Abstract
Isolated neutron stars that are asymmetric with respect to their spin axis are possible sources of detectable continuous gravitational waves. This paper presents a fully coherent search for such signals from eighteen pulsars in data from LIGO and Virgo's third observing run (O3). For known pulsars, efficient and sensitive matched-filter searches can be carried out if one assumes the gravitational radiation is phase-locked to the electromagnetic emission. In the search presented here, we relax this assumption and allow both the frequency and the time derivative of the frequency of the gravitational waves to vary in a small range around those inferred from electromagnetic observations. We find no evidence for continuous gravitational waves, and set upper limits on the strain amplitude for each target. These limits are more constraining for seven of the targets than the spin-down limit defined by ascribing all rotational energy loss to gravitational radiation. In an additional search, we look in O3 data for long-duration (hours–months) transient gravitational waves in the aftermath of pulsar glitches for six targets with a total of nine glitches. We report two marginal outliers from this search, but find no clear evidence for such emission either. The resulting duration-dependent strain upper limits do not surpass indirect energy constraints for any of these targets.
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1. Introduction
Continuous gravitational waves (CWs) are quasi-monochromatic signals expected to be ever-present in the data of gravitational-wave (GW) detectors such as Advanced LIGO (Aasi et al. 2015a) and Advanced Virgo (Acernese et al. 2015). While the observation of transient GWs from compact binary coalescences has become nearly commonplace (Abbott et al. 2021a), CWs have yet to be detected as of the third observing run (O3). One of the most enticing and commonly sought after sources of CWs is a rapidly spinning, asymmetric neutron star (NS); see Sieniawska & Bejger (2019) and Haskell & Schwenzer (2020) for recent reviews. In the case of a triaxial NS, CW emission occurs at twice the rotation frequency of the star.
Many NSs are observed as pulsars by radio, X-ray, or γ-ray telescopes (Lorimer & Kramer 2012). Pulsars can often be timed extremely precisely—in the best cases, the arrival of new pulses can be predicted to within tens of nanoseconds. This precision can enable exciting science, including sensitive tests of general relativity (Wex & Kramer 2020), placing constraints on the equation of state of the dense matter inside NSs (Lattimer & Prakash 2004; Kramer & Wex 2009; Ho et al. 2015), and using deviations in timing residuals of pulsars to search for a stochastic gravitational-wave background (Verbiest et al. 2016; Arzoumanian et al. 2020; Goncharov et al. 2021). Detecting GWs from spinning NSs would add a completely new messenger to the study of these extreme objects (Glampedakis & Gualtieri 2018; Haskell & Schwenzer 2020).
In this paper, we search in LIGO–Virgo O3 data (taken in 2019–2020) for CWs from NSs that have been observed as pulsars and precisely timed in either the radio or X-ray bands. We identify 18 promising candidates for which the observed spin-down (negative frequency derivative) implies indirect limits on CW emission that fall within a factor of 3 of the expected sensitivity of the search. Some other analyses assume the phase of CWs to be locked to the rotational phase of the crust of the star as observed by electromagnetic (EM) telescopes (Aasi et al. 2014; Abbott et al. 2017a,2018, 2019a, 2020b, 2021b, 2021e; Nieder et al. 2020; Ashok et al. 2021). Here, we relax that assumption and allow for the frequency of rotation—and its derivative—to differ from the EM-observed values by a small factor: the so-called "narrowband" search approach (Abbott et al. 2008, 2017b, 2019b; Aasi et al. 2015b; Nieder et al. 2020; Ashok et al. 2021). We use two separate analysis pipelines, the 5n-vector (Astone et al. 2014; Mastrogiovanni et al. 2017) and the frequency-domain 
-statistic (Jaranowski et al. 1998; Wette et al. 2018a) pipelines, to perform phase-coherent searches for CWs on O3 data over our widened parameter space. Using two separate pipelines allows us to cross-check results between pipelines, compare limits set by the two methods, and increase confidence in any potential detection by requiring both pipelines to see the same signal.
A scenario where GW and EM emission have similar, but slightly different phase evolution is plausible, e.g., when there is differential rotation between the rigid crust and superfluid parts of the star. Possible observational evidence for this comes, for example, from pulsar glitches (Link et al. 1992, 2000; Lyne et al. 2000; Fuentes et al. 2017; Haskell 2017): sudden spin-up events that are often followed by an exponential relaxation back to the simple spin-down scenario. In many models (Haskell & Melatos 2015), the superfluid and non-superfluid components build up a lag with respect to one another, and a glitch represents the sudden recoupling of the two components (Anderson & Itoh 1975; Lyne et al. 2000).
Glitches are also directly relevant for GW searches: first, some of our CW search targets glitched during O3. For these, we perform separate phase-coherent searches covering the parts of O3 before and after the glitches. Second, it is also possible that glitches trigger increased GW emission in their aftermath (van Eysden & Melatos 2008; Bennett et al. 2010; Prix et al. 2011; Melatos et al. 2015; Singh 2017; Yim & Jones 2020). Hence, we also perform additional searches for long-duration transient CW-like signals with durations from a few hours to four months after observed glitches during (or shortly before) O3, covering nine glitches from six pulsars.
The rest of this paper is structured as follows. We discuss our selection of target pulsars and the EM observations that we use to guide our search in Section 2, and the O3 GW data set in Section 3. In Section 4, we describe our methods of analysis. We present the results of the two CW searches in Section 5. We then cover the detailed setup and the results of the search for GW emission in the aftermath of pulsar glitches in Section 6. In Section 7, we conclude the paper with a discussion of the results obtained by all three analyses, as well as their astrophysical implications. Throughout the rest of the paper, we will often refer to the analyses using the 
-statistic and 5n-vector pipelines search over all data as the two "CW" searches, and the search for long-duration transients after glitches as the "transient search."
2. Electromagnetic Data and Target Selection
We start with timing solutions from EM observations of pulsars (also referred to as ephemerides), and search in a narrow band in frequency and spin-down, as further described below in Section 4. The widths of the frequency and spin-down search bands are usually larger than the uncertainty in the timing solutions. The observations we use were made in radio and X-ray bands, and were provided by the following observatories: the Canadian Hydrogen Intensity Mapping Experiment (CHIME) (Bandura et al. 2014; Amiri et al. 2021), University of Tasmania's Mount Pleasant Observatory 26 m telescope, the 42 ft telescope and Lovell telescope at Jodrell Bank, the MeerKAT telescope (observations made as part of the MeerTime project; see Bailes et al. 2020), the Nançay Decimetric Radio Telescope, the Neutron Star Interior Composition Explorer (NICER) (Gendreau et al. 2016), and the UTMOST timing program with the Molonglo Observatory Synthesis Telescope (MOST) (Bailes et al. 2017; Jankowski et al. 2018; Lower et al. 2020). The Tempo (Nice et al. 2015) and Tempo2 (Edwards et al. 2006) timing packages were used to fit the model parameters and provide timing solutions.
We select our targets for the CW searches in a manner similar to that of Abbott et al. (2019b), based on the sensitive frequency band of the Advanced LIGO and Virgo detectors and availability of precise ephemerides over the duration of O3 from EM observations. We have analyzed all isolated pulsars, except for one, with a rotation frequency between 10 and 350 Hz and for which the spin-down limit falls within a factor of 3 of the expected sensitivity of the full network over the course of O3. This frequency range includes all the high-value targets identified in Abbott et al. (2021e) for which it could be possible to go below the spin-down limit. Pulsars in this frequency range would produce CWs between 20 and 700 Hz. The only pulsar we do not analyze that satisfies these criteria is PSR J0537–6910, which was analyzed in detail using a narrowband approach to search for r-mode emission in Abbott et al. (2021c) and using a targeted approach in Abbott et al. (2021b). Searches lasting up to 120 days during inter-glitch periods for this pulsar are performed by the post-glitch transient search presented in Sections 4.4 and 6.1.
We estimate the expected sensitivity as
where S(f) is the power spectral density as a function of frequency for the LIGO Hanford, LIGO Livingston, and Virgo detectors (H1, L1, and V1, respectively), and Tobs is the observing time assuming 11 months of data with respective duty cycles of 75%, 77%, and 76%. The factor Θ ∼ 30 encodes the scaling of typical narrowband searches but is a function of the number of templates employed in the analysis (Astone et al. 2014).
On the other hand, the spin-down limit is an indirect upper limit (UL) on the GW amplitude assuming all of a pulsar's rotational energy loss comes in the form of GWs (Jaranowski et al. 1998; Prix 2009). It is given by
where I38 is the star's moment of inertia in units of 1038 kg m2, fspin is its rotation frequency, and 
is the absolute value of its spin-down rate (Abbott et al. 2019b). We have computed spin-down limits according to the most recent distance estimates given in the ATNF catalog (Manchester et al. 2005), version 1.64, and extrapolating the rotational frequencies and spin-down rates to O3. For several pulsars, we have used more recent distance estimates from the literature.
In Table 1, we present a list of targets along with the ranges of GW frequency and spin-down parameters and the corresponding numbers of templates covered for the two pipelines used to conduct the CW searches. We discuss our parameter range choices for each pipeline in Section 4. The observatory yielding the timing solution we use for each source is noted in the far right column.
