Searches for Continuous Gravitational Waves from Fifteen Supernova Remnants and Fomalhaut b with Advanced LIGO

TL;DR

This paper reports on directed searches for continuous gravitational waves from 16 candidate neutron stars, including supernova remnants and Fomalhaut b, using Advanced LIGO data, resulting in no detections but setting stringent upper limits.

Contribution

First comprehensive search for continuous gravitational waves from these specific neutron star candidates with advanced LIGO, establishing new upper limits on gravitational wave strain and neutron star properties.

Findings

No astrophysical signals detected.

Set upper limits on gravitational wave strain at 1e-25.

Constrained neutron star ellipticity and r-mode amplitudes.

Abstract

We describe directed searches for continuous gravitational waves from sixteen well localized candidate neutron stars assuming none of the stars has a binary companion. The searches were directed toward fifteen supernova remnants and Fomalhaut~b, an extrasolar planet candidate which has been suggested to be a nearby old neutron star. Each search covered a broad band of frequencies and first and second time derivatives. After coherently integrating spans of data from the first Advanced LIGO observing run of 3.5--53.7 days per search, applying data-based vetoes and discounting known instrumental artifacts, we found no astrophysical signals. We set upper limits on intrinsic gravitational wave strain as strict as $1\times10^{-25}$, on fiducial neutron star ellipticity as strict as $2\times10^{-9}$, and on fiducial $r$-mode amplitude as strict as $3\times10^{-8}$.

Tables

Table 1. Table 1: Targeted objects and astronomical parameters used in each search

SNR	parameter	Other name	RA+dec	$D$	$a$
(G name)	space		(J2000)	(kpc)	(kyr)
1.9$+$0.3		—	174846.9$-$271016	8.5	0.1
15.9$+$0.2		—	181852.1$-$150214	8.5	0.54
18.9$-$1.1		—	182913.1$-$125113	2	4.4
39.2$-$0.3		3C 396	190404.7$+$052712	6.2	3
65.7$+$1.2		DA 495	195217.0$+$292553	1.5	20
93.3$+$6.9		DA 530	205214.0$+$551722	1.7	5
111.7$-$2.1		Cas A	232327.9$+$584842	3.3	0.3
189.1$+$3.0	wide	IC 443	061705.3$+$222127	1.5	3
189.1$+$3.0	deep	IC 443	061705.3$+$222127	1.5	20
266.2$-$1.2	wide	Vela Jr.	085201.4$-$461753	0.2	0.69
266.2$-$1.2	deep	Vela Jr.	085201.4$-$461753	0.9	5.1
291.0$-$0.1		MSH 11$-$62	111148.6$-$603926	3.5	1.2
330.2$+$1.0		—	160103.1$-$513354	5	1
347.3$-$0.5		—	171328.3$-$394953	0.9	1.6
350.1$-$0.3		—	172054.5$-$372652	4.5	0.6
353.6$-$0.7		—	173203.3$-$344518	3.2	27
354.4$+$0.0	wide	—	173127.5$-$333412	5	0.1
354.4$+$0.0	deep	—	173127.5$-$333412	8	0.5
—	wide	Fomalhaut b	225739.1$-$293720	0.011	316
—	deep	Fomalhaut b	225739.1$-$293720	0.02	3000

