Postprocessing methods used in the search for continuous gravitational-wave signals from the Galactic Center

The search for continuous gravitational-wave signals requires the development of techniques that can effectively explore the low-significance regions of the candidate set. In this paper we present the methods that were developed for a search for continuous gravitational-wave signals from the Galactic Center [1]. First, we present a data-selection method that increases the sensitivity of the chosen data set by 20%–30% compared to the selection methods used in previous directed searches. Second, we introduce postprocessing methods that reliably rule out candidates that stem from random fluctuations or disturbances in the data. In the context of [J. Aasi et al., Phys. Rev. D 88, 102002 (2013)] their use enabled the investigation of marginal candidates (three standard deviations below the loudest expected candidate in Gaussian noise from the entire search). Such low-significance regions had not been explored in continuous gravitational-wave searches before. We finally present a new procedure for deriving upper limits on the gravitational-wave amplitude, which is several times faster with respect to the standard injection-and-search approach commonly used.

Figure 1

The histogram shows the mismatch distribution for the search template grid of Eq. (3). The average mismatch is $⟨m⟩\approx 15\%$.

Figure 2

The distributions of the average expected detection statistic over segments, $\overline{\mathrm{E}[2\mathcal{F}]}$. ${}^{—}$ denotes the average over a population of signals with a fixed fiducial GW amplitude and random polarization parameters ($\cos \iota$ and $\psi$). The figure shows that a large fraction of the segments in set $\mathcal{D}$ has high $\overline{\mathrm{E}[2\mathcal{F}]}$ values compared to the other sets. We also compare for each set the average value of $\overline{\mathrm{E}[2\mathcal{F}]}$ over the 630 segments, $⟨\overline{\mathrm{E}[2\mathcal{F}]}⟩$. The higher that number, the more sensitive the data set is.

Figure 3

The antenna pattern functions $\mathcal{P}(t_k)=F_{+}(t_k{)}^2+F_{\times }(t_k{)}^2$ at the midtime $t_k$ of each SFT of set $\mathcal{D}$ ($k$ indicates the order number of the SFTs and $t_k$ is on the $x$ axis). Red color shows the values for H1, black those for L1. The right plot is a zoom of the left plot for a duration of 2.9 days within the first weeks of S5. As described above, the segments are selected to maximize the expected $2\mathcal{F}$ value which explains the distribution of points along the solid curves. The maximum and minimum values of $\mathcal{P}(t)$ for the Hanford detector, just above 0.9 and just below 0.08, are due to the fact that at the latitude of Hanford ($46^{\circ }17^{\prime }{8}^{\prime \prime }$) the GC (which has a declination of $-29^{\circ }{0}^{\prime }28^{\prime \prime }$) can never reach the zenith.

Figure 4

The histograms illustrate the effect of the relaxed known-lines cleaning procedure in the context of the GC search: the black, dashed histogram contains the fraction of candidates that pass the strict known-lines cleaning, and the gray, solid histogram shows the fraction of candidates that pass the relaxed known-lines cleaning.

Figure 5

The left plot shows the averaged outcome of 200 searches that were performed on fake data with parameters randomly distributed over the search parameter space. As explained in the text, the detection statistic values are all normalized to the maximum of their respective search and then averaged over the 200 searches in each grid point. The grid points represent distances from the maximum. The right plot gives the standard deviation for each data point of the left plot. The black box denotes the cluster box. All candidates within this box are ascribed to the loudest representative candidate in the center by the clustering procedure.

Figure 6

The plot shows the fraction of candidates with $⟨2\mathcal{F}⟩$ values larger than half of maximum as a function of distance in frequency (top) and spin-down (bottom), from the maximum. The gray shaded areas denote the boundaries of the cluster box in each dimension.

Figure 7

The distribution of $⟨2\mathcal{F}⟩$ values before and after removing the contribution from the loudest among the different segments (a) for real candidates and (b) for a set of signal injections. Removing the contribution of the loudest segment shifts the distribution of $⟨2\mathcal{F}⟩$ values from just above to below the significance threshold at $⟨2\mathcal{F}⟩=4.77$ (vertical line). This clearly shows that most candidates have significant $⟨2\mathcal{F}⟩$ values only due to a single segment that contributes an extremely large $2\mathcal{F}$ value. The distribution of $⟨2\mathcal{F}⟩$ values from a set of continuous signal injections does not change: the contribution of all segments to the average value is comparable.

