An Einstein@home search for continuous gravitational waves from Cassiopeia A

We report the results of a directed search for continuous gravitational-wave emission in a broad frequency range (between 50 and 1000 Hz) from the central compact object of the supernova remnant Cassiopeia A (Cas A). The data comes from the sixth science run of LIGO and the search is performed on the volunteer distributed computing network Einstein@Home. We find no significant signal candidate, and set the most constraining upper limits to date on the gravitational-wave emission from Cas A, which beat the indirect age-based upper limit across the entire search range. At around 170 Hz (the most sensitive frequency range), we set 90% confidence upper limits on the gravitational wave amplitude $h_0$ of $\sim\!\!~2.9\times 10^{-25}$, roughly twice as constraining as the upper limits from previous searches on Cas A. The upper limits can also be expressed as constraints on the ellipticity of Cas A; with a few reasonable assumptions, we show that at gravitational-wave frequencies greater than 300~Hz, we can exclude an ellipticity of $\gtrsim\!\!~10^{-5}$.


I. INTRODUCTION
Isolated neutron stars with non-axisymmetric asymmetries are thought to be one of the best sources for continuous gravitational-wave emission. We report the results of a directed search for continuous gravitationalwave emission from the central compact object of the supernova remnant Cassiopeia A (Cas A) with the Laser Interferometer Gravitational-Wave Observatory (LIGO). Directed searches, in which the source and therefore the sky position are specified, are generally more sensitive than all-sky surveys. The reason is that typically fewer templates are needed for directed searches than for allsky surveys; this results in a smaller trials factor and hence in a smaller weakest detectable signal at fixed detection confidence.
At an age of a few hundreds of years, Cas A is one of the youngest known supernova remnants [1]. Its young age means that any asymmetries in the central compact object that were produced at birth are likely still present. Based on X-ray observations, the central compact object is most likely a neutron star with a low surface magnetic field strength [2]. No pulsed electromagnetic emission has been observed from the central object, so its spin parameters are unknown.
Assuming the central object is a neutron star, its asymmetries would be expected to continuously proa email: sylvia.zhu@aei.mpg.de b email: maria.alessandra.papa@aei.mpg.de duce slowly evolving and nearly monochromatic gravitational waves (e.g., [3]). We perform a search for this gravitational-wave emission from Cas A using data from the sixth LIGO science run with the volunteer distributed computer network Einstein@Home [4].
For the remainder of this paper, when we refer to Cas A, we are referring to the central compact object.

A. Data used in this search
The two LIGO interferometers are located in the US in Hanford, Washington and Livingston, Louisiana, a separation distance of 3000 km [5]. The last science run of initial LIGO, S6, took place between July 2009 and October 2010 [6]. For this analysis, we only use data taken between 06 February 2010 (GPS 949461068) and 20 October 2010 (GPS 971629632), selected for the best sensitivity [7].
Unlike what we did for previous Einstein@Home searches [8,9], we do not perform any upfront line cleaning to remove known artefacts.

B. The search set-up
We perform a semi-coherent search and rank the results according to the line-robust statistic O SGL [10], in a manner similar to [8,9]. The basic ingredient is the averaged F statistic [11,12], F, computed using the Global Correlation Transform (GCT) method [13,14]. In a stack-slide search, the time series data are partitioned into i = 1...N seg segments of length T coh each. The data from every segment are match-filtered against a set of signal templates each specified by a set of parameters (the signal frequency f , the first-order spindownḟ , the second-order spindownf , and the sky position) to produce values of the detection statistic F i for each segment (the coherent step). These F i values are then combined to produce an average value of the statistic across the N seg segments, 2F, which is the core statistic that we use in these analyses (the incoherent step). The values of the signal template parameters f ,ḟ , andf are given by a predetermined grid. The f grid spacing (i.e., the separation between two adjacent values of f in the search) is kept the same for the coherent and incoherent steps, while the spacings for theḟ andf grids for the incoherent summing are finer by factors of 90 and 60, respectively. The search parameters are summarized in Table I and were derived using the optimisation scheme described in [15] assuming a run duration of 6 months on Einstein@Home.

