Search for Gravitational Waves from Scorpius X-1 in LIGO O3 Data with Corrected Orbital Ephemeris

The low-mass X-ray binary (LMXB) Scorpius X-1 (ScoX1), which is presumed to consist of a neutron star (NS) of mass ≈1.4M_⊙ in a binary orbit with a companion star of mass ≈0.4M_⊙ (Steeghs & Casares 2002), is a very promising potential source of continuous gravitational waves (GWs), generated by the spin of the NS (Bildsten 1998; Watts et al. 2008). As such, it has been the target of a number of searches (Abbott et al. 2007,2017a, 2017b, 2017c, 2019a, 2019b, 2021, 2022b, 2022a; Aasi et al. 2015a; Meadors et al. 2017; Zhang et al. 2021) using data from the Advanced LIGO GW detectors (Aasi et al. 2015b), which have conducted three observing runs (O1, O2, and O3), the last two in coordination with Advanced Virgo (Acernese et al. 2015). As the spin frequency of ScoX1 is unknown, searches typically cover a wide range of the intrinsic GW signal frequency f₀, which in the simplest model is twice the spin frequency. Some of these searches, notably the cross-correlation method (Dhurandhar et al. 2008; Whelan et al. 2015) and the Viterbi method (Suvorova et al. 2016, 2017), are sensitive to aspects of the signal model, notably the parameters of the binary orbit. Hence, a campaign of electromagnetic observations and analyses known as "Precision Ephemerides for Gravitational-Wave Searches" (PEGS) has produced a series of updates to the orbital ephemeris for ScoX1 (Galloway et al. 2014; Wang et al. 2018; Killestein et al. 2023), which have been used to choose the parameter-space region covered in the GW searches. The most recent update, known as PEGS IV (Killestein et al. 2023), in addition to producing a more refined ephemeris also corrected errors in the previously published PEGS I (Galloway et al. 2014) and PEGS III (Wang et al. 2018) ephemerides. ¹¹ This means that the parameter-space region searched by the analysis of O1 data in Abbott et al. (2017c), which used elements of PEGS I and PEGS III, as well as the O2 and O3 analyses in Abbott et al. (2019b, 2022b, 2022a), Zhang et al. (2021), which used PEGS III, did not overlap with the likely regions of parameter space according to the PEGS IV ephemeris, nor with the revised PEGS I and PEGS III ephemerides published in Killestein et al. (2023). ¹² This paper presents a reanalysis of the LIGO O3 data (Abbott et al. 2023) according to the method of Abbott et al. (2022a), but with the parameter space determined by the PEGS IV ephemeris.

The signal model for GWs from an LMXB is a continuous signal with nearly constant intrinsic amplitude and frequency, Doppler-modulated by the relative motion of the source and the detector. As the Doppler modulation depends upon the extrinsic parameters of the system, including the sky position and orbital parameters of the binary system, accurate ranges of values for those parameters are an important input into GW searches. Since the sky location of ScoX1 is precisely known (Bradshaw et al. 1999; Abbott et al. 2007), and the orbital eccentricity is believed to be small (Steeghs & Casares 2002; Wang et al. 2018; Killestein et al. 2023), the important residual uncertainty is in the orbital velocity, period, and phase of the NS. The projected orbital velocity K₁ of the NS is usually described for GW searches in terms of the projected semimajor axis $a\sin i={K}_{1}{P}_{\mathrm{orb}}/(2\pi )$ of the orbit, measured in light-seconds. ¹³ Estimation of K₁ is particularly difficult for ScoX1 (Galloway et al. 2014), and the best constraint remains that of Wang et al. (2018), 40 km s⁻¹ ≲ K₁ ≲ 90 km s⁻¹. The orbital phase is generally described by the time at which the system reaches some reference point in its orbit. For GW searches, this is typically the time of ascension t_asc, when the NS crosses the ascending node, moving away from the observer. This is one-quarter of a period before the other typically quoted reference time, of inferior conjunction of the companion star. Given a value of t_asc and P_orb, an equivalent time of ascension in a later epoch ${t}_{\mathrm{asc}}^{\prime} ={t}_{\mathrm{asc}}+{n}_{\mathrm{orb}}{P}_{\mathrm{orb}}$ can be obtained by adding an integer number n_orb of orbits (Whelan et al. 2015). For analysis of LIGO O3 data, it is convenient to choose a ${t}_{\mathrm{asc}}^{\prime}$ value in the middle of the observing run, which lasted from 2019 April 1 to 2020 March 27, corresponding to GPS time 1,238,166,018 to 1,269,363,618.

