Searches for Continuous Gravitational Waves from Young Supernova Remnants in the Early Third Observing Run of Advanced LIGO and Virgo

The BSD-directed search pipeline is a hierarchical semicoherent method based on the FH transform (Antonucci et al. 2008; Astone et al. 2014b). A previous search using the BSD-directed search pipeline, pointing to the Galactic center in Advanced LIGO O2, was reported in Piccinni et al. (2020). The pipeline descibed in this section is based on the BSD framework, i.e., a library of functions that allows the user to freely select a subset of the detector strain data (in both frequency and time domain), starting from a collection of basic files (BSD files) in a special data format. All the properties of the framework are described in Piccinni et al. (2018), and here we only remind the reader that the standard format of the BSD files, containing an opportunely down-sampled complex time series, covers a 10 Hz frequency band and ∼1 month of data. For the purpose of this search, where the actual signal frequency is unknown, each BSD file is partially corrected for the Doppler modulation in each 1 Hz frequency subband using its central frequency (see Piccinni et al. 2020 for more details). From this partially corrected time series, a collection of time–frequency peaks (called "peakmaps") is obtained, by choosing all the local maxima above a given threshold from equalized spectra (Astone et al. 2005). The equalization is given by the square modulus of the periodogram divided by the average spectrum. In this way also narrow peaks are kept. This peakmap is the input of the FH transform, which maps each time–frequency peak into the intrinsic source frequency and spin-down $({f}_{0},{\dot{f}}_{0})$ plane at a given reference time. The resolution of a single FH map is the size of the bins in the template grid

$$\begin{eqnarray}&&\delta {f}_{\mathrm{FH}}=\displaystyle \frac{1}{{T}_{\mathrm{coh}}{K}_{f}},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\delta {\dot{f}}_{\mathrm{FH}}=\displaystyle \frac{1}{{T}_{\mathrm{coh}}{T}_{\mathrm{obs}}{K}_{\dot{f}}},\end{eqnarray} \tag{ 2 }$$

where T_coh is the coherence time, while T_obs is the observational time. The parameters K_f and ${K}_{\dot{f}}$ are the overresolution factors as described in Astone et al. (2014b), here chosen as K_f = 10 and ${K}_{\dot{f}}=2$. The coherence time T_coh scales with the maximum frequency of the band as $1/\sqrt{{f}_{\max }}$, and hence the frequency and spin-down bin sizes in Equations (1) and (2) change for each 10 Hz band. For a source with age t_age, the spin-down range is defined as $-{f}_{\max }/{t}_{\mathrm{age}}\leqslant \dot{f}\leqslant 0.1{f}_{\max }/{t}_{\mathrm{age}}$, where f_max is the maximum frequency in each 10 Hz band. In this analysis, the age of the source affects the parameter space investigated, with a wider spin-down range covered when the source is younger. When possible, we use the youngest age estimate available in the SNRcat catalog (Ferrand & Safi-Harb 2012; SNR 2020). On the other hand, according to the age of the source, we can consider the effects of the second-order spin-down as negligible or not (a discussion is reported in Appendix B.1). In this search, we investigate a frequency band of [10, 600] Hz for targets with assumed t_age ≤ 3 kyr and a wider range of [10, 1000] Hz for older sources. We remind the reader of the subtle difference when talking about the source age estimates (which is most of the time inferred from the SNR age) and the characteristic age of the star (which is unknown because they have no observed electromagnetic pulsations). The maximum coherence time used is 17.8 hr for the frequency band [10, 20] Hz and a minimum of 2.5 hr for [990, 1000] Hz. We search both positive and negative $\dot{f}$ to allow for the possibility of unexpected spin-up. A summary of the parameter space investigated for each source is shown in Table 2.

Table 2. Sources Searched in the BSD Analysis (Section 4.1) and the Parameter Space Covered

Source	Minimum tage (kyr)	Tcoh (hr)	f (Hz)	$\dot{f}$ (Hz s−1)
		(@100 Hz)		(@100 Hz)
G65.7+1.2, G189.1+3.0, G266.2–1.2	3	8	(10, 600)	(−1.06 × 10−9, 1.06 × 10−10)
G353.6–0.7	27	8	(10, 1000)	(−1.17 × 10−10, 1.17 × 10−11)
G18.9–1.1	4.4	8	(10, 1000)	(−7.13 × 10−10, 7.13 × 10−11)
G39.2–0.3	4.7	8	(10, 1000)	(−6.75 × 10−10, 6.75 × 10−11)
G93.3+6.9	5	8	(10, 1000)	(−6.34 × 10−10, 6.34 × 10−11)

Note. The coherence time and the spin-down/up range scale with the maximum frequency in each 10 Hz frequency band. For each source, we report the T_coh and spin-down/up range used for the frequency band [90, 100] Hz where f_max = 100 Hz.

