Model-based cross-correlation search for gravitational waves from Scorpius X-1

We consider the cross-correlation search for periodic gravitational waves and its potential application to the low-mass x-ray binary Sco X-1. This method coherently combines data not only from different detectors at the same time, but also data taken at different times from the same or different detectors. By adjusting the maximum allowed time offset between a pair of data segments to be coherently combined, one can tune the method to trade off sensitivity and computing costs. In particular, the detectable signal amplitude scales as the inverse fourth root of this coherence time. The improvement in amplitude sensitivity for a search with a maximum time offset of one hour, compared with a directed stochastic background search with 0.25-Hz-wide bins, is about a factor of 5.4. We show that a search of one year of data from the Advanced LIGO and Advanced Virgo detectors with a coherence time of one hour would be able to detect gravitational waves from Sco X-1 at the level predicted by torque balance over a range of signal frequencies from 30 to 300 Hz; if the coherence time could be increased to ten hours, the range would be 20 to 500 Hz. In addition, we consider several technical aspects of the cross-correlation method: We quantify the effects of spectral leakage and show that nearly rectangular windows still lead to the most sensitive search. We produce an explicit parameter-space metric for the cross-correlation search, in general, and as applied to a neutron star in a circular binary system. We consider the effects of using a signal template averaged over unknown amplitude parameters: The quantity to which the search is sensitive is a given function of the intrinsic signal amplitude and the inclination of the neutron-star rotation axis to the line of sight, and the peak of the expected detection statistic is systematically offset from the true signal parameters. Finally, we describe the potential loss of signal-to-noise ratio due to unmodeled effects such as signal phase acceleration within the Fourier transform time scale and gradual evolution of the spin frequency.

Figure 1

Plot of ${\delta }_{T_{\mathrm{sft}}}(f-f^{\prime })$ defined in (2.17) which determines the signal contribution to a given frequency bin of a short Fourier transform (SFT) of duration $T_{\mathrm{sft}}$ according to (2.16). Since the spacing between frequency bins is $\delta f=1/T_{\mathrm{sft}}$, there will be, for a given signal frequency $f_K$, one bin whose value of ${\kappa }_{Kk}=(f_k-f_K)/\delta f$ lies between each pair of vertical solid lines.

Figure 2

Plot of $\frac{{\mathcal{A}}_{+}^2+{\mathcal{A}}_{\times }^2}{2}$ and $\frac{{\mathcal{A}}_{+}^2-{\mathcal{A}}_{\times }^2}{2}$, the coefficients of the two contributions to $\mathrm{Re}{\mathrm{\Gamma }}_{KL}$ in (3.8). The factor $\frac{{\mathcal{A}}_{+}^2+{\mathcal{A}}_{\times }^2}{2}$ is also equal to $\frac{2}{5}\frac{(h_0^{\mathrm{eff}}{)}^2}{h_0^2}$ where $(h_0^{\mathrm{eff}}{)}^2$ is the combination of $h_0$ and $\cos \iota$ approximately measured by the cross-correlation statistic, as shown in, e.g., Eq. (3.9).

Figure 3

Plot of weighted averages ${⟨\mathrm{\Delta }{t}_{\alpha }^2⟩}_{\alpha }$ and ${⟨{\sin }^2\frac{\pi \mathrm{\Delta }{t}_{\alpha }}{P_{\mathrm{orb}}}⟩}_{\alpha }$ appearing in the metric components (4.31) as a function of maximum allowed lag time $T_{\max }$. The dotted lines show the approximate values (4.33) and (4.34) neglecting the variation of the weighting factor. The solid line (labeled HLV) shows the value for a search using detectors at the LIGO Hanford, LIGO Livingston, and Virgo sites, assuming a source at the sky position of Sco X-1, and that all detectors have the same sensitivity at the relevant frequency, and all sidereal times are evenly sampled. The dashed line (HL) shows the same thing for a search using only the LIGO detectors at Hanford and Livingston. The actual weighted averages (and therefore the number of templates needed to cover the parameter space) are less than the approximate ones, because the geometrical factor $({\mathrm{\Gamma }}_{\alpha }^{\mathrm{ave}}{)}^2$ weights smaller lag times more.

Figure 4

The optimal SFT length $T_{\mathrm{sft}}$, defined in (5.34) and (5.31), as a function of frequency, for a signal with the most likely orbital parameters for Sco X-1, as given in Table 1, assuming that $d=3$, i.e., the density of points in parameter space grows as the third power of the coherence time $T_{\max }$. This is appropriate for a search over, e.g., frequency $f_0$, projected semimajor axis $a_p$, and time of ascension $t_{\mathrm{asc}}$ (when the uncertainty in the period $P_{\mathrm{orb}}$ is small enough that a single value may be assumed), in the case where $T_{\max }$ is small compared with $P_{\mathrm{orb}}$. The solid line represents a more optimistic scenario where the average cosine appearing in the second term of (5.31) is approximately unity, which should also be the case if $T_{\max }\ll P_{\mathrm{orb}}$. The dashed line represents a worst-case scenario where the average is approximately zero. The optimal SFT length maximizes the expected SNR in (5.30) and represents a balance between two competing effects: if $T_{\mathrm{sft}}$ is too large, phase acceleration will lead to a loss in SNR compared to the ideal formula (3.19); if $T_{\mathrm{sft}}$ is too small, the large number of SFT pairs in the computation will lead to a restriction on the possible $T_{\max }$ achievable at fixed computing cost, and reduce the ideal SNR itself.