Table 2. Table 2: Derived parameters used in each search

SNR	parameter	$f_{\min }$	$f_{\max }$	$T_{\mathrm{span}}$	$T_{\mathrm{span}}$	Start of span	H1	L1	Duty	$h_0^{\mathrm{age}}$
(G name)	space	(Hz)	(Hz)	(s)	(days)	(UTC, 2015)	SFTs	SFTs	factor	$(\times {10}^{-25})$
1.9$+$0.3		38	1332	336 307	3.9	Nov 30 03:53:08	156	141	0.79	8.4
15.9$+$0.2		72	538	887 744	10.3	Nov 25 13:39:16	369	304	0.68	3.6
18.9$-$1.1		45	987	1 133 255	13.1	Nov 21 00:00:40	462	346	0.64	5.4
39.2$-$0.3		98	295	1 965 780	22.8	Nov 28 00:47:19	641	647	0.59	2.1
65.7$+$1.2		53	794	1 932 067	22.4	Dec 14 04:52:40	774	555	0.62	3.4
93.3$+$6.9		41	1215	1 051 764	12.2	Nov 25 12:39:16	385	354	0.63	5.9
111.7$-$2.1		31	1998	775 855	9.0	Nov 26 20:58:03	317	294	0.71	12.6
189.1$+$3.0	wide	37	1547	803 419	9.3	Nov 26 12:43:17	331	296	0.70	8.7
189.1$+$3.0	deep	50	805	1 933 867	22.4	Dec 14 04:52:40	775	555	0.62	3.4
266.2$-$1.2	wide	19	1998	462 616	5.4	Nov 28 02:17:19	191	213	0.79	136
266.2$-$1.2	deep	32	1998	799 819	9.3	Nov 26 12:43:17	329	294	0.70	11.2
291.0$-$0.1		42	987	788 409	9.1	Nov 26 18:28:03	322	295	0.70	5.9
330.2$+$1.0		53	731	851 744	9.9	Nov 25 23:39:16	349	302	0.69	4.5
347.3$-$0.5		27	1998	578 325	6.7	Nov 28 05:17:19	237	253	0.76	19.9
350.1$-$0.3		42	1038	637 577	7.4	Nov 28 02:17:19	257	271	0.75	6.5
353.6$-$0.7		132	275	3 762 662	43.5	Nov 21 02:30:40	1339	1078	0.58	1.4
354.4$+$0.0	wide	36	1677	301 250	3.5	Nov 28 02:17:19	125	152	0.83	14.4
354.4$+$0.0	deep	62	635	790 209	9.1	Nov 26 17:58:03	323	295	0.70	4.0
Fomalhaut b	wide	19	1998	2 492 267	28.8	Sep 18 20:08:24	955	799	0.63	116
Fomalhaut b	deep	22	1998	4 639 371	53.7	Nov 19 23:13:10	1626	1295	0.57	20.7

Equations

$$f(t)=f+\dot{f}(t-t_0)+\frac{1}{2}\ddot{f}(t-t_0{)}^2,$$

$$h_0^{\mathrm{age}}=1.26\times 10^{-24}\left(\frac{\text{mbox}3.30kpc}{D}\right){\left(\frac{\text{mbox}300yr}{a}\right)}^{1/2},$$

$$-\frac{f}{a}\le \dot{f}\le -\frac{1}{6}\frac{f}{a};$$

$$2\frac{{\dot{f}}^2}{f}\le \ddot{f}\le 7\frac{{\dot{f}}^2}{f}.$$

$$ϵ=9.5\times 10^{-5}\left(\frac{h_0}{10^{-24}}\right)\left(\frac{D}{\text{mbox}1kpc}\right){\left(\frac{\text{mbox}100Hz}{f}\right)}^2.$$

$$\alpha =0.028\left(\frac{h_0}{10^{-24}}\right){\left(\frac{\text{mbox}100Hz}{f}\right)}^3\left(\frac{D}{\text{mbox}1kpc}\right).$$

Full text

gravitational waves — stars: neutron — supernova remnants

††software: LALSuite (LIGO Scientific Collaboration, 2018)

\AuthorCollaborationLimit

=3000

1 Introduction

With the detections of several binary black hole mergers (Abbott et al., 2016a, b, 2017a, 2017b, 2017c) and one binary neutron star merger (Abbott et al., 2017d) seen also in electromagnetic waves (Abbott et al., 2017e), Advanced LIGO and Virgo have spectacularly inaugurated the field of gravitational wave (GW) astronomy. While the binary neutron star merger has had far-reaching implications for our knowledge of neutron star matter (De et al., 2018; Abbott et al., 2018), a continuous GW signal could teach us even more—not just about bulk properties but internal magnetic fields, the extent and strength of crystalline phases, and potentially other microphysics of extreme matter (Owen, 2009; Glampedakis & Gualtieri, 2017).

Young isolated neutron stars are promising sources of continuous GWs. The spin-downs of young pulsars are rapid enough to include significant continuous GW emission, as shown by the latest GW search for known pulsars (Abbott et al., 2017f, g). Theoretical arguments suggest that $r$-modes (oscillations dominated by the Coriolis force) might remain unstable and detectable in neutron stars up to a few thousand years old (Owen, 2010, and references therein). Most young supernova remnants do not contain known pulsars (Green, 2014). On the other hand, many of these SNRs contain small pulsar wind nebulae, central compact objects, or other well localized non-pulsing candidate neutron stars. Also, some of these SNRs are young enough that a neutron star could not have been kicked far, and thus the star can be considered well localized even if it is not seen at all. GW searches directed at single sky positions can significantly improve on the sensitivities of all-sky surveys, even while needing to cover a wide band of possible GW frequencies and first and second time derivatives due to lack of pulsations from the object (Wette et al., 2008). This makes non-pulsing isolated neutron stars attractive targets for continuous GW searches if they are well localized.