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The first set of candidates is selected from a final FH map, which is the sum of all the single monthly based FH maps spanning the same frequency and spin-down ranges. These candidates are independently selected in each detector, including Virgo, using the ranking procedure of Astone et al. (2014b), where candidates with the highest FH number count are kept. At a later stage, coincidences are calculated between the candidate sets from the two LIGO detectors using a coincidence distance defined as

$$\begin{eqnarray}&&d=\sqrt{{\left(\displaystyle \frac{{\rm{\Delta }}f}{\delta {f}_{\mathrm{FH}}}\right)}^{2}+{\left(\displaystyle \frac{{\rm{\Delta }}\dot{f}}{\delta {\dot{f}}_{\mathrm{FH}}}\right)}^{2}},\end{eqnarray} \tag{ 3 }$$

where Δf and ${\rm{\Delta }}\dot{f}$ are the differences between the candidate parameters in each data set. A candidate is then selected when the coincidence distance is below a given threshold distance, d_thr, in this search chosen equal to 4. The choice of the window size has been widely discussed in Astone et al. (2014b), using injected simulated signals.

The coincidence step has been applied first to the pair of LIGO candidates. At a later stage, the same coincidence criterion has been applied between the HL coincident candidates and the most significant Virgo candidates. Candidates found in triple coincidence were discarded after applying the postprocessing methods described in Appendix A. However, we cannot conclude with certainty that a pair of LIGO candidates are nonastrophysical if they have d < d_thr but are not seen in Virgo data, because Virgo is less sensitive than LIGO. For this reason we also postprocessed all the candidates found in coincidence between H and L only.

Surviving candidates are further investigated through a follow-up process described in Appendix A. Also, we apply a threshold to the Critical Ratio (CR) ρ_CR, which measures the statistical significance of a candidate based on the number count associated with the pixel of the FH map where the candidate lies. The threshold ρ_CR,thr is chosen as the mean ρ_CR plus one standard deviation of the CR distribution across the candidates excluding those due to known instrumental lines (Appendix A.1.1) and with an inconsistent significance among the two detectors (Appendix A.1.2). For the targets G65.7+1.2, G189.1+3.0, and G266.2–1.2, we use ρ_CR,thr = 4.7; for G18.9–1.1 and G93.3+6.9, we use ρ_CR,thr = 4.6; and for G353.6–0.7 and G39.2–0.3, we use ρ_CR,thr = 4.5. The threshold chosen here is less stringent than in Piccinni et al. (2020), where the threshold was ≈6.5, corresponding to the probability of picking an average of one false candidate over the total number of points in the parameter space, under the assumption of Gaussian noise. For this work, a lower CR threshold is picked since we are using some new postprocessing methods, described in Appendix A, which allow us to follow up a higher number of candidates, given the low computational cost of each step.

An HMM is an efficient search algorithm capable of handling both spin-down and spin wandering. Previous searches for young SNRs using an HMM (Sun et al. 2018) were conducted in the Advanced LIGO O2 data, but no evidence for a GW signal was reported (Millhouse et al. 2020).

An HMM models a time-varying signal with underlying hidden (i.e., unobservable) parameters by treating the hidden parameters as links in a Markov chain, with each hidden parameter linked to an observable through a likelihood statistic. Given an observed sequence, the goal is to infer the most probable hidden sequence. For a set of N_T observations at discrete times $\{{t}_{0},{t}_{1},...,{t}_{{N}_{T}-1}\}$, the corresponding discrete states $\{q({t}_{0}),q({t}_{1}),...,q({t}_{{N}_{T}-1})\}$ (chosen from N_Q possible hidden states $\{{q}_{1},...,{q}_{{N}_{Q}}\}$) form a Markov chain with transition probabilities from t_k to t_k+1 defined by ${A}_{{q}_{i}{q}_{j}}=P[q({t}_{k+1})={q}_{j}| q({t}_{k})={q}_{i}]$. For this search, we choose ${A}_{{q}_{i}{q}_{i}}={A}_{{q}_{i\pm 1}{q}_{i}}=1/3$ and all other ${A}_{{q}_{i}{q}_{j}}=0$, allowing the frequency to remain static or wander up or down one bin for each time step. This allows us to track both spin-down and stochastic spin wandering, which may cause spin-up. Strictly speaking, spin-down is expected to be more rapid than spin-up due to spin wandering, but the exact values of ${A}_{{q}_{i}{q}_{j}}$ have minimal effect on the performance of an HMM, provided they capture the behavior of the signal in a broad sense (Quinn & Hannan 2001; Suvorova et al. 2016). We assume a uniform prior over the initial state, i.e., ${\rm{\Pi }}[q({t}_{0})]={N}_{Q}^{-1}$. The observations are denoted $\{o({t}_{0}),o({t}_{1}),...,o({t}_{{N}_{T}-1})\}$ and are connected to q(t_k ) through unknown parameters. We call the probability of observing o(t_k ) given some state q(t_k ) the emission probability ${L}_{o({t}_{k})q({t}_{k})}=P[o({t}_{k})| q({t}_{k})]$. Given some observed sequence O, we can then infer the most likely hidden sequence Q* by maximizing

$$\begin{eqnarray}&&P({Q}^{* }| O)={\rm{\Pi }}[q({t}_{0})]\prod _{k=1}^{{N}_{T}-1}{L}_{o({t}_{k})q({t}_{k})}{A}_{q({t}_{k})q({t}_{k-1})}.\end{eqnarray} \tag{ 4 }$$

The Viterbi algorithm is an efficient implementation of the inference step, using dynamic programming to sample and discard unfavorable paths at each time step (Viterbi 1967; Suvorova et al. 2016).