Table 1. Setup Parameters for Fully Coherent CW Search Pipelines
| Name | R.A. | Decl. | f | Δf (5v) |
|
|
 (5v) |
|
|
|
|
| References |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| × 10−13 | × 10−15 | × 10−15 | × 10−23 | × 10−23 | × 107 | × 107 | |||||||
| (Hz) | (Hz) | (Hz) | (Hz s−1) | (Hz s−1) | (Hz s−1) | (Hz s−2) | (Hz s−2) | ||||||
| J0534+2200 bg | 05h34m31 97 |
| 59.241 | ... | 0.24 | −7370.0 | ... | 2900.0 | 2360.0 | 5.2 | ... | 983.7 | a |
| J0534+2200 ag | 05h34m3197 |
| 59.241 | 0.24 | 0.24 | −7370.0 | 2300.0 | 2900.0 | 2360.0 | 5.2 | 498.0 | 9362.0 | a |
| J0711–6830 | 07h11m5418 |
| 364.234 | 0.72 | 1.5 | −0.00989 | 2.1 | 0.004 | 0.0 | 0.0 | 4.391 | 45.85 | b |
| J0835–4510 | 08h35m2052 |
| 22.371 | 0.089 | 0.089 | −313.0 | 130.0 | 120.0 | 504.0 | 0.0 | 32.9 | 281.6 | c |
| J1101–6101 | 11h01m4496 |
| 31.846 | 0.13 | 0.13 | −45.3 | 19.0 | 18.0 | 0.0 | 0.0 | 7.026 | 61.52 | d |
| J1105–6107 | 11h05m2571 |
| 31.644 | 0.13 | 0.13 | −79.7 | 32.0 | 32.0 | 554.0 | 35.0 | 11.64 | 266.0 | e |
| J1809–1917 | 18h09m4313 |
| 24.166 | 0.097 | 0.097 | −74.4 | 32.0 | 30.0 | 3.7 | 0.14 | 8.886 | 152.3 | a |
| J1813-1749 bg | 18h13m3511 |
| 44.703 | ... | 0.18 | −1290.0 | ... | 510.0 | 0.0 | 0.0 | ... | 92.08 | d |
| J1813-1749 full | 18h13m3511 |
| 44.703 | 0.18 | 0.18 | −1290.0 | 520.0 | 510.0 | 0.0 | 0.0 | 266.4 | 2270.0 | d |
| J1828–1101 | 18h28m1885 |
| 27.754 | 0.11 | 0.11 | −57.0 | 23.0 | 23.0 | 4.23 | 9.4 | 7.481 | 1162.0 | a |
| J1833–0827 | 18h33m4026 |
| 23.449 | 0.094 | 0.094 | −25.2 | 11.0 | 10.0 | −0.463 | 0.082 | 2.873 | 242.2 | e |
| J1838–0655 | 18h38m313 |
| 28.363 | 0.11 | 0.11 | −199.0 | 83.0 | 80.0 | 67.1 | 7.8 | 27.13 | 555.4 | d |
| J1856+0245 | 18h56m5091 |
| 24.714 | 0.099 | ... | −189.0 | 79.0 | ... | 0.0 | 0.0 | 22.42 | ... | a |
| J1913+1011 | 19h13m2034 |
| 55.694 | 0.22 | 0.22 | −52.5 | 23.0 | 21.0 | −0.321 | 2.6 | 15.02 | 1951.0 | a |
| J1925+1720 | 19h25m2706 |
| 26.434 | 0.11 | 0.11 | −36.6 | 15.0 | 15.0 | 3.93 | 6.3 | 4.538 | 469.2 | a |
| J1928+1746 | 19h28m4255 |
| 29.097 | 0.12 | 0.12 | −55.7 | 23.0 | 22.0 | 0.0 | 0.0 | 7.845 | 68.41 | a |
| J1935+2025 | 19h35m4194 |
| 24.955 | 0.092 | 0.1 | −189.0 | 79.0 | 76.0 | 95.0 | 3.4 | 20.99 | 581.3 | a |
| J1952+3252 | 19h52m5821 |
| 50.587 | 0.2 | 0.2 | −74.9 | 32.0 | 30.0 | 2.92 | 0.012 | 18.61 | 3908.0 | f |
| J2124–3358 | 21h24m4384 |
| 405.588 | 0.91 | 1.6 | −0.0169 | 2.1 | 0.0068 | 0.0 | 0.0 | 5.595 | 48.06 | g |
| J2229+6114 | 22h29m657 |
| 38.709 | 0.15 | 0.16 | −590.0 | 240.0 | 260.0 | 1170.0 | 0.0 | 107.3 | 1021.0 | f |
Note. This table lists frequency f and spin-down (
, 
) parameters corresponding to twice the rotational parameters in the pulsar ephemerides at the start of the O3 run (2019 April 1 15:00 UTC, MJD 58574.000), as discussed in Section 4.1. Right ascension (R.A.) and declination (decl.) are given in J2000 coordinates. In the column headings, 
stands for the 
-statistic search discussed in Section 4.3 and 5v stands for the 5n-vector method discussed in Section 4.2. The suffixes "bg" and "ag" refer to "before" and "after" glitch, respectively, while "full" corresponds to the full run in the case of PSR J1813–1749. Observatories are indicated by letters in the rightmost column, as follows: (a) Jodrell Bank 42ft telescope and Lovell telescope; (b) MeerKAT telescope (part of the MeerTime project) (Bailes et al. 2020); (c) University of Tasmania's Mt. Pleasant Observatory 26 m telescope; (d) NICER (Gendreau et al. 2016); (e) MOST (Bailes et al. 2017; Jankowski et al. 2018; Lower et al. 2020); (f) CHIME (part of the CHIME pulsar project, Bandura et al. 2014; Amiri et al. 2021); (g) Nançay Decimetric Radio Telescope. This table is available online in a machine-readable format (LIGO Scientific Collaboration et al. 2022).
Download table as: ASCIITypeset image
Two pulsars on our list of CW search targets, PSR J0534+2200 (the Crab) and PSR J1105–6107, glitched during O3 (Shaw et al. 2021), on 23 July 2019 and 9 April 2019, respectively. One other, PSR J1813–1749, shows marginal evidence for a glitch on or close to 3 August 2019 (Ho et al. 2020b).
For this reason, we perform two separate searches, before and after the estimated glitch epoch, which are identified in Table 1 with the suffixes "bg" and "ag" for "before glitch" and "after glitch," respectively.
For PSR J0534+2200, we use O3 data until the last observation before the glitch. The analysis after the glitch uses data from 10 days after the glitch epoch, accounting for the estimated relaxation time, until the end of O3.
For PSR J1105–6107, we only search after the glitch, because the glitch occurs nearly at the start of the run. The after-glitch search uses data from two days after the estimated glitch until the end of O3. For PSR J1813–1749, we perform two separate searches: one fully coherent search across the full O3 duration, assuming the pulsar did not glitch, and a search before the glitch. No search after the glitch is performed, because the glitch occurs near the end of the EM timing model (although a post-glitch transient search is conducted assuming the timing model remains valid within uncertainties; see below). Further details about these glitches will also be discussed in Section 6.
For the post-glitch transient search, we have used information from the glitch catalogs maintained at ATNF (Hobbs et al. 2021) and Jodrell Bank (Espinoza et al. 2011; Shaw et al. 2021; Basu et al. 2022). We have selected six pulsars with GW frequencies f > 15 Hz and with glitches observed during (or shortly before) O3: PSR J0534+2200, PSR J0537–6910, PSR J0908–4913, PSR J1105–6107, PSR J1813–1749, and PSR J1826–1334. Another pulsar within our frequency band of interest, PSR J2021+3651, was observed to glitch during O3 by Jodrell Bank, but the glitch time uncertainty is too large to make our transient search setup feasible. As mentioned before, for PSR J1813–1749 it is not certain if a glitch actually occurred (Ho et al. 2020b), but we perform an opportunistic search here with its assumed parameters.
The ephemerides for PSR J0534+2200 and PSR J1826–1334 were provided by Jodrell Bank; a detailed discussion of the PSR J0534+2200 glitch is given in Shaw et al. (2021). Ephemerides for PSR J0908–4913 and PSR J1105–6107 are derived from UTMOST radio observations; these are discussed further in Lower et al. (2019, 2020). A potential ambiguity in timing solutions from periodically scheduled surveys like UTMOST, particularly affecting glitch size estimates, was discussed by Dunn et al. (2021), but for these two targets it has been confirmed that the provided values are correct. For PSR J1813–1749 and PSR J0537–6910, we use NICER X-ray observations as reported in Ho et al. (2020a, 2020b) and Abbott et al. (2021b). Details for all targets will be listed in Section 6.1.
3. GW Data
We analyze GW strain data taken at the LIGO Hanford Observatory (H1), LIGO Livingston Observatory (L1), and Virgo (V1), during their O3 observing run. O3 consisted of two separate sections, separated by a month-long commissioning break. O3a ran from 1 April 2019, 15:00 UTC until 1 October 2019, 15:00 UTC. O3b ran from 1 November 2019, 15:00 UTC, to 27 March 2020, 17:00 UTC. See Buikema et al. (2020) and Acernese et al. (2019) for general descriptions of the respective performances of the LIGO and Virgo detectors during O3. The calibration of this data set and its uncertainty budget are described in Sun et al. (2020, 2021) and Acernese et al. (2022a).
Data preparation for all searches starts with removing times of large transient noise with known causes, referred to as "CAT1" vetoes (Davis et al. 2021; Acernese et al. 2022b), from calibrated strain data.
This procedure removed 0.7 days out of 246 days of H1 data, 2.2 days out of 256 days of L1 data, and 0.46 days out of 251 days of V1 data.
The data are then cleaned to remove some known monochromatic or quasi-monochromatic detector artifacts that can be effectively monitored by external sensors, such as signals injected for calibration, or contamination from power mains (Davis et al. 2019; Sun et al. 2020, 2021; Acernese et al. 2022a, 2022b; Viets & Wade 2021). The H1 and L1 data near the 60 Hz mains power frequency are also cleaned using a nonlinear filtering method outlined in Vajente et al. (2020).
The data from L1 and H1 are then "gated" to remove further large transient artifacts, which removes an additional 1.0 day of data from L1 and 0.62 days of data from H1 (Davis et al. 2021; Zweizig & Riles 2021).
There are some differences in how each search pipeline uses this data, as will also be discussed in detail in Section 4. The 5n-vector search creates Short Fourier Databases (SFDBs) from the time-domain data. SFDBs consist of a collection of short-duration (1024 s long) fast Fourier transforms (FFTs), overlapped by half and Tukey-windowed with a window shape parameter of α = 0.001. During the construction of the SFDBs, an extra time-domain cleaning (see, e.g., Astone et al. 2005, in addition to those discussed in the previous paragraph) is applied to identify delta-like time-domain disturbances. The 
-statistic and transient pipelines create 1800 s Tukey-windowed short Fourier transforms (SFTs) with a window shape parameter of α = 0.001, and then extract a narrow frequency range around the frequency band for each pulsar.
Additional narrow lines with identified instrumental origin are listed in Goetz et al. (2021) and Piccinni et al. (2021); we use these for pipeline-level cleaning or post-processing vetoes, as discussed in later sections.
4. Signal Model and Search Methods
4.1. Signal Model
With all three pipelines, we search for quasi-monochromatic GW signals with fixed intrinsic amplitude at approximately twice the pulsar rotation frequency, f ≈ 2fspin, allowing for some deviation in the narrowband search approach. The general frequency evolution model for such GWs, following the standard model for a pulsar's fspin as a function of time 
at the pulsar, is
For most pulsars, no glitches have been observed during O3, and hence only the first term for the long-time spin-down evolution is relevant:
with a reference time Tref and up to N frequency derivatives f(k). We will also use 
, 
, and 
for the first three derivatives in this paper. The three search methods in this paper have different maximum values of N that they can handle, as discussed in Appendix A.
For glitching pulsars, the additional term is
(or multiple such terms for pulsars with multiple glitches), where Θ(x) is the Heaviside step function, Tgl is the glitch time, 
with M ≤ N are the permanent jumps in the frequency and its derivatives, δ
fR is the part of the frequency jump that relaxes back, and τR is the relaxation time. Relaxation is not necessarily observed for all glitches. The glitch term is typically a small correction on top of 
.
Building on this frequency evolution model, the full GW signal received by a detector is
where F+,×(t; α, δ, ψ) is the detector response to + and × polarized GWs received at the detector at time t, coming from R.A. α and decl. δ with polarization angle ψ (Jaranowski et al. 1998). Timing corrections for translating from 
to t are discussed below.