Directed GW searches for isolated neutron stars have been published targeting SNRs (Abadie et al., 2010, 2011; Aasi et al., 2015; Sun et al., 2016; Zhu et al., 2016; Abbott et al., 2017h) and promising locations including the galactic center (Abadie et al., 2011; Aasi et al., 2013; Abbott et al., 2017h) and the core of a nearby globular cluster—where multi-body interactions might effectively rejuvenate some neutron stars’ continuous GW emission (Abbott et al., 2017i). The only such search of data from advanced interferometers so far (Abbott et al., 2017h) employed methods from stochastic background searches which, while quick to implement, are not as sensitive as continuous wave search methods.

Here we present the first directed continuous wave searches for isolated non-pulsing neutron stars in data from the first Advanced LIGO observing run (O1). We used an extension of the coherent data analysis pipeline used in Abadie et al. (2010) and Aasi et al. (2015), to which this paper is a sequel. The improved noise curve (with respect to initial LIGO and Virgo) means that we can search more targets with sensitivity beating the indirect upper limit on GW emission due to energy conservation (Wette et al., 2008) based on the age of the neutron star (similar to the spin-down limit for known pulsars). We include not only more supernova remnants, but also the directly imaged exoplanet candidate Fomalhaut b, which has been proposed to be an old nearby neutron star (Neuhäuser et al., 2015)—close enough that it is an attractive target in spite of being much older than the others. We do not include SN 1987A because it is so young that the possible spin-down parameter space is too large to cover with a coherent wide band search and a reasonable computational cost.

2 Searches

2.1 Methods

These searches were based on the multi-interferometer $\mathcal{F}$-statistic (Jaranowski et al., 1998; Cutler & Schutz, 2005). The $\mathcal{F}$-statistic accounts for the modulation of the signal due to the daily rotation of the detectors by adding the outputs of sinusoidal matched filters in quadrature. For these searches the frequency evolution of each filter, in the reference frame of the solar system barycenter, was given by

[TABLE]

where $t_{0}$ is the beginning of the observation and the frequency derivatives are evaluated at that time and in a slight abuse of notation we use a simple $f$ for $f(t_{0}).$ Hence these filters are designed to detect neutron stars without binary companions whose spin-down is not too fast (requiring third or higher frequency derivatives) or too irregular (having significant timing noise or glitches) during the observation. In stationary Gaussian noise, $2\mathcal{F}$ is drawn from a $\chi^{2}$ distribution with four degrees of freedom, which for loud signals makes the amplitude signal-to-noise ratio roughly $\sqrt{\mathcal{F}/2}.$ If a signal is present, the $\chi^{2}$ is noncentral.

We used data from LIGO O1, but none from Virgo because that interferometer was down for upgrades during O1. At the frequencies to which LIGO was most sensitive (about 100-300 Hz), the strain noise amplitude was about 3–4 times lower than in the sixth LIGO science run (S6) (Abbott et al., 2016c). However there were many more spectral lines due to instrumental artifacts than in S6, which complicated the analysis. We used the calibration described in Abbott et al. (2017f), which is an update of the first O1 calibration described in Abbott et al. (2017j). Hence, as in Abbott et al. (2017f), our upper limits on strain are uncertain by at least 14%. Like many other continuous GW searches, ours used data in the form of short Fourier transforms of duration 1800 s, high pass filtered and Tukey windowed to reduce artifacts, recording only frequencies up to 2 kHz.

While each search targeted a specific direction (right ascension and declination), each had to cover a broad band of frequencies and first and second derivatives. That is, a bank of signal templates was required, constructed to cover the parameter space $(f,\dot{f},\ddot{f})$ with sufficient density (Whitbeck, 2006; Wette et al., 2008). We chose coverage such that the maximum loss of power signal-to-noise ratio due to mismatch between the signal and the nearest template (Owen, 1996; Brady et al., 1998) was no worse than 20%, a common choice in continuous GW analyses. Given the parameter choices described below, this resulted in $10^{12}$–$10^{13}$ templates for most searches, with the Cas A search getting more than $10^{14}$ since it was allocated ten times the computing cycles of each other search.

