Upper limits on gravitational waves from Scorpius X-1 from a model-based cross-correlation search in Advanced LIGO data

We present the results of a semicoherent search for continuous gravitational waves from the low-mass X-ray binary Scorpius X-1, using data from the ﬁ rst Advanced LIGO observing run. The search method uses details of the modeled, parametrized continuous signal to combine coherently data separated by less than a speci ﬁ ed coherence time, which can be adjusted to trade off sensitivity against computational cost. A search was conducted over the frequency range 25 – 2000 Hz , spanning the current observationally constrained range of binary orbital parameters. No signi ﬁ cant detection candidates were found, and frequency-dependent upper limits were set using a combination of sensitivity estimates and simulated signal injections. The most stringent upper limit was set at 175 Hz , with comparable limits set across the most sensitive frequency range from 100 to 200 Hz . At this frequency, the 95% upper limit on the signal amplitude h 0 is ´ - 2.3 10 25 marginalized over the unknown inclination angle of the neutron star ’ s spin, and ´ - 8.0 10 26 assuming the best orientation ( which results in circularly polarized gravitational waves ) . These limits are a factor of 3 – 4 stronger than those set by other analyses of the same data, and a factor of ∼ 7 stronger than the best upper limits set using data from Initial LIGO science runs. In the vicinity of 100 Hz , the limits are a factor of between 1.2 and 3.5 above the predictions of the torque balance model, depending on the inclination angle; if the most likely inclination angle of 44 ° is assumed, they are within a factor of 1.7.


Introduction
Rotating neutron stars (NSs) are the primary expected source of continuous, periodic gravitational waves (GWs) for groundbased GW detectors. Targets include known pulsars (Aasi et al. 2014a), non-pulsating NSs in supernova remnants (Wette et al. 2008;Abadie et al. 2010;Aasi et al. 2015a), and unknown isolated (Aasi et al. 2016;Abbott et al. 2016a) or binary NSs (Aasi et al. 2014b). A particularly promising source is an accreting NS in a low-mass X-ray binary (LMXB); accretion torque spins up the NS into the frequency band of the detectors, and the accretion can generate an asymmetric mass or current quadrupole that acts as the source for the GWs (Watts et al. 2008). An approximate equilibrium between the accretion spinup and GW spin-down, as well as other spin-down torques can produce a signal that is nearly periodic in the NSʼs rest frame, and then Doppler-shifted due to the orbital motion of the NS and the motion of the detector on the surface of the Earth. Such an equilibrium scenario would produce a relation between the observed accretion-induced X-ray flux of the LMXB and the expected strength of the GWs. ScorpiusX-1 (ScoX-1), the most luminous LMXB, is therefore a promising potential source of GWs (Papaloizou & Pringle 1978;Wagoner 1984;Bildsten 1998). ScoX-1 is presumed to consist of an NS of mass »  M 1. 4 in a binary orbit with a companion star of mass »  M 0.4 (Steeghs & Casares 2002). Some of the parameters inferred from observations of the system are summarized in Table 1.
Several methods were used to search for ScoX-1 in data from the Initial LIGO science runs of 2002-2011: Abbott et al. (2007a) performed a fully coherent search (Jaranowski et al. 1998) on six hours of data from the second science run. Starting with the fourth science run, results for ScoX-1 were reported (Abbott et al. 2007b;Abadie et al. 2011) as part of a search for stochastic signals from isolated sky positions (Ballmer 2006). In the fifth science run, a search (Aasi et al. 2015b) was done for Doppler-modulated sidebands associated with the binary orbit (Messenger & Woan 2007;Sammut et al. 2014). In the sixth science run, ScoX-1 was included in a search (Aasi et al. 2014b) principally designed for unknown binary systems (Goetz & Riles 2011), and this method 149 Deceased, March 2016. 150 Deceased, March 2017. 151 Deceased, February 2017. 152 Deceased, December 2016.
