Gravitational waves from Scorpius X-1: A comparison of search methods and prospects for detection with advanced detectors

The low-mass X-ray binary Scorpius X-1 (Sco X-1) is potentially the most luminous source of continuous gravitational-wave radiation for interferometers such as LIGO and Virgo. For low-mass X-ray binaries this radiation would be sustained by active accretion of matter from its binary companion. With the Advanced Detector Era fast approaching, work is underway to develop an array of robust tools for maximizing the science and detection potential of Sco X-1. We describe the plans and progress of a project designed to compare the numerous independent search algorithms currently available. We employ a mock-data challenge in which the search pipelines are tested for their relative proficiencies in parameter estimation, computational efficiency, robustness, and most importantly, search sensitivity. The mock-data challenge data contains an ensemble of 50 Scorpius X-1 (Sco X-1) type signals, simulated within a frequency band of 50–1500 Hz. Simulated detector noise was generated assuming the expected best strain sensitivity of Advanced LIGO [1] and Advanced VIRGO [2] ($4\times {10}^{-24}{\mathrm{Hz}}^{-1/2}$). A distribution of signal amplitudes was then chosen so as to allow a useful comparison of search methodologies. A factor of 2 in strain separates the quietest detected signal, at $6.8\times {10}^{-26}$ strain, from the torque-balance limit at a spin frequency of 300 Hz, although this limit could range from $1.2\times {10}^{-25}$ (25 Hz) to $2.2\times {10}^{-26}$ (750 Hz) depending on the unknown frequency of Sco X-1. With future improvements to the search algorithms and using advanced detector data, our expectations for probing below the theoretical torque-balance strain limit are optimistic.

Figure 1

The distribution of SNRs for the open and closed simulated signals.

Figure 2

A comparison of detected signal properties for each search pipeline. We plot the values of $h_0$, optimal SNR, and estimated ${\log }_{10}(p)$ value for the detected (color) and nondetected (grey) signals from each pipeline. The 3rd panel shows the ${\log }_{10}(p)$ values from the search pipelines which were able to estimate reliable values, with the black horizontal dashed line representing the detection threshold of ${\log }_{10}(p)=-2$. Note that the TwoSpect and CrossCorr pipelines generated nominal $p$-values, but as they were known not to be quantitatively accurate (see Sec. pp3-s1 and Sec. pp4-s2), they are not shown here. The full list of detected and nondetected signals is given in Appendix pp1

Figure 3

The upper and lower plots show the detection efficiencies for each pipeline as a function of $h_0$ and multidetector optimal SNR, respectively. The shaded regions represent the 50% uncertainty (interquartile range) after marginalizing over the parameters of a basic sigmoid function $f(x)=(1+e^{-\alpha (\log (x)-\beta )}{)}^{-1}$ using the posterior distribution generated from the $50\mathrm{detection}/\mathrm{nondetection}$ results from the closed signal bands. The posterior was constructed using a Jeffreys prior on $\alpha$ and $\beta$ so that the inferred efficiencies are all less than unity, even for the CrossCorr pipeline, which detected all 50 closed signals. However, the exact turnover point of the CrossCorr efficiency curve is more uncertain and less robust against changes of the fitting procedure than the others.

Figure 4

Comparison of signal amplitude estimates (from detected signals) and upper limits (from nondetected signals) from each pipeline as a function of intrinsic signal frequency. The top panel shows the 95% confidence upper limits on $h_0$ for nondetected signal from each pipeline that provided such results. The black crosses indicate the true value of $h_0$. In the bottom panel, the solid symbols in the top panel show the fractional errors in $h_0$ estimates, and open symbols show the quoted one-sigma error bars, again divided by the true $h_0$ value. The uncertainty in $h_0$ is comparable for all pipelines which provided estimates, since all were dominated by the unknown value of $\cos \iota$. The complete details of the amplitude estimates and upper limits can be found in Appendix pp1.

Figure 5

Comparisons of parameter estimation for detected signal from each pipeline as a function of intrinsic signal frequency. The solid symbols in each plot show the difference between the true and estimated values of intrinsic signal frequency, projected semimajor axis, and time of passage of the ascending node for the detected signals for pipelines which report those quantities. The open symbols show the quoted one-sigma error bars corresponding to each estimate. The complete details of the parameter estimates can be found in Appendix pp1. The causes of notable outliers seen in the estimation of the orbital semimajor axis by TwoSpect are discussed in Sec. 6b.

Figure 6

Distributions of estimated parameter offsets relative to their estimated measurement uncertainties for the detected MDC signals. Only TwoSpect and CrossCorr return an estimate of the projected semimajor axis, and only CrossCorr returns an estimate of time at the ascending node. None of the pipelines return period estimates.

Figure 7

Distribution of offsets between $h_0$ upper limits and actual values nondetected MDC signal. The Polynomial search returns only frequency estimation for detection and does not return an $h_0$ upper limit, while CrossCorr had no nondetections among the 50 closed signals.

Figure 8

Simulated correction of ${\widehat{Y}}_{\mathrm{tot}}$ for the Radiometer search across the standard 0.25 Hz-wide bin for a variety of frequencies. Bin center falls at the frequency offset of 0 Hz. The traces, from bottom to top, display the simulation for 40 (black, solid), 290 (blue, dashed), 538 (red, dot-dashed), 1000 (magenta, solid), 1076 (cyan, dashed), 1290 (black, dot-dashed), and 1500 Hz (blue, solid). Below 538 Hz note that at the bin boundary, where only half of the SNR is in this bin, the correction factor is two. The correction factor is one when the signal is completely within the bin. $\mathrm{\Delta }f$ is larger for 290 Hz than for 40 Hz, thus the frequencies at which the injection is completely within the chosen bin (and thus has a correction factor of one) are fewer. Above 538 Hz, it is not possible to have the entire injection within a single frequency bin $df$. At most, $df/\mathrm{\Delta }f$ of the injection can be within a single bin, resulting in a correction factor that is always greater than one. Above 1076 Hz, ${\widehat{Y}}_{\mathrm{tot}}$ is always $df/\mathrm{\Delta }f$ of the injection.

Figure 9

Frequency bin correction factor necessary to account for signals spanning multiple bins in the Radiometer search. The solid blue line denotes the correction factor and the dashed cyan lines denote its $1\sigma$ uncertainty. At low frequencies, it is statistically infrequent for a signal to span multiple frequency bins, resulting in a correction factor close to one. At high frequencies, signals always span at least two frequency bins which is reflected by the larger correction factor. The increasing uncertainty with increasing frequency below 538 Hz reflects that increasing $\mathrm{\Delta }f$ results in fewer occurrences of a simulation falling completely within a frequency bin which results in a broadened distribution of the simulated trials. The decreasing uncertainty with increasing frequency above 538 Hz is affected by two factors. One is that increasing $\mathrm{\Delta }f$ increases the occurrences among the simulated trials of completely spanning the frequency bin. The other factor is that the range of correction factors for a particular frequency bin lessens with increasing frequency (and hence increasing $\mathrm{\Delta }f$).

Figure 10

Ratio of measured and corrected to injected $h_0$ for the Radiometer search for the open signals. An extra empirical factor of $\sqrt{3}$ has been included in the uncertainties. When the measured and the injected $h_0$ agree to within one standard deviation, the error bars for the data (red) should intersect the blue line which has a value of one.

Figure 11

Example of $T_{\max }$ values for the CrossCorr search divided according to orbital parameter space. The “inner regions,” which are more likely to contain the signal parameters, use a longer coherence length $T_{\max }$.