Gravitational-wave Constraints on the Equatorial Ellipticity of Millisecond Pulsars

CW searches can be divided into three main types. Targeted searches look for signals from known pulsars whose rotational phase is accurately determined from electromagnetic observations, considerably simplifying the search. Directed searches look for signals from small sky areas, such as supernova remnants, where a neutron star is believed to reside, but for which no timing solution exists, so that a wide range of rotational parameters needs to be searched over. All-sky searches look for signals over all sky directions and also over a wide range of rotational parameters. Many searches of these three types have already been carried out, using LIGO and Virgo data. For recent examples, see Abbott et al. (2019a, 2019f, 2019g). No detections have been made, and consequently upper limits have been set on the strengths of such signals.

In this paper we report new results of targeted searches for CW signals from five pulsars, using the most recent LIGO and Virgo data sets. Specifically, we use data from the first and second observing runs (O1 and O2), together with data from the first half of the third observing run (O3a), allowing us to set improved upper limits compared to other recent searches (e.g., Abbott et al. 2019a).

It is possible to carry out such searches for many more (several hundred) known pulsars (Abbott et al. 2019a). We report results here for pulsars of particular interest. Specifically, we target three older, recycled pulsars, two of which are millisecond pulsars and one of which is only mildly recycled, that are believed to have undergone periods of accretion, and two very young pulsars: Crab and Vela. We search for the older pulsars, and particularly the recycled pulsars, because the signal amplitude is proportional to the square of the frequency, and therefore only small distortions are necessary to make a detection possible (see Equation (4)). The young pulsars are interesting because their rapid spin-down means that only a small fraction of their spin-down energy need go into the gravitational-wave channel for a detection to be possible. Here we obtain direct gravitational-wave observational limits that are at or below the spin-down limits for two of the recycled pulsars. This is the first time the spin-down limit has been equaled or surpassed for a recycled pulsar. As such, this represents a significant milestone for gravitational-wave astronomy.

The structure of this paper is as follows. In Section 1.2 we describe the signal models we used. In Section 2 we discuss the analysis methods used in the searches. In Section 3 we describe both the gravitational-wave data we used, and also the radio pulsar data that was used to produce the timing solutions on which the gravitational-wave searches were based. In Section 4 we describe our results, which are then discussed in Section 5. Finally, in Section 6, we draw some conclusions.

We will assume gravitational-wave emission that is tied closely to the rotational phase of the star. In the simplest case of a triaxial star spinning steadily about a principal moment-of-inertia axis, the gravitational-wave emission is at exactly twice the star's spin frequency.

There are several mechanisms, however, that can produce slightly different signals. Free precession of the star can produce a small frequency offset between the gravitational-wave and (twice) the spin frequency, and also produce a lower harmonic, at or close to the spin frequency (Zimmermann & Szedenits 1979; Jones & Andersson 2002). In most cases, free precession would modulate the observed radio pulsar frequency, a phenomenon not commonly observed in the pulsar population. However, as noted by Jones (2010), the presence of a superfluid component within the star with a spin axis misaligned from that of the main rotation can produce this dual-harmonic emission, while leaving no imprint on the radio emission. Another possibility is that the dominant gravitational-wave emission is produced by a solid core (Glendenning 1996; Owen 2005) whose spin frequency is slightly greater than that of the crust, again leading to a small mismatch between the gravitational and (twice) the radio pulsar frequency; see Abbott et al. (2008).

With these considerations in mind, we follow previous CW analyses and carry out three different sorts of searches within this paper. The simplest search assumes a single gravitational-wave component, at exactly twice the observed spin frequency, as deduced from radio pulsar observations. We carry out "dual-harmonic searches," allowing for emission at both one and two times the spin frequency. And we also carry out searches allowing for a small mismatch between the electromagnetic and gravitational signal frequencies, so-called "narrowband" searches.

The basic form of the waveform used in dual-harmonic searches is described in detail in Jones (2015), and used to perform searches in Pitkin et al. (2015), and Abbott et al. (2017d, 2019a). We refer the reader to these papers, and in particular Section 1.1 and Appendix A of Abbott et al. (2017d). We reproduce the main results here for completeness.