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was subsequently improved to search directly for ScoX-1 (Meadors et al. 2016) and applied to Initial LIGO data (Meadors et al. 2017). A mock data challenge (Messenger et al. 2015) was conducted to compare several of the methods to search for ScoX-1, and the most sensitive (detecting all 50 simulated signals in the challenge, and 49 out of the 50 "training" signals) was the cross-correlation (CrossCorr) method (Dhurandhar et al. 2008;Whelan et al. 2015) used in the present analysis. 153 The Advanced LIGO detectors (Aasi et al. 2015c) carried out their first observing run (O1) from 2015 September 12 to 2016 January 19 (Abbott et al. 2016b). Searches for transient signals were carried out in near-real time and resulted in the observation of the binary black hole (BBH) mergers GW 150914 (Abbott et al. 2016c) and GW 151226 (Abbott et al. 2016d), and the possible BBH merger LVT 151012 (Abbott et al. 2016b), as well as upper limits on the rates and strengths of other sources (Abbott et al. 2016e, 2017a(Abbott et al. 2016e, , 2017b. Searches for persistent stochastic or periodic sources were conducted using data from the full duration of the run and include searches for isotropic and anisotropic stochastic signals (Abbott et al. 2017c(Abbott et al. , 2017d) and a variety of known and unknown NSs (Abbott et al. 2017e). So far, two analyses including searches for GWs from Sco X-1 besides the current one have been released: Abbott et al. (2017d) included the direction of ScoX-1 in their directed unmodeled search for persistent GWs, and Abbott et al. (2017f) performed a directed search for ScoX-1 using a hidden Markov model.

Model of GWs from Sco X-1
The modeled GW signal from a rotating NS consists of a "plus" polarization component, , and a "cross" polarization component, . The signal recorded in a particular detector will be a linear combination of + h and h × determined by the detector's orientation as a function of time. The two polarization amplitudes are which is equal to h 0 2 for circular polarization and h 8 0 2 for linear polarization (Messenger et al. 2015; note that this differs by a factor of 2.5 from the definition of ( ) h 0 eff 2 in Whelan et al. 2015).
It has been suggested (Papaloizou & Pringle 1978;Wagoner 1984;Bildsten 1998) that an LMXB may be in an equilibrium state where the spin-up due to accretion is due to the spin-down due to GWs. In that case, the GW amplitude can be related to the accretion rate, as inferred from the X-ray flux F X (Watts et al. 2008): For ScoX-1, using the observed X-ray flux =F 3.9 X ---10 erg cm s 7 2 1 from Watts et al. (2008), and assuming that the GW frequency f 0 is twice the spin frequency n s (as would be the case for GWs generated by triaxiality in the NS), the torque Notes. Uncertainties are s 1 unless otherwise stated. There are uncertainties (relevant to the present search) in the projected velocity amplitude K 1 of the NS, the orbital period P orb , and the time T asc at which the neutron star crosses the ascending node (moving away from the observer), measured in the solar system barycenter. The orbital eccentricity of ScoX-1 is believed to be small (Steeghs & Casares 2002;Wang 2017) and is ignored in this search. The inclusion of eccentric orbits would add two search parameters that are determined by the eccentricity and the argument of periapse (Messenger 2011;Leaci & Prix 2015). a The sky position (as quoted in Abbott et al. 2007a and derived from Bradshaw et al. 1999) is determined to the microarcsecond, and therefore can be treated as known in the present search. b The inclination i of the orbit to the line of sight, from the observation of radio jets in Fomalont et al. (2001), is not necessarily the same as the inclination angle ι of the neutron star's spin axis, which determines the degree of polarization of the GW in Equation (1). c The value of the projected orbital velocity K 1 is difficult to determine experimentally, and previous works used a value from Abbott et al. (2007a), which was derived with some assumptions from Steeghs & Casares (2002) and equivalent to  -40 5km s 1 . The broader range listed here comes from Doppler tomography measurements and Monte Carlo simulations in Wang (2017), which show K 1 to be weakly determined beyond the constraint that   , which was used to determine the parameter range in Table 2. d The time of ascension T asc , at which the neutron star crosses the ascending node (moving away from the observer), measured in the solar system barycenter, is derived from the time of inferior conjunction of the companion given in Galloway et al. (2014) by subtracting P 4 orb . It corresponds to a time of 2008 June 17 16:06:20 UTC and can be propagated to other epochs by adding an integer multiple of P orb , which results in increased uncertainty in T asc and correlations between P orb and T ; asc see Figure   Recent works (Haskell et al. 2015a(Haskell et al. , 2015b have cast doubt on the ubiquity of the GW torque balance scenario in light of other spin-down mechanisms; the torque balance level remains an important benchmark for search sensitivity, and the detection or non-detection at or below that level would provide insight into the behavior of accreting NSs.

