Results of the deepest all-sky survey for continuous gravitational waves on LIGO S6 data running on the Einstein@Home volunteer distributed computing project

We report results of a deep all-sky search for periodic gravitational waves from isolated neutron stars in data from the S6 LIGO science run. The search was possible thanks to the computing power provided by the volunteers of the Einstein@Home distributed computing project. We find no significant signal candidate and set the most stringent upper limits to date on the amplitude of gravitational wave signals from the target population. At the frequency of best strain sensitivity, between $170.5$ and $171$ Hz we set a 90% confidence upper limit of ${5.5}^{-25}$, while at the high end of our frequency range, around 505 Hz, we achieve upper limits $\simeq {10}^{-24}$. At $230$ Hz we can exclude sources with ellipticities greater than $10^{-6}$ within 100 pc of Earth with fiducial value of the principal moment of inertia of $10^{38} \textrm{kg m}^2$. If we assume a higher (lower) gravitational wave spindown we constrain farther (closer) objects to higher (lower) ellipticities.

Results of the deepest all-sky survey for continuous gravitational waves on LIGO S6 data running on the Einstein@Home volunteer distributed computing project B. P. Abbott et al. * (LIGO Scientific Collaboration and Virgo Collaboration) (Received 6 July 2016;published 18 November 2016) We report results of a deep all-sky search for periodic gravitational waves from isolated neutron stars in data from the S6 LIGO science run. The search was possible thanks to the computing power provided by the volunteers of the Einstein@Home distributed computing project. We find no significant signal candidate and set the most stringent upper limits to date on the amplitude of gravitational wave signals from the target population. At the frequency of best strain sensitivity, between 170.5 and 171 Hz we set a 90% confidence upper limit of 5.5 × 10 −25 , while at the high end of our frequency range, around 505 Hz, we achieve upper limits ≃10 −24 . At 230 Hz we can exclude sources with ellipticities greater than 10 −6 within 100 pc of Earth with fiducial value of the principal moment of inertia of 10 38 kg m 2 . If we assume a higher (lower) gravitational wave spin-down we constrain farther (closer) objects to higher (lower) ellipticities. DOI: 10.1103/PhysRevD.94.102002

I. INTRODUCTION
In this paper we report the results of a deep all-sky Einstein@Home [1] search for continuous, nearly monochromatic gravitational waves (GWs) in data from LIGO's sixth science (S6) run. A number of all-sky searches have been carried out on LIGO data, [2][3][4][5][6][7][8][9][10][11], of which [5,7,10] also ran on Einstein@Home. The search presented here covers frequencies from 50 Hz through 510 Hz and frequency derivatives from 3.39 × 10 −10 Hz=s through −2.67 × 10 −9 Hz=s. In this range we establish the most constraining gravitational wave amplitude upper limits to date for the target signal population.

II. LIGO INTERFEROMETERS AND THE DATA USED
The LIGO gravitational wave network consists of two observatories, one in Hanford (WA) and the other in Livingston (LA) separated by a 3000-km baseline [12]. The last science run (S6) [13] of this network before the upgrade towards the advanced LIGO configuration [14] took place between July 2009 and October 2010. The analysis in this paper uses a subset of this data: from GPS 949469977 (2010 Feb 6 05∶39:22 UTC) through GPS 971529850 (2010 Oct 19 13∶23:55 UTC), selected for good strain sensitivity [15]. Since interferometers sporadically fall out of operation ("lose lock") due to environmental or instrumental disturbances or for scheduled maintenance periods, the data set is not contiguous and each detector has a duty factor of about 50% [16].
As done in [7], frequency bands known to contain spectral disturbances have been removed from the analysis. Actually, the data has been substituted with Gaussian noise with the same average power as that in the neighboring and undisturbed bands. Table III identifies these bands.