If we denote the signals at one and two times the spin frequency as h₂₁(t) and h₂₂(t), respectively, we have

$$\begin{eqnarray}\begin{array}{rcl}{h}_{21} & = & -\displaystyle \frac{{C}_{21}}{2}\left[{F}_{+}^{D}(\alpha ,\delta ,\psi ;t)\sin \iota \cos \iota \cos \left({\rm{\Phi }}(t)+{{\rm{\Phi }}}_{21}^{C}\right)\right.\\ & & +\,\left.{F}_{\times }^{D}(\alpha ,\delta ,\psi ;t)\sin \iota \sin \left({\rm{\Phi }}(t)+{{\rm{\Phi }}}_{21}^{C}\right)\right],\end{array}\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}\begin{array}{rcl}{h}_{22} & = & -{C}_{22}\left[{F}_{+}^{D}(\alpha ,\delta ,\psi ;t)(1+{\cos }^{2}\iota )\cos \left(2{\rm{\Phi }}(t)+{{\rm{\Phi }}}_{22}^{C}\right)\right.\\ & & +\,\left.2{F}_{\times }^{D}(\alpha ,\delta ,\psi ;t)\cos \iota \sin \left(2{\rm{\Phi }}(t)+{{\rm{\Phi }}}_{22}^{C}\right)\right].\end{array}\end{eqnarray} \tag{ 2 }$$

In these equations, C₂₁ and C₂₂ are dimensionless constants that give the amplitudes of the components. The angles (α, δ) are the R.A. and decl. of the source, while the angles (ι, ψ) specify the orientation of the star's spin axis relative to the observer. The quantities ${{\rm{\Phi }}}_{21}^{C},{{\rm{\Phi }}}_{22}^{C}$ are phase angles. The functions ${F}_{+}^{D}$ and ${F}_{\times }^{D}$, known as the antenna or beam functions, describe how the two polarization components of the signal project onto the detector (see, e.g., Jaranowski et al. 1998). The quantity Φ(t) is the rotational phase of the source.

The special and familiar case of single harmonic emission from a steadily spinning triaxial star is obtained by setting C₂₁ = 0, leaving only the higher-frequency component. In this case, the amplitude is more conventionally parameterized as the dimensionless h₀, the amplitude of the (circularly polarized) signal that would be received if the star lay directly above or below the plane of the detector, with its spin axis pointing directly toward (or away from) the detector, so that h₀ = 2C₂₂. Such triaxial stars are often colloquially described as having "mountains," or having a dimensionless equatorial ellipticity defined in terms of its principal moments of inertia $({I}_{{xx}},{I}_{{yy}},{I}_{{zz}})$:

$$\begin{eqnarray}&&\epsilon \equiv \displaystyle \frac{| {I}_{{xx}}-{I}_{{yy}}| }{{I}_{{zz}}},\end{eqnarray} \tag{ 3 }$$

with the understanding that the star spins about the z-axis. The gravitational-wave amplitudes and equatorial ellipticities are then related by

$$\begin{eqnarray}&&{h}_{0}=\displaystyle \frac{16{\pi }^{2}G}{{c}^{4}}\displaystyle \frac{{I}_{{zz}}\epsilon {f}_{\mathrm{rot}}^{2}}{d},\end{eqnarray} \tag{ 4 }$$

where f_rot is the rotational frequency and d is the star's distance. Yet another quantity that is often quoted is the mass quadrupole Q₂₂, a quantity with the same dimension as the moment of inertia, and one that appears directly in the mass quadrupole formalism for calculating gravitational-wave amplitudes:

$$\begin{eqnarray}&&{Q}_{22}={I}_{{zz}}\epsilon \sqrt{\displaystyle \frac{15}{8\pi }}.\end{eqnarray} \tag{ 5 }$$

When applying these formulae, we will use a fiducial value ${I}_{{zz}}^{\mathrm{fid}}={10}^{38}$ kg m² for the moment of inertia.

We quote our results in terms of the ratio between minimum gravitational-wave detectable amplitude and the spin-down limit, which is given by:

$$\begin{eqnarray}&&{h}_{0,\mathrm{sd}}=\displaystyle \frac{1}{d}{\left(\displaystyle \frac{5{{GI}}_{{zz}}}{2{c}^{3}}\displaystyle \frac{| {\dot{f}}_{\mathrm{rot}}| }{{f}_{\mathrm{rot}}}\right)}^{1/2},\end{eqnarray} \tag{ 6 }$$

which comes from the assumption that all rotational energy lost by the pulsar powers the gravitational-wave emission. This limit is surpassed when the minimum detectable gravitational-wave amplitude h₀ is smaller than ${h}_{0,\mathrm{sd}}$.

We also make a distinction between intrinsic and observed spin-downs of the pulsars we analyze. The observed spin-downs are affected by the transverse velocity of the source (Shklovskii 1970), and can differ substantially from the intrinsic ones (see Table 2). So when possible, we use the intrinsic spin-down to calculate the spin-down limit.