CrossCorr Search Method
The CrossCorr method was presented in Dhurandhar et al. (2008) and refined for application to ScoX-1 in Whelan et al. (2015). It was applied to simulated Advanced LIGO data in a mock data challenge (Messenger et al. 2015;Y. Zhang et al. 2017, in preparation). It was originally developed as a modelbased improvement of the directional stochastic search of Ballmer (2006), which has been used to set limits on gravitational radiation from specific sky directions including ScoX-1 (Abbott et al. 2007b;Abadie et al. 2011). The method allows data to be correlated up to an adjustable coherence time T max . The data are split into segments of length T sft between 240 and 1400 s (depending on frequency) and Fourier transformed. In a given data segment or short Fourier transform (SFT), the signal is expected to be found in a particular Fourier bin (or bins, considering the effects of spectral leakage). The signal bins are determined by the intrinsic frequency and the expected Doppler shift, which is in turn determined by the time and detector location, as well as the assumed orbital parameters of the LMXB. If the SFTs are labelled by the index K, L, etc., which encodes both the detector in question and the time of the SFT, and z K is the appropriately normalized Fourier data in the bin(s) of interest, the CrossCorr statistic has the form This includes the product of the data from SFTs K and L, where KL is in a list of allowed pairs , defined by < K L and , the times of the two different data segments should differ by no more than some specified lag time T max , which we also refer to as the coherence time. The complex weighting factors W KL are chosen (according to Equations (2.33)-(2.36) and (3.5) of Whelan et al. 2015) to maximize the expected statistic value subject to the normalization r = ( ) Var 1. The expected statistic value is then (this is the quantity called  ave in Whelan et al. 2015) and h 0 eff is the combination of h 0 and i cos defined in Equation (2), S K is constructed from the noise power spectrum and G KL ave from the antenna patterns for detectors K and L at the appropriate times, N det is the number of detectors participating in the search, T obs is the observing time per detector, and the factor of 0.903 arises from spectral leakage, assuming we consider contributions from all Fourier bins. (See Equation (3.19) of Whelan et al. 2015 for more details.) Increasing T max increases the sensitivity of the search, but also increases the computing cost. In order to maximize the chance for a potential detection, a range of choices for T max was used for different values of signal frequency and orbital parameters. The method used longer coherence times in regions of parameter space where (1) the detectable signal level given the frequency-dependent instrumental noise was closer to the expected signal strength from torque balance, (2) the cost of the search was lower due to template spacing, i.e., at lower frequencies and a i sin values, or (3) the signal had higher prior probability of being found, i.e., closer to the most likely value of T asc . This is illustrated in Figure 2. The full set of coherence times used ranges from 25,290 s for 25-50 Hz (for the most likely T asc and smallest a i sin values) to 240 s at frequencies above 1200 Hz. The search was performed using a bank of template signals laid out in hypercubic lattice in the signal parameters of intrinsic frequency f 0 , projected semimajor axis a i sin , time of ascension T asc , and (where appropriate) orbital period P orb . The range of values in each direction, motivated by Table 1 and Figure 1, is shown in Table 2. The lattice spacing for the initial search was chosen to correspond to a nominal metric mismatch (fractional loss of signal-to-noise ratio (S/N) associated with a one-lattice-spacing offset in a given direction, assuming quadratic approximation) of 25% in each of the four parameters, using the metric computed in Whelan et al. (2015). The lattice was constructed (and spacing computed) for each of the 18 orbital parameter space cells shown in Figure 2 in each 0.05 Hz-wide frequency band. This resulted in a total of9 10 5 -2 10 8 detection statistics per 0.05 Hz, as detailed in Table 3.

