probability regions for the location and inclination angle of the spinning NSBH software injection (Sec. 3b2). (Left) The binary is localized well on the sky. (Right) In this case, the true value lies outside of the 90% credible interval of the joint distance-inclination marginalized probability density function.

Figure 11

Posterior probability distributions for the dimensionless spin magnitude of the heavier (left) and lighter (right) components of the binary from the spinning NSBH software injection (Sec. 3b2), as inferred in the model ST (Table 1), full-spin STPN; the true values are shown with vertical red lines.

Figure 12

(Left) Posterior probability distributions for the chirp mass $\mathcal{M}$ of the blind injection (Sec. 3c) for signal models at 2.5 pN. The injected value is marked with a vertical line. (Right) Overlay of 90% probability regions for the joint posterior distribution on the component masses $m_1$, $m_2$ of the binary. The bias introduced by an analysis with a model which disallows spin is clear.

Figure 13

Joint posterior probability regions for the location and inclination angle of the blind injection (Sec. 3c). (Left) The sky location is constrained to several distinct regions lying along a half circle on the sky. (Right) Again, the characteristic V-shape degeneracy in distance and inclination is evident.

Figure 14

Posterior probability distributions for the dimensionless spin magnitude (left) and tilt angle (angle between the spin vector and the orbital angular momentum, measured at 40 Hz, right) of the heavier component of the binary from the blind injection (Sec. 3c), as inferred with model full-spin STPN at 2.5 pN order in phase (Table 1); the true values are shown with vertical red lines. The large difference in masses allows for the spin of the massive component to be measured. On the other hand, the spin of the light component is unconstrained.

Figure 15

Overlay of 90% probability regions for the joint posterior distribution on the component masses $m_1$, $m_2$ of the binary for the blind injection (Sec. 3c), as inferred with model ST_25 (Table 1), full-spin STPN (at 2.5 pN order in phase), using data from the Hanford detector only, the Livingston detector only or the whole Hanford-Livingston-Virgo network.

Figure 16

Overlay of 90% probability regions for the joint posterior probability distribution of the component masses $m_1$, $m_2$ of the binary (left) and the distance and inclination (right) for a series of signal injection times. Both sets of distributions show a spread comparable to the spread observed between different waveform models, as shown in Figs. 1 and 2.

Figure 17

The plots show in red ten overlaid noise power spectral densities (top left: LIGO-Hanford; top right: LIGO-Livingston; bottom: Virgo), each computed over 1024 s of data, and separated by 10 s. In black is the PSD used in Figs. 1 and 2. The PSDs were used in the analyses whose results are shown in Figs. 16 and 18 and provide an indication of the fluctuation of the noise in the instruments over a time span of minutes.

Figure 18

(Left) Posterior probability distributions for the chirp mass $\mathcal{M}$ of the nonspinning BBH injection for signal model TF2 (Table 1), TaylorF2 at 3.5 pN order, for the hardware injection (Sec. 3a1) and three software injections of the same parameters injected 100, 500 and 1000 seconds before the hardware injection (see Sec. 3d). The injected value is marked with a vertical red line. (Right) Overlay of 90% probability regions for the joint posterior distribution on the inclination and distance of the binary. Different realizations of the noise, rather than differences between hardware and software injections, are the likely reason for the variations in recovered PDFs.