Figure 5

Expected sensitivity (3.25) for a search of one year of coincident data from either the two LIGO detectors (labeled HL) or the three $\mathrm{LIGO}+\mathrm{Virgo}$ detectors (labeled HLV), at design sensitivity. The value plotted is the observable $h_0$ at 5% false dismissal probability, assuming an overall false alarm probability of 5% and a trials factor of $10^8$ for a single-template false alarm probability of $5\times {10}^{-10}$ (i.e., see Sec. 3c and Table 3). The three curves in each set are, from top to bottom, for $T_{\max }=6\min$, 60 min and 10 hr. They are compared to the signal strength (6.2) predicted by the torque balance argument [12].

Figure 6

Relative scaling of expected computing cost per logarithmic frequency interval for a search of one year of coincident data from either the two LIGO detectors (labelled HL) or the three $\mathrm{LIGO}+\mathrm{Virgo}$ detectors (labelled HLV). The three curves in each set, are, from top to bottom, for $T_{\max }=10\mathrm{hr}$, 60 min and 6 min. The calculation assumes that the computing cost scales with the number of SFT pairs times the number of points in parameter space. It also assumes that the optimal SFT length $T_{\mathrm{sft}}$ given by (5.34) and (5.31) has been chosen at each frequency, and that we are searching over frequency and two orbital parameters. The approximate scaling is as $f_0^4{T}_{\max }^4$, so for instance a $T_{\max }=60\min$ search from 100 to 200 Hz would consume the same resources as a $T_{\max }=6\min$ search from 1000 to 2000 Hz. For reference, the mock data analysis in [28], which was accomplished in approximately 20,000 CPU-days, covered a set of roughly logarithmically-spaced frequency bands totaling 250 Hz spread from 50 Hz to 1375 Hz at a range of $T_{\max }$ values from 9 to 90 min.

Figure 7

The general Tukey window $w_{\beta }(\theta )$ as defined in (a12) for a generic value of the parameter $\beta \in [0,1]$, where $\beta$ is the fraction of the window length taken up by the transitions from 0 to 1 and back.

Figure 8

Specific versions of the general Tukey window $w_{\beta }(\theta )$ as defined in (a12): the rectangular window $w^{\text{rect}}(\theta )=w_0(\theta )$, a canonical ($\beta =\frac{1}{2}$) Tukey window $w_{1/2}(\theta )$, and the Hann window $w^{\text{Hann}}(\theta )=w_1(\theta )$.

Figure 9

The window leakage function ${\xi }^w(\kappa )=T_{\mathrm{sft}}^{-1}{\delta }_{T_{\mathrm{sft}}}^w(\kappa /T_{\mathrm{sft}})$ defined in (a2) for the windows shown in Fig. 8. The explicit formulas are given in (a14) for the rectangular window, (a16) for the canonical ($\beta =\frac{1}{2}$) Tukey window, and (a15) for the Hann window. Note that the version for rectangular-windowed data is just ${\xi }^{\text{rect}}(\kappa )=T_{\mathrm{sft}}^{-1}{\delta }_{T_{\mathrm{sft}}}(\kappa /T_{\mathrm{sft}})=\mathrm{sinc}(\kappa )=\frac{\sin \pi \kappa }{\pi \kappa }$, which is the finite-time delta function plotted in Fig. 1.

Figure 10

The leakage factor $⟨({\mathrm{\Xi }}^w{)}^2⟩$ appearing in (a11) for a search using between one and six bins from each SFT, assuming a general Tukey window from the family (a12). We see that, for any number of bins, the most sensitive search is when $\beta =0$, i.e., for rectangular windows. In particular, when a single bin is used from each SFT, we have $⟨{\mathrm{\Xi }}^2⟩=0.774$ for rectangular windowing ($\beta =0$), $⟨({\mathrm{\Xi }}_{\beta }^w{)}^2⟩=0.699$ for a canonical ($\beta =\frac{1}{2}$) Tukey window, and $⟨({\mathrm{\Xi }}^{\text{Hann}}{)}^2⟩=0.601$ for Hann windowing ($\beta =1$). Note that the $\beta =0$ value on each curve is just the corresponding “cumulative” number from Table 2.

Figure 11

Eigenvalues $\{{\omega }_K\}$ of the weights matrix $\mathbf{W}$ defined in (2.35) for two scenarios. On the left, we show one day of observation with the LLO, LHO and Virgo detectors, assuming equal sensitivity, with $T_{\mathrm{sft}}=900\mathrm{s}$ and $T_{\max }=3600\mathrm{s}$. On the right, we show one year of observation under the same conditions, constructed as the union of 365 such days, spread throughout the year. In both cases, the start and end of each day include data gaps of 900–1800 s, randomly and independently generated for each detector.

Figure 12

False alarm probabilities for the cross-correlation statistic in the day-long and year-long scenarios considered in Fig. 11, using the explicit formula (b9) as well as numerical integration of (b10), along with the probabilities we would get if we assumed the statistic to be Gaussian. For a day-long observation (with three detectors, $T_{\mathrm{sft}}=900\mathrm{s}$ and $T_{\max }=3600\mathrm{s}$), both methods give comparable results, but the Gaussian approximation is invalid for single-template false alarm probabilities below about $10^{-2}$. Note that for large signal values, a single exponential term dominates. For a year-long observation, practical calculation with (b9) is impossible due to underflow issues. The numerical integration of (b10) becomes unstable for false alarm probabilities below $10^{-12}$, but not before quantifying deviations from the Gaussian approximation even for a year-long observation.