Follow-up of Candidates
Although the detection statistic ρ is normalized to have zero mean and unit variance in Gaussian noise, the trials factor Figure 1. Range of search parameters T asc and P orb . The ellipses show curves of constant prior probability corresponding to s 1 , s 2 , and s 3 (containing 39.3%, 86.5%, and 98.9% of the prior probability, respectively), including the effect of correlations arising from the propagation of the T asc estimate from Table 1 to the mid-run value in Table 2. The search region is chosen to include the s 3 ellipse, with the range of T asc within s 1 receiving a deeper search, as illustrated in Figure 2. The inner and outer regions contain 68.0% and 99.5% of the prior probability, respectively. Note that the apparent inefficiency in searching unlikely regions of T asc -P orb space is mitigated by the fact that the search does not typically resolve P orb , resulting in only one value being included in the search. associated with the large number of templates at different points in parameter space results in numerous candidates with  r 6. A follow-up was performed whenever ρ exceeded a threshold of 6.5 for < < f 25 Hz 400 Hz 0 , 6.2 for < < f 400 Hz 600 Hz 0 , and 6.0 for < < f 600 Hz 2000 Hz 0 . These thresholds were chosen in light of the number of templates searched (cf . Table 3) as a function of frequency. For each 5 Hz band, the threshold at which the expected number of Gaussian outliers was 0.1 (Figure 3). For simplicity, the three thresholds (6.5, 6.2, and 6.0) were chosen to be close to or slightly below these threshold values. As a result, the number of expected Gaussian outliers per 5 Hz was between 0.06 and 0.92. Table 3 shows the total expected number of outliers in each range of frequencies under the Gaussian assumption. Since the noise was not Gaussian, the actual number of signals followed up was significantly larger, as also shown in Table 3.
The follow-up procedure was as follows: 1. Data contaminated by known monochromatic noise features ("lines") in each detector were excluded from the search from the start. In most cases, the timedependent orbital Doppler modulation of the expected signal meant that a narrow line only affected data relevant to a subset of the SFTs from the run. Pairs involving these SFTs needed to be excluded from the sum in Equation (5) and the normalization in Equation (7). The impact of this is illustrated in Figure 6 (in Appendix A), which shows the reduction in the sensitivity J from the omission of pairs from Equation (7). 2. Because a strong signal generally led to elevated statistic values over a range of frequencies, all of the candidates within 0.02 Hz of a local maximum were "clustered" together, with the location of the maximum determining the parameters of the candidate signal. These are known as the "level 0" results. 3. A "refinement" search was performed in a´13 13 13 13 grid in f 0 , with the same T max as the original search, and a i sin , T asc , and P orb centered on the original candidate, with a grid spacing chosen to be one-third of the original spacing (with appropriate modifications for P orb depending on whether that parameter was resolved in the original search). This procedure produces a grid that covers ±2 grid spacings of the original grid and has a mismatch of approximately´» ( ) 25% 1 3 2.8% 2 . The results of this refinement stage are known as "level 1." 4. A deeper follow-up was done on the level 1 results, with T max increased to4 its original value. According to the the theoretical expectation in Equation (7), this should approximately double the statistic value ρ for a true signal. Since this increase in coherence time also produces a finer parameter space resolution, the density of the grid was again increased by a further factor of 3 in each direction (resulting in a mismatch of approximatelý´» ( ) ( ) 25% 1 3 4 3 4.9% 2  2 ), 154 and the size of the grid was´´13 13 13 13. The results of this followup stage are known as "level 2." Signals whose detection statistic ρ decreases at this stage are dropped from the follow-up. 5. Surviving level 2 results were followed up by once again quadrupling the coherence time T max to16 the original value, and increasing the density by a factor of 3 in each direction, for an approximate mismatch of25%´» ( . Again, true signals are expected to approximately double their statistic values, and the grid is modified as at level 2. The results of this round of follow-up are known as "level 3." 