III. THE SEARCH
The search described in this paper targets nearly monochromatic gravitational wave signals as described for example by Eqs. 1-4 of [7]. Various emission mechanisms could generate such a signal as reviewed in Sec. II A of [11]. In interpreting our results we will consider a spinning compact object with a fixed, nonaxisymmetric mass quadrupole, described by an ellipticity ϵ.
We perform a stack-slide type of search using the GCT (Global Correlation Transform) method [17,18]. In a stackslide search the data is partitioned in segments and each segment is searched with a matched-filter method [19]. The results from these coherent searches are combined by summing (stacking) the detection statistic values from the segments (sliding), one per segment (F i ), and this determines the value of the core detection statistic: There are different ways to combine the single-segment F i values, but independently of the way that this is done, this type of search is usually referred to as a "semicoherent search". So stack-slide searches are a type of semicoherent search. Important variables for this type of search are the coherent time baseline of the segments T coh , the number of segments used N seg , the total time spanned by the data T obs , the grids in parameter space and the detection statistic used to rank the parameter space cells. For a stack-slide search in Gaussian noise, N seg × 2F follows a χ 2 4N seg chi-squared distribution with 4N seg degrees of freedom. These parameters are summarized in Table I. The grids in frequency and spin-down are each described by a single parameter, the grid spacing, which is constant over the search range. The same frequency grid spacings are used for the coherent searches over the segments and for the incoherent summing. The spin-down spacing for the incoherent summing, δ _ f, is finer than that used for the coherent searches, δ _ f c , by a factor γ. The notation used here is consistent with that used in previous observational papers [20] and in the GCT methods papers cited above.
The sky grid is the union of two grids: one is uniform over the projection of the celestial sphere onto the equatorial plane, and the tiling (in the equatorial plane) is approximately square with sides of length with m sky ¼ 0.3 and τ E ≃ 0.021 s being half of the light travel time across the Earth. As was done in [7], the skygrids are constant over 10 Hz bands and the spacings are the ones associated through Eq. (2) to the highest frequency f in the range. The other grid is limited to the equatorial region (0 ≤ α ≤ 2π and −0.5 ≤ δ ≤ 0.5), with constant right ascension α and declination δ spacings equal to dð0.3Þ-see Fig. 1. The reason for the equatorial "patching" with a denser sky grid is to improve the sensitivity of the search: the sky resolution actually depends on the ecliptic latitude and the uniform equatorial grid underresolves particularly in the equatorial region. The resulting number of templates used to search 50 mHz bands as a function of frequency is shown in Fig. 2. The search is split into work-units (WUs) sized to keep the average Einstein@Home volunteer computer busy for about six hours. Each WU searches a 50 mHz band, the entire spin-down range and 13 points in the sky, corresponding to 4.9 × 10 9 templates out of which it returns only the top 3000. A total of 12.7 million WUs are necessary to cover the entire parameter space. The total number of templates searched is 6.3 × 10 16 .

A. The ranking statistic
The search was actually carried out in separate Einstein@Home runs that used different ranking statistics to define the top-candidate-list, reflecting different stages in the development of a detection statistic robust with respect to spectral lines in the data [21]. In particular, three ranking statistics were used: the average 2F statistic over the segments, 2F , which in essence at every template point is the likelihood of having a signal with the shape given by the template versus having Gaussian noise; the line-veto statisticÔ SL which is the odds ratio of having a signal versus having a spectral line; and a general line-robust statistic,Ô SGL , that tests the signal hypothesis against a Gaussian noise þ spectral line noise model. Such a statistic can match the performance of both the standard average 2F statistic in Gaussian noise and the line-veto statistic in presence of single-detector spectral disturbances and statistically outperforms them when the noise is a mixture of both [21].
We combine the 2F -ranked results with theÔ SL -ranked results to produce a single list of candidates ranked according to the general line-robust statisticÔ SGL . We now explain how this is achieved. Alongside the detection statistic value and the parameter space cell coordinates of each candidate, the Einstein@Home application also returns the single-detector 2F X values ("X" indicates the detector). These are used to compute, for every candidate of any run, theÔ SGL through Eq. 61 of [21] lnÔ with the angle-brackets indicating the average with respect to detectors (X) andF whereô X LG is the assumed prior probability of a spectral line occurring in any frequency bin of detector X,p L is the line prior estimated from the data, N det ¼ 2 is the number of detectors, andô SL is an assumed prior probability of a line being a signal (set arbitrarily to 1; its specific value does not affect the ranking statistic). Following the reasoning of Eq. 67 of [21], with N seg ¼ 90 we set c Ã ¼ 20.64 corresponding to a Gaussian false-alarm probability of 10 −9 and an average 2F transition scale of ∼6 (F ð0Þ Ã ∼ 3). Theô X LG values are estimated from the data as described in Sec. VI. A of [21] in 50-mHz bands with a normalized-SFT-power threshold P X thr ¼ P thr ðp FA ¼ 10 −9 ; N X SFT ∼ 6000Þ ≈ 1.08. For every 50 mHz band the list of candidates from the 2F -ranked run is merged with the list from theÔ SL -ranked run and duplicate candidates are considered only once. The resulting list is ranked by the newly computedÔ SGL and the top 3000 candidates are kept. This is our result-set, and it is treated in a manner that is very similar to [3].