For our purposes, the hidden state is the true GW frequency and the observable is the value of the ${ \mathcal F }$-statistic, calculated coherently over a block of duration T_coh and width (in the frequency domain) ${\left(2{T}_{\mathrm{coh}}\right)}^{-1}$. The ${ \mathcal F }$-statistic is a maximum likelihood filter for a CW signal of frequency f with time derivatives $\dot{f}$, $\ddot{f}$, etc. (for more details on the ${ \mathcal F }$-statistic, please see Jaranowski et al. 1998). In this search, we compute the ${ \mathcal F }$-statistic as a function of f only and account for spin-down by choosing ${T}_{\mathrm{coh}}\propto {\left|{\dot{f}}_{0}^{\max }\right|}^{-1/2}$ (as in Sun et al. 2018), where ${\dot{f}}_{0}^{\max }$ is the maximum $\dot{f}$ within T_coh, such that the signal should wander by at most one frequency bin per time step.

We choose our parameter space according to the detectability of a potential signal. First, we estimate the maximum expected GW strain for a neutron star at distance D with characteristic age t_age and a principle moment of inertia I_zz using

$$\begin{eqnarray}\begin{array}{rcl}{h}_{0}^{\mathrm{age}} & = & 2.27\times {10}^{-24}\left(\displaystyle \frac{1\,\mathrm{kpc}}{D}\right){\left(\displaystyle \frac{1\,\mathrm{kyr}}{{t}_{\mathrm{age}}}\right)}^{1/2}\\ & & \times {\left(\displaystyle \frac{{I}_{\mathrm{zz}}}{{10}^{38}\,\mathrm{kg}\,{{\rm{m}}}^{2}}\right)}^{1/2}\end{array}\end{eqnarray} \tag{ 5 }$$

and assuming purely gravitational spin-down (Wette et al. 2008). We also estimate the minimum detectable strain using an analytic estimate of the 95% confidence sensitivity for a semicoherent search, given by (Wette et al. 2008; Sun et al. 2018)

$$\begin{eqnarray}&&{h}_{0}^{\mathrm{est}}={\rm{\Theta }}{S}_{n}{\left(f\right)}^{1/2}{\left({T}_{\mathrm{obs}}{T}_{\mathrm{coh}}\right)}^{-1/4},\end{eqnarray} \tag{ 6 }$$

where S_n (f) is the noise amplitude spectral density. The statistical threshold Θ is defined by the location in parameter space and typically lies in the range 30≲ Θ ≲ 40. Following previous studies for CWs with an HMM, we take Θ = 35 (Wette et al. 2008; Sun et al. 2018). The frequency range for each source is defined by ${h}_{0}^{\mathrm{est}}\lt {h}_{0}^{\mathrm{age}}$. The parameter space for each source, including T_coh, is summarized in Table 3, and the process for defining the parameter space is described in Appendix B.2.

Table 3. Sources Searched in the Single-harmonic Viterbi Analysis (Section 4.2) and the Parameter Space Covered

Source	Minimum tage (kyr)	D (kpc)	Tcoh (hr)	f (Hz)	$\dot{f}$ (Hz s−1)
G1.9+0.3	0.10	8.5	1.0	(31.56, 121.7 )	$\left(-3.858\times {10}^{-8},3.858\times {10}^{-8}\right)$
G15.9+0.2	0.54	8.5	1.0	(44.03, 657.1)	$\left(-3.858\times {10}^{-8},3.858\times {10}^{-8}\right)$
G18.9–1.1	4.4	2	1.9	(31.02, 1511 )	$\left(-1.507\times {10}^{-8},1.507\times {10}^{-8}\right)$
G39.2–0.3	3.0	6.2	2.8	(62.02, 459.2)	$\left(-1.968\times {10}^{-8},1.968\times {10}^{-8}\right)$
G65.7+1.2	20	1.5	4.7	(35.10, 1128 )	$\left(-3.149\times {10}^{-9},3.149\times {10}^{-9}\right)$
G93.3+6.9	5.0	1.7	1.9	(30.00, 1668 )	$\left(-1.335\times {10}^{-8},1.335\times {10}^{-8}\right)$
G111.7–2.1	0.30	3.3	1.0	(25.71, 365.1)	$\left(-3.858\times {10}^{-8},3.858\times {10}^{-8}\right)$
G189.1+3.0	3.0	1.5	1.4	(26.13, 2000 )	$\left(-1.968\times {10}^{-8},1.968\times {10}^{-8}\right)$
G266.2–1.2	0.69	0.2	1.0	(18.36, 839.6)	$\left(-3.858\times {10}^{-8},3.858\times {10}^{-8}\right)$
G291.0–0.1	1.2	3.5	1.0	(31.97, 1460 )	$\left(-3.858\times {10}^{-8},3.858\times {10}^{-8}\right)$
G330.2+1.0	1.0	5	1.1	(36.57, 1039 )	$\left(-3.858\times {10}^{-8},3.858\times {10}^{-8}\right)$
G347.3–0.5	1.6	0.9	1.0	(21.74, 1947 )	$\left(-3.858\times {10}^{-8},3.858\times {10}^{-8}\right)$
G350.1–0.3	0.60	4.5	1.0	(31.96, 730.1)	$\left(-3.858\times {10}^{-8},3.858\times {10}^{-8}\right)$
G353.6–0.7	27	3.2	10	(77.86, 318.3)	$\left(-2.295\times {10}^{-9},2.295\times {10}^{-9}\right)$
G354.4+0.0	0.10	5	1.0	(25.72, 121.7)	$\left(-3.858\times {10}^{-8},3.858\times {10}^{-8}\right)$

Note. The parameter space for each of the 15 sources is derived using the age and distance estimates in the second and third columns.