6. Unknown instrumental lines in a single detector are likely to produce strong correlations between SFTs from that detector. To check for this, at each stage of follow-up, level 1 and beyond, the CrossCorr statistic r HH was calculated using only data from LIGO Hanford Observatory (LHO), and the statistic r LL using only data from LIGO Livingston Observatory (LLO). If we write r HL as Notes. Ranges for T asc and P orb are chosen to cover s 3 of the observational uncertainties, as illustrated in Figure 1 Table 1, and corresponds to a time of 2015 November 13 02:03:07 UTC, near the middle of the O1 run. (This is useful when constructing the lattice to search over the orbital parameter space, as noted in Whelan et al. 2015.) The increase in uncertainty is due to the uncertainty in P orb . Since, for example, [ ] E HH , we expect true signals to have higher overall detection statistics ρ than the single-detector statistics r HH and r LL . We therefore veto any candidate for which r r > HH or r r > LL at any level of follow-up. This is responsible for the reduction of candidates from level 0 to level 1 seen in Table 3. A total of 127 candidates survive level 3 of the follow-up. To check whether any of them represent convincing detection candidates, we plot in Figure 4 the ratio by which the S/N increases from level 1 to level 2, and from level 2 to level 3. We also plot the corresponding ratios for all of the candidates surviving level 2 (the16 original T max follow-up is not available for candidates that fail level 2), and also for the simulated signal injections described in Appendix A. We see that none of the candidates come close to doubling their S/N at either stage; in fact, none of them even double their S/N from level 1 to level 3. We empirically assess the follow-up procedure with the injections, and find that their S/Ns generally increase by slightly less than the naïvely expected factor of 2 (perhaps because of the increasing mismatch at later follow-up levels). We do see that the injected signals (at least those that survive level 2 follow-up and appear on the plot) nearly all increase their S/N noticeably more than any of the candidates from the search. Also note that of the 666 injected signals (out of 754) that produced ρ values above their respective thresholds, 652 survived all levels of follow-up. (There were four vetoed at level 1, four at level 2, and six at level 3 of the follow-up.) All but a handful of those 652 (between one and four, depending on the stringency of the criterion) are well-separated from the bulk of the search results in Figure 4. We thus conclude that our follow-up procedure is relatively robust, and that there are no convincing detection candidates from the search.
The signal model in this search assumes that the GW frequency f 0 in the NSʼs reference frame is constant. In practice, the Notes. For each range of frequencies, this table shows the minimum and maximum coherence time T max used for the search, across the different orbital parameter space cells (see Figure 2), the threshold in signal-to-noise ratio (S/N) ρ used for follow-up, the total number of templates, and the number of candidates at various stages of the process. (See Section 4 for detailed description of the follow-up procedure.) a This is the number of candidates that would be expected in Gaussian noise, given the number of templates and the follow-up threshold. b This is actual number of candidates (after clustering) that crossed the S/N threshold and were followed up. c This is the number of candidates remaining after refinement. All of the candidates "missing" at this stage have been removed by the single-detector veto for unknown lines. d This is the number of candidates remaining after each one has been followed up with a T max equal to4 the original T max for that candidate. (True signals should approximately double their S/N; any candidates whose S/N goes down have been dropped.) All of the signals present at this stage are shown in Figure 4, which also shows the behavior of the search on simulated signals injected in software. e This is the number of candidates remaining after T max has been increased to16 its original value. Figure 3. Selection of follow-up threshold as a function of frequency. If the data contained