B. Identification of undisturbed bands
Even after the removal of disturbed data caused by spectral artifacts of known origin, the statistical properties of the results are not uniform across the search band. In what follows we concentrate on the subset of the signalfrequency bands having reasonably uniform statistical properties. This still leaves us with the majority of the search parameter space while allowing us to use methods that rely on theoretical modeling of the significance in the statistical analysis of the results. Our classification of "clean" vs "disturbed" bands has no pretence of being strictly rigorous, because strict rigor here is neither useful nor practical. The classification serves the practical purpose of discarding from the analysis regions in parameter space with evident disturbances and must not dismiss real signals. The classification is carried out in two steps: a visual inspection and a refinement on the visual inspection.
The visual inspection is performed by three scientists who each look at various distributions of the detection statistics over the entire sky and spin-down parameter space in 50 mHz bands. They rank each band with an integer score 0,1,2,3 ranging from "undisturbed" (0) to "disturbed" (3). A band is considered "undisturbed" if all three rankings are 0. The criteria agreed upon for ranking are that the distribution of detection statistic values should not show a visible trend affecting a large portion of the f − _ f plane and, if outliers exist in a small region, outside this region the detection statistic values should be within the expected ranges. Figure 3 shows theÔ SGL for three bands: two were marked as undisturbed and the other as disturbed. One of the bands contains the f − _ f parameter space that harbors a fake signal injected in the data to verify the detection pipelines. The detection statistic is elevated in a small region around the signal parameters. The visual inspection procedure does not mark as disturbed bands with such features.
Based on this visual inspection 13% of the bands between 50 and 510 Hz are marked as "disturbed". Of these, 34% were given by all visual inspectors rankings smaller than 3, i.e. they were only marginally disturbed. Further inspection "rehabilitated" 42% of these. As a result of this refinement in the selection procedure we exclude from the current analysis 11% of the searched frequencies (Table IV). Figure 4 shows the highest values of the detection statistic in half-Hz signal-frequency bands compared to the expectations. The set of candidates that the highest detection statistic values are picked from, does not include the 50 mHz signal-frequency bands that stem entirely from fake data, from the cleaning procedure, or that were marked as disturbed. In this paper we refer to the candidates with the highest value of the detection statistic as the loudest candidates.
The loudest expected value over N trials independent trials of 2F is determined 1 by numerical integration of the probability density function given, for example, by Eq. 7 of [20]. For this search, we estimate that N trials ≃ 0.87N templ , with N templ being the number of templates searched.
As a uniform measure of significance of the highest 2F value across bands that were searched with different values of N trials we introduce the critical ratio CR defined as the deviation of the measured highest 2F from the expected value, measured in units of the standard deviation The highest and most significant detection statistic value from our search is 2F ¼ 8.6 at a frequency of about 52.76 Hz with a CR ¼ 29. This is due to a fake signal. The second highest value of the detection statistic is 7.04 at a frequency of about 329.01 Hz corresponding to a CR of 4.6. The second highest-CR candidate has a 2F of 6.99, is FIG. 3. On the z-axis and color-coded is theÔ SGL in three 50 mHz bands. The top band was marked as "undisturbed". The middle band is an example of a "disturbed band". The bottom band is an example of an "undisturbed band" but containing a signal, a fake one, in this case. Sorting loudest candidates from half-Hz bands according to detection statistic values is not the same as sorting them according to CR. The reason for this is that the number of templates is not the same for all half-Hz bands. This is due to the grid spacings decreasing with frequency (Eq. (2) and to the fact that, as previously explained, some 50 mHz bands have been excluded from the current analysis and hence some half-Hz bands comprise results from fewer than ten 50 mHz bands. Figure 7 gives the fill-level of each half-Hz band, i.e. how many 50 mHz bands have contributed candidates to the analysis out of ten. We use the CR as a measure of the significance because it folds in correctly the effect of varying number of templates in the half-Hz bands.
After excluding the candidate due to the fake signal, in this data we see no evidence of a signal: the distribution of p values associated with every measured half-Hz band loudest is consistent with what we expect from noise-only across the measured range (Fig. 8). In particular we note two things: 1) the two candidates at CR ¼ 4.6 and CR ¼ 4.8 are not significant when we consider how many half-Hz bands we have searched, and 2) there is no population of low significance candidates deviating from the expectation of the noise-only case. The p value for the loudest measured in any half-Hz band searched with an FIG. 5. Highest values of the significance (CR) in every half-Hz band as a function of band frequency. Since the significance folds in the expected value for the loudest 2F and its standard deviation, the significance of the loudest in noise does not increase with frequency. Our results are consistent with this expectation. which contribute to the results in every half-Hz band. As explained in the text, some bands are excluded because they are all from fake data or because they are marked as disturbed by the visual inspection. The list of excluded bands is given in Table IV. FIG. 8. p values for the loudest in half-Hz bands of our data (histogram bars) and expected distribution of pure noise data for reference (black markers). effective number of independent trials N trials ¼ 0.87N trials is obtained by integrating Eq. 6 of [20] between the observed value and infinity.