In the case of the narrowband search, a range of frequencies and spin-down rates is searched over, centered on the rotationally derived values, allowing for fractional deviations of up to a maximum value. For emission close to 2f_rot, this corresponds to ranges in search frequency f_GW and its first time derivative ${\dot{f}}_{\mathrm{rot}}$ of:

$$\begin{eqnarray}&&1-\delta \lt \displaystyle \frac{{f}_{\mathrm{GW}}}{2{f}_{\mathrm{rot}}}\lt 1+\delta ,\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&1-\delta \lt \displaystyle \frac{{\dot{f}}_{\mathrm{GW}}}{2{\dot{f}}_{\mathrm{rot}}}\lt 1+\delta .\end{eqnarray} \tag{ 8 }$$

Previous narrowband searches used values of δ of the order of $\sim { \mathcal O }({10}^{-4})$ motivated partly by astrophysical considerations for the gravitational-wave emission mechanism. In fact, Equations (7) and (8) can take into account the possibility that the gravitational wave is emitted by a free precessing biaxial neutron star (Jones & Andersson 2002) or the possibility that the star crust and core are linked by a torque that would enforce corotation. In the previous cases, the gravitational wave emitted would be a nearly monochromatic signal emitted at a slightly different frequency and spin-down with respect to the one observed from electromagnetic observations. Section 2 below gives further details of how these signal models are used by the various data analysis methods. We note that the values of δ chosen for the present search are sufficient to cover a parameter range roughly an order of magnitude greater than what is expected astrophysically by the above mechanisms.

As described in Dupuis & Woan (2005), for each pulsar, this method preprocesses the raw gravitational-wave strain, which is then used as the input to a Bayesian parameter estimation code (Pitkin et al. 2017). The parameter estimation uses a nested sampling algorithm, as implemented in the LALInference package (Veitch & Vecchio 2010; Veitch et al. 2015), to infer the unknown gravitational-wave parameters of the expected signal, which depend on the signal model described Section 1.2. In contrast to the previous searches for the l = m = 2 mode using this method (e.g., Aasi et al. 2014; Abbott et al. 2017d, 2019a), which have directly inferred the gravitational-wave amplitude h₀ for each signal, we now parameterize the amplitude in terms of the mass quadrupole Q₂₂ and pulsar distance d as in Equations (4) and (5). The distances are given Gaussian prior probability distributions, with mean and standard deviation values taken from the distance estimates for the pulsars (see Table 2). The Q₂₂ prior distribution is chosen to be flat over the range $[0,5\times {10}^{37}]$ kg m², and zero outside this range. This is not a physically motivated range, but is chosen to be more than an order of magnitude larger than the largest upper limit found in Abbott et al. (2019a).

In the gravitational-wave analysis, we assume that the signal evolution is affected by a glitch in the same way as that observed with the electromagnetic pulses, except that each glitch may introduce a phase offset between the electromagnetic and gravitational-wave signals. These unknown phase offset parameters are included in the parameter inference. Three of the Crab pulsar glitches described in Section 3.2.2 occurred between O2 and O3, so it would be impossible to use our gravitational-wave data to distinguish different phase offsets for each of these glitches. Therefore, only one phase offset parameter is required to account for the three glitches. During this work an error was found and fixed in the analysis when accounting for the glitch behavior during the parameter inference stage. This led to the time-domain Bayesian results for the Crab and Vela pulsar from Abbott et al. (2019a) being updated to those now given in Abbott et al. (2020a).

As described in Section 3.2.2, for the Vela pulsar, we have a coherent timing model over only the period of O3a. Therefore, we have to combine the results from an analysis on O1 and O2 data with that from O3a in a semi-coherent manner. This also means that we do not need to account for the Vela pulsar glitch between O2 and O3a with the inclusion of an additional phase offset. Because of the bug described above, an analysis of combined O1 and O2 data used in Abbott et al. (2019a) was repeated for this work, but with the corrected code and (for the single harmonic search) with parameter inference on Q₂₂ and distance instead of h₀. For the single harmonic search, the joint posterior on Q₂₂ and ι was fitted with a multivariate Gaussian mixture model (using the BayesianGaussianMixture function within scikit-learn; Pedregosa et al. 2011), allowing a maximum of 20 components. This mixture model was then used as the prior on these parameters when analyzing O3a data. For the dual-harmonic search, the mixture model was fitted to the joint C₂₁, C₂₂, and ι posterior.