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We split the data into N_band frequency subbands of width 2 Hz to ensure that loud, non-Gaussian noise artifacts (e.g., lines) are confined to one subband and do not affect the whole analysis. We overlap the frequency subbands by 0.57 Hz, ensuring that any signal corresponding to a rapidly spinning down neutron star can always be contained in a single subband.

For each subband, we apply the Viterbi algorithm outlined above and obtain N_Q frequency paths ending in N_Q different bins with associated likelihoods ${ \mathcal L }$. Alternative implementations of Viterbi (including Suvorova et al. 2016; Sun et al. 2018) used a Viterbi score as their detection statistic (see Section 4.3). This statistic generally requires N_T ≪ N_Q . Millhouse et al. (2020) demonstrated that this statistic fails to identify an injected (or real) path for N_T ∼ N_Q because the score is calculated for the optimal path relative to other paths in the band. If most of the paths overlap, the optimal path is similar to other paths in the band. In this search, we have a minimum T_coh = 1 hr (N_T = 4391, N_Q = 14,400), which is sufficient for almost one-third of Viterbi paths to converge over T_obs and consequently lower than the sensitivity of the Viterbi score. To maintain the search sensitivity with N_T ∼ N_Q , we use the log-likelihood ${ \mathcal L }$ as our detection statistic. Using the process outlined in Appendix B.2, we determine the 1% false-alarm threshold for each source and denote the corresponding likelihood ${{ \mathcal L }}_{\mathrm{th}}$. We follow up all unique frequency paths with ${ \mathcal L }\gt {{ \mathcal L }}_{\mathrm{th}}$ using the procedure described in Appendix A and find no CW candidates that cannot be described by nonastrophysical noise.

Methods in Sections 4.1 and 4.2 assume that the star rotates about one of its principal axes of the moment of inertia, and hence the GWs are emitted at 2f_⋆. This assumption is based on the fact that the phenomenon of free precession is not clearly observed in the population of known pulsars (Jones 2010). However, the superfluid interior of a star pinned to the crust along an axis nonaligned with any of its principal axes could allow the star to emit GWs at both f_⋆ and 2f_⋆, even without free precession (Jones 2010; Bejger & Królak 2014; Melatos et al. 2015). The dual-harmonic emission mechanism motivates searches combining the two frequency components of a signal to improve signal-to-noise ratio. The HMM tracking scheme described in Section 4.2 has been extended to track two frequency components simultaneously (Sun et al. 2019). The signal model considered in this section consists of both f_⋆ and 2f_⋆ components, given by (Jaranowski et al. 1998; Sun et al. 2019)

$$\begin{eqnarray}&&{h}_{2+}=\displaystyle \frac{1}{2}{h}_{0}(1+{\cos }^{2}\iota ){\sin }^{2}\theta \cos 2{\rm{\Phi }},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{h}_{2\times }={h}_{0}\cos \iota {\sin }^{2}\theta \sin 2{\rm{\Phi }},\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{h}_{1+}=\displaystyle \frac{1}{8}{h}_{0}\sin 2\iota \sin 2\theta \sin {\rm{\Phi }},\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{h}_{1\times }=\displaystyle \frac{1}{4}{h}_{0}\sin \iota \sin 2\theta \cos {\rm{\Phi }},\end{eqnarray} \tag{ 10 }$$

where ι is the inclination angle of the source, θ is the wobble angle between the star's rotation axis and its principal axis of the moment of inertia, and Φ is the GW signal phase observed at the detector. In general, when precession and triaxiality of the star are included, emission occurs at other frequencies too (Zimmermann & Szedenits 1979; Van Den Broeck 2005; Lasky & Melatos 2013).

In this analysis, the HMM formulation generally follows the description in Section 4.2, with three major updates. First, two different coherent times of T_coh = 12 and 9 hr are selected for three sources with t_age ≳ 20 kyr and four sources with t_age ≲5 kyr, respectively. Second, two frequency components are tracked simultaneously. The GW signal for each frequency component is assumed to be monochromatic over T_coh. The signal power in each frequency bin is computed by the two-component ${ \mathcal F }$-statistic, denoted by ${{ \mathcal F }}_{1}({f}_{i})+{{ \mathcal F }}_{2}(2{f}_{i})$, where ${{ \mathcal F }}_{1}$ and ${{ \mathcal F }}_{2}$ are the ${ \mathcal F }$-statistic outputs computed in two separate frequency bands, and f_i is the frequency value in the ith bin. We use Δf = 1/(4T_coh) and 2Δf = 1/(2T_coh) as frequency bin sizes when computing ${{ \mathcal F }}_{1}$ and ${{ \mathcal F }}_{2}$, respectively, such that both the f_⋆ and 2f_⋆ signal components stay in one bin for each time interval T_coh. Third, we assume that the signal frequency evolution is dominated by secular spin-down and can be approximated by a negatively biased random walk. The unknown spin-down rate lies in the range between zero and the maximum estimated spin-down rate and can vary over time. Hence, we use a transition probability matrix ${A}_{{q}_{i-1}{q}_{i}}={A}_{{q}_{i}{q}_{i}}=1/2$, with all other entries being zero. The full frequency band is divided into 1 Hz and 1.5 Hz subbands for T_coh = 12 hr and T_coh = 9 hr, respectively, to parallelize computing. The detection statistic used in this analysis requires that the number of frequency bins in each subband (with bandwidth B) is significantly larger than the total number of tracking steps (i.e., 2BT_coh ≫ T_obs/T_coh). Thus, for T_coh = 9 hr, we choose a 0.5 Hz wider subband such that the requirement is satisfied. More details are provided in Appendix B.3.