The parameter-space ranges used for searches of O2 and O3 data (Abbott et al. 2019b, 2022b, 2022a; Zhang et al. 2021) were generated using the PEGS III ephemeris (Wang et al. 2018). Killestein et al. (2023) subsequently published the improved PEGS IV ephemeris, also documenting calibration errors in PEGS III. The values of P_orb and t_asc in these ephemerides are summarized in Table 1, which shows that the t_asc range originally published in Wang et al. (2018) is inconsistent with current estimates of that parameter. In Figure 1, we show the plausible ranges of P_orb and ${t}_{\mathrm{asc}}^{\prime}$ (propagated to the middle of O3), along with the region of parameter space searched in the cross-correlation analysis (Abbott et al. 2022a), and in the reanalysis presented in this paper. We see that the parameter-space region searched in the original O3 search is inconsistent with the PEGS IV ephemeris, but the region searched in the present reanalysis has about 1/3 the area in parameter space, allowing a search to be performed more quickly at the same sensitivity.

Zoom In Zoom Out Reset image size

Table 1. Orbital Parameters in the PEGS III and PEGS IV Ephemerides

Parameter	PEGS III a	PEGS IV b
Porb (s)	68,023.86 ± 0.043	68,023.91 ± 0.017
tasc (GPS s) c	974,416,624 ± 50	1,078,153,676 ± 33
${t}_{\mathrm{asc}}^{\prime}$ (GPS s) d	1,255,015,049 ± 185	1,255,015,866 ± 55
$\mathrm{Corr}({P}_{\mathrm{orb}},{t}_{\mathrm{asc}}^{\prime} )$ e	0.96	0.80

Notes. Uncertainties are 1σ.

^a Values in this column are inferred from Wang et al. (2018). ^b Values in this column are inferred from Killestein et al. (2023). ^c The time of ascension t_asc, at which the NS 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 by subtracting P_orb/4. The values quoted in this row are those for which the correlations in the P_orb and t_asc uncertainties are negligible, and correspond to 2010 November 21 23:16:49 UTC and 2014 March 6 15:07:40 UTC, respectively. ^d The time of ascension ${t}_{\mathrm{asc}}^{\prime}$ after propagating t_asc forward by n_orb orbits (4125 and 2600, respectively), corresponding to times of 2019 October 13 15:17:11 UTC and 2019 October 13 15:30:48 UTC, near the middle of the O3 run. The uncertainty is obtained by combining the uncertainty in ${t}_{\mathrm{asc}}^{\prime}$ in quadrature with n_orb times the uncertainty in P_orb (Whelan et al. 2015). ^e The correlation between the uncertainties in P_orb and ${t}_{\mathrm{asc}}^{\prime} ={t}_{\mathrm{asc}}+{n}_{\mathrm{orb}}{P}_{\mathrm{orb}}$, given uncorrelated uncertainties in P_orb and t_asc.

References. Wang et al. (2018), Killestein et al. (2023).

Download table as:  ASCIITypeset image

We present here the results of a reanalysis of the LIGO O3 data (Abbott et al. 2023) using the cross-correlation search (Whelan et al. 2015), with the revised PEGS IV ephemeris of Killestein et al. (2023). Full details of the analysis pipeline are given in Abbott et al. (2022a). We highlight here changes made in light of the revised ephemeris.

The search was performed over a range of signal frequencies from 25 to 1600 Hz. Since the search is tunable, with the coherence time ${T}_{\max }$ chosen to balance computing cost and sensitivity, different ${T}_{\max }$ values are chosen across signal frequency and orbital parameter space to roughly optimize the chance of detecting a signal. The same coherence times were used as in Abbott et al. (2022a), with $240\,{\rm{s}}\leqslant {T}_{\max }\leqslant {\rm{18,720}}\,{\rm{s}}$. As in Abbott et al. (2022a), the search covered a range of projected semimajor axes $1.44\,\mathrm{lt}-{\rm{s}}\leqslant a\sin i\leqslant 3.25\,\mathrm{lt}-{\rm{s}}$. ¹⁴ The ${t}_{\mathrm{asc}}^{\prime}$–P_orb space was covered using the sheared period coordinate of Wagner et al. (2022), which for the reanalysis was defined as

$$\begin{eqnarray}&&\tilde{P}={P}_{\mathrm{orb}}-2.47\times {10}^{-4}({t}_{\mathrm{asc}}^{\prime} -{\rm{1,255,015,866}}).\end{eqnarray} \tag{ 1 }$$

