Semianalytic sensitivity estimates for catalogs of gravitational-wave transients

I investigate the sensitivity of gravitational-wave searches by analyzing the response of matched filters in stationary Gaussian noise. In particular, I focus on the ability to analytically model the distribution of observed filter responses maximized over coalescence phase and/or a template bank as well as the response of statistics defined for a network of detectors. Semianalytic sensitivity estimates derived assuming stationary Gaussian noise are compared to sensitivity estimates obtained from real searches processing real noise, which is neither perfectly stationary nor perfectly Gaussian. I find that semianalytic estimates are able to reproduce real search sensitivity for the LIGO-Virgo-KAGRA Collaboration’s third observing run with high fidelity. I also discuss how to select computational speed-ups (hopeless signal-to-noise ratio cuts) and make predictions for the fourth observing run using projected detector sensitivities.

Figure 1

Distributions of observed filter responses from a single detector maximized over the template bank (left: ${\rho }_{h,\mathcal{R}}$ and right: ${\rho }_{h,\varphi }$) for several ${\rho }_{\mathrm{opt}}$. (red, solid) Measured distributions from simulations and (blue, dashed) predicted distributions of ${\rho }_{\mathcal{R}}$ and ${\rho }_{\varphi }$ with the optimal template. Note that, for small ${\rho }_{\mathrm{opt}}$, the distributions can differ significantly. The agreement is much better for large ${\rho }_{\mathrm{opt}}$. Regardless, the tails to large ${\rho }_{h,\mathcal{R}}$ and ${\rho }_{h,\varphi }$ always agree reasonably well. See Fig. 2 for survival functions at fixed detection thresholds as a function of ${\rho }_{\mathrm{opt}}$.

Figure 2

Survival function of the response maximized over the template bank (top, ${\rho }_{h,\mathcal{R}}$ and bottom, ${\rho }_{h,\varphi }$) as a function of ${\rho }_{\mathrm{opt}}$. (red, solid) Measured survival functions with (shaded) $1\sigma$ uncertainty and (blue, dashed) predicted survival functions for the optimal template.

Figure 3

Survival functions at a fixed detection threshold as a function of ${\rho }_{\mathrm{net},\mathrm{opt}}$ for a two-detector network with (left) all signal power in a single detector and (right) equal signal power in each detector. Colors and line styles are the same as Fig. 2. As expected, the survival functions are larger at each value of ${\rho }_{\mathrm{net},\mathrm{opt}}$ for multidetector networks compared with single detector networks, particularly when both ${\rho }_{\mathrm{thr}}$ and ${\rho }_{\mathrm{opt}}$ are small. However, there is no significant dependence on the balance of signal power between detectors. Compare to Figs. 2 and 4.

Figure 4

Survival functions at fixed detection thresholds as a function of ${\rho }_{\mathrm{net},\mathrm{opt}}$ for a three-detector network assuming all detectors have equal individual optimal signal-to-noise ratios. Colors and line styles are the same as Fig. 2. Compare to Figs. 2 and 3.

Figure 5

The distribution of primary masses assumed within Sec. 5. This distribution captures the key features identified in Farah et al. [23] and Abbott et al. [6]: a high rate of low-mass systems followed by a sharp dropoff to a relatively flat distribution until a gentle steepening at the highest masses.

Figure 6

Distributions of detected redshifts ($z$) assuming a reference astrophysical population with the additional requirement that $30M_{\odot }\le m_{1,\mathrm{src}}\le 60M_{\odot }$. (gray) Real injections selected based on GstLAL’s FAR ($\le 1/\mathrm{year}$), (solid blue) semianalytic selection based on ${\rho }_{\mathrm{net},\varphi }$ with optimized ${\rho }_{\mathrm{thr}}$, (solid orange) semianalytic selection based on ${\rho }_{\mathrm{net},\mathrm{opt}}$ with optimized ${\rho }_{\mathrm{thr}}$, and (dashed green) semianalytic selection based on ${\rho }_{\mathrm{net},\mathrm{opt}}$ with the same ${\rho }_{\mathrm{thr}}$ as ${\rho }_{\mathrm{net},\varphi }$.

Figure 7

Differences in the cross entropy [Eq. (36)] between the real distribution of detected redshifts in Fig. 6 ($30M_{\odot }\le m_{1,\mathrm{src}}\le 60M_{\odot }$) and semianalytic models based on (blue) ${\rho }_{\mathrm{net},\varphi }$ and (orange) ${\rho }_{\mathrm{net},\mathrm{opt}}$ for various ${\rho }_{\mathrm{thr}}$.

Figure 8

Mass-dependence of the optimized ${\rho }_{\mathrm{thr}}$ for models based on ${\rho }_{\mathrm{net},\varphi }$ as a function of detector-frame chirp mass (${\mathcal{M}}_{\det }$) for (red circles) GstLAL [24, 25, 26, 27], (blue squares) MBTA [28, 29], (green upward triangles) PyCBC-broad and (purple sideways triangles) PyCBC-BBH [30, 31, 32, 33, 34, 35], and (gray stars) cWB [36, 37, 38]. Shading approximates the precision of the fit to real injections as a function of ${\rho }_{\mathrm{thr}}$ (see Fig. 7).

Figure 9

Detected distributions of additional single-event parameters for events with $30M_{\odot }\le m_{1,\mathrm{src}}\le 60M_{\odot }$ with (black) real injections selected based on GstLAL’s FAR ($\le 1/\mathrm{year}$) and (blue) an optimized semianalytic model with a selection based on ${\rho }_{\mathrm{net},\varphi }$ ($\ge 10.0$).

Figure 10

Representative PSDs used to generate the semianalytic sensitivity estimates used in this work. O3 PSDs are the same ones that were used to generate hopeless signal-to-noise ratio cuts in Ref. [9]. Projections for low and high O4 sensitivity are from Ref. [39]. As of the time of this writing, Virgo has not yet joined O4 and the LIGOs have low-frequency sensitivities that are closer to their O3 values than the projections for O4.

Figure 11

Semianalytic approximations of the detection probability assuming a fixed population with different detection thresholds (${\rho }_{\mathrm{thr}}$) and hopeless cuts (${\rho }_{\mathrm{cut}}$) on ${\rho }_{\mathrm{net},\mathrm{opt}}$. Top: the detection probability modeled as a cut on ${\rho }_{\mathrm{net},\varphi }$. Bottom: the difference between the approximation to the detection probability with vs without a hopeless cut. Shaded regions approximate $1\sigma$ uncertainty from the finite number of Monte Carlo samples. Calibration uncertainty is thought to limit the precision in the rate inferred for a fixed population to within $O(1\%)$ relative uncertainty.

Figure 12

Left: detection probability as a function of redshift for several detection thresholds (top, ${\rho }_{\mathrm{thr}}=8$ and bottom, ${\rho }_{\mathrm{thr}}=9$) and hopeless cuts (blue, orange, green, red: ${\rho }_{\mathrm{cut}}=5$, 6, 7, 8). Right: ratios of the estimated detection probability with a hopeless cut to the detection probability without any hopeless cut. Separate Monte Carlo sets are drawn within each bin (uniformly spaced in comoving volume and source-frame time), and shaded regions approximate $1\sigma$ uncertainty from the finite Monte Carlo sample size. Black lines show the amount by which truncation error in estimates of $P(\det |z)$ could be compensated by redshift evolution of the form $dN/dz\sim (1+z{)}^{\kappa -1}(d{V}_c/dz)$.