Seven sources in the top half of Table 1 with an assumed age of t_age ≳ 3 kyr are searched using this method. Due to the fact that two frequency bands are combined, this method is susceptible to noise features present in either band. Coherent times shorter than ∼5 hr and, correspondingly, wider Δf can further degrade the sensitivity. Hence, we do not search the other eight sources with t_age ≲ 3 kyr that require a much shorter T_coh. The parameter space covered for each source is listed in Table 4. The ${\dot{f}}_{\star }$ range covered in this analysis is hence $| {\dot{f}}_{\star }| \in [0,1/(4{T}_{\mathrm{coh}}^{2})]$. The frequency range is determined as follows. For all seven sources, we fix the minimum frequency at 50 and 100 Hz for f_⋆ and 2f_⋆, respectively. We do not search below 50 Hz because the number of instrumental lines in each 1 Hz band significantly increases at low frequencies and the optimal Viterbi paths would be dominated by noise artifacts. The maximum frequency is set by the assumed minimum characteristic age of the source, t_age (the second column in Table 4), assuming $| {\dot{f}}_{\star }| ={f}_{\star }{\left(n-1\right)}^{-1}{t}_{\mathrm{age}}^{-1}$ (Sun et al. 2018; Abbott et al. 2019e), where $n={f}_{\star }{\ddot{f}}_{\star }/{\dot{f}}_{\star }^{2}$ is the braking index, with ${\ddot{f}}_{\star }$ being the second time derivative of f_⋆. We assume that the spin-down of the star is dominated by gravitational radiation due to a nonzero ellipticity, i.e., n = 5.

Table 4. Sources Searched in the Dual-harmonic Viterbi Analysis (Section 4.3) and the Parameter Space Covered

Source	Minimum tage (kyr)	Tcoh (hr)	f⋆ (Hz)	${\dot{f}}_{\star }$ (Hz s−1)
G65.7+1.2	20	12	(50, 338)	(− 1.34 × 10−10, 0)
G189.1+3.0	20	12	(50, 338)	(− 1.34 × 10−10, 0)
G353.6–0.7	27	12	(50, 457)	(− 1.34 × 10−10, 0)
G18.9–1.1	4.4	9	(50, 132)	(− 2.38 × 10−10, 0)
G39.2–0.3	3	9	(50, 90)	(− 2.38 × 10−10, 0)
G93.3+6.9	5	9	(50, 150)	(− 2.38 × 10−10, 0)
G266.2–1.2	5.1	9	(50, 153)	(− 2.38 × 10−10, 0)

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We use the Viterbi score S as the detection statistic in the dual-harmonic search, which indicates the significance of the optimal Viterbi path obtained in each subband compared to all other paths in that band at the final step of the tracking. Given that the condition N_T ≪ N_Q is generally satisfied with the choices of T_coh in this method, the issue described in Section 4.2 with short T_coh ∼ 1 hr does not happen. The full mathematical definition of S is given in Sun et al. (2019). We determine a threshold corresponding to 1% false-alarm probability S_th = 5.47 and S_th = 5.33 for T_coh = 12 hr and T_coh = 9 hr, respectively, obtained from Monte Carlo simulations in Gaussian noise and verified in real O3a data. The results obtained from simulations in O3a interferometric noise are consistent with the Gaussian noise thresholds.

Surviving candidates are all compatible with noise fluctuations, and no evidence of their presence is found in Virgo O3a and/or in the full LIGO O3 data. We compute the constraints on the strain amplitude using a well-established method used in Piccinni et al. (2020) and described in Dreissigacker et al. (2018). The sensitivity curve is obtained from the 95% confidence level upper limits of 10 randomly selected frequency subbands of 1 Hz each for targets in the [10, 1000] Hz frequency band, and nine subbands for the remaining targets. The ${h}_{0}^{95 \% }$ in the subbands is computed with the frequentist approach, i.e., injecting 50 signals with a given amplitude h₀ and computing the corresponding detection efficiency. The injections are done for each source, assuming the same sky position as the selected source for each injection. The spin-down and polarization parameters ($\cos \iota$ and ψ) are randomly chosen from their uniform distributions. We repeat the injections in a given subband using 618 values of h₀ in the interval [1.3 × 10⁻²⁶, 3 × 10⁻²³]. The detection efficiency for a given amplitude h₀ is given by the fraction of injections recovered. The actual ${h}_{0}^{95 \% }$ corresponding to a detection efficiency of 0.95 is derived from the sigmoidal fit of the detection efficiency curve versus the injected amplitude.