The range of ${t}_{\mathrm{asc}}^{\prime}$ values was set to 1,255,015,866 ± 3 × 55, while the P_orb values were constrained to lie in an ellipse centered on $(\tilde{P},{t}_{\mathrm{asc}}^{\prime} )=({\rm{68,023.91}}\,{\rm{s}},{\rm{1,255,015,866}})$ with semiaxes of 3.3 × 0.010 s for $\tilde{P}$ and 3.3 × 55 s for ${t}_{\mathrm{asc}}^{\prime}$. The boundaries of this region in $({P}_{\mathrm{orb}},{t}_{\mathrm{asc}}^{\prime} )$ space are shown in solid black lines in Figure 1. Note that the software bug that led to the slightly misaligned definition of $\tilde{P}$ used in Abbott et al. (2022a) was fixed before the reanalysis, so the search region now lines up with the PEGS IV prior uncertainty region, as seen in Figure 1. As in Abbott et al. (2022a), the $\tilde{P}$ coordinate was unresolved for most analysis jobs, and a single template was sufficient to cover the range 68,023.91 ± 3.3 × 0.010 s.

As in Abbott et al. (2022a), the search was carried out at a nominal parameter-space mismatch of 0.25. Due to the reduced parameter ranges in the more precise PEGS IV ephemeris, this required fewer templates than in the original analysis. As a consequence, the threshold for follow-up, which is set using the expected number of false alarms from Gaussian noise at a particular frequency, could be reduced, as shown in Figure 2. In the present search, we followed up candidates with a signal-to-noise ratio (S/N) above 6.1 from 25 Hz < f₀ < 400 Hz, 6.0 for 400 Hz < f₀ < 600 Hz, and 5.6 for 600 Hz < f₀ < 1600 Hz, compared to 6.3, 6.2, and 5.8, respectively, in the original analysis of Abbott et al. (2022a).

Zoom In Zoom Out Reset image size

The detection candidates that exceeded the follow-up threshold were subjected to a hierarchical follow-up using successively finer grids and longer coherence times. At each stage, the candidates for which the S/N from a search using only one detector (LIGO Hanford Observatory, hereafter LHO; or LIGO Livingston Observatory, hereafter LLO) exceeded the S/N obtained from the full data and were rejected as likely narrowband instrumental features (lines). The initial results were known as level (0); level (1) used the same coherence time ${T}_{\max }$ and only refined the grid, while level (2) and level (3) each successively quadrupled ${T}_{\max }$ relative to the previous level, which would ideally double the S/N of a signal. If a candidate's S/N went down from one level of follow-up to the next, it was discarded. Table 2 shows the numbers of candidates surviving each level of follow-up. A total of 24 candidates survived level (3) of follow-up. For each of the candidates surviving level (2) (so that level (3) follow-up was run), Figure 3 shows the ratios of S/Ns at successive levels.

Zoom In Zoom Out Reset image size

Table 2. Summary of Numbers of Templates and Candidates

f0(Hz)		${T}_{\max }$(s)		ρ	Number of Templates	Expected Gauss. False Alarms b	Follow-up Level
Min	Max	Min	Max	Thresh a			0 c	1 d	2 e	3 f
25	50	10,080	18,720	6.1	2.24 × 109	1.2	62	34	5	0
50	100	8160	14,280	6.1	9.65 × 109	5.1	134	115	41	0
100	150	6720	10,920	6.1	1.71 × 1010	9.1	148	146	81	3
150	200	5040	8640	6.1	1.79 × 1010	9.5	151	151	74	1
200	300	2400	4800	6.1	1.30 × 1010	6.9	60	60	22	8
300	400	1530	3060	6.1	6.51 × 109	3.5	24	24	5	3
400	600	720	2160	6.0	9.96 × 109	9.8	312	199	20	9
600	800	360	360	5.6	7.34 × 108	7.9	8	8	0	0
800	1200	300	300	5.6	1.69 × 109	18.2	234	65	2	0
1200	1600	240	240	5.6	1.65 × 109	17.7	328	61	7	0

Notes. For each range of GW frequencies, this table shows the minimum and maximum coherence time ${T}_{\max }$ used for the search, the threshold in the S/N ρ used for follow-up, the total number of templates, and the number of candidates at various stages of the process. Compare to Table 3 of Abbott et al. (2022a).