The amplitude of each GW polarization can be written as
where ι is the inclination angle of the pulsar, ϕ0 is the initial phase, h0 is an overall dimensionless strain amplitude as discussed below, and Φ(t; {f(k)}, α, δ) is the integrated phase from the timing model of Equation (3).
To convert from 
at the source to t at the detector, several timing corrections typically need to be addressed: proper motion of the source, binary orbital motion, and the motion of the detectors around the solar system barycenter. Proper motions between the pulsar and the solar system can usually be ignored (Covas 2020) for these searches, and in this analysis we do not cover sources in binary orbits as in, e.g., Abbott et al. (2021d, 2021e, 2022), Ashok et al. (2021). However, the sky position dependent correction between solar system barycenter and detector frame is a crucial part of the analysis methods described in the following, and is implemented in all cases (Jaranowski et al. 1998).
For GWs from a non-axisymmetric star, the amplitude h0 can be written as
where d is the distance to the NS, f is the GW frequency, and Izz
is the principal moment of inertia. The ellipticity, 
, is given by
In summary, our CW signal model depends on two sets of parameters: the frequency-evolution parameters λ = {f(k), α, δ}, and the amplitude parameters 
. The signal model for the long-duration transient search is also based on the one discussed here, but with an additional window function in time, as will be discussed below in Section 4.4.
For both CW pipelines, we search over a narrow range of GW frequencies and first spin-down terms centered on their respective central estimates from the source ephemeris: f = 2fspin(1 ± κ) and 
, where we take κ = 2 × 10−3. This value is intended to allow for physical offsets between the frequency of EM-observed pulsar rotation and the GW emission process, as discussed in the introduction and in more detail in Abbott et al. (2008, 2019b).
The frequency and spin-down changes due to glitches offer evidence of how large of a physical offset we might expect. While glitches with 
are observed occasionally, this value of κ encompasses most glitches that have been observed (see, e.g., Fuentes et al. 2017; Lower et al. 2021), and increasing this value significantly to encompass all observed glitch sizes would not be computationally feasible. The full widths we allow for both parameters, Δf = 4κ
fspin and 
, are shown in Table 1 for each target we consider. Higher-order frequency derivatives are handled differently by each pipeline, which we discuss in the next two subsections and in Appendix A.
4.2. 5n-vector Narrowband Pipeline
The 5n -vector narrowband pipeline uses the 5n-vector method of Astone et al. (2010, 2012) to search for CWs in a narrow frequency and spin-down range around the source ephemerides. This method searches for the characteristic modulations imprinted by the Earth's rotation on a GW signal, splitting it into five harmonics. The search is applied for the n detectors present in the network. For more details on the implementation of the method we use here, see Mastrogiovanni et al. (2017).
The starting point of the analysis is a collection of 1024 s long FFTs built as explained in the previous section. For each target, we extract a frequency band large enough to contain the Doppler modulation of every template that we consider. When higher-order frequency derivatives are used to fit the EM data, the corresponding GW spin-down terms (including 
) are fixed to twice the values provided in the source ephemerides.
For each target, we then correct for the Doppler modulation in the time domain using a nonuniform downsampling to 1 Hz. Using this time series, we then apply two matched filters (for the two CW polarizations) and construct a detection statistic S coherently combining the matched filter results from all detectors (Mastrogiovanni et al. 2017) for each template in the f and 
space. The detection statistic is defined as
where A+/× are the Fourier components of 
and H+/× are GW amplitude estimators built from the Fourier components of the data and A+/×. The template bank bin size in frequency is given by the inverse of the time-series duration and the spin-down bin size is given by its inverse squared.
The final step is to select the local maximum of the detection statistic in every 10−4 Hz band over all spin-down values considered. Within this set of points in the parameter space, we select as outliers those with a p-value (defined as the tail probability of the detection statistic noise-only distribution) below a 1% threshold, taking into account the number of templates applied in the f–
space. The noise-only distribution of the detection statistic used to estimate the threshold for candidate selection is built using all the templates excluded from the analysis in each 10−4 Hz band. The detection statistic value corresponding to the threshold for candidate selection is extrapolated from the noise-only distribution using an exponential fit of the tail.
Finally, to compute the 95% confidence level ULs 
on the CW amplitude in the case of no detection, we perform software injections with simulated GW signals in each 10−4 Hz sub-band to estimate the value of h0 at which we achieve 95% detection efficiency at a false-alarm probability of 1%.
4.3. Frequency-domain 
-statistic Pipeline
We also use the frequency-domain 
-statistic pipeline to search for CWs in a narrow frequency and spin-down range around the source ephemerides. The 
-statistic is a matched filter that is maximized over the amplitude parameters 
of the quasi-monochromatic signals (Jaranowski et al. 1998; Cutler & Schutz 2005); see Tenorio et al. (2021a) for a review covering its applications.
For each target, we search over the range of GW frequency f and spin-down 
, with matched-filter templates chosen using the multidetector 
-statistic metric (Prix 2007; Wette et al. 2008), which we call through the LALSuite (LIGO Scientific Collaboration 2021) program lalapps_Weave (Wette et al. 2018a). If the second spin derivative 
is nonzero, we also search over the range 
, where 
is the 1 σ uncertainty given in the source ephemeris. Higher-order frequency derivatives, up to f(4), are fixed to the EM-measured values. Because of constraints on the pipeline infrastructure, if the source ephemeris for a target includes derivatives above f(4), we refit the arrival times with fewer frequency derivatives. This results in slightly different ephemerides being used for this search and for the 5n-vector search. We discuss targets for which this is the case in Appendix A.
We place templates with a maximum matched-filter mismatch from a putative source of 0.02, which leads to a very dense grid compared to the one used by the 5n-vector search. This can be seen by comparing the final two (non-reference) columns of Table 1. The analysis uses a set of 1800 s Tukey-windowed SFTs over the full data span for all three detectors, and produces a detection statistic, which we denote 
, for each matched-filter template. See Section 3 for a full discussion of data processing and data quality cuts made before generation of the SFTs.
Following Tenorio et al. (2022), we construct 104 randomly chosen batches of templates from our search results for each pulsar, and fit a Gumbel distribution to the distribution of the maxima of these batches. We can then propagate this Gumbel distribution based on the total number of templates to estimate the tail probability (p-value) of the largest template across the full search for that target. We outline our implementation of this method more fully in Appendix B.1.
If any templates have a p-value of less than 1% and corresponding large 
values, we perform follow-up studies of these templates. A list of frequencies of known narrow spectral artifacts, fl, and their nominal widths, wl, are collated for all three detectors (Goetz et al. 2021; Piccinni et al. 2021). If ∣fl − flarge∣ < wl, where flarge is the frequency of the large-
template, then we veto that template. We also look at the single-detector 
value calculated for each detector. If the individual 
value for such a template in a single detector is larger than the 
value calculated using the whole network, then we also veto that large-
template (Keitel et al. 2014; Leaci 2015). Any remaining large-
templates with p < 1% are then flagged for further follow-up, and could be considered candidates for detection (subject to further studies).
In the absence of a detection, we set ULs at 95% confidence, 
, for each target using the software-injection scheme set out in, e.g., Abbott et al. (2007).
4.4. Post-glitch Transient Search
The search for long-duration transient CW-like signals from glitching pulsars is motivated by the idea that some of the excess energy liberated in a glitch could correspond to transient changes in the quadrupole moment of the star and be radiated away in GWs (van Eysden & Melatos 2008; Bennett et al. 2010; Prix et al. 2011; Melatos et al. 2015; Singh 2017; Yim & Jones 2020) during the post-glitch relaxation phase.
To search for such signals, we use the transient 
-statistic method introduced by Prix et al. (2011) and previously applied to LIGO O2 data in searches for GWs from glitches in the Crab and Vela pulsars (Keitel et al. 2019). The idea is to search for long-duration transients that are "CW-like" in the sense of following the standard quasi-monochromatic CW signal model from Section 4.1 with only an additional window function ϖ(t; t0, τ) in time applied:
where the transient parameters 
consist of the window shape, signal start time t0, and a duration parameter τ. As in Keitel et al. (2019), we limit ourselves here to the simplest case of rectangular windows, meaning the signal is exactly a standard CW that starts at time t0 and is cut off at time t0 + τ, with no additional amplitude evolution. Some form of amplitude decay would be expected for a realistic signal linked to post-glitch relaxation (Yim & Jones 2020). As demonstrated in Prix et al. (2011) and Keitel et al. (2019), the signal-to-noise ratio (S/N) loss from using rectangular windows in a search for exponentially decaying signals is mild, while using exponentially windowed search templates would mean a much higher computational cost (Keitel & Ashton 2018).
The search uses the same underlying 
-statistic library functions in LALSuite (LIGO Scientific Collaboration 2021) as the CW search described in Section 4.3, analyzing SFT data with the lalapps_ComputeFstatistic_v2 program that has been used before in, e.g., searches for CWs from supernova remnants (Aasi et al. 2015c; Abbott et al. 2019c; Lindblom & Owen 2020). For each target, we perform a search at a fixed sky location over a grid in f(k) parameters, with the number of spin-down terms depending on the pulsar ephemerides, going up to at most a 
term. At each grid point λ, the standard multidetector 
-statistic is calculated over the maximum duration of interest. Intermediate per-SFT quantities from this calculation, the so-called 
-statistic atoms, are kept. As described in Prix et al. (2011), partial sums of these atoms are evaluated in a loop over a range of {t0, τ}. The resulting matrix of 
statistics gives the likelihoods of transient CW-like signals in each time range. We marginalize 
over uniform priors on t0 and τ, obtaining the Bayes factor BtS/G for transient CW-like signals against Gaussian noise as our detection statistic. As demonstrated by Prix et al. (2011), BtS/G improves detection efficiency compared to taking the maximum of 
, and as demonstrated by Tenorio et al. (2022), it also produces cleaner background distributions on real data. The computational cost of these searches scales linearly with the number of λ templates and with the product of the numbers of t0 and τ values, and is dominated by the partial summing steps over the latter two parameters. See Prix et al. (2011) and Keitel & Ashton (2018) for details.
As this is only the second search thus far for long-transient GWs of this type (after Keitel et al. 2019), we describe its practical setup in more detail in Section 6.1.
5. CW Search Results
As described below, we find no significant candidates, although both the 
-statistic and 5n-vector CW searches find outliers requiring further follow up after the vetoes in Section 4 were applied. Hence, we set observational ULs on the GW strain from each target, constraining GW emission below the spin-down limit on 7 of the 18 target pulsars. All UL results are shown in Figure 1, and listed in Table 2. We give details of the results from both pipelines, including outliers that were followed up, in the rest of this section.