The time-domain ${ \mathcal F }$/${ \mathcal G }$-statistic method uses the ${ \mathcal F }$ and ${ \mathcal G }$ statistics developed in Jaranowski et al. (1998) and Jaranowski & Królak (2010). The ${ \mathcal F }$-statistic is used when the amplitude, phase, and polarization of the signal are unknown, whereas the ${ \mathcal G }$-statistic is applied when only amplitude and phase are unknown, and the polarization of the signal is known (as described in Section 2.4). The methods have been used in several analyses of LIGO and Virgo data (Abadie et al. 2011; Aasi et al. 2014; Abbott et al. 2017d).

In this method a signal is detected in the data if the value of the ${ \mathcal F }$- or ${ \mathcal G }$-statistic exceeds a certain threshold corresponding to an acceptable false-alarm probability. We consider the false-alarm probability of 1% for the signal to be significant. The ${ \mathcal F }$- and ${ \mathcal G }$-statistics are computed for each detector and each inter-glitch period separately. The results from different detectors or different inter-glitch periods are then combined incoherently by adding the respective statistics. When the values of the statistics are not statistically significant, we set upper limits on the amplitude of the gravitational-wave signal.

The frequency-domain 5n-vector method has been introduced in Astone et al. (2010, 2012) and used in several analyses of LIGO and Virgo data (Abadie et al. 2011; Aasi et al. 2014; Abbott et al. 2017d, 2019a). It is also the basis of the narrowband pipeline described in Section 2.5. In this paper it has been applied to a subset of three pulsars: J0711−6830, the Crab pulsar, and the Vela pulsar.

In contrast to past analyses—which used resampling—the barycentric, spin-down, and Einstein delay corrections are done by heterodyning the data, using the band sampled data framework (Piccinni et al. 2019). This significantly reduces the computational cost of the analysis, which drops from about half of a CPU-day to a few CPU-minutes per source per detector. A detection statistic, based on the matched filter among the 5n-vectors of the data and the signal, is obtained and used to estimate the significance of an analysis result. Upper limits are computed using the approach first introduced in Aasi et al. (2014).

As in Abbott et al. (2019a), two independent analyses have been done assuming that the emission takes place at two times the star rotation frequency and at the rotation frequency (according to the model described in Jones 2010). While performing this analysis, we identified an incorrect choice for the range of amplitudes used to inject simulated signals in the O2 analysis of the pulsar J0711−6830; see Abbott et al. (2020a) for more details. This affects only the upper limit computation at the rotation frequency for J0711−6830, and the corrected value is given in Abbott et al. (2020a).

As with previous analyses, all of the pipelines produce results for the Crab and Vela pulsars based on two different assumptions. The first is that the orientation of the pulsar is unknown, so a uniform prior over the inclination and polarization angle space is used. The second uses estimates of the source orientation based on X-ray observations of the pulsar wind nebulae tori (Ng & Romani 2004, 2008), which are included in the pipelines as narrow priors on inclination and polarization angle (effectively defining the polarization state of the signal), as given in Table 3 of Abbott et al. (2017d).

The 5n-vector narrowband pipeline described in Mastrogiovanni et al. (2017) uses the 5n-vector method of Astone et al. (2010, 2012) and expands it to a narrow frequency and spin-down range around the source ephemerides values. This pipeline has previously been applied to the O1 and O2 data sets in Abbott et al. (2017e, 2019b) permitting the analysis of pulsars for which ephemerides were not accurately known.

In contrast to Abbott et al. (2019b), we now combine the matched filter's results between the detectors using weight factors computed from the power spectral density: each data set is weighted inversely by the median noise power in the analyzed frequency band. This allows the analysis to depend most strongly on the most sensitive data set. The final step is to select the local maximum of a detection statistic every 10⁻⁴ Hz over the spin-down values considered. Within this set of points in the parameter space, we select as outliers those with a p-value below a 0.1% threshold (taking into account the number of trials).

This method targets pulsars J0711−6830, Crab, and Vela. For J0711−6830 and Vela, we analyzed 6 months of data, so the frequency and spin-down resolutions were $6.5\times {10}^{-8}\,\mathrm{Hz}$ and $4.3\times {10}^{-15}\,\mathrm{Hz}\,{{\rm{s}}}^{-1}$, respectively. For Crab, the resolutions were $1.0\times {10}^{-7}\,\mathrm{Hz}$ and $1.1\times {10}^{-14}\,\mathrm{Hz}\,{{\rm{s}}}^{-1}$ since we considered only data preceding the glitch (∼115 days). The narrowband resolutions relate to the natural discretization step of the discrete Fourier transform. The resolution ensures that a nearly monochromatic gravitational-wave signal, emitted in the explored parameter space, is subject to a maximum loss of signal-to-noise ratio of ∼36% (Ransom et al. 2002). Note that, in order to reduce this loss, a half-bin interpolation of the Fourier transform is implemented in the code.