All searches ran on the Atlas computing cluster at the Max Planck Institute for Gravitational Physics (Albert Einstein Institute) in Hanover, Germany using the same tag (S6SNRSearch) of the LALSuite software package (LIGO Scientific Collaboration, 2018) as in Aasi et al. (2015) although the controlling scripts were upgraded. Most searches used roughly $10^{5}$ core hours (split into roughly $3\times 10^{4}$ batch jobs) and Cas A used more than $10^{6}$ (split into roughly $3\times 10^{5}$ jobs). The splitting into jobs was used in the vetoes and other post-processing described in Sec. 2.4. Post-processing for each search used at most of order ten percent of the core hours dedicated to the search. Several terabytes of search results were written to disk.

2.2 Target List

Our choice of targets required that a search of fixed computational cost be sensitive enough to detect the strongest continuous GW signal consistent with broad conservation of energy considerations. As introduced by Wette et al. (2008) for the CCO in supernova remnant Cas A, the strongest possible signal based on the age $a$ and distance $D$ of the source,

[TABLE]

is analogous to the spin-down limit for known pulsars and indicates the strongest possible intrinsic strain produced by an object whose unknown spin-down is entirely due to GW emission and has been since birth. The fiducial parameters (for Cas A) show that this limit can be high enough to be interesting. The intrinsic strain $h_{0}$ (Jaranowski et al., 1998) characterizes the GW metric perturbation without reference to any particular orientation or polarization, and hence is typically a factor 2–3 greater than the strain response measured by the interferometers. The indirect limit $h_{0}^{\mathrm{age}}$ is slightly different for $r$-mode emission (Owen, 2010) than for the mass quadrupole source tacitly assumed above and in most of the literature, but we neglect this small difference. Due to uncertainties in the neutron star mass and equation of state, $h_{0}^{\mathrm{age}}$ is uncertain by of order 50%, which we also neglect.

To choose directions to search, we started from the Green catalog of supernova remnants (Green, 2014). We picked x-ray point sources (CCOs or candidate CCOs), small PWNe, and in some cases relatively young SNRs where any neutron star could not yet have moved far. We selected only targets with age and distance estimates so that we could evaluate $h_{0}^{\mathrm{age}}.$ In some cases there is a wide range of estimates in the literature, leading to significant differences in $h_{0}^{\mathrm{age}}.$ In most cases we used the most optimistic estimates, yielding the highest $h_{0}^{\mathrm{age}}$ but also the most difficult search over the widest band of frequency and spin-down parameters. In addition to this wide search using the optimistic age and distance, we did a deep search using the most pessimistic age and distance in cases where the strain sensitivity would improve over the wide search by a factor of roughly $\sqrt{2}.$

The resulting targets and chosen parameters are shown in Table 1. We now briefly summarize each target and the provenance of the parameters used for it.

G1.9+0.3—Currently the youngest known SNR in the galaxy (Reynolds et al., 2008). Several arguments favor it being a Type Ia (Reynolds et al., 2008), which would leave no neutron star behind, but this is not definite and the remnant’s youth makes it an interesting target on the chance that it is not Type Ia. We used the position of the center of the remnant from the discovery paper (Reich et al., 1984). At maximum kick velocity any neutron star could have moved only a few arcseconds, which is not an issue for our searches. The age and distance shown are from the “rediscovery” paper (Reynolds et al., 2008), though the latter is a nominal galactic center distance.

G15.9+0.2—The CCO was discovered in Chandra data by Reynolds et al. (2006). We used the lower limit on age and the galactic center distance estimate from the same paper, though both quantities may be significantly greater (Klochkov et al., 2016).

G18.9–1.1—The position is that of the Chandra point source discovered by Tüllmann et al. (2010). Age and distance estimates are from the previous ROSAT and ASCA observations of Harrus et al. (2004).

G39.2–0.3—Also known as 3C 396. The PWN and embedded point source were found by Olbert et al. (2003) in Chandra data, the point source being localized to within $2^{\prime\prime}$ in spite of the PWN. Su et al. (2011) estimate the age and distance, the latter based on the tangent point of the spiral arm.

G65.7+1.2—Also known as DA 495. Arzoumanian et al. (2008) found the Chandra point source in the PWN. The quoted distance (Kothes et al., 2004) and minimum age (Kothes et al., 2008) are derived slightly inconsistently due to assumed distances to the galactic center. We did not attempt to resolve the inconsistency, though we did choose the distance from the former paper since it uses the more commonly accepted galactic center distance. The latter paper (and others) also argue that the distance could be several times higher.

G93.3+6.9—Also known as DA 530. The position and age are from Jiang et al. (2007) and the distance estimate is from Foster & Routledge (2003). Jiang et al. (2007) find no Chandra point source, but the X-ray intensity of the faint candidate PWN falls off on a scale of $6^{\prime\prime},$ which qualifies as a point source for our purposes.