IV. UPPER LIMITS
The search did not reveal any continuous gravitational wave signal in the parameter volume that was searched. We hence set frequentist upper limits on the maximum gravitational wave amplitude consistent with this null result in half-Hz bands: h 90% 0 (f). h 90% 0 (f) is the GW amplitude such that 90% of a population of signals with parameter values in our search range would have produced a candidate louder than what was observed by our search. This is the criterion hereafter referred to as "detection".
Evaluating these upper limits with injection-andrecovery Monte Carlo simulations in every half-Hz band is too computationally intensive. So we perform them in a subset of 50 bands and infer the upper limit values in the other bands from these. The 50 bands are evenly spaced in the search frequency range. For each band j ¼ 1…50, we measure the 90% upper limit value corresponding to different detection criteria. The different detection criteria are defined by different CR values for the assumed measured loudest. The first CR bin, CR 0 , is for CR values equal to or smaller than 0, the next bins are for i < CR i ≤ ði þ 1Þ with i ¼ 1…5. Correspondingly we have h 90%;j 0;CR i for each band. For every detection criteria and every band we determine the sensitivity depth [22], and by averaging these sensitivity depths over the bands we derive a sensitivity depth for every detection criteria: D 90% We use these to set upper limits in the bands k where we have not performed injection-and-recovery simulations as where CR i ðkÞ is the significance bin of the loudest candidate of the kth band and S h ðf k Þ the power spectral density of the data (measured in 1= ffiffiffiffiffiffi Hz p ). The values of the sensitivity depths range between D 90% ffiffiffiffiffiffi Hz p Þ. The uncertainties on the upper limit values introduced by this procedure are ≃10% of the nominal upper limit value. We represent this uncertainty as a shaded region around the upper limit values in Fig. 9. The upper limit values are also provided in tabular form in the FIG. 9. 90% confidence upper limits on the gravitational wave amplitude of signals with frequency within half-Hz bands, from the entire sky and within the spin-down range of the search. The light red markers denote half-Hz bands where the upper limit value does not hold for all frequencies in that interval. A list of the excluded frequencies is given in the Appendix. Although not obvious from the figure, due to the quality of the data we were not able to analyze the data in some half-Hz bands, so there are some points missing in the plot. For reference we also plot the upper limit results from two searches: one on the same data (Powerflux) [2] and on contemporary data from the Virgo detector (frequency Hough) [4]. The Powerflux points are obtained by rescaling the best (crosses) and worst-case (dots) upper limit values as explained in the text. It should be noted that the Powerflux upper limits are set at 95% rather than 90% but refer to 0.25 Hz bands rather than half-Hz. Appendix in Table II. We do not set upper limits in half-Hz bands where the results are entirely produced with fake data inserted by the cleaning procedure described in Sec. II. Upper limits for such bands will not appear in Table II nor in Fig. 9. There also exist 50 mHz bands that include contributions from fake data as a result of the cleaning procedure or that have been excluded from the analysis because they were marked as disturbed by the visual inspection procedure described in Sec. III B. We mark the half-Hz bands which host these 50 mHz bands with a different colour (light red) in Fig. 9. In Table IV in the Appendix we provide a complete list of such 50-mHz bands because the upper limit values do not apply to those 50-mHz bands. Finally we note that, due to the cleaning procedure, there exist signal frequency bands where the search results might have contributions from fake data. We list these signal-frequency ranges in Table V. For completeness this table also contains the cleaned bands of Table IV, under the column header "all fake data".