C. The detection and ranking statistics
The 2F statistic gives a measure of the likelihood that the data resembles a signal versus Gaussian noise; therefore, signals are expected to have high values of 2F. However, line disturbances in the data can also result in high values of 2F. The line-robust statistic, O SGL , was designed to address this by testing the signal hypothesis against a composite noise model comprising a combination of Gaussian noise and a single-detector spectral line. The O SGL parameters are tuned as described in [10] using simulations so that the detection efficiency of O SGL performs as well as 2F in Gaussian noise and better in the presence of lines. For this search, the value of c (related to the tuning parameter in the choice of prior, see [10]) is set to be 34.8, which corresponds to a Gaussian false-alarm probability of 10 −9 .
The O SGL distribution even in Gaussian noise is not known analytically. Therefore, although we use the O SGL toplists to find the best signal candidates, we still use 2F as the detection statistic for ascertaining a candidate's significance. For a stack-slide search with N seg segments, the N seg × 2F distribution in Gaussian noise follows a chi-squared distribution with 4N seg degrees of freedom [16].

D. The parameter space
Since the spin parameters of Cas A are unknown, our search encompasses a large range of possible gravitational-wave frequencies f ; namely, from 50 to 1000 Hz. For a given value of f , theḟ andf ranges are given by the following specifications: (1) where τ NS is the fiducial age of the neutron star, taken to be 300 years. As discussed in [3], this choice of τ NS is on the young end of the age estimates, which yields a larger search parameter space than other, less conservative choices. The searched parameter space at each value of f is a rectangle in theḟ −f plane, and the search volume increases quadratically with f ( Fig. 1). Compared to [3,17,18], the largest magnitude of the first order spindown parameter is the same, corresponding to a conservative assumption (in the sense that it allows for the broadest range of first order spindown values) on the average braking index at fixed age of the object. The range of the second order spindown is constructed differently here than in [3,18] in that it does not depend onḟ . The highest searched value off is nḟ 2 max /f , with n = 2 being the instantaneous braking index. The searches mentioned above took this as the lower boundary of thef range and set the upper boundary at n = 7. Our choice does not search such a broad range off values, and is driven by ease of set-up of the search. Observational data on braking indexes supports this choice [19].
We estimate that searching over third order spindown values is not necessary. We do this by counting how many templates are needed to cover the third order spindown range. The 3rd order spindown template extent ∆ . ( For this search we set m = 0.2 and γ 3 3.89 × 10 5 [20]. The template extent of Eq. 3 is mg 33 where g ij is the inverse of the phase metric [21]. The 3rd order spindown range, consistent with the choices of Eq. 2, is 6f /τ 3 NS . With τ NS = 300 years, we find that we do not need more than a single template to cover the 3rd order spindown range; therefore, we do not need to add a 3rd order spindown dimension. The location of Cas A is known to within ∼ 1", which is smaller than the sky resolution of our search. Hence, we only search a single sky position (right ascension = 23h 23m 28s, declination = 58 • 58' 43").

E. Distribution of the computational load
The search runs on volunteer computers in the Ein-stein@Home network, and is split into 9.2 million work units (WUs), with each WU designed to run for about six hours on a modern PC. A single WU encompasses a 50-mHz range in f and the entire range off at the start value of f , along a single slice out of theḟ range. The results from WUs that search over the same 50-mHz range are combined into a single band, and these multiple WUs together cover the entireḟ range at that value of f . Each WU searches through approximately 5 × 10 10 templates, and returns two lists of results corresponding to the 3000 templates with the highest values of the 2F and O SGL statistics (described in section II C), called the toplists. The total number of templates included in this search is 4.99 × 10 17 .

