Multiwaveform cross-correlation search method for intermediate-duration gravitational waves from gamma-ray bursts

Gamma-ray bursts (GRBs) are flashes of $\gamma$-rays thought to originate from rare forms of massive star collapse (long GRBs) or from mergers of compact binaries (short GRBs) containing at least one neutron star (NS). The nature of the postexplosion/postmerger remnant [NS versus black hole (BH)] remains highly debated. In $\sim 50\%$ of both long and short GRBs, the temporal evolution of the x-ray afterglow that follows the flash of $\gamma$-rays is observed to “plateau” on timescales of $\sim {10}^2–{10}^4\mathrm{s}$ since explosion, possibly signaling the presence of energy injection from a long-lived, highly magnetized NS (magnetar). The cross-correlation algorithm (CoCoA) proposed by Coyne et al. [Phys. Rev. D 93, 104059 (2016)] aims to optimize searches for intermediate-duration (${10}^2–{10}^4\mathrm{s}$) gravitational waves (GWs) from GRB remnants. In this work, we test CoCoA on real data collected with ground-based GW detectors. We further develop the detection statistics on which CoCoA is based to allow for multiwaveform searches spanning a physically motivated parameter space, so as to account for uncertainties in the physical properties of GRB remnants.

Figure 1

GW signal frequency (left) and amplitude (right) as a function of time for the waveforms used in this study (see Corsi and Mészáros [26], and also Sec. 3 and Table 1). The thick black portions of the CM09short/long waveforms represent the $256/1024$-s-long segments where the sliding average of the signal strain is maximized. These portions of the CM09long/short signals are used in this study to allow for direct comparison with the results presented in [31] (see text for further discussion).

Figure 2

PSDs of the LIGO O2 data used in this analysis (orange and green for LH and LL, respectively). We also plot the PSDs of the simulated colored Gaussian noise (blue) and of the simulated white Gaussian noise (black) that we use for comparison. The vertical dashed and dotted lines mark the frequency range spanned by the CM09long GW signal used in these tests (see Sec. 3 for discussion).

Figure 3

We show how CM09long time-frequency tracks with start times 14 s (or $7\times \mathrm{\Delta }{T}_{\mathrm{SFT}}$) apart never overlap. Each $\mathrm{\Delta }{T}_{\mathrm{SFT}}$ is represented with a black rectangle, and each of three nonoverlapping CM09long tracks are plotted with a different color as an example. By calculating $\rho$ along each of the nonoverlapping CM09long tracks that can be fitted in a given stretch of real detector data, we maximize the number of independent realizations of $\rho$ populating the statistical distributions shown in Fig. 4.

Figure 4

CoCoA tests on real O2 data from two detectors (LL and LH) in the stochastic (top), matched-filter (center), and semicoherent (with four coherent segments; bottom) limits. Gray histograms are background distributions normalized by the variance in Eq. (2.18) in the stochastic limit, and by $C_{\chi }$ in Eq. (2.24) for nonstochastic searches. Black dashed lines are a normalized Gaussian with zero mean (top); a central ${\chi }^2$ with 2 d.o.f. (center); a central ${\chi }^2$ with $2N_{\mathrm{coh}}=8$ d.o.f. (bottom). Red histograms are normalized (by $C_{\chi }$, $C_{\chi }/N_{\mathrm{coh}}$ and the variance for the matched-filter, semicoherent and stochastic limits, respectively) distributions of real data when CM09long is injected at a distance such that $\mathrm{FAP}=0.1\%$ and $\mathrm{FDP}=50\%$, assuming optimal source orientation. Red lines are a normalized Gaussian with mean as in Eq. (2.17) (top); a noncentral ${\chi }^2$ with 2 d.o.f. (center) and noncentrality parameter $\lambda$ as in Eq. (2.25); a noncentral ${\chi }^2$ with $2N_{\mathrm{coh}}=8$ d.o.f. (bottom) and noncentrality parameter $\lambda$.

Figure 5

Effect of timing uncertainties on a multitrial CoCoA search: results for the background statistic. We use simulated LIGO data with (colored) O2 PSD and set $\mathrm{\Delta }{T}_{\mathrm{SFT}}=0.25\mathrm{s}$. In order to keep the number of trials fixed, for searches with $t_{\mathrm{unc}}=120\mathrm{s}$ we take $N_{\mathrm{on}}=60$, and for searches with $t_{\mathrm{unc}}=2\mathrm{s}$ we take $N_{\mathrm{on}}=1$. We compare recovered results in the absence of a signal (gray histograms) to the analytical expectations derived in Sec. 5c (black lines). To match the recovered results, we define an effective number of trials for Eq. (5.6) that accounts for dependencies between trials (red lines). In the first row we search for CM09long, only varying the start time of the waveform for each trial, taking identical values of $T_{\mathrm{coh}}=4\mathrm{s}$, but using different values of $N_{\mathrm{on}}$. In the first column we search for the same waveform and use identical values of $N_{\mathrm{on}}=60$, but show different values of $T_{\mathrm{coh}}$. Lastly in the second column we search with identical values of $N_{\mathrm{on}}=1$ and $T_{\mathrm{coh}}=4\mathrm{s}$, but search for different waveforms (CM09long/CM09short).