Given that the sensitivity to h₀ is proportional to $\sqrt{{S}_{n}(f)}$, which is the noise amplitude spectral density, we compute the Normalized Upper Limit (NUL), ${h}_{\mathrm{NUL}}({f}_{i})={h}_{0}^{95 \% }({f}_{i})/\sqrt{{S}_{n}({f}_{i})}$, in each of the randomly chosen subbands. We remark that it is the inverse of the more widely used "sensitivity depth" (Behnke et al. 2015). Since the NUL values should follow a linear trend, given by the dependence of the coherence time used in each 10 Hz band, we extrapolate the NUL values of the remaining bands with a linear fit of the NUL versus frequency. In this way we can translate the NUL values, interpolated from the linear fit for each 1 Hz band, into the ${h}_{0}^{95 \% }(f)$ curve. The final ${h}_{0}^{95 \% }(f)$ curve is then obtained for each detector, by multiplying the NUL values extrapolated from the linear fit in each 1 Hz band by the corresponding value of $\sqrt{{S}_{n}(f)}$ in that band, i.e.,

$$\begin{eqnarray}&&{h}_{0}^{95 \% }(f)={h}_{\mathrm{NUL}}(f)\sqrt{{S}_{n}(f)}.\end{eqnarray} \tag{ 11 }$$

The sensitivity plots are presented in Figure 1, where we also report the indirect age-based limit from Equation (5) (solid line) for each target. The best sensitivity is below the indirect age-based limit for all the sources. In particular for G65.7+1.2, G189.1+3.0, and G266.21.2/Vela Jr., this happens for the full frequency band analyzed, except for the most disturbed regions, and for all the detectors. The difference in sensitivity among the analyzed targets is caused by the different antenna pattern response due to different sky locations of the sources, even when the same coherence time is used for multiple sources. We present different curves for each detector; the combined ${h}_{0}^{95 \% }(f)$ result would correspond to the one for the less sensitive LIGO detector. The best sensitivity at 95% confidence level occurs at the Livingston detector at h₀ ≈ 7.8 ×10⁻²⁶ near 200 Hz for G65.7+1.2 and at h₀ ≈ 7.7 × 10⁻²⁶ for G39.2–0.3 in the same bucket region.

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We report no evidence of CWs in the single-harmonic Viterbi search. In this section, we estimate the sensitivity of this search across 9 of the 15 sources. We estimate the sensitivity first using Equation (6) and assume that this is a reasonable representation of the key parameters determining the sensitivity, i.e., that between sources the sensitivity of the search is predominantly determined by T_coh. So we determine the sensitivity for T_coh = 1 hr using G266.2–1.2 and G347.3–0.5 and assume that the variation in sky position for other targets with the same T_coh has a negligible effect on sensitivity. This assumption has been validated through detailed simulations. For each source we set limits on, we inject 100 simulated signals with fixed h₀ and randomly select f and ${\dot{f}}_{0}$ into five frequency subbands, selected at random from a set of bands with no known lines, and which returned <2 unique paths with ${ \mathcal L }\gt {{ \mathcal L }}_{\mathrm{th}}$ in the original search. We then apply the Viterbi algorithm to each injection. We repeat this for 5–10 values of h₀. Each set of N_I = 100 injections forms a binomial distribution, with each injection and search acting as a Bernoulli trial with a probability of success (efficiency) p. We infer the value of p given s successes for each h₀ given using the Wilson interval (Wilson 1927)

$$\begin{eqnarray}\begin{array}{rcl}p & \approx & \displaystyle \frac{s+\tfrac{1}{2}{\left(1-{\alpha }_{F}/2\right)}^{2}}{{N}_{I}+{\left(1-{\alpha }_{F}/2\right)}^{2}}\\ & & \pm \displaystyle \frac{1-{\alpha }_{F}/2}{{N}_{I}+{\left(1-{\alpha }_{F}/2\right)}^{2}}\sqrt{\displaystyle \frac{s({N}_{I}-s)}{{N}_{I}}+\displaystyle \frac{{\left(1-{\alpha }_{F}/2\right)}^{2}}{4}},\end{array}\end{eqnarray} \tag{ 12 }$$

where α_F is the false-alarm probability. For each frequency band, we fit a sigmoid curve (as in Banagiri et al. 2019) to the set of h₀ and the corresponding p using the Bayesian inference package Bilby (Ashton et al. 2019) with a uniform prior over the sigmoid parameters. We sample the posterior and, for each sample, determine the ${h}_{0}^{95 \% }$ as the h₀ corresponding to p = 95%. We take the average ${h}_{0}^{95 \% }$ of this population to be the 95% frequentist confidence upper limit in that frequency band. For each frequency band, we calculate $a={h}_{0}^{95 \% }/{h}_{0}^{\mathrm{est}}$ at the appropriate frequency, where ${h}_{0}^{\mathrm{est}}$ is estimated by Equation (6). Lastly, we find the mean a across the five frequency bands and calculate the sensitivity across the full frequency band as ${h}_{0}^{95 \% }={{ah}}_{0}^{\mathrm{est}}$, plotted as the curves in Figure 2. We overplot the age-based limit from Equation (5) (dashed line) for each target. Our search is more sensitive than the age-based limit for all targets except G18.9-1.1, G39.2-0.3, G330.2+1.0, and G353.6-0.7, despite G353.6-0.7 having the smallest detectable strain in this search, 2.64 × 10⁻²⁵ at 172 Hz. The targets with the poorest overall sensitivity (those with short T_coh) place the tightest constraints relative to the age-based spin-down limit.