Figure 8

The histogram shows the distribution of the $⟨2\mathcal{F}⟩$ values of the 1138 candidates that are investigated in the follow-up search. All candidates with values larger than 6 can be ascribed to a pulsar signal hardware injection that was performed during S5.

Figure 9

The two histograms show the distances (absolute value) between the parameters of the loudest template and the parameters of the injected signal measured in frequency and spin-down bins of the initial search [i.e. with the spacing of Eq. (3)]. The counts refer to values of the distances between the right and left edges of each bin.

Figure 10

This histogram shows the mismatch distribution of the follow-up search. The average value is $⟨m_{\text{coh}}⟩=1\%$. Only in a small fraction of cases ($\lesssim 1\%$) can the mismatch be as high as 4.7%.

Figure 11

This histogram shows the distribution of the maximum $2{\mathcal{F}}_{\text{coh},\mathrm{G}}^{*}$ resulting from a coherent follow-up search. A thousand searches with grids defined by Eq. (14) were performed, with different realizations of pure Gaussian noise (black dashed line). The mean value measured on this distribution is $35\pm 3$ (dashed gray line). Based on the total number of templates $N_{\text{coh}}$, the predicted mean is $E[2{\mathcal{F}}_{\text{coh},\mathrm{G}}^{*}]=40\pm 4$ (solid gray line). From the simulation data the number of effective independent templates can be estimated to $N_{\text{eff}}\sim 0.1N_{\text{coh}}$ (black line). The solid black line is the estimated probability density function for the maximum under the assumption of $N_{\text{eff}}$ independent templates.

Figure 12

False dismissal: The plot shows the distribution of $\mathcal{R}$ for 1000 different realizations of CW signals in Gaussian noise with parameters randomly distributed within the search parameter space. The threshold is set to ${\mathcal{R}}_{\text{thr}}=0.5$ (gray line). Only four of the 1000 signals have $\mathcal{R}<{\mathcal{R}}_{\text{thr}}$, implying a false-dismissal rate of $0.4\%$.

Figure 13

False alarm: The least significant candidates that are considered in the follow-up have $2\mathcal{F}$ values at the significance threshold. Those candidates are expected to result in a value of $2{\mathcal{F}}_{\text{coh}}^{0.77}=105.83$ in the follow-up search if they are due to a CW. This plot shows the distribution of $\mathcal{R}$ values resulting from a follow-up search on 1000 different realizations of Gaussian noise. None of the candidates satisfied the condition $2{\mathcal{F}}_{\text{coh}}^{*}>52.9$, which implies a Gaussian false-alarm rate of $\le 0.1\%$.

Figure 14

Here we show the percentage of excluded parameter space per upper-limit set as a function of frequency. The gray vertical lines denote the frequencies that host known detector artifacts and which are cleaned strictly. The 1 Hz lines are not shown. The periodical variations of the excluded parameter space are due to the 1 Hz harmonics. We remind the reader that the searched frequency band starts at 78 Hz and ends at 496 Hz.

Figure 15

The cumulative distribution of the excluded parameter space per set (black line). The distribution shows a steep increase towards the 13% threshold (gray line). After reaching that threshold, the distribution has a shallow knee up to the 100% loss. As explained in the text, the shape of this distribution simplifies the choice of the threshold.

Figure 16

The plot shows $C(h_0)$ for different $h_0$ values for one example set. By fitting a straight line to the data points in a small enough region around the 90% confidence value the 3% error on the confidence can be translated into an uncertainty of $\lesssim 5\%$ on $h_0^{90\%}$.

Figure 17

The plot shows the distributions of the $⟨2\mathcal{F}⟩$ values of the surviving candidates after each of the postprocessing steps. The vertical solid line denotes the significance threshold at 4.77. The dotted line denotes the candidates that survive the known-lines cleaning. The clustering (solid, dark gray line) lowers the histogram evenly over the different $⟨2\mathcal{F}⟩$ values, as one would expect it. The $\mathcal{F}$-statistic consistency veto (bold, gray line) removes high $⟨2\mathcal{F}⟩$ outliers and lowers the right tail. The application of the permanence veto removes many of the low-significance candidates. All candidates with $⟨2\mathcal{F}⟩$ values above $\sim 6$ survive this step. However, these can all be ascribed to a hardware injection that was performed during S5. The remaining group of 59 “real” CW candidates are then ruled out by the coherent follow-up search and the $\mathcal{R}$ veto. Six of the candidates that can be ascribed to the hardware injection would pass this veto.