Figure 1. The red solid, blue dashed, and purple dotted curves show the expected sensitivities for H1, L1, and V1, respectively, using Equation (1). The blue pentagons indicate the median 95% CL ULs from the 5n-vector search across all 10−4 Hz sub-bands for each source. The black crosses indicate 95% CL ULs from the 
-statistic search, which are set across the full search range for each target. The orange triangles indicate the spin-down limit, hsd, with error bars that reflect uncertainty in the distance to each source. In a few cases, the error bars are smaller than the size of the markers. We discuss and compare these limits in more detail in Section 7. We do not show uncertainties on ULs here, although we discuss uncertainties due to the UL-setting method as well as calibration uncertainties in Section 7. The data for this figure are available online (LIGO Scientific Collaboration et al. 2022).
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Standard image High-resolution imageTable 2. UL Results on CW Strain from the 5n-vector (5v) and 
-statistic (
) Pipelines.
| Name | f |
 (5v) |
 ( ) | hsd |
|
|
(5v) |
( ) |
sd
| d | σd |
|---|---|---|---|---|---|---|---|---|---|---|---|
| (Hz) | ×10−26 | ×10−26 | ×10−26 | (5v) | ( ) | ×10−5 | ×10−5 | ×10−5 | (kpc) | (kpc) | |
| J0534+2200 bg | 59.24 | ... | 8.79 | 143.0 | ... | 0.061 | ... | 4.6 | 76.0 | 2.00 a | 0.5 |
| J0534+2200 ag | 59.24 | 5.2 | 6.09 | 143.0 | 0.036 | 0.043 | 2.7 | 3.2 | 76.0 | 2.00 a | 0.5 |
| J0711–6830 | 364.23 | 2.29 | 2.69 | 1.25 | 1.8 | 2.2 | 0.0017 | 0.002 | 0.00095 | 0.11 | 0.041 |
| J0835–4510 | 22.37 | 36.2 | 39.5 | 341.0 | 0.11 | 0.12 | 19.0 | 21.0 | 180.0 | 0.28 b | 0.02 |
| J1101–6101 | 31.85 | 8.96 | 10.5 | 4.25 | 2.1 | 2.5 | 59.0 | 68.0 | 28.0 | 7.00 | 2.7 |
| J1105–6107 | 31.64 | 8.94 | 10.1 | 17.1 | 0.52 | 0.59 | 20.3 | 23.0 | 38.0 | 2.36 | 0.92 |
| J1809–1917 | 24.17 | 21.7 | 26.1 | 13.7 | 1.6 | 1.9 | 110.0 | 140.0 | 73.0 | 3.27 | 1.3 |
| J1813–1749 full | 44.70 | 4.61 | 5.85 | 21.9 | 0.21 | 0.27 | 10.0 | 13.0 | 64.0 | 6.20 c | 2.4 |
| J1813–1749 bg | 44.70 | ... | 9.61 | 21.9 | ... | 0.44 | ... | 21.0 | 64.0 | 6.20 c | 2.4 |
| J1828–1101 | 27.75 | 14.8 | 16.3 | 7.67 | 1.9 | 2.1 | 87.0 | 96.0 | 45.0 | 4.77 | 1.9 |
| J1833–0827 | 23.45 | 22.9 | 29.3 | 5.88 | 3.9 | 5.0 | 180.0 | 230.0 | 46.0 | 4.50 | 1.8 |
| J1838–0655 | 28.36 | 13.2 | 15.3 | 10.2 | 1.3 | 1.5 | 100.0 | 120.0 | 79.0 | 6.60 d | 0.9 |
| J1856+0245 | 19.35 | 20.5 | ... | 11.2 | 1.8 | ... | 200.0 | ... | 110.0 | 6.32 | 2.46 |
| J1913+1011 | 55.69 | 3.47 | 4.37 | 5.36 | 0.65 | 0.81 | 4.9 | 6.1 | 7.5 | 4.61 | 1.8 |
| J1925+1720 | 26.43 | 14.3 | 17.6 | 5.93 | 2.4 | 3.0 | 98.0 | 120.0 | 41.0 | 5.06 | 2.0 |
| J1928+1746 | 29.10 | 10.5 | 13.5 | 8.15 | 1.3 | 1.7 | 51.0 | 66.0 | 39.0 | 4.34 | 1.7 |
| J1935+2025 | 24.95 | 19.2 | 23.7 | 15.3 | 1.3 | 1.5 | 130.0 | 170.0 | 110.0 | 4.60 | 1.8 |
| J1952+3252 | 50.59 | 3.73 | 4.93 | 10.3 | 0.36 | 0.48 | 4.1 | 5.5 | 11.0 | 3.00 e | 2.0 |
| J2124–3358 | 405.59 | 2.4 | 2.9 | 0.374 | 6.4 | 7.7 | 0.0057 | 0.0068 | 0.00095 | 0.44 f | 0.05 |
| J2229+6114 | 38.71 | 5.78 | 7.23 | 33.1 | 0.17 | 0.22 | 11.0 | 14.0 | 63.0 | 3.00 g | 2.0 |
Note. We show the ULs on strain amplitude at 95% confidence for both pipelines, 
, the spin-down limit for each target hsd along with the implied limit on ellipticity for each pipeline, and the limit on ellipticity implied by the spin-down limit sd. Finally, we also show the ratio of the 95% confidence UL and the spin-down limit. We surpass the spin-down limit for seven targets: PSR J0534+2200, PSRJ0835–4510, PSR J1105–6107, PSR J1813–1749, PSR J1913+1011, PSR J1952+3252, and PSR J2229+6114. We do not show uncertainty on ULs here, although we discuss uncertainty due to the UL-setting method as well as calibration uncertainty in Section 7. Distance estimates d and their uncertainties σd
are from dispersion measures (with a fiducial uncertainty of 40% from Yao et al. 2017) unless noted with one of the following letters: (a) Kaplan et al. (2008); (b) Dodson et al. (2003); (c) Camilo et al. (2021); (d) Gotthelf & Halpern (2008); (e) Verbiest et al. (2012); (f) Reardon et al. (2021); (g) Halpern et al. (2001). This table and a machine-readable file are available online (LIGO Scientific Collaboration et al. 2022).
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When discussing ULs in the following section, it should be noted that physically meaningful constraints on the GW energy emission are set when the ULs are lower than the spin-down limit (i.e., when the limit has been surpassed). Ellipticity constraints for pulsars whose spin-down limit is not surpassed would imply a 
higher than the one observed.
Finally, distance estimates used for the spin-down limits are given in Table 2. These are either from the ATNF pulsar catalog (based on dispersion measures) or from the literature. For the pulsar PSR J1813–1749, different models of the electron density in the Galaxy yield different distance estimates (Camilo et al. 2021). We have used the more optimistic estimate d = 6.2 ± 2.4 kpc, although a different model gives a more pessimistic estimate of d = 12 kpc.
5.1. 5n-vector Narrowband Pipeline
The 5n-vector pipeline found outliers only for two targets: PSR J1828–1101 and PSR J1838–0655. Figures 2 and 3 show the distribution of outliers in the searched frequency band (marginalized over the spin-down plane), along with the power spectral density (PSD) for each of the three detectors in that band. For PSR J1828–1101, we find 10 outliers around 27.7116 Hz with p-values between 10−5 and 10−2, and for PSR J1838–0655, we find 13 outliers around 28.3134 Hz with p-values between 10−3 and 10−2. The nominal p-values we report here assume underlying Gaussian noise, an assumption that can break down in the presence of instrumental artifacts.
Figure 2. Top panel: Median PSD across the run in each frequency bin for Hanford (red, solid), Livingston (blue, dashed), and Virgo (purple, dotted), for the frequency band searched for PSR J1828–1101. There is an obvious disturbance caused by a resonance in one of the suspended optics at H1, which is marked with the vertical dashed black line. Its estimated width is given by the shaded region. The line is clearly associated with large values of the detection statistics in the two other panels. Middle panel: S-statistic obtained for PSR J1828–1101 in the narrow frequency range explored and marginalized over the spin-down values. The horizontal dashed line indicates the threshold corresponding to a 1% false-alarm rate. The orange markers indicate the outliers found. Bottom panel: 
obtained for PSR J1828–1101 in the narrow frequency range explored. The horizontal dashed line indicates the threshold corresponding to a 1% false-alarm rate. The orange markers indicate the outliers found and vetoed using the known lines veto.
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Figure 3. Top panel: Median PSD across the run in each frequency bin for Hanford (red, solid), Livingston (blue, dashed), and Virgo (purple, dotted) for the frequency band searched for PSR J1838–0655. There is an obvious disturbance (of unknown origin) at H1 which is associated with large values of the detection statistics in the two other panels. Middle panel: S-statistic obtained for PSR J1838–1101 in the narrow frequency range explored and marginalized over the spin-down values. The horizontal dashed line indicates the threshold corresponding to a 1% false-alarm rate. The orange markers indicate the outliers found. Bottom panel: 
obtained for PSR J1838–1101 in the narrow frequency range explored. The horizontal dashed line indicates the threshold corresponding to a 1% false-alarm rate. The orange markers indicate the outliers found. Follow-ups of these outliers are discussed in Appendix C.
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Standard image High-resolution imageThese outliers are all due to noise disturbances identified in H1 data. For PSR J1828–1101, the PSDs for each of the three detectors around the outliers are shown in the top panel of Figure 2, while the S-statistic, defined in Equation (10), is shown in the middle panel. The outliers are produced by a known noise line at 27.71 Hz with width of 0.02 Hz caused by a resonance in one of the suspended optics at H1. For PSR J1838–0655, the outliers are due to an identified broad noise line of unknown origin (Goetz et al. 2021). The top panel of Figure 3 shows the PSD for each of the three detectors around these outliers. As can be seen, various broad noise disturbances are present in H1 data in the frequency band of PSR J1838–0655. Even if the origin of these noise lines is unknown, we can confidently exclude them as astrophysical CW signals because, if they were real, they would have been present also in the L1 data, which are more sensitive. In Section 5.2, we also perform a dedicated follow-up for the candidates due to these noise disturbances showing that they are only visible in H1 data.
The median ULs across the 10−4 Hz sub-bands, 
, for each pulsar are shown in column 3 of Table 2 and plotted with blue pentagons in Figure 1. The ratio of the UL with hsd is shown in column 6 of Table 2, and the limits on the ellipticity of the pulsar are shown in column 8. We discuss and compare these limits to those of the 
-statistic pipeline, to previous searches for the same targets, and to the spin-down limits, in Sections 5.3 and 7.
5.2. Frequency-domain -statistic Pipeline
The 
-statistic pipeline identified outliers with p-value < 0.01 that could not be vetoed with the methods described in Section 4.3 for two targets: PSR J1813–1749 bg and PSR J1838–0655. As described in Section 4.3, a first round of vetoes was made where we rejected outliers that were close to known narrow spectral artifacts. This resulted in vetoing outliers in PSR J1828–1101, which are shown in the bottom panel of Figure 2 for completeness and comparison with the 5n-vector search. We also vetoed outliers with a 
value that was larger when running on data from a single detector than when running on the full network. In Figures 3 and 4, we show the PSD in the top panel in the search band for the remaining outliers for PSR J1838–0655 and PSR J1813–1749 bg, respectively, while the search pipeline statistic, the 
-statistic discussed in Section 4.3, is shown in the bottom panel(s).