V. CONCLUSIONS
Our upper limits are the tightest ever placed for this set of target signals. The smallest value of the GW amplitude upper limit is 5.5 × 10 −25 in the band 170.5-171 Hz. Figure 9 shows the upper limit values as a function of search frequency. We also show the upper limits from [2], another all-sky search on S6 data, rescaled according to [23] to enable a direct comparison with ours. Under the assumption that the sources are uniformly distributed in space, our search probes a volume in space a few times larger than that of [2]. It should however be noted that [2] examines a much broader parameter space than the one presented here. The Virgo VSR2 and VSR4 science runs were contemporary to the S6 run and more sensitive at low frequency with respect to LIGO. The Virgo data were analyzed in search of continuous signals from the whole sky in the frequency range 20-128 Hz and a narrower spindown range than that covered here, with j _ fj ≤ 10 −10 Hz=s [4]. Our sensitivity is comparable to that achieved by that search and improves on it above 80 Hz.
Following [24], we define the fraction x of the spin-down rotational energy emitted in gravitational waves. The star's ellipticity necessary to sustain such emission is where c is the speed of light, G is the gravitational constant, f is the GW frequency and I the principal moment of inertia of the star. Correspondingly, x _ f is the spin-down rate that accounts for the emission of GWs, and this is why we refer to it as the GW spin-down. The gravitational wave amplitude h 0 at the detector coming from a GW source like that of Eq. (14), at a distance D from Earth is Based on this last equation, we can use our GW amplitude upper limits to bound the minimum distance for compact objects emitting continuous gravitational waves under different assumptions on the object's ellipticity (i.e. FIG. 10. Gravitational wave amplitude upper limits recast as curves in the f − xj _ fj plane for sources at given distances and having assumed I ¼ 10 38 kg m 2 . f is the signal frequency and xj _ fj is the gravitational-wave spin-down, i.e. the fraction of the actual spin-down that accounts for the rotational energy loss due to GW emission. Superimposed are curves of constant ellipticity ϵðf; _ fjI ¼ 10 38 kg m 2 ). The dotted line at j _ fj max indicates the maximum magnitude of searched spin-down.
gravitational wave spin-down). This is shown in Fig. 10. We find that for most frequencies above 230 Hz our upper limits exclude compact objects with ellipticities of 10 −6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 10 38 kg m 2 I q (corresponding to GW spin-downs between 10 −12 Hz=s and 10 −11 Hz=s) within 100 pc of Earth. Both the ellipticity and the distance ranges span absolutely plausible values and could not have been excluded with other measurements.
We expect the methodology used in this search to serve as a template for the assessment of Einstein@Home run results in the future, for example the next Einstein@Home run, using advanced LIGO data that is being processed as this paper is written. Results of searches for continuous wave signals could also be mined further, probing subthreshold candidates with a hierarchical series of follow-up searches. This is not the topic of this paper and might be pursued in a forthcoming publication.        (Table continued) 3. 50 mHz signal-frequency bands that did not contribute to results Signal frequency ranges where the results might have contributions from fake data. When the results are entirely due to artificial data, the band is listed in the "all fake data" column; bands where the results comprise contributions from both fake and real data are listed in the other three columns. The "mixed, left" and "mixed, right" columns are populated only when there is a matching "all fake data" entry, which highlights the same physical cause for the fake data, i.e., the cleaning. The "mixed, isolated" column lists isolated ranges of mixed data. The list of input data frequencies where the data was substituted with artificial noise are given in Table I 190.763 190.813 D 192.363 192.613 D 193.613 194.313 D 196.963 197.013 D 197.713 197.763 D (Table continued) 197.863 197.963 D 198.113 198.163 D 198.563 198.663 D 199.213 199.313 D 199.513 199.613 D 199.813 200