G111.7–2.1—Also known as Cas A. The position of the CCO is from the Chandra “first light” observation (Tananbaum, 1999), the distance is from Reed et al. (1995), and the age is from Fesen et al. (2006).

G189.1+3.0—Also known as IC 443. The position is that of the Chandra point source found by Olbert et al. (2001) embedded in the PWN. This object is often studied, with a wide range of distance and age estimates in the literature. We used Petre et al. (1988) for an optimistic age estimate. Our pessimistic age estimate is not quite the most extreme in the literature, but rather a best fit for a pessimistic scenario from relatively recent modeling (Swartz et al., 2015). We did not use the most optimistic distance quoted, but rather the assumed association with the I Gem cluster from Fesen & Kirshner (1980) which is common in the literature.

G266.2–1.2—Also known as Vela Jr. The position is that of the CCO found by Pavlov et al. (2001). We used Iyudin et al. (1998) for the most optimistic age and distance estimates. The pessimistic age estimate is from Allen et al. (2015), which was published too recently for the previous paper in this series (Aasi et al., 2015). Allen et al. (2015) also discuss the possible association of several surrounding objects with the nearer concentration of the Vela Molecular Ridge, at a spread of distances providing our pessimistic distance estimate (Liseau et al., 1992) and rendering the more pessimistic ones unlikely.

G291.0–0.1—Also known as MSH 11$-$62. The position and age are from the Chandra point source discovery paper (Slane et al., 2012). The distance is from Moffett et al. (2001). The age and distance are derived in slightly inconsistent ways, but rather than attempt to repeat the calculations we used the numbers quoted in the literature.

G330.2+1.0—The CCO was discovered by Park et al. (2006) in Chandra data with sub-arcsecond position accuracy. We used a distance estimate from radio observations (McClure-Griffiths et al., 2001) and an age estimate from the x-ray spectrum (Park et al., 2009).

G347.3–0.5—Mignani et al. (2008) obtained the sub-arcsecond position from archival Chandra data, although the CCO had been identified in ASCA data earlier (Slane et al., 1999). We used the distance from Cassam-Chenaï et al. (2004) and the age from the proposed identification with a possible SN 393 (Wang et al., 1997). Although this identification may be problematic given the inferred properties of such a supernova, other age estimates are comparable (Fesen et al., 2012).

G350.1–0.3—Position and distance estimates are from the discovery paper of the CCO candidate by Gaensler et al. (2008). The age is from Chandra observations Lovchinsky et al. (2011).

G353.6–0.7—Halpern & Gotthelf (2010) identified the most likely of several candidate CCOs. The age estimate (Tian et al., 2008) makes this CCO candidate the only one that is almost certainly too old for $r$-modes, although we still set upper limits on $r$-mode amplitude. The distance estimate is also from Tian et al. (2008). We used the first-observation position contained in the name of the candidate CCO rather than the slightly better Chandra position reported by Halpern & Gotthelf (2010); the roughly $1^{\prime\prime}$ difference is not significant for GW integration times used in this paper.

G354.4+0.0—All parameters are from the discovery paper (Roy & Pal, 2013). No associated point source has been detected yet, but if the remnant’s age is correct any young neutron star should be within roughly $20^{\prime\prime}$ of the center (whose location we used for the GW search). Such a position error is not significant for the integration times used here.

Fomalhaut b—Considered an extrasolar planet candidate since its discovery in a visible light image (Kalas et al., 2008), this has been proposed based on a lack of infrared detection to be a serendipitous discovery of a nearby neutron star (Neuhäuser et al., 2015). Parameters are taken from Neuhäuser et al. (2015), with the maximum distance an attempt to balance the uncertainties in the scenarios discussed there. After this search was run, Poppenhaeger et al. (2017) searched for and did not find the object with Chandra. If the object is a neutron star, this somewhat reduced the possible distance and significantly increased the minimum age.