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The constraints obtained in this search are for a random-walk signal model including spin-down and spin wandering. The random-walk signal model (including spin-down and spin wandering) and the range of ${\dot{f}}_{0}$ searched (up to ${\dot{f}}_{0}^{\max }=3.9\times {10}^{-8}$ Hz s⁻¹ for T_coh = 1 hr) mean that the ${h}_{0}^{95 \% }$ for this search is less stringent than for the other pipelines in this and other papers, which use a different signal model (e.g., Taylor expansion) and smaller range of $\dot{f}$. For G65.7+1.2, one of the injections at just over 1000 Hz appears to be on a noise spike despite known noise features being filtered out; however, the scale factor obtained for that band is consistent with the other four bands tested.

No evidence of CWs is found in the dual-harmonic Viterbi search. We empirically derive the sensitivity by estimating the signal strain ${h}_{0}^{95 \% }$ in each frequency subband (as a function of f_⋆), such that a signal with ${h}_{0}\geqslant {h}_{0}^{95 \% }$ can be detected on 95% or more occasions. Since this pipeline considers a signal model with both f_⋆ and 2f_⋆ components, we use f_⋆ instead of the GW frequency to avoid confusion. Note that the sensitivity on the strain h₀ quoted in this pipeline is based on a different signal model from the other two pipelines (see Equations (7)–(10)). Here we assume the special scenario θ = 45° and $\cos \iota =0$, i.e., signals at both f_⋆ and 2f_⋆ are linearly polarized. In this scenario, tracking the two frequency bands simultaneously offers the most significant sensitivity improvement from searching a single band, compared to other choices of θ and $\cos \iota$ (Sun et al. 2019).

Figure 3 shows ${h}_{0}^{95 \% }$ in all subbands and a set of frequentist upper limits obtained through injections in a handful of randomly selected sample subbands (orange crosses). The procedure to produce these results is as follows. First, we derive the frequentist upper limit in one sample subband, starting from 112.5 Hz for f_⋆ and 225 Hz for 2f_⋆. A set of 200 synthetic signals are injected into the O3a data at random sky positions in that subband with a fixed h₀. We use fixed θ = 45° and $\cos \iota =0$. The other source parameters, including f_⋆, ${\dot{f}}_{\star }$, the polarization angle, and the initial phase, are randomly drawn from their uniform distributions. The corresponding detection rate is calculated. This process is repeated with different h₀ values with step size 1 × 10⁻²⁶ and 2 × 10⁻²⁶ in the regions where the detection rate is roughly above and below 50%, respectively. With all the injected h₀ values and the corresponding detection rates, ${h}_{0}^{95 \% }$ is obtained through a sigmoidal fit. The ${h}_{0}^{95 \% }$ value found in the sample subband is 2.9 × 10⁻²⁵ and 3.2 × 10⁻²⁵ for T_coh = 12 and 9 hr, respectively. Next, we use these values obtained in the sample subband (starting from 112.5 Hz for f_⋆ and 225 Hz for 2f_⋆) to analytically calculate ${h}_{0}^{95 \% }$ in the full frequency band (blue circles), using the scaling (Sun et al. 2019)

$$\begin{eqnarray}&&{h}_{0}^{95 \% }(f)\propto {\left(\displaystyle \frac{{S}_{n}(f){S}_{n}(2f)}{{S}_{n}(f)+{S}_{n}(2f)}\right)}^{1/2},\end{eqnarray} \tag{ 13 }$$

where S_n is the effective power spectral density calculated from the harmonic mean of the two detectors over all the 30-minute SFTs collected from 2019 September 1 to October 1 (GPS time 1251331218–1253923218). Finally, in order to verify the analytical scaling, the simulation procedure in the first step is repeated in several other randomly selected subbands, indicated by the orange crosses. The ${h}_{0}^{95 \% }$ values obtained empirically in those sample subbands agree to <1.5% with the analytic sensitivity estimates. In the full frequency band searched, the best ${h}_{0}^{95 \% }$ values for T_coh = 12 and 9 hr are 2.88 × 10⁻²⁵ at f_⋆ =158.75 Hz and 3.17 × 10⁻²⁵ at f_⋆ = 123.75 Hz, respectively. These results are obtained from randomized sky positions and hence apply to all sources using the same T_coh. In the dual-harmonic Viterbi pipeline, the sensitivity is dominated by the length of T_coh (for a fixed T_obs) rather than the sky position of the source. Additional spot checks validate that the difference between the empirical ${h}_{0}^{95 \% }$ values obtained from a fixed sky location and those from randomized sky positions is negligible.