F. Semiautomatic identification of disturbances
When the noise is purely Gaussian, the 2F distribution is well-modelled and the significances of signal candidates can be determined in a straight-forward manner. However, disturbances generate deviations from the expected distribution. In order to meaningfully use the same statistical analysis on all of the candidates, the disturbed 50-mHz bands must be excluded from the search. Previous searches [8,9] relied on a visual inspection of the full data set in order to identify the disturbed bands, which is a very time consuming endeavor. Here, we introduce a semiautomatic method that greatly reduces the number of bands that need to be visually inspected.
We use two indices to identify bands that cannot automatically be classified as undisturbed: 1) the density of toplist candidates in that band and 2) their average 2F. We classify as undisturbed those bands whose maximum density and average 2F are well within the bulk distribution of the values for these quantities in the neighbouring frequency bands, and mark the remainder as potentially disturbed and in need of visual inspection.
The size of the toplist and the frequency grid spacing are fixed. Therefore, when a disturbance is present in a 50-mHz band, the toplists within that band disproportionately include templates in the parameter space near the disturbance. We look for evidence of disturbances in the 50-mHz bands using a method that mimics and replaces the visual inspection used in previous searches [8]: For a given band, we calculate the density of candidates in a 10 × 10 grid in f -ḟ space, and take the maximum density as an indicator of how disturbed the band is likely to be. Since disturbances also manifest as deviations in the 2F distribution, we use the mean of 2F as an additional indicator of how much a band is disturbed. A visual representation of these concepts is shown in Fig. 2.
Because the search volume increases with f , both the mean of 2F and the candidate density vary with frequency. To account for this effect, we compare the observed maximum density and mean 2F from each band with the distribution of maximum density and mean 2F values in sets of 200 contiguous 50-mHz bands (10 Hz). These constitute our reference distributions.
Since the majority of bands are undisturbed, the reference distributions are composed of a well-defined bulk (from the undisturbed bands) with tails (disturbed bands), as illustrated in Fig. 3. We define the "bulk" of each distribution by eye, and then mark the bands that fall outside of this bulk on either side as being potentially disturbed; we generally expect disturbed bands to be in the upper ends of the distributions (that is, to have particularly large values of maximum density and mean 2F) but also include bands in the lower ends so as not to miss any unexpected disturbed behavior. We proceed with a full visual inspection only of this potentially disturbed subset. Fig. 3 shows the reference distributions for the bands between 90 and 100 Hz. These are typical examples and illustrate how the "by eye" definition of the bulk of the distributions is not subtle. When selecting the bulk, we err on being conservative: when in doubt, we label bands as being potentially disturbed, as these will be re-inspected later.
If a signal were present, it would not be excluded because of the automated procedure. On the one hand, if it were so weak that the band would not be marked as disturbed, the band would automatically be included in the analysis. On the other hand, if it were strong enough that the automated procedure mark its band as being potentially disturbed, then it would be visually inspected by a human who would recognise the signature of a signal and not discard the band.
This method still requires human input in two steps: first, to define the bulk of the reference distributions; and second, to inspect the subset of potentially disturbed bands. However, the "calibration" work necessary for de-termining the bulk of the reference distributions only requires the inspection of 2 distributions every 10 Hz rather than multiple distributions every 50 mHz. Furthermore, the bands that do not pass the undisturbed-classification criteria and require visual inspection are only 15% of the total set. Overall, this procedure still cuts down the required time from multiple days with multiple people to a few hours by a single person.
This procedure requires minimal tuning and relies only on the assumption that the reference distributions are predominantly undisturbed. This has so far been our experience on all the LIGO data sets that we have inspected. We are confident that this method can be applied to other sets of gravitational wave data.
When we compare this method against a full visual inspection of a few search frequency ranges (50 to 100 Hz, 450 to 500 Hz, and 950 to 1000 Hz), it identifies ∼ 95% of the disturbed bands and misses only the most marginal disturbances. After we apply this method to the entire frequency range, we exclude a total of 1991 50-mHz bands as being disturbed (∼ 10%); these are listed in Table S2.