2.3 Parameter Space

After sky position, the key parameters for each search were the GW frequency band $(f_{\min},f_{\max})$ and time span of integration $T_{\mathrm{span}}.$ As in Aasi et al. (2015), these parameters were determined in an iterative process intended to produce a search more sensitive than $h_{0}^{\mathrm{age}}$ over as wide a frequency band as possible for a fixed computational cost. Due to Doppler shifts and several features of the analysis, we capped the maximum frequency at 1998 Hz rather than the 2 kHz in the SFTs. The cost, approximated as proportional to $a^{-1.1}f_{\max}^{2.2}T_{\mathrm{span}}^{4},$ was kept comparable to Abadie et al. (2010) for most targets, but Cas A was allocated ten times as many computational cycles due to its status as youngest known neutron star in the galaxy. Due to some inaccuracy in the power-law fit used for computational cost as a function of the key parameters, the computational cost and sensitivity varied by up to 20–30% from these goals. For a given frequency $f,$ as in Abadie et al. (2010) and Aasi et al. (2015), we searched

[TABLE]

and for a given $\dot{f}$ we searched

[TABLE]

These ranges and the computational cost fixed $f_{\min},$ $f_{\max},$ and $T_{\mathrm{span}}$ for each search.

We then chose the start time of each search by the same method as Abadie et al. (2010) and Aasi et al. (2015), minimizing the harmonic mean of the strain noise power spectral density during the span over the frequency band $(f_{\min},f_{\max}).$ Neglecting the small effect of the declination of the target, this corresponds to maximizing the search sensitivity for a fixed $T_{\mathrm{span}}$—which is roughly a fixed computational cost. Hence the algorithm chose spans when both interferometers had good noise performance and little down time, usually later in O1. The resulting search parameters are described in Table 2.

We applied the same consistency checks as in previous searches: For each search we checked using the parameter space metric (Whitbeck, 2006; Wette et al., 2008) that neglect of the third frequency derivative in Eq. (1) did not significantly reduce $2\mathcal{F},$ even in the worst case (G1.9+0.3). We also checked that the position uncertainties of the targets also did not significantly reduce $2\mathcal{F}.$ A simple approximation (Whitbeck, 2006) suggests that the sky resolution of these searches is an arcminute or two at 2 kHz and a 10 day integration, and it scales inversely with $f_{\max}$ and $T_{\mathrm{span}}.$ We spot checked this with injection studies and found it to be accurate. Given the integration times in Table 2, even the worst position uncertainty ($20^{\prime\prime}$ for G354.4+0.0) is well within bounds for a single directed search. Finally, we checked that the standard 1800 s SFT duration did not diminish sensitivity to signals with $\dot{f}$ high enough that the frequency could move to another SFT frequency bin over the duration of the SFT. This effect was negligible except for SNR G1.9+0.3, where it could reduce the sensitivity (raise the detectable $h_{0}$) by of order 10% at frequencies above 1 kHz.

2.4 Post-processing

Each search recorded a list of candidates with high values of $2\mathcal{F},$ which was then pared using two automated vetoes designed for instrumental artifacts, as used in Abadie et al. (2010) and Aasi et al. (2015). The “Fscan veto” used a normalized spectrogram formed from the SFTs to detect and veto spectral lines and nonstationary noise. Its implementation and parameters were the same as in Aasi et al. (2015) except that we fixed a bug in the old code whereby the Doppler shift due to the Earth’s orbital motion was not applied. (This bug allowed more noise lines to pass the automated vetoes and require manual scrutiny, but had a negligible effect on the false dismissal rate.) The “interferometer consistency veto” ruled out candidates for which a single-interferometer $2\mathcal{F}$ exceeded the two-interferometer $2\mathcal{F}$ for the same event, indicating a disturbance present in only one interferometer. It also vetoed entire search jobs if the number of candidates vetoed was high enough. This veto was also applied in the same way as in Aasi et al. (2015), except that the threshold for vetoing an entire search job was 5% of the templates in that job. Unlike in previous papers in this series, we also vetoed a list of known instrumental spectral lines compiled from studies of the interferometers (Covas et al., 2018).

After these steps, including fixing the Doppler bug, the searches still had almost two thousand jobs containing non-vetoed outliers above the 95% confidence level for Gaussian noise. All of these jobs were examined by hand. As in Aasi et al. (2015), two plots were made and inspected for each job. (See Figure 1 of that paper for illustrative examples.) In case of a real or injected signal the first plot, of $2\mathcal{F}$ vs. frequency for all loud candidates in the job, would show a $\delta$-function like spike even for very loud signals, as verified by studying hardware injections. The candidates generally showed broad bands of high noise, occupying a fraction of order unity of the search job frequency band except for a handful which occupied a few percent of the search band. These few candidates, which were still of order one hundred times broader than a real signal would be, were verified to be hardware injected test signals detectable in the wrong sky location due to their huge amplitudes. The second plot for each search job containing candidates was a semilog histogram of loud candidates, which on inspection typically showed the tail of a $\chi^{2}(4)$ distribution with the wrong amplitude, indicative of a broad band disturbance in the noise spectrum. See Aasi et al. (2015) for examples and further details.