For each pulsar, we analyze a gravitational-wave frequency and spin-down range set to within 0.4% of the ephemerides frequency and spin-down. This corresponds ²¹⁵ to $\delta \sim 2\,\times {10}^{-3}$ in Equations (7) and (8). With respect to the O2 narrowband search, this corresponds to a volume explored in the frequency/spin-down range that is four times larger. We report the frequency and spin-down bands explored in Table 1.

Table 1. Frequency/Spin-down Ranges Explored in the 5n-vector Narrowband Search

Pulsar	${\rm{\Delta }}{f}_{\mathrm{GW}}$	${\rm{\Delta }}{\dot{f}}_{\mathrm{GW}}$	nf	${n}_{\dot{f}}$
	(Hz)	(Hz s−1)
J0534+2200 a (Crab)	0.24	$3.0\times {10}^{-12}$	$3.8\times {10}^{6}$	270
J0711−6830	0.72	$8.4\times {10}^{-15}$	$1.2\times {10}^{7}$	3
J0835−4510 (Vela)	0.10	$1.4\times {10}^{-13}$	$1.4\times {10}^{6}$	33

Notes. Second and third columns: frequency and spin-down ranges explored, respectively. Fourth and fifth columns: number of values in frequency and number of spin-down values considered, respectively. The total number of templates per pulsar is ${n}_{f}\times {n}_{\dot{f}}$.

^a Only data before the glitch reported in Shaw et al. (2019) are considered.

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Finally, for computing the 95% confidence level upper limits on the gravitational-wave amplitude h₀, we use the procedure described in Abbott et al. (2019b) to inject several simulated gravitational-wave signals in each 10⁻⁴ Hz sub-band. For each sub-band, we set the upper limit at the strain amplitude for which 95% of the injected signals are recovered.

We use a combination of data from the first, second, and third observing runs of the Advanced LIGO (Aasi et al. 2015) and Virgo (Acernese et al. 2015) gravitational-wave detectors. For O1 and O2, only data from the LIGO Hanford (H1) and LIGO Linvingston (L1) detectors have been used, while for O3, data from both LIGO detectors and the Virgo (V1) detector have been used. The O1 data cover the period from 2015 September 11 to 2016 January 19, with duty factors of ∼51% and ∼60% for L1 and H1, respectively. The O2 data cover the period from 2016 November 30 to 2017 August 25, with duty factors of ∼57% and ∼59% for L1 and H1, respectively (including commissioning breaks). For O3, a period from 2019 April 1 to 2019 October 1 was designated O3a, prior to a one month commissioning break. O3a had duty factors of ∼76%, ∼71%, and ∼76% for L1, H1, and V1, respectively.

The data and subsequent upper limits are subject to uncertainty in the calibration of the instruments. The calibration uncertainty varies in amplitude and phase over the course of a run. We do not account for these variations in our results (see below), but we expect them to have a negligible impact on the results. For more details of the O1 and O2 data and calibration used in these searches, see the discussions in Abbott et al. (2017d, 2019a). The full raw strain data from the O1 and O2 runs are publicly available from the Gravitational Wave Open Science Center ²¹⁶ (Vallisneri et al. 2015; Abbott et al. 2019h). For the LIGO O3a data set, the time-domain Bayesian and ${ \mathcal F }/{ \mathcal G }$-statistic methods use the "C01" calibration for LIGO, while the 5n-vector methods use the "C00" calibration. The C01 calibration has estimated maximum amplitude and phase uncertainties of 7% and 4 deg, respectively (Sun et al. 2020), while the C00 estimates are 8% and 5 deg. For the Virgo O3a data set, all of the pipelines use the "V0" calibration with estimated maximum amplitude and phase uncertainties of 5% and 3 deg, respectively.

For the Bayesian analysis, we estimate that the statistical uncertainty on the upper limits due to the use of a finite number of posterior samples is on the order of 1%. For the $5n$-vector analysis, the statistical uncertainty on the upper limits has been estimated to be 1%–3% depending on the target.