G. Analysis of undisturbed bands
The 2F distribution in Gaussian noise only depends on the number of effectively independent templates searched (N ). However, the grid spacings are chosen to maximize signal recovery, so the N templates are not fully independent. The observed distribution is instead described by an effective number of templates N eff < N . The value of N eff is obtained by fitting the distribution of the loudest candidates (i.e., the highest values of 2F).
We divide the entire set of 50-mHz bands across our search frequency range into 2000 partitions of approximately equal parameter space volume, which results in ∼ 2 × 10 14 templates per partition. In order to create these partitions, we calculate an exact partitioning of the total search volume and divide the full range of 50-mHz bands so that the number of templates in each partition best matches the number of templates in the exact partitioning. Since the number of templates in a band grows with frequency, the frequency width spanned by each partition decreases with increasing frequency. This can be clearly seen in the left panels of Fig. 4. In this way, since each partition contains roughly the same number of search templates, the expected loudest candidate in each partition is the same and drawn from the same underlying distribution, defined by N eff . For this search, we find that N eff /N ≈ 0.65. Fig. 4 shows both the 2F (top) and the critical ratio CR (bottom) for the loudest candidates. We define CR as where 2F meas is the measured value of the loudest, 2F exp the expected value of the loudest, and σ 2F is the expected standard deviation for the loudest over a partition. The loudest candidate over the entire search is in the 620.85 Hz band and has a 2F value of 8.77; this is also the most significant candidate, with a CR of 4.56. However, if we consider the entire searched parameter space rather than just the partition at 620.85 Hz, the CR value of the most significant candidate drops to < 0; i.e., the expected loudest is actually higher than the loudest that we observe. This tells us that our search has not revealed any gravitational wave signal from Cas A in the targeted waveform parameter space, as even the template that most resembles a signal has a statistical significance that is well within the expectations due to random chance. We convert the CR values of the loudest candidates to p-values to represent the chance probability of finding a partition-loudest candidate as significant as or more significant than what was measured in the search. The results are plotted in Fig. 5, along with the expected distribution of p-values in Gaussian noise. There is a small systematic deviation from the expected distribution which arises from a subtle difference between the O SGL and 2F toplists and is not due to any physical effect.

III. UPPER LIMITS
We find no candidates with CR > 5 and no excess in the p-value distribution. Therefore, we set frequentist 90% upper limits on the continuous gravitationalwave strain h 90% 0 in our search range using the process described in previous works [8,9], which we summarise below.
The h 90% 0 in a partition is the gravitational-wave amplitude at which 90% of a population of signals with parameters within the partition would produce a more significant candidate than the most significant candidate measured by the search in that partition. We determine h 90% 0 by injecting signals at fixed amplitudes bracketing the h 90% 0 level, then running the search on these injections and counting how many injections were recovered (i.e., how many produced a candidate more significant than the loudest measured by the actual search). Because this injection-and-recovery procedure is time-consuming, we perform it on only a subset of twenty representative partitions -uniformly distributed in frequency in the search range -rather than the full set of 2000, and use these results to derive the upper limits in all the other partitions.
For each of the twenty injection partitions, we fit a sigmoid to the detection efficiency (the fraction of recovered injections) as a function of injection amplitude to determine both the value of h 90% 0 and the 1-σ uncertainty on h 90% 0 . We determine the h 90%,j 0,CRi in each of the injection partitions corresponding to different detection criteria binned by CR, with CR i = [0, 1, 2, 3, 4, 5]. For each CR i , we derive the corresponding sensitivity depths For each of the remaining partitions, at frequencies around f k , we derive the upper limit as where CR i (f k ) is the significance bin of the loudest candidate of the partition at f k and S h (f k ) the power spectral density of the data. D 90% CRi 70 Hz −1/2 for this search. Our upper limits are plotted in Fig. 6 in red with 1σ uncertainties in gray, and provided in tabular form as Supplemental Material. The uncertainties in h 90% 0 that we report here are propagated from the statistical uncertainties in fitting the recovery. The partitions containing disturbed bands (which were not included in the analysis) are marked with open circles.
The upper limit value near 170 Hz, where the detectors are the most sensitive, is 2.9 × 10 −25 . This value is roughly two times lower than the previous most constraining upper limit on Cas A [18], plotted in blue, which also used S6 data. Our upper limits are also more than twice as constraining as an earlier Cas A search, plotted in green [3], which ran on S5 data 1 . Our upper limits beat the so-called indirect age-based limit [17] across the vast majority of the frequency range.