No candidates survived inspection of these plots, and therefore we conclude that no astrophysical signal was detected.

3 Upper limits

Our method of setting upper limits was almost the same as in previous papers (Abadie et al., 2010; Aasi et al., 2015). In each 1 Hz band searched, we estimated the value of $h_{0}$ that would be detected 95% of the time by our search (assuming random variation of other signal parameters such as inclination of the star’s rotation axis to the line of sight) at a louder value than the loudest $2\mathcal{F}$ actually recorded by the search in that band. We made an initial estimate from a semianalytic integration of the expected $2\mathcal{F}$ distribution. Then we injected simulated signals with different values of $h_{0}$ near this value to refine the location of the 95% confidence (5% false dismissal) threshold. We reduced the number of injections per band to 1000 (from 6000 in previous papers) due to the computational cost of setting upper limits on wider bands.

For each search we pared the list of upper limits on $h_{0}$ versus frequency. We dropped bands where the injections indicated the false dismissal rate was more than 5% and we dropped $\pm 1$ Hz bands around harmonics of the 60 Hz power mains up to 300 Hz.

The resulting upper limits on $h_{0},$ in 1 Hz frequency bands, are plotted in Figs. 1–5. Each curve has roughly the same shape as the amplitude spectral density of the strain noise. The line of dots near the top of each plot corresponds to bands where no upper limit was set. Some features such as the “violin modes” of the interferometer test mass suspension (roughly 500 Hz and harmonics) are evident. The horizontal line in each plot is $h_{0}^{\mathrm{age}},$ the strain the search was intended to beat. In some cases the estimate of sensitivity made before performing the search was wrong by of order 10%, so the upper limits (lower dots) do not always lie below the line.

Upper limits on $h_{0}$ can be converted to upper limits on fiducial neutron-star ellipticity $\epsilon=|I_{xx}-I_{yy}|/I_{zz}$ (where $I_{ab}$ is the moment of inertia) using (e.g. Wette et al., 2008)

[TABLE]

This number assumes $I_{zz}=10^{45}$ g cm${}^{2}.$ Uncertainties in the mass, radius, and neutron star equation of state make the conversion from $h_{0}$ to $\epsilon$ uncertain by a factor of two or more. This fiducial ellipticity can be converted to the true shape of the star (Johnson-McDaniel, 2013) or other quantities (Owen, 2010). We plot upper limits on $\epsilon$ for a selection of searches in the left hand panel of Fig. 6. We do not plot the indirect limits on $\epsilon$ and $\alpha$ derived from $h_{0}^{\mathrm{age}}$ since they are close to the direct upper limits on the scale of the plot. We do not plot the remaining searches because their upper limits are close to those of the searches plotted. The great differences between curves are mainly due to the distances to the sources; hence Fomalhaut b has the best upper limits—of order $10^{-9}$ at high frequencies.

Upper limits on $h_{0}$ can be converted to the common $r$-mode amplitude parameter $\alpha$ (Lindblom et al., 1998) via (Owen, 2010)

[TABLE]

This number assumes a fiducial set of stellar parameters described in Owen (2010) and is uncertain by a factor of up to about three depending in the neutron star mass and equation of state. We plot upper limits on $\alpha$ for a selection of searches in the right hand panel of Fig. 6. Again, the differences between curves are mainly due to the source distances. The best upper limits, apart from Fomalhaut b which is almost certainly too old for active $r$-modes, are of order $10^{-6}$ at high frequencies for Vela Jr.

4 Discussion

These are the first directed searches of Advanced LIGO data using continuous wave analysis methods. These searches have improved on previous directed searches by covering wider parameter ranges and more targets, and by setting better upper limits on targets searched previously. Our upper limits on $h_{0}$ approach $2\times 10^{-25}$ for many targets and approach $1\times 10^{-25}$ for one—about a factor of 3 improvement on Aasi et al. (2015), due mainly to the improvement in the detectors. And our upper limits beat the indirect limit $h_{0}^{\mathrm{age}}$ over bands of 1–2 kHz for more targets than were ever published before. [Searches for some of these targets in less sensitive S6 data for the purpose of testing code were described in an unpublished thesis (Idrisy, 2015).] As with previous data runs, we improved on the sensitivity of all-sky wide-band searches (Abbott et al., 2018) but did not match the sensitivity of searches for known pulsars with full timing solutions (Abbott et al., 2017f). As before, the directed searches described here also have the caveats that there might be no neutron star present in some cases, any neutron star might be spinning too slowly to be detected, and a neutron star spinning at a detectable frequency might glitch—the latter phenomenon in a CCO is now an observation (Gotthelf & Halpern, 2018) rather than a surmise, and would somewhat reduce the sensitivity of these searches (Ashton et al., 2017). Even with some longer integration times here, timing noise is not an issue unless these objects are orders of magnitude noisier than known pulsars (Ashton et al., 2015).