Besides calibration uncertainties, the detectors' data sets are polluted by several noise disturbances. Some of these disturbances are qualitatively visible as spikes or other deviations from smoothness in the noise power spectral densities (PSDs) for L1, H1, and V1 in Figure 1 with respect to the five pulsars frequency searched.

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The timing solutions used in our gravitational-wave searches have been derived from electromagnetic observations of pulsars. These pulsars' basic properties are given in Table 2, and are further explained in the next subsections.

Table 2. Properties of the Pulsars in This Search

Pulsar	frot	${\dot{f}}_{\mathrm{rot}}$	${\dot{f}}_{\mathrm{rot}}^{\mathrm{int}}$	Distance	Spin-down
	(Hz)	(Hz s−1)	(Hz s−1)	(kpc)	luminosity (W)
Young pulsars
J0534+2200 (Crab)	29.6	$-3.7\times {10}^{-10}$	⋯	2.0 ± 0.5 a	$4.5\times {10}^{31}$
J0835−4510 (Vela)	11.2	$-2.8\times {10}^{-11}$ b	⋯	${0.287}_{-0.017}^{+0.019}$ c	$6.9\times {10}^{29}$
Recycled pulsars
J0437−4715	173.7	$-1.7\times {10}^{-15}$	$-4.1\times {10}^{-16}$	0.15679 ± 0.00025 d	$2.8\times {10}^{26}$
J0711−6830	182.1	$-4.9\times {10}^{-16}$	$-4.7\times {10}^{-16}$	0.110 ± 0.044 e	$3.4\times {10}^{26}$
J0737−3039A	44.1	$-3.4\times {10}^{-15}$	⋯	${1.15}_{-0.16}^{+0.22}$ f	$5.9\times {10}^{26}$

Notes. If an intrinsic rotation period derivative ${\dot{P}}_{\mathrm{rot}}^{\mathrm{int}}$ is available from the Australia Telescope National Facility (ATNF) Pulsar Catalog (Manchester et al. 2005), and is significantly different from the observed value, then this is converted into an intrinsic frequency derivative via ${\dot{f}}_{\mathrm{rot}}^{\mathrm{int}}=-{f}_{\mathrm{rot}}^{2}{\dot{P}}_{\mathrm{rot}}^{\mathrm{int}}$ and is quoted here. For J0437−4715 and J0711−6830, this intrinsic frequency derivative will be used to calculate the spin-down luminosity and the spin-down limits in Table 3.

^a Kaplan et al. (2008). ^b The ${\dot{f}}_{\mathrm{rot}}$ value given here is for the observation span used in this work; however, the spin-down limit shown in Table 3 uses the long-term value of ${f}_{\mathrm{rot}}=-1.57\times {10}^{-11}$ Hz s⁻¹ as given in the ATNF Pulsar Catalog (Manchester et al. 2005). ^c This distance is from Dodson et al. (2003), although the Bayesian analysis described in Section 2.1 uses a symmetric distance uncertainty of 0.288 ± 0.018 kpc. ^d Reardon et al. (2016). ^e This distance is based on dispersion measure from the Yao et al. (2017) model, with a 40% uncertainty assumed. ^f This distance is from Deller et al. (2009), although the Bayesian analysis described in Section 2.1 uses a symmetric distance uncertainty of 1.18 ± 0.19 kpc.

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3.2.1. Recycled Pulsars

3.2.2. Young Pulsars

The results from the targeted searches for all five pulsars are summarized in Table 3 with the three different pipelines presented together for ease of comparison.

Table 3. Limits on Gravitational-wave Amplitude, and Other Derived Quantities, for the Three Targeted Searches