no signal and only Gaussian noise, each template in parameter space would have some chance of producing a statistic value exceeding a given threshold. Within each 0.05 Hz frequency band, the total number of templates was computed and used to find the threshold at which the expected number of Gaussian outliers above that value would be 0.1 (short blue lines). For simplicity, the actual follow-up threshold was chosen near or below that level, producing thresholds of 6. equilibrium in an LMXB will be only approximate, and the intrinsic frequency will vary stochastically with time. Whelan et al. (2015) estimated the effect of spin wandering under a simplistic random-walk model in which the GW frequency underwent a net spin-up or spin-down of magnitude |˙| f drift , changing on a timescale T drift . The fractional loss of S/N was estimated as where T run is the duration of the observing run from the start to end, not considering duty factors (in contrast to the T obs appearing in Equation (7) (1998). Similar values have been used in the first ScoX-1 mock data challenge (Messenger et al. 2015) and other work on Sco X-1 (Leaci & Prix 2015;Whelan et al. 2015). 155 In the O1 run, where the run duration was =T 1.12 10 s run 7 , the theoretical fractional loss of S/N will be -- Since our largest initial T max value is 25,290 s, the impact on the initial search and the upper limit of spin wandering at or below this level would be negligible. Note that even spin wandering, which posed no complication for the initial search, could potentially be a limitation for the follow-up procedure, where T max is increased by a factor of 4 at level 2 and a factor of 16 at level 3. In any event, the impact depends on the level of spin wandering present, which is still an area of open research.

Upper Limits
In the absence of a detection, we set upper limits on the strength of gravitational radiation from ScoX-1, as a function of frequency. We used as a detection statistic r max , the maximum statistic value observed in a 0.05 Hz band. We produced frequentist 95% upper limits via a combination of theoretical considerations and calibration with simulated signals, as explained in detail in Appendix A. The starting point was a Bayesian upper limit constructed using the expected statistical properties of the detection statistic and corrected for the reduction of sensitivity due to known lines. A series of simulated signal injections was then performed and used to estimate a global adjustment factor to estimate the amplitude at which a signal would have a 95% chance of increasing the r max value in a band.
The procedure produced two sets of upper limits: a limit on h 0 including marginalization over the unknown inclination angle ι, and an unmarginalized limit on the quantity h 0 eff defined in Equation (2) to which the search is directly sensitive. The h 0 eff upper limit can also be interpreted as a limit on h 0 subject to the assumption of circular polarization (optimal spin orientation corresponding to i cos =±1). It can be converted to a limit assuming linear polarization i = cos 0 by multiplying by = 8 2.83. If we assume that the NS spin is aligned with the binary orbit (as one would expect for an NS spun up by accretion), i » »  i 44 , we obtain a limit on h 0 , which is the h 0 eff upper limit multiplied by 1. 35.
We show the marginalized and unmarginalized upper limits of this search in Figure 5, along with the other upper limits on ScoX-1 set with O1 data: the unmodeled stochastic radiometer (Ballmer 2006) results of Abbott et al. (2017d) and the directed search results of Abbott et al. (2017f) using Viterbi tracking of a hidden Markov model (Suvorova et al. 2016) to expand the applicability of the sideband search (Messenger & Woan 2007;Sammut et al. 2014;Aasi et al. 2015b) over the whole run. The present results improve on these by a factor of 3-4, yielding a marginalized limit of ´h 2.3 10 0 25 and an unmarginalized limit of ´h 8.0 10 0 eff 26 at the most sensitive signal frequencies between around 100 Hz and 200 Hz. The marginalized 95% upper limits from Initial LIGO data (Abadie et al. 2011;Aasi et al. 2015b;Meadors et al. 2017) were all around -1.5 10 25 , so we have achieved an overall improvement of a factor of 6-7 from Initial LIGO to Advanced LIGO's first observing run, a combination of decreased detector noise and algorithmic improvements.