Most of our upper limits on $\epsilon$ and $\alpha$ are competitive with the largest numbers predicted by theory. The maximum $\epsilon$ for “mountains” supported by elastic stresses of normal neutron star matter is probably $10^{-5}$–$10^{-6}$ (Horowitz & Kadau, 2009; Johnson-McDaniel & Owen, 2013; Baiko & Chugunov, 2018), and for many of our searches upper limits are in this region over hundreds of Hz. The maximum $\alpha$ (nonlinear saturation amplitude) for $r$-modes is probably of order $10^{-3}$ (Bondarescu et al., 2009), and for many of our searches upper limits beat this over hundreds of Hz. Mountains supported by an internal magnetic field can produce $\epsilon$ of order $10^{-4}(B/10^{15}\mbox{ G})^{2}$ where $B$ is the poloidal part of the field (e.g. Ciolfi & Rezzolla, 2013). Since, unlike elastic mountains, magnetic mountains are likely to be within about an order of magnitude of this limit for a given internal field, depending on its configuration, our upper limits on $\epsilon$ translate into rough limits on internal magnetic field—if a neutron star is present and spinning rapidly enough to emit GWs in band.

More data from Advanced LIGO and Advanced Virgo is now available, with more live time and lower noise amplitude than before. The detectable values of intrinsic strain, ellipticity, and $r$-mode amplitude are proportional to the noise amplitude and to the inverse square root of the live time. This makes more targets feasible for directed searches at greater sensitivity, increasing the chances of a detection of continuous GWs. Such searches will be done in the near future.

The authors gratefully acknowledge the support of the United States National Science Foundation (NSF) for the construction and operation of the LIGO Laboratory and Advanced LIGO 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. The authors gratefully acknowledge the Italian Istituto Nazionale di Fisica Nucleare (INFN), the French Centre National de la Recherche Scientifique (CNRS) and the Foundation for Fundamental Research on Matter supported by the Netherlands Organisation for Scientific Research, for the construction and operation of the Virgo detector and the creation and support of the EGO consortium. The authors also gratefully acknowledge research support from these agencies as well as by the Council of Scientific and Industrial Research of India, the Department of Science and Technology, India, the Science & Engineering Research Board (SERB), India, the Ministry of Human Resource Development, India, the Spanish Agencia Estatal de Investigación, the Vicepresidència i Conselleria d’Innovació, Recerca i Turisme and the Conselleria d’Educació i Universitat del Govern de les Illes Balears, the Conselleria d’Educació, Investigació, Cultura i Esport de la Generalitat Valenciana, the National Science Centre of Poland, the Swiss National Science Foundation (SNSF), the Russian Foundation for Basic Research, the Russian Science Foundation, the European Commission, the European Regional Development Funds (ERDF), the Royal Society, the Scottish Funding Council, the Scottish Universities Physics Alliance, the Hungarian Scientific Research Fund (OTKA), the Lyon Institute of Origins (LIO), the Paris Île-de-France Region, the National Research, Development and Innovation Office Hungary (NKFIH), the National Research Foundation of Korea, Industry Canada and the Province of Ontario through the Ministry of Economic Development and Innovation, the Natural Science and Engineering Research Council Canada, the Canadian Institute for Advanced Research, the Brazilian Ministry of Science, Technology, Innovations, and Communications, the International Center for Theoretical Physics South American Institute for Fundamental Research (ICTP-SAIFR), the Research Grants Council of Hong Kong, the National Natural Science Foundation of China (NSFC), the Leverhulme Trust, the Research Corporation, the Ministry of Science and Technology (MOST), Taiwan and the Kavli Foundation. The authors gratefully acknowledge the support of the NSF, STFC, MPS, INFN, CNRS and the State of Niedersachsen/Germany for provision of computational resources.

This paper has been assigned document number LIGO-P1800333-v7. Data files for the figures are available at https://dcc.ligo.org/P1800333/public.