Pulsar name	${h}_{0}^{\mathrm{sd}}$	Analysis	${C}_{21}^{95 \% }$	${C}_{22}^{95 \% }$	${h}_{0}^{95 \% }$	${Q}_{22}^{95 \% }$	${\epsilon }^{95 \% }$	${h}_{0}^{95 \% }/{h}_{0}^{\mathrm{sd}}$
(J2000)	(10−26)	method	(10−26)	(10−27)	(10−26)	(1032 kg m2)
Young pulsars a
J0534+2200 b (Crab)	140	Bayesian	12.7(7.9)	6.3(5.6)	1.5(1.2)	6.6(5.7)	$8.6(7.4)\times {10}^{-6}$	0.010(0.009)
		${ \mathcal F }$/${ \mathcal G }$-statistic	8.9(6.2)	7.9(7.1)	1.9(1.5)	7.9(6.3)	$10(8.1)\times {10}^{-6}$	0.014(0.011)
		5n-vector	15.9(12.4)	⋯	3.0(2.9)	12.6(12.1)	$16.3(15.7)\times {10}^{-6}$	0.021(0.021)
J0835−4510 (Vela)	330	Bayesian	1100(980)	120(84)	22(17)	91(73)	$12.0(9.5)\times {10}^{-5}$	0.067(0.052)
		${ \mathcal F }$/${ \mathcal G }$-statistic	1470(1370)	116(48)	23(12)	96(50)	$12.4(6.4)\times {10}^{-5}$	0.070(0.036)
		5n-vector	1700(1400)	⋯	24(24)	100(102)	$13.0(13.2)\times {10}^{-5}$	0.073(0.073)
Recycled pulsars
J0437−4715	0.79	Bayesian	2.2	4.1	0.78	0.0074	$9.5\times {10}^{-9}$	0.99
		${ \mathcal F }$/${ \mathcal G }$-statistic	2.1	7.2	0.86	0.0082	$11.0\times {10}^{-9}$	1.1
		5n-vector	⋯	⋯	⋯	⋯	⋯	⋯
J0711−6830	1.2	Bayesian	2.6	3.5	0.82	0.0064	$8.3\times {10}^{-9}$	0.68
		${ \mathcal F }$/${ \mathcal G }$-statistic	2.4	9.4	0.98	0.0059	$7.7\times {10}^{-9}$	0.82
		5n-vector	2.9	⋯	0.91	0.0053	$7.2\times {10}^{-9}$	0.76
J0737−3039A	0.62	Bayesian	5.9	3.3	0.69	0.80	$1.0\times {10}^{-6}$	1.1
		${ \mathcal F }$/${ \mathcal G }$-statistic	3.0	1.2	0.99	1.10	$1.4\times {10}^{-6}$	1.6
		5n-vector	⋯	⋯	⋯	⋯	⋯	⋯

Notes. Parameters for the pulsars can be found in Table 2.

^a For J0534+2200 and J0835−4510, the results in parentheses are those when using restricted priors on the pulsar orientation. ^b For the $5n$-vector results, only data from the O3a run were used for all three pulsars.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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No evidence for a signal was observed from any of the five pulsars in either of the harmonics studied, so as with previous analyses, we present the results as 95% credible upper limits on the gravitational-wave amplitudes: C₂₁ and C₂₂ for the dual-harmonic search, and h₀ for the single harmonic search. For the single harmonic search, using Equations (4) and (5), we place equivalent limits on the mass quadrupole Q₂₂ and fiducial ellipticity . The posterior distributions on Q₂₂ for the Bayesian analysis, along with the point estimate upper limits from the ${ \mathcal F }/{ \mathcal G }$-statistic and 5n-vector pipelines, are shown in Figure 2 for the three recycled pulsars and Figure 3 for Crab and Vela. In Figure 3 the upper limits are shown using both restricted-orientation and unrestricted priors as described in Section 2.4.

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Despite the analysis pipelines being largely independent, and the statistical procedures used to derive upper limits being different, there is a broad agreement among the different pipelines. One source of differences, however, comes from the pipelines not all using the same data sets. The 5n-vector search analyzed only O3a data while the Bayesian and ${ \mathcal F }/{ \mathcal G }$-statistic search coherently (or semi-coherently in the case of the Vela pulsar) combined data from O1, O2, and O3a. The methods of combining data for the Vela pulsar analysis are discussed in Section 2.

Another source of differences is that the Bayesian analysis does not assume a fixed distance, but instead includes it as a parameter to be estimated from the data. Therefore, limits on Q₂₂ and h₀ are computed marginalizing over the distance, rather than assuming a fixed value. In general, the distance posterior distributions match their priors well but with a small bias toward larger values when the distance priors are wide. However, for J0711−6830 the bias is more obvious with the peak in the distance posterior being at a value approximately 20% larger than that of the prior. This biasing of the distance is due to our choice of a flat prior on Q₂₂, which is not an uninformative distribution; i.e., there is much more prior weight for large Q₂₂ values, disfavoring smaller distances.

In the discussions below, we will generally refer to the most stringent, i.e., lowest, limit from the different searches and will discuss only the single harmonic (l = m = 2 mass quadrupole) and unrestricted-orientation priors results in detail.

4.1.1. Recycled Pulsars

4.1.2. J0737−3039A

4.1.3. Young Pulsars

The narrowband search found no evidence for GW emission from J0711−6830, the Crab pulsar, or the Vela pulsar, although several analysis outliers were found for two of these pulsars.