We also plot for comparison the torque balance level predicted by Equation (4). The marginalized limits on h 0 come closest to this level at 100 Hz, where they are within a factor of 3.4 of this theoretical level. In terms of h 0 eff , the torque balance level depends on the unknown value of the inclination ι. For the most optimistic case of circular polarization ( i cos =±1), our unmarginalized limit is a factor of 1.2 above the torque balance level near 100 Hz. Assuming linear polarization puts our limits within a factor of 3.5 of this level, and the most likely , because those candidates do not survive level 2 follow-up and are therefore not subjected to level 3 follow-up. From Equation (5) and Equation (7), the naïve expectation is that the S/N will roughly double each time T max is quadrupled. Empirically, the follow-ups of injections do not show exactly that relationship, but the vast majority do show significant increases in S/N, which are not seen in any of the follow-ups of search candidates, leading to the conclusion that no convincing detection candidates are present. 155 For comparison, the maximum spin wandering that could be tracked by the Viterbi analysis of Abbott et al. (2017f)  value of i = 44 corresponds to an upper limit curve a factor of 1.7 above the torque balance level, again near 100 Hz.

Outlook for Future Observations
We have presented the results of a search for GWs from ScoX-1 using data from Advanced LIGO's first observing run. The upper limits on the GW amplitude represent a significant improvement over the results from Initial LIGO and are within a factor of 1.2-3.5 of the benchmark set by the torque balance model, depending on assumptions about system orientation. Future observing runs (Abbott et al. 2016f) are expected to produce an improvement in the detector strain sensitivity of 2.5. An additional enhancement will come with longer runs, as the amplitude sensitivity of the search scales as T obs 1 4 . Algorithmic improvements that allow larger T max with the same computing resources will also lead to improvements, as the sensitivity scales as T max 1 4 as well. A promising area for such an improvement is the use of resampling (Patel et al. 2010) to reduce the scaling of computing cost with T max (G. D. Meadors et al. 2017, in preparation). (A similar method is used in the proposed semicoherent search described in Leaci & Prix 2015.) These anticipated instrumental and algorithmic improvements make it likely that search sensitivities will surpass the torque balance level over a range of frequencies (as projected in Whelan et al. 2015), and suggest the possibility of a detection during the advanced detector era, depending on details of the system such as GW frequency, inclination of the NS spin to the line of sight, and how close the system is to GW torque balance.

A.2. Correction for Known Lines
Although we calculate a single J value for each of the 18 search regions for a given 0.05 Hz band and use it in the calculation, the search can in principle have a different J value for each template. This is because of the correction which omits data contaminated by Doppler-modulated known instrumental lines from the sum in Equation (7), a process that depends on the signal frequency f 0 as well as the projected orbital semimajor h 0 eff 2 defined in Equation (7). For a given signal frequency f 0 , data are removed at some times due to the timevarying Doppler shift which depends on the orbital parameter asini. The effect is to lower J relative to the value it would have if the lines were not removed; this "sensitivity ratio" goes to zero if all of the data relevant to a signal frequency f 0 are removed by the line. The first three plots contain illustrations of the percentiles of this ratio, taken over intervals of width 0.05 Hz. (There is a range of values in each frequency interval because of its finite width, and the range of asini values which determine the magnitude of the Doppler modulation.) Note that the broad line at 300 Hz (a harmonic of the 60 Hz AC power line) effectively nullifies the search at that frequency. The last plot shows the fifth percentile of the sensitivity ratio in 0.05 Hz intervals across the whole sensitivity band. axis a i sin . In each 0.05 Hz band, we estimate the distribution of the ratio of the actual J to the band-wide J; the percentiles of this distribution are illustrated in Figure 6. We divide by the fifth percentile of this distribution (shown in the last panel of Figure 6) to produce corrected h 0 2 and ( ) h 0 eff 2 upper limits.