^a This is the threshold for initiating follow-up, i.e., to produce a level (0) candidate. ^b This is the number of candidates that would be expected in Gaussian noise, given the number of templates and the follow-up threshold. ^c This is the actual number of candidates (after clustering), which crossed the S/N threshold and were followed up. ^d 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. ^e This is the number of candidates remaining after each has been followed up with a ${T}_{\max }$ equal to 4 × 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 3, which also shows the behavior of the search on simulated signals injected in software. ^f This is the number of candidates remaining after ${T}_{\max }$ has been increased to 16 × its original value.

Download table as:  ASCIITypeset image

For the most part, we reproduce the results of Abbott et al. (2022a), that the outliers of the search do not increase their S/N upon follow-up in the way that simulated signals do. There is one possible exception: a candidate at a frequency of 510.71 Hz, which has S/N ρ of 6.08 at level (0), 6.48 at level (1), 9.58 at level (2), and 10.41 at level (3). Note that, while this candidate would not have made the follow-up threshold in the original search, which was 6.2 for frequencies between 400 and 600 Hz (as compared with 6.0 in the follow-up), there is no such outlier in the original search with the PEGS III ephemeris; all of the templates searched with 510.70 Hz < f₀ < 510.72 Hz produced S/N ρ < 4.5 in the original search.

We have strong reasons to believe the outlier in the reanalysis is an instrumental artifact. The outlier is located in a frequency range heavily contaminated by violin modes in both LIGO detectors (Davis et al. 2021). The cross-correlation analysis excludes data at frequencies contaminated by known lines (Goetz et al. 2021) from the analysis. For LIGO Hanford Observatory (LHO), this includes data from 510.71527 to 510.72653 Hz, and for LLO, this includes data from 507.89972 to 516.25972 Hz. The relative amplitude of possible Doppler modulation for ScoX1 is ≲4 × 10⁻⁴, so a signal with intrinsic frequency 510.7 Hz could be received at the detector with a frequency from ∼510.5 to ∼510.9 Hz. Thus LLO is completely excluded from the analysis of this candidate, as indicated by the blue single detector square in Figure 3. As a consequence, the S/N using only LHO data is the same as the S/N from the full search, and the unknown line veto cannot be applied to this candidate.

As an additional investigation, we reran the follow-up with no data excluded. The S/N of the candidate dropped to 4.95, while the S/N using only LHO increased to 65.90. (The S/N using only LLO was 2.45.) I.e., without the known line veto to eliminate LLO data, the outlier would have been eliminated as a possible candidate by the unknown line veto.

This outlier was further scrutinized by a multistage Markov Chain Monte Carlo follow-up using the method described in Tenorio et al. (2021) with the PyFstat package (Ashton & Prix 2018; Keitel et al. 2021; Ashton et al. 2022). The setup was identical to that of Abbott et al. (2022a): Templates were placed adaptively around the outlier to compute the semicoherent ${ \mathcal F }$-statistic (Jaranowski et al. 1998; Cutler & Schutz 2005) using a decreasing number of coherent segments (660, 330, 92, 24, 4, and 1), which correspond to a coherence time ranging from half a day to the full observing run. A Bayes factor was computed using the ${ \mathcal F }$-statistic values of consecutive stages corresponding to the loudest template. The signal hypothesis assesses the consistency of these values, while comparing against the noise hypothesis checks the inconsistency of the final value with the background distribution. The resulting Bayes factor is significantly lower than expected for a signal detectable by this search. Moreover, the semicoherent ${ \mathcal F }$-statistic accumulation of the outlier in the LHO data suggests an instrumental origin.

We have reanalyzed the LIGO O3 data with the cross-correlation pipeline, using the corrected and improved PEGS IV ephemeris of Killestein et al. (2023). Having found no credible detection candidates, we reproduce the null result of Abbott et al. (2022a), that there is no GW signal from ScoX1 detectable at the level of sensitivity of that search. We do not produce an upper limit from the reanalysis, but as approximate sensitivity computations for the two searches agree, we conclude that the upper limits published in Abbott et al. (2022a) remain valid.

Since the cross-correlation analysis of this paper, as in Abbott et al. (2022a), does not explicitly consider a signal with stochastically varying frequency ("spin wandering")—although it is somewhat robust to it (Whelan et al. 2015)—there is information to be gained from reanalysis of the O3 data using a hidden Markov model as in Abbott et al. (2022b).