For J0711−6830 there were 19 outliers. Sixteen outliers clustered at the boundaries of the analyzed frequency band and were due to artifacts created by the band extraction close to the integer frequency of 364 Hz. These artifacts are created due to subsample processes at 1 Hz. The remaining three outliers, labeled as C17, C18, and C19 (see Figure 4, top panel), were found by the narrowband pipeline with a p-value of $\sim 1.2\times {10}^{-7}$, which when rescaled for the number of trials corresponds to a p-value of $\sim 5.0\times {10}^{-4}$. In order to assess the significance of the outliers, we performed a narrowband search for J0711−6830 using the same setup for the rotational parameters but using an "off-source" sky position. This procedure would effectively blind the analysis to the presence of a possible astrophysical signal, thus allowing us to build an empirical noise-only distribution of the detection statistic, which can be used to reassess the outliers' significances. From this analysis we found that the p-values of two of the three outliers were above the narrowband ceiling of 0.1% (C17 and C19), while the value for C18 was 0.06%. This test indicated that by re-assigning the significance with the off-source method, two of the outliers would not have passed the narrowband threshold for candidate selection and hence could be due to low-level noise instrumental artifacts.

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The three outliers were followed-up by two of the targeted pipelines. For the three outliers, the time-domain Bayesian pipeline found a strong preference for the hypothesis that the data (with a bandwidth of 1/60 Hz centered on the outlier) was consistent with Gaussian noise compared to a hypothesis that it also contained signals coherent between detectors. Specifically, the Bayes factors recovered for the signal versus noise hypothesis were <10⁻⁴; thus they are likely the noise origin for all of these outliers. Additionally, for C17 and C19, the pipeline recovered a maximum posterior on h₀ of $1.6\times {10}^{-26}$ and $8.5\times {10}^{-27}$. As argued in the case of the Vela pulsar in Section 4.1.3, this can be related to the presence of instrumental noise contributions.

The 5n-vector targeted pipeline also performed a follow-up of the most significant of the three outliers (C18), with frequency ∼364.25 Hz, using software injections with an amplitude set to that of one of the recovered outliers. The pipeline found that the distribution of the software injection's detection statistic was compatible with the value displayed by the outlier. More precisely, for a set of 50 injected signals, it found an average critical ratio $\overline{\mathrm{CR}}=7.0$ with a standard deviation of 3.2, to be compared with a value 8.5 found for the actual analysis candidate. In the absence of any signal, the noise average critical ratio was found to be $\overline{\mathrm{CR}}=0.3\pm 1.3$.

Given that the previous tests did not conclusively establish the noise origin of the outliers, we performed a narrowband analysis using the full O3 LIGO data set (C00 calibration). If the outliers were due to a real continuous wave signal, we would have expected to see them as more significant in this analysis. Figure 4 compares the detection statistics obtained from the narrowband analysis using only O3a data and the full O3 data set. We see that the outliers found in the O3a run are no longer present when using the full run, which is inconsistent with an astrophysical signal.

Finally, for the Vela pulsar, we found four outliers, but these are due to noise disturbances in the data; see Figure 5. One of the candidates was due to the left sidebands of a known H1 disturbance at 22.347 Hz together with a noise disturbance known in V1 at 22.333 Hz. The other three outliers were due to the right sidebands of the H1 disturbance and an L1 broadband noise disturbance around 22.365 Hz.

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Hence we computed the 95% confidence level upper limits on the gravitational-wave amplitudes h₀ and the corresponding limits on the fiducial ellipticities, as shown in Figure 6. For pulsar J0711−6830 we obtain median values of the upper limit on the amplitude h₀ and the ellipticity over the analyzed frequency band of $2.6\times {10}^{-26}$ and $2.0\times {10}^{-8}$, respectively. Unfortunately, the narrowband pipeline does not surpass the spin-down limit for this pulsar.

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For the Crab and Vela pulsars, we obtain median values for the upper limit on h₀ of $8.1\times {10}^{-24}$ and $3.90\times {10}^{-25}$, while the corresponding median upper limits on the fiducial ellipticities are $4.4\times {10}^{-5}$ and $2.0\times {10}^{-4}$, respectively. These upper limits are factors of 1.6 and 2.25 better than the upper limit obtained with O2 data in Abbott et al. (2019b). This improvement has been partially made possible by the inclusion of Virgo data and the slightly improved sensitivities in O3 for the two LIGO detectors. Another major contribution, however, comes from the new PSD-weighted analysis that can account for data having different PSDs.