A.3. Empirical Adjustment from Software Injections
We performed a series of re-analyses of the data with a total of 754 simulated signals ("software injections") added to the data stream to validate the upper limits including the known line correction. The signals were generated over signal frequencies from 25 to 500 Hz, some with h 0 set to some multiple of the marginalized 95% upper limit h 0UL , and others with h 0 eff set to some multiple of the unmarginalized 95% upper limit h 0UL eff . We defined "recovery" of the injection as an increase in the maximum detection statistic r max compared to the results with no signal present. (Follow-up of the recovered injections that crossed the relevant ρ threshold was also performed as a way of testing our follow-up procedure, as described in Section 4.) We find that the fraction of signals of each type recovered when the injection is done at the upper limit level to be slightly below the expected 95%. 156 This is to be expected, as there are various approximations in the method, such as the tolerated mismatch in the initial parameter space grid and the acceptable loss of S/N due to finite-length SFTs, which should lead to an S/N slightly less than that predicted by Equation (6).
To estimate empirically the amount by which the upper limits should be scaled to produce a 95% injection recovery efficiency, we apply the method described in Whelan (2015) and used to produce the efficiency curves in Messenger et al. (2015). crosses 95%. A total of eight sets of injections were performed, four with h 0 at a specified multiple of h 0UL , and four with h 0 eff at a specified multiple of h 0UL eff . The multipliers were 1.0, 1.1, 1.2, and a random value between 1.1 and 2.0 chosen from a loguniform distribution. For the unmarginalized h 0 eff upper limit, we use all eight sets of injections, 754 in total and find the expectation value of the efficiency crosses 95% at . This factor has been applied to h 0UL eff to produce the upper limits in Figure 5.
For the marginalized h 0 upper limit, we must restrict ourselves to the four injection sets which specified h h 0 0UL . This is because our search is primarily sensitive to h 0 eff , and specifying h 0 eff while choosing the inclination angle ι randomly implies anticorrelations between h 0 and i | | cos . Signals with  (14); the 5th, 25th, 50th, 75th, and 95th percentiles are calculated from these distributions as a function of x 0 . The x 0 value at which the posterior expectation e ( |{ } { }) E D x , i i 0 crosses 0.95 is used as a correction factor by which we multiply h0 UL to produce the final marginalized upper limit shown in Figure 5. At right, we do the same thing for h 0 eff , using the full set of 754, and derive a correction factor by which to multiply the unmarginalized upper limit h 0UL eff . Note that in the h 0 search the value of x 0 corresponding to  = 0.95 0 is less accurately determined than that for the h 0 eff search, both because of the smaller number of applicable injections and because the detection efficiency depends more weakly on h 0 . high h 0 values will tend to be those with unfavorable polarization, and therefore not be any easier to detect. Using the 376 applicable injections, we estimate the 95% efficiency at » h h 1.44 0 0UL and use this factor when generating the final upper limit shown in Figure 5. Note that this is less well determined than the factor for the unmarginalized h 0 eff upper limit. This is both because of the smaller number of injections used and because h 0 correlates less well with detectability than h 0 eff . However, the upper limit curve for h 0 is very close to the unmarginalized upper limit assuming linear polarization ( i = cos 0), which is consistent with the expectation that the 95% upper limit will be dominated by this worse-case scenario.

Appendix B Results with a Constrained Semimajor Axis
As noted in Table 2, the range of a i sin values searched was chosen based on preliminary information from Wang (2017), which constrained the projected orbital velocity K 1 to lie between 10 and -90 km s 1 . This was subsequently refined to between 40 and -90 km s 1 . For comparison, we recomputed the upper limits, discarding the results of searches with asini1.44lt-s, corresponding to the nine bottom-most parameter space cells shown in Figure 2. The results were not significantly different (for instance, they were barely noticeable on plots like Figure 5), but for illustration we plot in Figure 8 the ratio of the two sets of upper limits. A bigger impact of the refined parameter space will be seen in future runs, when computing resources can be concentrated on the allowed range of a i sin values. The results are generally comparable; we plot the ratio of the upper limits rather than reproducing the curves in Figure 5, because the changes in the latter would barely be noticeable. The step-like features that are visible are due to the details of the search (such as T max values) being different in different frequency ranges listed in Table 3.