The O4 run of Advanced LIGO, Advanced Virgo, and KAGRA (Akutsu et al. 2021) is scheduled to begin in May 2024 and run for approximately 18 months (LIGO-Virgo-KAGRA Collaboration 2023). The improved sensitivity of the detectors will enable more sensitive searches for GWs from ScoX1, and the greater precision of the PEGS IV ephemeris will enable the search to be done more efficiently.

We wish to thank the members of the LIGO-Virgo-KAGRA Collaboration continuous waves group for useful feedback. J.T.W., J.K.W., and K.J.W. were supported by NSF grants PHY-1806824 and PHY-2110460. J.T.W. and J.K.W. thank Goethe University Frankfurt. R.T., D.K., and A.M.S. are supported by the Spanish Ministerio de Ciencia e Innovación and the Spanish Agencia Estatal de Investigación grant PID2019-106416GB-I00/AEI/MCIN/10.13039/501100011033, European Union NextGenerationEU funds (PRTR-C17.I1), the Comunitat Autònoma de les Illes Balears through the Direcció General de Política Universitaria i Recerca with funds from the Tourist Stay Tax Law 2017-006 (PRD2018/24, PDR2020/11), the Conselleria de Fons Europeus, Universitat i Cultura del Govern de les Illes Balears, the FEDER Operational Program 2021–2027 of the Balearic Islands, and EU COST Actions CA18108 and CA17137. R.T. is supported by the Spanish Ministerio de Universidades (ref. FPU 18/00694). J.C. was supported by NSF grant PHY-1764464. E.G. is grateful for the support of the John Edwards Leadership Fund. D.K. is supported by the Spanish Ministerio de Ciencia, Innovación y Universidades (ref. BEAGAL 18/00148), cofinanced by the Universitat de les Illes Balears and grants RED2022-134204-E and RED2022-134411-T. T.L.K. is supported by the UK Science and Technology Facilities Council (STFC), grant No. ST/T506503/1. A.N. was supported by NSF awards PHY-1912598 and PHY-1944412 and thanks the University of Washington, Bothell. D.S. is supported by STFC, grant Nos. ST/T007184/1, ST/T003103/1, and ST/T000406/1.

The authors are grateful for computational resources provided by LIGO Laboratory, supported by National Science Foundation grants PHY-1626190 and PHY-1700765, by the Open Science Grid (OSG) Consortium (Pordes et al. 2007; Sfiligoi et al. 2009; OSG 2006), which is supported by the National Science Foundation awards #2030508 and #1836650, and by the Digital Research Alliance of Canada. ¹⁵ Scripts and data pertaining to the original O3 analysis are available at https://dcc.ligo.org/LIGO-T2200419/public.

The analysis reported in this paper also used the segment lists of Goetz & Riles (2023) and the self-gating procedure documented in Zweizig & Riles (2021).

This research has made use of data available from the Gravitational Wave Open Science Center, ¹⁶ a service of LIGO Laboratory, the LIGO Scientific Collaboration, the Virgo Collaboration, and KAGRA. LIGO Laboratory and Advanced LIGO are funded by the United States National Science Foundation (NSF) as well as the Science and Technology Facilities Council (STFC) of the United Kingdom, the Max-Planck-Society (MPS), and the State of Niedersachsen/Germany for support of the construction of Advanced LIGO and construction and operation of the GEO600 detector. Additional support for Advanced LIGO was provided by the Australian Research Council. Virgo is funded, through the European Gravitational Observatory (EGO), by the French Centre National de Recherche Scientifique (CNRS), the Italian Istituto Nazionale di Fisica Nucleare (INFN), and the Dutch Nikhef, with contributions by institutions from Belgium, Germany, Greece, Hungary, Ireland, Japan, Monaco, Poland, Portugal, Spain. KAGRA is supported by Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan Society for the Promotion of Science (JSPS) in Japan; National Research Foundation (NRF) and Ministry of Science and ICT (MSIT) in Korea; Academia Sinica (AS) and National Science and Technology Council (NSTC) in Taiwan.

Software: LALSuite (LIGO Scientific Collaboration 2018), LatticeTiling (Wette 2014), PyFstat (Ashton & Prix 2018; Keitel et al. 2021; Ashton et al. 2022), ptemcee (Foreman-Mackey et al. 2013; Vousden et al. 2016), numpy (Harris et al. 2020), matplotlib (Hunter 2007), scipy (Virtanen et al. 2020), SwigLAL (Wette 2020).

This paper has been assigned LIGO document No. LIGO-P2300042-v8.