IV. CONCLUSIONS
The upper limits on the gravitational wave strain from Cas A translate into constraints on the shape of Cas A. band is plotted in the blue histogram, and the expectation in Gaussian noise is shown in the black scatter points. We do not find any excess in our search. The small systematic deviation in our data from the expected is caused by a subtle difference in the OSGL-and 2F-rankings.
As described in [23], a neutron star's mass distribution can be described by the ellipticity , where = |I xx − I yy | I zz (8) and I zz is the principal moment of inertia of the star around its rotational axis. If a neutron star at a distance D and spinning at a frequency f /2 has a nonaxisymmetric distortion , then it will produce a continuous gravitational wave with a frequency f and amplitude h 0 . These quantities are related to each other as follows: Eq. 9 shows how we can re-express the constraints on the gravitational wave amplitude as constraints on the ellipticity. We take the distance to Cas A to be 3.4 kpc [1] and I zz to be 10 38 kg m 2 . These constraints on source ellipticity are shown in Fig. 7. For instance, if Cas A is emitting gravitational waves at around 200 Hz (and, therefore, spinning at a frequency of 100 Hz), its ellipticity should be less than a few times 10 −5 , since we would have been able to detect gravitational waves produced by larger ellipticities.
The maximum ellipticity is the ellipticity necessary to sustain emission at the spindown limit, i.e., when all of the lost rotational energy is radiated as gravitational waves. This spindown ellipticity is whereḟ is twice the spin-frequency derivative.
The highest spin-down ellipticity for an object emitting gravitational waves at a frequency f that our search could have detected can be computed from Eq. 10 by settingḟ = f /300 years. For an isolated system, ifḟ is twice the spin-frequency derivative, larger ellipticities would violate energy conservation. For this reason we only highlight the region between the ellipticity upper limit curve and the spindown ellipticity curve as excluded by the search. However, we note that systems in general could have ellipticities larger than the spindown ellipticity if the gravitational waveḟ (the apparentḟ ) differs from the intrinsic one due to, for example, radial motion.

V. ACKNOWLEDGMENTS
The authors thank the Einstein@Home volunteers who have supported this work by donating compute cycles of their machines. Maria Alessandra Papa, Bruce Allen and Xavier Siemens gratefully acknowledge the support from NSF PHY Grant 1104902. All the post-processing computational work for this search was carried out on the ATLAS super-computing cluster at the Max-Planck-Institut für Gravitationsphysik / Leibniz Universität Hannover. We also acknowledge the Continuous Wave Group of the LIGO Scientific Collaboration for useful discussions. This document has been assigned LIGO Laboratory document number LIGO-P1600212. upper limits from this search (90% UL) 1σ uncertainty from this search indirect age-based limit S5 Cas A search (95% UL) [3] S6 "Nine Young SNRs" (95% UL) [18] FIG. 6. 90% confidence strain amplitude upper limits in each of the 2000 partitions. The results for partitions that contain only undisturbed 50-mHz bands are plotted in the filled red circles, while the results for partitions with disturbed 50-mHz bands are plotted in the open red circles. We also plot the 95% confidence upper limits from two previous searches on Cas A in green and blue. Our upper limits beat the so-called indirect age-based limit [17] across the band.