Figure 4. Top panel: Median PSD across the run in each frequency bin for Hanford (red), Livingston (blue), and Virgo (purple) for the frequency band searched for PSR J1813–1749. There is a small bump near a known line in H1 (marked with the vertical dashed line). However, this is distinct from the spike in 
values shown in the bottom panel. Bottom panel: 
obtained for PSR J1813–1749 bg in the narrow frequency range explored. The horizontal dashed line indicates the threshold corresponding to a 1% false-alarm rate. The orange markers indicate the outliers found. We were unable to identify this outlier as being due to the known line, but after follow-up detailed in Appendix C, it is clear that these outliers are caused by an artifact that affects the first two weeks of O3.
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Standard image High-resolution imageThe outliers for PSR J1813–1749 bg are associated with an artifact that contaminates the first two weeks of O3. Although it is nearby, it lies outside the nominal width of the line specified in Goetz et al. (2021), which can be seen in Figure 4, and therefore it is unlikely to be explicitly caused by that line. However, in Appendix C, we perform a follow-up study that shows that the outliers in this band increase in significance when running on data only from the first two weeks of the run. This behavior is inconsistent with a CW signal, and so we veto these outliers.
The remaining outliers from PSR J1838–0655, shown in Figure 3, are in the same frequency range as those for the 5n-vector pipeline. These outliers are all near a clear disturbance in H1, and the 
value for all of them is larger when running only on H1 data than when running only on L1 data. However, L1 is significantly more sensitive than H1 in the frequency bands of interest. In Appendix C, we detail a follow-up study on the remaining outliers. We show, based on a realistic set of injections, that for a true signal it is very unlikely to have a larger 
value when running only on H1 data than when running only on L1 data, due to the difference in the sensitivity for the two detectors.
One pulsar, PSR J1856+0245, was in a frequency band that had significant instrumental disturbances, and so we could not reliably estimate the background distribution and do not report ULs (or search results) for it. We discuss this case in Appendix B.1.
Given that there are no remaining candidates that exceed our follow-up threshold, we set ULs at 95% confidence on the strain amplitude, h0, of a potential CW signal produced by our targets. The UL, calculated across the full parameter space, 
, for each pulsar is shown in column 4 of Table 2 and plotted with black crosses in Figure 1. The ratio 
is shown in column 7 of Table 2, and the limit on the ellipticity of each pulsar is shown in column 9. We discuss these limits in detail, and in the context of indirect limits and the 5n-vector results, in Section 7.
5.3. Comparison between Pipelines
The ULs from the 
-statistic pipeline are, on average, 19% higher than those set by the 5n-vector pipeline. They are not directly comparable, however, as the 5n-vector pipeline results are the median across 10−4 Hz sub-bands, while the 
-statistic limits are set across the whole parameter space for each pulsar. A more equitable comparison is to compare the largest UL across all sub-bands for the 5n-vector, which is on average 9% lower than the ULs set by the 
-statistic pipeline.
A direct comparison would be very involved due to the differing pre-treatment of the data and methods used by the two pipelines. However, there are a few clear methodological differences that could explain the difference in ULs, which we highlight below.
The most obvious difference is in the effective "trials factors" used in generating the thresholds for follow-up, which are then used for the threshold of "detection" when performing the software injections used to calculate ULs. Looking at a larger number of templates reduces the effective mismatch between templates and a signal, but also increases the likelihood of finding a larger detection statistic due to a noise fluctuation. The 
-statistic search uses significantly more templates than the 5n-vector pipeline in nearly all cases. Fitting the relationship between the ratio of templates used in the two pipelines to the ratio of their ULs, we find that, for an equal number of templates, the 
-statistic search would have limits ∼ 5% higher than the maximum UL compared to the 5n-vector search. This is consistent with the scaling found in Astone et al. (2014). Inevitably, these differences are a reflection of the full range of choices made for the two pipelines, which also include how thresholds for follow-up are set (e.g., the method outlined in Appendix B.1 versus extrapolating the tail of the background distribution), the cleaning procedures used in preparing the data, and the densities of the templates used to perform the searches.
6. Post-glitch Transient Search
6.1. Search Setup
For all six targets in the post-glitch transient searches, we use SFTs of duration 1800 s (as discussed in Section 3) from both LIGO detectors and Virgo, except for PSR J0908–4913 and PSR J1826–1334. The frequency bands searched for these two targets are affected by broadband instrumental disturbances in Virgo. For this analysis, we additionally clean narrow instrumental lines with known instrumental origin (Goetz et al. 2021; Piccinni et al. 2021) before the searches, by replacing affected SFT bins with Gaussian noise matching the PSD in the surrounding range.
Observed parameters and search setups for the five pulsars that have a single known glitch during our time of interest are summarized in Table 3. For the "Big Glitcher" PSR J0537–6910 (Middleditch et al. 2006; Ho et al. 2020a), which was previously considered in searches for persistent CWs in the inter-glitch periods by Fesik & Papa (2020) and Abbott et al. (2021b, 2021c), we perform four separate searches following four observed glitches as illustrated in Figure 5. Corresponding parameters are listed in Table 4.
Figure 5. Relative timing of the O3 observing run, the glitches that NICER observed from PSR J0537–6910 (Ho et al. 2020a; Abbott et al. 2021b), and the four post-glitch long-duration transient searches we perform for this target.
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Standard image High-resolution imageTable 3. Parameters and Transient Search Setups for Five of the Glitching Pulsars
| J0534+2200 | J0908–4913 | J1105–6107 | J1813–1749 | J1826–1334 | |
|---|---|---|---|---|---|
| d [pc] | 2000 ± 500 a | 1000 ± 390 | 2360 ± 920.4 | 6200 ± 2400 b | 3606 ± 1406 |
| R.A. | 05h34m3197 | 09h08m3547 | 11h05m2571 | 18h13m3511 | 18h26m1316 |
| Decl. |
|
|
|
|
|
| Tgl [s] | 1247930924 | 1254619602 | 1238824009 | 1248825618 | 1264809618 |
| ΔTgl [ days] | 0.003 | 4.488 | 1.997 | 1 | 21 |
| Δfgl/f | 1.24 × 10−8 | 2.21 × 10−8 | 1.17 × 10−6 | 1.34 × 10−7 | 2.48 × 10−6 |
| 1.52 × 10−4 | 6.73 × 10−4 | 3.53 × 10−3 | ... | 6.82 × 10−3 |
| ... | ... | ... | ... | ... |
| Tref [s] | 1247845509 | 1256655802 | 1238651637 | 1248739243 | 1262996751 |
| f [Hz] | 59.204190 | 18.722385 | 31.628009 | 44.679639 | 19.691348 |
| Δf [Hz] | 0.059234 | 0.018732 | 0.031644 | 0.044702 | 0.019701 |
 [Hz s−1] | − 7.3707 × 10−10 | − 2.6526 × 10−12 | − 7.9786 × 10−12 | − 1.2863 × 10−10 | − 1.4650 × 10−11 |
 [Hz s−1] | 7.3670 × 10−13 | 2.6513 × 10−15 | 3.5865 × 10−14 | 1.3200 × 10−13 | 1.1358 × 10−13 |
 [Hz s−2] | 2.3631 × 10−20 | ... | 6.2723 × 10−21 | ... | 8.9844 × 10−22 |
 [Hz s−2] | 2.3643 × 10−23 | ... | 1.1809 × 10−21 | ... | 7.6842 × 10−24 |
 [Hz s−3] | ... | ... | − 5.1143 × 10−28 | ... | 4.2734 × 10−29 |
 [Hz s−3] | ... | ... | 7.9821 × 10−29 | ... | 2.3250 × 10−30 |
| Nλ | 3917139 | 507911 | 61497 | 7120630 | 22753 |
Notes. This table contains information about the five pulsars with a single post-glitch transient analysis. See Table 4 for the special case of PSR J0537–6910. Distances are taken from the ATNF catalog unless indicated otherwise with a footnote, and all other parameters are derived from the ephemerides as discussed in the text. Δfgl, 
, and 
are the glitch step sizes. For each search, the reference time Tref is chosen to match the earliest SFT data timestamp in 
, and the GW frequency and spin-down parameters f, 
, etc, are extrapolated to this time. Here, Δf, 
, etc., refer to their corresponding search bandwidths. Nλ
is the total number of frequency-evolution parameters covered by each search.
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Table 4. Parameters of the Four Searches Targeting Glitches of PSR J0537–6910
| Glitch 5 | Glitch 6 | Glitch 7 | Glitch 8 | |
|---|---|---|---|---|
| Tgl [s] | 1237420818 | 1243555218 | 1258243218 | 1263513618 |
| ΔTgl [ days] | 5 | 8 | 3 | 5 |
| Δfgl/f | 1.49 × 10−7 | 4.36 × 10−7 | 1.22 × 10−7 | 3.88 × 10−7 |
| 4.77 × 10−4 | 4.34 × 10−4 | 1.11 × 10−3 | 1.22 × 10−3 |
| 2.00 × 10−1 | −1.34 × 10−1 | 4.88 | −4.32 |
| Tref [s] | 1238166018 | 1242864932 | 1257985627 | 1263081781 |
| f [Hz] | 123.767904 | 123.766064 | 123.760042 | 123.758055 |
| Δf [Hz] | 0.123830 | 0.123828 | 0.123822 | 0.123820 |
 [Hz s−1] | −3.9990 × 10−10 | −3.9980 × 10−10 | −4.0034 × 10−10 | −4.0017 × 10−10 |
 [Hz s−1] | 3.9970 × 10−13 | 3.9960 × 10−13 | 7.3810 × 10−13 | 7.3493 × 10−13 |
 [Hz s−2] | 1.9990 × 10−20 | 1.4420 × 10−20 | 9.7927 × 10−20 | −9.4572 × 10−20 |
 [Hz s−2] | 2.0000 × 10−23 | 3.7517 × 10−21 | 1.4036 × 10−19 | 1.1735 × 10−19 |
 [Hz s−3] | ... | ... | ... | ... |
 [Hz s−3] | ... | ... | ... | ... |
| Nλ | 593911 | 11474334 | 15837589 | 21543992 |
Note. Here, d = (49.59 ± 0.55) kpc (Pietrzyński et al. 2019), the R.A. is 05h37m4742, and the decl. is 
for all four searches, while the frequency and spin-down parameters are extrapolated to the Tref corresponding to each search. See Table 3 for details on the listed parameters.