Figure 6

Horizon distances at 50% FDP and a FAP of 1% for a source located at the sky position of GW170817, and for a search of CM09long with $t_{\mathrm{unc}}=120\mathrm{s}$ and different values of $N_{\mathrm{on}}$. The computational time of a search scales with $N_{\text{trials}}$. CoCoA distance horizons are compared with those of a single-trial stochastic search on a perfectly matching waveform (no temporal or physical uncertainties; blue dashed lines).

Figure 7

Horizon distances at 50% FDP and a FAP of 1% for a source located at the sky position of GW170817, and for a search of CM09short with $t_{\mathrm{unc}}=120\mathrm{s}$ and different values of $N_{\mathrm{on}}$. The computational time of a search scales with $N_{\text{trials}}$. CoCoA distance horizons are compared with those of a single-trial stochastic search on a perfectly matching waveform (no temporal or physical uncertainties; blue dashed lines).

Figure 8

Horizon distances at 50% FDP and a FAP of 1% for a source located at the sky position of GW170817, and for searches of CM09long (left) and CM09short (right) with $t_{\mathrm{unc}}=2\mathrm{s}$ and $N_{\mathrm{on}}=1$. CoCoA distance horizons are compared with those of a single-trial stochastic search on a perfectly matching waveform (no temporal or physical uncertainties; blue dashed lines).

Figure 9

Horizon distances at 50% FDP and 1% FAP, and the sky position of GW170817 for a search of 26 sets of magnetar physical parameters obtained by shifting of $\mathrm{\Delta }B={10}^{12}\mathrm{G}$, $\mathrm{\Delta }M=5\times {10}^{-2}{M}_{\odot }$, and $\mathrm{\Delta }R=0.2\mathrm{km}$, the values of ($B$, $M$, $R$) for CM09long/CM09short. The search assumes $t_{\mathrm{unce}}=120\mathrm{s}$ and $N_{\mathrm{on}}=60$ for CM09long (left), and $t_{\mathrm{unc}}=2\mathrm{s}$ and $N_{\mathrm{on}}=1$ for CM09short (right), giving 9 trials for each choice of physical parameters and 234 total trials. CoCoA distance horizons are compared with those of a single-trial stochastic search on a perfectly matching waveform (no temporal or physical uncertainties; blue dashed lines).

Figure 10

Example of detection efficiency ($1-\text{FDP}$) vs distance $d$ for a semicoherent search of waveforms with small changes in physical parameters to CM09long when CM09long is injected as in Sec. 6b. In this example, $N_{\mathrm{on}}=60$, and $T_{\mathrm{coh}}=4\mathrm{s}$ which is the coherence time that produces the largest distance found from a search of CM09long as in Fig. 9. The distance corresponding to a 50% FDP is marked in green. The error in distance is taken by first finding the error in efficiency through the DKW inequality. Efficiency errors are then connected through the sigmoid function to find errors in distance at the chosen FDP (50% in this example). See Sec. 6a or [31] for more discussion.

Figure 11

The combined background distribution of the ${\tilde{\rho }}_{\max }$ statistic for a search on O2 data that considers Bar1–Bar6, the waveforms with all combinations of $(\beta ,M,R,B)$ one step away from those of Bar1–Bar6, and nine choices of $t_{\mathrm{on}}$, for a total number of trials of $N_{\text{trial}}=481$. The gray histogram shows the results of the search over 9.1 days of real O2 data before the time of the merger of GW170817. This background is used to set the FAP threshold for the comparison to the STAMP test. The pink histogram shows the background when using simulated O2-like Gaussian noise similar to what is used in Sec. 5. The thresholds for a FAP of 1% for both the real and simulated data are show using black and red dashed lines, respectively.

Figure 12

Horizon distances for STAMP as in [20] compared to the ones of a CoCoA stochastic search. CoCoA (dark and light blue), even with the less sensitive stochastic limit, is more sensitive than STAMP (red and green). However, the gained sensitivity comes at the expense of computational cost. A CoCoA search with $t_{\mathrm{unc}}=8.5$ days (light blue), like that used by the STAMP search presented in [20], produces a number of trials 6 orders of magnitude larger. Similarly, a CoCoA search with $t_{\mathrm{unc}}=2\mathrm{s}$ (dark blue) produces a number of trials only about a factor of 2 smaller than a STAMP search with a much longer timing uncertainty of $t_{\mathrm{unc}}=8.5$ days. We note that the Coherent Wave Burst (cWB) pipeline [36] has also produced upper limits for the GW170817 postmerger search that are comparable to the STAMP ones shown here (see [20]).