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We derive the ranges of frequency evolution parameters λ and transient parameters 
to be covered in each search from the ephemerides for that pulsar, the uncertainty δ
Tgl in the glitch time Tgl, and the availability of GW data relative to Tgl. We set a maximum signal duration of 
to be searched for each glitch, guided both by observed glitch recovery times being typically on shorter timescales (Lyne et al. 2000; Espinoza et al. 2011; Yu et al. 2013; Yim & Jones 2020; Lower et al. 2021) and by computational cost constraints (Keitel & Ashton 2018). Signal start times t0 cover 
, and the reference time Tref for the frequency and spin-down parameter grids (again assuming GW emission near twice the rotation period) is set to the earliest SFT timestamp in this range. On each side of the nominal f(k) values extrapolated to Tref, the grids cover the maximum of (i) 0.001 times the nominal value (this is one quarter of the width used by the CW searches as described in Section 4.1), (ii) the 1σ fit uncertainties from the ephemerides propagated to Tref, and (iii) the glitch step size in that parameter. We place the grids using the metric from Prix (2007) and the template placement algorithm from Wette (2014), as implemented in lalapps_ComputeFstatistic_v2 and lalpulsar (LIGO Scientific Collaboration 2021), with a mismatch parameter of 0.2. The algorithm can add some additional grid points outside the nominal ranges to reduce mismatch near the edges. But for targets with 
and higher terms in their timing solution, we strictly limit template placement in all spin-down terms to the nominal range, to limit computational cost. Our resolution in 
is dt0 = d
τ =2 TSFT = 3600 s. Due to implementation details of the 
-statistic (Prix 2015), the minimum duration is 
.
In some special cases, we modified the default setup to match the availability of GW data, explaining some apparent anomalies in the configurations listed in Tables 3 and 4. The glitch from PSR J0908–4913 happened in October of 2019, during the maintenance break between O3a and O3b, a month for which no GW data are available. We still search for the standard set of τ up to 120 days after this glitch, but fix t0 to the first available SFT timestamp after the break—any templates with a different t0 in the regular band around the glitch time would yield identical detection statistics—and are insensitive to shorter-duration signals in this case. For PSR J1826–1334, O3 ended 84 days after the nominal glitch time, so we shorten the analysis accordingly.
For PSR J0537–6910, as illustrated in Figure 5 and following the numbering from Ho et al. (2020a), we perform searches covering the durations from each of glitches 5–7 until the next glitch, and for glitch 8 we search until the end of O3. Glitch 5 happened in late March of 2019, shortly before the start of O3, and hence, as for PSR J0908–4913, t0 is fixed to the data start time.
6.2. Results
To determine whether or not there are interesting outliers in the results of each search, for most targets we again follow the procedure described in Tenorio et al. (2022) to empirically estimate the distribution of the expected highest outlier. For two targets, PSR J1105–6107 and PSR J1826–1334, the number of templates is too low for this method to deliver robust results. Instead, for these targets, we use spatial off-sourcing (Isi et al. 2020) to estimate the background distribution and set a threshold. The specific implementations for both cases are explained in Appendix B.2.
For eight of the nine searches, there are no candidates above threshold. The background distribution from the search after the eighth glitch of PSR J0537–6910 is less clean and requires an additional notching step, as discussed in Appendix B.2. We then find two marginal outliers at about 2 and 1 standard deviations, respectively, from the mean of the estimated extreme-value distribution (4% and 14% tail probability). The best-fit signal durations are about 60 and 45 days, respectively. Appendix D lists the full parameters of these outliers and describes additional analyses to follow them up.
In summary, we cannot associate either outlier with any known spectral artifact or single-detector feature in the data. They pass several consistency tests and vetoes, and behave as weak CW-like candidates are expected to in a fully phase-coherent targeted follow-up (Dupuis & Woan 2005; Pitkin et al. 2017) over the corresponding subsets of O3 data. Nevertheless, we do not consider these as promising detection candidates, for two main reasons: first, they would imply a signal with far higher amplitude than allowed by the estimated glitch energy for this target (see discussion of ULs below). Second, their significance is low, compared also with the 1% tail probability thresholds used by the two CW pipelines in this paper. This is reinforced by follow-up 5n-vector fixed-duration analyses, which recover local S-statistic peaks at the parameters of each outlier, but not as the loudest candidates in the band. Still, we conservatively use a higher threshold (from including the outliers in the background estimate with no notching) in the UL procedure for PSR J0537–6910 glitch 8.
Having identified no convincing detection candidates above threshold for any of the targets, we set ULs on the GW strain after each glitch. As in the search, we assume CW-like signals that are constant over the signal duration τ, and we evaluate the ULs as a function of τ. To do so, we simulate signals following Equation (11) with λ parameters and t0 drawn uniformly from within the search ranges, several discrete steps in τ and h0, and the remaining amplitude parameters randomized over their natural ranges. We add these signals to the original SFTs without the spectral cleaning (as described in Section 6.1), redo the cleaning step to ensure any signals falsely dismissed due to cleaning are properly accounted for, and perform small searches over a portion of the original λ grid (0.1 mHz in f and the closest spin-down points) and the full 
range. The 
ULs are obtained from a sigmoid fit to the recovery fraction 
above the BtS/G threshold from the main search, as a function of injected h0 at fixed τ. Results for all targets are shown in Figure 6.
Figure 6. Strain ULs 
obtained for the six glitching pulsars targeted with the long-duration transient search, as a function of signal duration τ (dashed curves with points representing the injection set). The four searches following glitches in PSR J0537–6910 are shown in separate panels, for readability. Uncertainties from finite injection sample size are at the 3%–5% level, comparable to or lower than amplitude uncertainty from detector calibration as discussed in Section 7. The indirect glitch excess energy ULs from Equation (12), originally derived in Prix et al. (2011) and based on the estimated size of each glitch and distance to the pulsar as listed in Tables 3 and 4, are shown for comparison (red shaded bands). The data for this figure are available online (LIGO Scientific Collaboration et al. 2022).
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Standard image High-resolution imageWe compare these empirical ULs to an indirect constraint derived in Prix et al. (2011) according to the two-fluid model of NS glitches (Lyne et al. 2000). In this model, the excess superfluid energy liberated in a glitch gives a UL to the total GW energy, which corresponds to a τ-dependent strain UL:
Qualitatively similar ULs also hold for other glitch models such as crust-cracking starquakes (Middleditch et al. 2006); see again Prix et al. (2011) for details. For the purpose of showing this indirect UL in Figure 6, we have assumed a fiducial moment of inertia 
kg m2 and distances to each target as listed in Tables 3 and 4.
As can be see in Figure 6, our observational ULs do not reach below the indirect energy constraint for any of the targets. While the sensitivity of O3 was improved over O2, none of the O3 targets had such a favorable combination of large frequency step size and small distance as the one from the Vela pulsar during O2 (Palfreyman 2016; Sarkissian et al. 2017; Palfreyman et al. 2018). Our closest result to the benchmark is for PSR J1105–6107, which had the second largest glitch of the selected targets in terms of Δfgl/f, and is at a closer distance and more favorable frequency than the only other target with a larger glitch (PSR J1826–1334). For this glitch, the closest the τ-dependent ULs get to the indirect energy constraint is within a factor of 1.6. As discussed, these empirical ULs are valid specifically for GW signals following the simple CW-like model of Equation (11), whereas many other models for GW emission following pulsar glitches are conceivable.
7. Conclusion
We have used data from the LIGO–Virgo observing run O3 to search for CW signals from 18 known pulsars, allowing for a mismatch between the frequencies of EM and GW emission, and for long-duration CW-like transients from six glitching pulsars. All statistically significant outliers were associated with instrumental disturbances and vetoed, and so we set ULs on the GW strain amplitude from each target.
The narrowband CW results obtained here complement the more constraining limits obtained from perfectly targeted searches on the same O3 data (Abbott et al. 2021e), where the GW frequency was assumed to be at exactly once or twice the EM-measured rotation frequency. With respect to the narrowband search done with the first half of O3 (Abbott et al. 2020b), we have improved our ULs by a factor of ∼ 35% for PSR J0534+2200 and ∼ 10% for PSR J0711–6830 and PSR J0835–4510, using respective parameter spaces ∼ 6.5, 1.2, and 7.1 larger for these three pulsars. For the remaining pulsars, which were also searched for using the same method in O2, the limits presented here are a factor of 2–3 lower than those set in Abbott et al. (2019b). We surpass the spin-down limits for seven pulsars, including PSR J1105–6107 and PSR J1913+1011, for the first time using the narrowband method.
The best upper limit set across the whole frequency band is h0 ≲ 2.3 × 10−26 for PSR J0711–6830 at roughly 364 Hz using the 5n-vector search. We also set the best limit on pulsar ellipticity with PSR J0711–6830 at ≲ 1.7 × 10−8. However, neither of these limits surpass the indirect spin-down limits for this pulsar, which are a factor of ∼2.2 lower than the upper limits for the search. For pulsars with 
, the best 
is h0 ≲ 3.5 × 10−26 for PSR J1913+1011 and the best ellipticity limit is ≲ 2.7 × 10−5 for PSR J0534+2200.
Our long-duration transient search results for signals following nine glitches in six pulsars significantly extend the target set for such searches over the two glitches from two pulsars previously covered with O2 data by Keitel et al. (2019), demonstrating the flexibility of the search method. We report two marginal outliers from the search for the last glitch of PSR J0537–6910 during O3, but do not consider them as significant detection candidates. We have set duration-dependent strain ULs for each post-glitch search and compared the results with the indirect glitch energy UL benchmark established in Prix et al. (2011), but have not reached this benchmark for any of these targets, with PSR J1105–6107 coming closest to it.
Although we do not explicitly show systematic error or statistical uncertainty on upper limits presented in Figures 1 and 6 or Table 2, their impact on these results is discussed here. Statistical uncertainty due to finite sampling when calculating limits is ∼ 3% for the 
-statistic CW search and 3%–5% for the post-glitch transient search. For the 5n-vector search, the uncertainty due to which part of the frequency band is sampled (i.e., typical distance from median UL across all 0.1 mHz sub-bands) is ∼2.4%. Meanwhile, the systematic error in the calibration of the O3 strain data is quantified in Sun et al. (2020, 2021) for LIGO and Acernese et al. (2022a) for Virgo. The error depends upon the time at which data were collected and upon the frequency band of the measurement. The maximum estimated systematic error in absolute magnitude of the strain across all frequencies and time is 7% for the LIGO detectors in O3a, 11% for the LIGO detectors in O3b, and 5% for Virgo, though typically the error is smaller than these levels. Additionally, calibration systematics are not correlated between detectors, and so it is likely that the absolute error in our combined measurements would be less than these values.
We expect that, with future upgrades of the LIGO–Virgo–KAGRA network (Abbott et al. 2020a), we can further improve GW results on the known pulsar population, and will be able to constrain emission scenarios and nuclear physics both for quiescent-state pulsars and those perturbed by glitches.
This material is based upon work supported by NSF's LIGO Laboratory, which is a major facility fully funded by the National Science Foundation. The authors also gratefully acknowledge the support of 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 GEO 600 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 Netherlands Organization for Scientific Research (NWO), 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 (AEI), the Spanish Ministerio de Ciencia e Innovación and Ministerio de Universidades, the Conselleria de Fons Europeus, Universitat i Cultura and the Direcció General de Política Universitaria i Recerca del Govern de les Illes Balears, the Conselleria d'Innovació Universitats, Ciència i Societat Digital de la Generalitat Valenciana and the CERCA Programme Generalitat de Catalunya, Spain, the National Science Centre of Poland and the European Union—European Regional Development Fund; Foundation for Polish Science (FNP), the Swiss National Science Foundation (SNSF), the Russian Foundation for Basic Research, the Russian Science Foundation, the European Commission, the European Social Funds (ESF), 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 French Lyon Institute of Origins (LIO), the Belgian Fonds de la Recherche Scientifique (FRS-FNRS), Actions de Recherche Concertées (ARC) and Fonds Wetenschappelijk Onderzoek—Vlaanderen (FWO), Belgium, the Paris Île-de-France Region, the National Research, Development and Innovation Office Hungary (NKFIH), the National Research Foundation of Korea, the Natural Science and Engineering Research Council Canada, Canadian Foundation for Innovation (CFI), the Brazilian Ministry of Science, Technology, and Innovations, 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, the United States Department of Energy, and the Kavli Foundation. The authors gratefully acknowledge the support of the NSF, STFC, INFN, and CNRS for provision of computational resources.
This work was supported by MEXT, JSPS Leading-edge Research Infrastructure Program, JSPS Grant-in-Aid for Specially Promoted Research 26000005, JSPS Grant-in-Aid for Scientific Research on Innovative Areas 2905: JP17H06358, JP17H06361 and JP17H06364, JSPS Core-to-Core Program A. Advanced Research Networks, JSPS Grant-in-Aid for Scientific Research (S) 17H06133 and 20H05639, JSPS Grant-in-Aid for Transformative Research Areas (A) 20A203: JP20H05854, the joint research program of the Institute for Cosmic Ray Research, University of Tokyo, National Research Foundation (NRF), Computing Infrastructure Project of KISTI-GSDC, Korea Astronomy and Space Science Institute (KASI), and Ministry of Science and ICT (MSIT) in Korea, Academia Sinica (AS), AS Grid Center (ASGC) and the Ministry of Science and Technology (MoST) in Taiwan under grants including AS-CDA-105-M06, Advanced Technology Center (ATC) of NAOJ, and Mechanical Engineering Center of KEK.
We would like to thank all of the essential workers who put their health at risk during the COVID-19 pandemic, without whom we would not have been able to complete this work.
We acknowledge that CHIME is located on the traditional, ancestral, and unceded territory of the Syilx/Okanagan people. We are grateful to the staff of the Dominion Radio Astrophysical Observatory, which is operated by the National Research Council of Canada. CHIME is funded by a grant from the Canada Foundation for Innovation (CFI) 2012 Leading Edge Fund (Project 31170) and by contributions from the provinces of British Columbia, Québec, and Ontario. The CHIME/FRB Project, which enabled development in common with the CHIME/Pulsar instrument, is funded by a grant from the CFI 2015 Innovation Fund (Project 33213) and by contributions from the provinces of British Columbia and Québec, and by the Dunlap Institute for Astronomy and Astrophysics at the University of Toronto. Additional support was provided by the Canadian Institute for Advanced Research (CIFAR), McGill University, and the McGill Space Institute thanks to the Trottier Family Foundation, and the University of British Columbia. The CHIME/Pulsar instrument hardware was funded by NSERC RTI-1 grant EQPEQ 458893-2014. This research was enabled in part by support provided by WestGrid (www.westgrid.ca) and Compute Canada (www.computecanada.ca). We acknowledge support from the Natural Sciences and Engineering Research Council of Canada (NSERC) funding reference #CITA 490888-16, the Canadian Institute for Advanced Research, and the UBC Four Year Fellowship (6456).
We acknowledge support from EPSRC/STFC fellowship (EP/T017325/1), ANID/FONDECYT grants 1171421 and 1211964, and NASA grants 80NSSC19K1444 and 80NSSC21K0091. This work is supported by NASA through the NICER mission and the Astrophysics Explorers Program, and uses data and software provided by the High Energy Astrophysics Science Archive Research Center (HEASARC), which is a service of the Astrophysics Science Division at NASA/GSFC and High Energy Astrophysics Division of the Smithsonian Astrophysical Observatory.
Software: Astropy (Robitaille et al. 2013; Price-Whelan et al. 2018), corner.py (Foreman-Mackey 2016), distromax (Tenorio et al. 2021c, 2022), GWpy (Macleod et al. 2021), LALSuite (LIGO Scientific Collaboration 2021), matplotlib (Hunter 2007), numpy (Harris et al. 2020), OctApps (Wette et al. 2018b), pandas (McKinney 2010; Reback et al. 2020), ptemcee (Foreman-Mackey et al. 2013; Vousden et al. 2015), PyFstat (Ashton & Prix 2018; Ashton et al. 2021; Keitel et al. 2021), scipy (Virtanen et al. 2020), SWIGLAL (Wette 2020), Tempo (Nice et al. 2015), Tempo2 (Edwards et al. 2006).
Appendix A: Differences in Pulsar Ephemerides Used for Different Pipelines
Due to variations in code infrastructure for the 
-statistic and 5n-vector pipelines, there were some cases where we used ephemerides with different parameterizations to search for the same source. In general, the lalapps_Weave code used for the 
-statistic CW search can search up to the fourth derivative in frequency, and the automatic template placement will work best if the reference time at which the frequency and spin-downs are defined is during or close to the time at which we are conducting the search. Similarly, the transient search (also based on the 
-statistic, but using the lalapps_ComputeFstatistic_v2 code) can search up to 
. On the other hand, the 5n-vector pipeline has no such limitations on the number of frequency derivatives.
In several cases, which we document below, the EM ephemerides used for the 5n-vector search included higher-order frequency derivatives to improve the phenomenological fit and reduce the effects of timing noise. In those cases, we refit the ephemerides using at most a fourth frequency derivative for the 
-statistic pipelines and we then used the simple heuristic from Equation (11) of Prix & Itoh (2005) to verify that the effect of the increased timing residuals on our matched-filter analysis was less than the maximum mismatch used to place templates for the search. The affected pulsars were:
- 1.PSR J0534+2200: The Crab pulsar was originally fit with 12 frequency derivatives, with an average timing residual of TRES = 135.0 μs, and this ephemeris was used for the 5n-vector search. For both-statistic-based pipelines we search with up to two frequency derivatives with an average timing residual of TRES = 482.382 μs.
- 2.PSR J0835–4510: The Vela pulsar was originally fit with seven frequency derivatives, with an average timing residual of TRES = 137.1 μs, and this ephemeris was used for the 5n-vector search. The-statistic CW pipeline ran with a search up to four frequency derivatives with an average timing residual of TRES = 133.2 μs.
- 3.PSR J1809–1917: This pulsar was originally fit with five frequency derivatives, with an average timing residual of TRES = 1766.4 μs, and this ephemeris was used for the 5n-vector search. The-statistic pipeline ran with a search up to four frequency derivatives with an average timing residual of TRES = 489.4 μs. In this case, it is likely that the timing residuals are smaller because the reference time used for the fits was moved to the middle of the observation time.
Appendix B: Settings for Background Estimation in -statistic Pipelines
B.1. CW Search
In this appendix, we outline the details of the method used for estimating the distribution of the expected largest outlier for the frequency-domain 
-statistic CW search. We then set a threshold corresponding to a p-value of 1% under the assumption that no signal is present in the data. We use the distromax method introduced by Tenorio et al. (2022), with some slight modifications due to the specific situation at hand.
We want to estimate the distribution of the maximum value of 
for the full search for a single pulsar, under the assumption that no signal is present. To be conservative, we will exclude bands corresponding to known disturbances when estimating this distribution, and as such we will have outliers that exceed our threshold to follow up that are associated with the disturbances. However, this means that we will not set our threshold accounting for the disturbances, which would generally push them upward and potentially cause a signal to be missed. The procedure is as follows:
- 1.Run the search and save the templates with the top 2 × 107 values of.
- 2.
- 3.Remove templates that are within 4 × 10−5 f of unidentified lines, and in the case where the line is specified as "broad" in Goetz et al. (2021), we remove templates within 4 × 10−4 f.
- 4.Perform a single iteration of the notching procedure from Tenorio et al. (2022), which removes templates with a frequency within 5 × 10−5 Hz of any template exceeding an internal threshold.
- 5.Split the remaining values into 104 random batches of size ∼ 2 × 103. Find the maximum in each of those batches, and fit a Gumbel distribution to that set of maxima.
- 6.Propagate that Gumbel distribution based on the number of batches (Tenorio et al. 2022), and the total number of templates (i.e., the number of templates used before we perform notching, because this is the number of templates used to perform the search) to obtain a distribution for the maximum of the full search for that individual pulsar.
What is left is a distribution on the maximum value of 
under the assumption that no signal is present, and we can then integrate this distribution to find the value of 
that corresponds to a p-value of 1%. Any of the 2 × 107 original templates with p < 1% are then subject first to the vetoing procedure described in Section 4.3, and if they pass this procedure, then they are flagged for more extensive follow-up.
With the exception of one pulsar, PSR J1856+0245, the notching procedure above removes less than 40% of the frequency band. In the case of PSR J1856+0245, the known and unknown lines notching procedure removes the full frequency band, and so we do not search for CWs from this pulsar with the 
-statistic pipeline.
B.2. Transient Search
The basic approach for most targets in this analysis is the same as for the CW search, following the distromax method introduced by Tenorio et al. (2022): we fit a Gumbel distribution to the measured maxima of our detection statistic BtS/G over random subsets ("batchmaxes"). We then propagate the parameters of this distribution considering the full number of templates. However, we then set the threshold less deep in the tails of the distribution than in the CW search, namely at the mean plus one standard deviation of the propagated Gumbel distribution. This same threshold is used both for candidate identification and later in the ULs procedure. The results from the seven transient searches where we apply this method are generally relatively clean, as indicated by the batchmax histograms and goodness of the Gumbel distribution fits, and so we do not employ any notching of disturbances across the board. However, for glitches 7 and 8 of PSR J0537–6910, additional features appear in the batchmax histograms and degrade the Gumbel fit. In both cases, with three iterations of the automated notching procedure from Tenorio et al. (2022), we can produce clean batchmax histograms. For glitch 7, the threshold on BtS/G is quite robust under notching (changing from 11.3 to 11.0), and no outliers are recovered. On the other hand, for glitch 8, the threshold is lowered sufficiently (from 12.5 to 10.5), which reclassifies the largest values at two frequencies as marginal outliers. Follow-up of these outliers is detailed in Appendix D.
For PSR J1105–6107 and PSR J1826–1334, the number of templates is too small to obtain robust Gumbel fits with the distromax method. Instead, we use the off-sourcing approach (Isi et al. 2020): we indirectly estimate the background distribution by rerunning the full search 1000 times on different sky positions but with otherwise identical settings. This is feasible in these cases because the two searches are very cheap, i.e., less than a single core hour per run. By off-sourcing on the sky, we sample combinations of the same data that are statistically independent from the templates used in the main search. The practical implementation here matches that of Tenorio et al. (2021b): we keep decl. fixed and change only R.A. α, excluding a part of the sky closer than 0.5π to the pulsar in order to ensure statistical independence. We then fit a Gumbel distribution to the set of the highest BtS/G values from each off-sourced search and again set the threshold at the mean plus one standard deviation of this distribution.
Appendix C: Follow-up of Remaining Outliers for CW Searches
In this section, we discuss follow-up of outliers found when searching in the direction of PSR J1838–0655 and PSR J1813–1749 (before glitch) with the 
-statistic. The same region of parameter space that produced outliers for PSR J1838–0655 in the 
-statistic search also produced outliers in the 5n-vector search.
The 
-statistic results of PSR J1813–1749, searching before the potential glitch, show an outlier at 44.62443216 Hz, close to a known line at H1 (see Figure 4). However, the outlier frequency is outside the nominal width of the line specified in Goetz et al. (2021), and is therefore unlikely to be caused by that specific artifact. On 16 April 2019, the frequencies of the calibration lines were changed to improve detector calibration, which altered the character of the persistent narrowband artifacts seen in H1 data (Sun et al. 2020). The known line in Figure 4 is an example of one such artifact. To test whether this candidate is caused by a separate but similar artifact, we run the search from the start of the run until only 16 April 16 2019. In Figure 7, by zooming in on the frequency range around the outliers, we show that quite a few templates have significantly higher values of 
than the outliers do for the full run. This behavior is inconsistent with a CW signal, and so we veto these outliers.
Figure 7.
values vs. frequency (blue circles) for a search from 2019 April 1–2019 April 16 around the 44.62443216 Hz outlier in the 
-statistic search for PSR J1813–1749 bg. The red line indicates the 
value associated with the outlier, while the vertical black dashed line shows its frequency. It is clear that the search on just 2 weeks of data produces larger 
values than the search on the full year of data, which is inconsistent with a true signal. The data for this figure are available online (LIGO Scientific Collaboration et al. 2022).
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Standard image High-resolution imageIn Figure 3, we show that the outliers associated with PSR J1838–0655 are near an unknown disturbance at H1. The 
values running only on H1 data are larger for all of these candidates than they are when running on only L1 data, despite L1 being more sensitive in this frequency band. To test whether this is reasonable for a signal, we use the 1454 software injections that were detected above the search threshold when calculating ULs. We plot the 
calculated using L1 on the y-axis and 
calculated using H1 on the x-axis of Figure 8 for all injections (blue to yellow varied colors), and for the outliers (red). The dashed line indicates where these two values are equal. Only 2 out of 1454 injections cross the red line, meaning that if we veto any candidates to the right of the red line, we incur a false dismissal of 0.1%. Therefore, we veto this outlier in both the 
-statistic and the 5n-vector searches.
Figure 8. Individual detector 
values calculated for software injections detected when calculating ULs (blue to yellow dots indicating low to high h0 injections), and outliers (red dots). The dashed line indicates where 
is equal for the two detectors. We see the red points clustered away from the software injections. Only two of the software injection recoveries fall on the right-hand side of the red line. The data for this figure are available online (LIGO Scientific Collaboration et al. 2022).
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Standard image High-resolution image
Appendix D: Follow-up of PSR J0537–6910 Glitch 8 Transient Search Outliers
Here, we discuss follow-up analyses of the marginal outliers recovered for the post-glitch transient search targeting glitch 8 of the Big Glitcher PSR J0537–6910. The parameters of the two outliers of interest are listed in Table 5. Another template with BtS/G marginally above threshold is not offset by more than a single bin in any parameter from outlier A; we therefore do not follow that one up separately.
Table 5. Outliers from the Transient Search after Glitch 8 of PSR J0537–6910
| f (Hz) |
 (10−10 Hz s−1) |
 (10−20 Hz s−2) |
 (s) | τML (s) |
|
| Tail Probability | |
|---|---|---|---|---|---|---|---|---|
| A | 123.7934283 | −3.997 | −8.39 | 1263844981 | 5209200 | 53.4 | 11.00 | 4% |
| B | 123.8640925 | −3.995 | −1.89 | 1263628981 | 3913200 | 53.2 | 10.86 | 14% |
Note. The transient parameter estimates 
and τML correspond to the location of the 
value in the 
map. The posterior estimates (assuming flat priors, as for the BtS/G statistic) are within 1 hour of the ML values—with the exception of τ for outlier A, which is about 7 hours longer than the ML value. The tail probability refers to the propagated Gumbel distribution following the distromax method with three notching iterations.
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First, we checked for any evidence that the outliers may be caused by instrumental artifacts. There are no spectral artifacts of identified or unidentified origin listed in the relevant frequency band in Goetz et al. (2021) for LIGO nor in Piccinni et al. (2021) for Virgo. Comparing the time-frequency tracks of the outliers with single-detector spectrograms (Weldon et al. 2019) also reveals no suspicious structures. As a multidetector consistency check (Keitel et al. 2014; Leaci 2015), we recomputed the 
maps at each outlier's λ points for each individual detector. The corresponding 
and BtS/G values and best-fit (t0, τ) pairs are collected in Table 6. Virgo does not recover these candidates, as expected because of its much lower sensitivity in this frequency range; instead, it returns unrelated short-duration peaks. But the H1 and L1 results are fully consistent with each other and with the multidetector result.
Table 6. Detector Consistency Follow-up for PSR J0537–6910 Glitch 8 Outliers
| Detector(s) |
 (s) | τML (s) |
|
|
 (s) | τMP (s) |
|---|---|---|---|---|---|---|
| Outlier A | ||||||
| HLV | 1263844981 | 5209200 | 53.4 | 11.00 | 1263843607 | 5182844 |
| H1 | 1263758883 | 5284800 | 29.0 | 5.14 | 1263758773 | 5217811 |
| L1 | 1264317301 | 4669200 | 32.1 | 5.62 | 1263952957 | 5052102 |
| V1 | 1252638018 | 57600 | 27.5 | 3.05 | 1240493379 | 282595 |
| Outlier B | ||||||
| HLV | 1263628981 | 3913200 | 53.2 | 10.86 | 1263628503 | 3912769 |
| H1 | 1263629283 | 2196000 | 31.2 | 4.80 | 1263629183 | 2197636 |
| L1 | 1263590101 | 3945600 | 33.6 | 5.27 | 1263589393 | 4108995 |
| V1 | 1240437618 | 3600 | 28.3 | 2.95 | 1247483923 | 5400 |
Note. These values are all obtained from the 
maps evaluated at the λ parameters as listed for each outlier in Table 5. The transient parameter estimates 
and τML correspond to the location of the 
value, while 
and τMP are maximum-posterior estimates.
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Other possible vetoes against an astrophysical origin include checking whether or not the detection statistics would increase when turning off Doppler modulation inside the 
-statistic code ("DMoff;" Zhu et al. 2017), or searching at a different sky position (off-sourcing; Isi et al. 2020). With a DMoff rerun of the full search band, we recover neither of the outliers as a local maximum, and the overall maximum at 
is lower, meaning that the outliers pass this check. For off-sourcing, we perform 1000 analyses at different α (at least 0.5π away from PSR J0537–6910) over 5 mHz wide sub-bands of the original search. This does not turn out to veto the outliers either, as we do not find extended sets of similar or larger BtS/G values at many other sky positions and similar frequencies, as could be expected from some types of instrumental artifacts. However, we can once again fit a Gumbel distribution to the maxima of these off-sourced searches and propagate to the corresponding expected distribution over the whole search bank while taking into account the missing trials factor of 25. This results in tail probabilities of 2% and 3% for the two outliers, which are lower than the 4% and 14% from distromax but still above the 1% threshold used by the two CW search pipelines.
We also followed up both outliers with four separate pipelines: (i) gridless follow-up with PyFstat (Ashton & Prix 2018; Ashton et al. 2021; Keitel et al. 2021; Tenorio et al. 2021b); (ii) the 
-statistic CW pipeline using lalapps_Weave, as described in Section 4.3; (iii) the 5n-vector method as described in Section 4.2; and (iv) the targeted time-domain Bayesian method (Dupuis & Woan 2005; Pitkin et al. 2017), as also used for the targeted searches in Abbott et al. (2021e). In (i), we use the same implementation of the transient 
-statistic as in the main search. But for (ii)–(iv), we treat the candidates as putative CW signals of fixed length, fixing the start time t0 and end time t0 + τ of these analyses based on the outlier parameters from Table 5.
The PyFstat package (Ashton et al. 2021; Keitel et al. 2021) contains an implementation of Markov Chain Monte Carlo (MCMC) gridless follow-up (Ashton & Prix 2018; Tenorio et al. 2021b) using the ptemcee sampler (Foreman-Mackey et al. 2013; Vousden et al. 2015) and the same 
-statistic code in LALSuite (Wette 2020; LIGO Scientific Collaboration 2021). We do not employ this here as a veto or for significance estimation, but only to check that the candidates can be recovered independently of the original search grid setup, when putting Gaussian priors on the λ parameters around their previously recovered values, and now also allowing variations in sky position. These PyFstat analyses recover consistent 
values and locations.
The lalapps_Weave pipeline also uses the same underlying 
-statistic implementation. The main difference is that this method does not search over different signal start times and durations; instead, the start time t0 and end time t0 + τ are fixed, and it only searches over the λ parameters. As expected, this follow-up recovers 
-statistic peaks consistent with the outlier parameters.
With the 5n-vector method and a similar follow-up setup as for lalapps_Weave, we again recover local peaks at the frequency of each outlier. But over the given data span, this follow-up also recovers other peaks with higher S-statistic values across the probed 0.45 Hz bands and spin-down ranges. Thus, it does not confirm the outliers as candidates of interest. For outlier A, there are 15 peaks with higher detection statistic (corresponding to 0.36% of the top level candidates in every 10−4 Hz sub-band and marginalized over the spin-down region). For outlier B, there are five peaks with higher detection statistics (0.15% of the top level candidates in every 10−4 Hz sub-band and spin-down region).
Finally, the targeted time-domain Bayesian method fixes the signal duration as well as the λ parameters, and performs nested sampling over the four amplitude parameters 
. For both outliers, this returned Bayes factors of about 500 in favor of a coherent signal in all three detectors, as opposed to noise or incoherent signals in the individual detectors. However, this fully targeted analysis does not provide any additional assessment of statistical significance. In addition, the estimated strain amplitudes h0 from this follow-up, 
and 
at 95% credible intervals, are a factor of 10 above the indirect glitch energy UL from Equation (12) assuming a signal from PSR J0537–6910 and with the NICER-observed glitch size.