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Assuming that the star's rotational kinetic energy loss is all radiated in GWs, the age-based indirect strain limits ${h}_{0}^{\mathrm{age}}$ can be calculated for each source by fixing θ = 45° (consistent with the scenario presented in Figure 3), setting t_age to the value in Table 4, and setting the distance to the minimum value in Table 1. Note that the ${h}_{0}^{\mathrm{age}}$ value derived explicitly for the dual-harmonic model with θ = 45° is a factor of ∼2 larger than the value calculated from Equation (5) (Zimmermann & Szedenits 1979; Wette et al. 2008). For five out of the seven sources, the indirect limits are much larger than the constraints obtained in this search across the full band. For G39.2–0.3, the ${h}_{0}^{95 \% }$ has beaten the indirect limit at most of the frequencies except for the noisy bands around 60 Hz. For G353–0.7, the ${h}_{0}^{95 \% }$ obtained from the search is close to ${h}_{0}^{\mathrm{age}}$ but has not reached it at any frequency. We emphasize that the sensitivity in the dual-harmonic Viterbi pipeline, and whether it beats the indirect limit, is not directly comparable to other methods owing to the model difference.

The sensitivity in terms of the GW strain amplitude can be converted into constraints on the fiducial ellipticity of the neutron star, (Jaranowski et al. 1998), and the r-mode amplitude parameter, α (Owen 2010). We first discuss the constraints obtained in the BSD and single-harmonic Viterbi pipelines. For ellipticity, we assume that the GW frequency f (equivalent to f₀ in the single-harmonic Viterbi pipeline) is at 2f_⋆, which aligns with the model of a perpendicular biaxial rotor considered in both pipelines. Given ${h}_{0}^{95 \% }$ (derived with a uniform prior on the $\cos \iota$), we constrain the ellipticity of the neutron star in terms of the GW frequency f = 2f_⋆ via (Jaranowski et al. 1998)

$$\begin{eqnarray}&&\epsilon =9.46\times {10}^{-5}\left(\displaystyle \frac{{h}_{0}}{{10}^{-24}}\right)\left(\displaystyle \frac{D}{1\,\mathrm{kpc}}\right){\left(\displaystyle \frac{100\,\mathrm{Hz}}{f}\right)}^{2},\end{eqnarray} \tag{ 14 }$$

assuming that the moment of inertia with respect to the rotation axis (I_zz for a perpendicular biaxial rotor) is 10³⁸ kg m². We can also convert ${h}_{0}^{95 \% }$ to limits on the amplitude of r-mode oscillations via (Owen 2010)

$$\begin{eqnarray}&&\alpha \simeq 0.028\left(\displaystyle \frac{{h}_{0}}{{10}^{-24}}\right)\left(\displaystyle \frac{D}{1\,\mathrm{kpc}}\right){\left(\displaystyle \frac{100\,\mathrm{Hz}}{f}\right)}^{3}.\end{eqnarray} \tag{ 15 }$$

Figures 4 and 5 present the constraints on and α obtained from the BSD and single-harmonic Viterbi pipelines, respectively. The most stringent constraints, ≲ 10⁻⁷ and α ≲ 10⁻⁵, come from the BSD pipeline (see Figure 4), where the values are converted using the ${h}_{0}^{95 \% }$ curve from the L detector. The results from the single-harmonic Viterbi pipeline, covering more targets and a wider parameter space, are presented in Figure 5.

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We also convert the ${h}_{0}^{95 \% }$ obtained in the dual-harmonic Viterbi pipeline to the 95% confidence constraint on the ellipticity of the star. Since GW emission is at both f_⋆ and 2f_⋆, Equation (14) can be written in terms of the spin frequency of the star f_⋆,

$$\begin{eqnarray}&&\epsilon =2.36\times {10}^{-5}\left(\displaystyle \frac{{h}_{0}}{{10}^{-24}}\right)\left(\displaystyle \frac{D}{1\,\mathrm{kpc}}\right){\left(\displaystyle \frac{100\,\mathrm{Hz}}{{f}_{\star }}\right)}^{2}.\end{eqnarray} \tag{ 16 }$$

Figure 6 shows the limits on as a function of f_⋆. Note that the results here are converted from the ${h}_{0}^{95 \% }$ values obtained for a specific scenario with source properties θ = 45° and $\cos \iota =0$, and hence the values in Figure 6 are not directly comparable to the results obtained in other conventional searches where θ = 90° and the emission is only at 2f_⋆. The signal model adopted in the dual-harmonic search cannot be interpreted as current quadrupole emission from an r-mode, so we do not infer r-mode amplitudes from ${h}_{0}^{95 \% }$.

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The strictest constraints on the intrinsic GW strain from the BSD pipeline are ${h}_{0}^{95 \% }\approx 7.7\times {10}^{-26}$ for G39.2–0.3 and ${h}_{0}^{95 \% }\approx 7.8\times {10}^{-26}$ for G65.7+1.2 near 200 Hz. The results obtained by the Viterbi pipelines set the first constraints on CWs that allow for spin wandering in the signal model. Note that the authors of Millhouse et al. (2020) conducted a search for 13 out of the 15 sources in Advanced LIGO O2 data using the single-harmonic Viterbi method but did not derive the constraints from the search sensitivity. Furthermore, the dual-harmonic Viterbi analysis provides the first results for these SNR sources derived considering two frequency harmonics simultaneously. The best constraints on the star's ellipticity obtained at frequencies f ≳ 100 Hz reach < 10⁻⁶ for most of the sources, reaching below the rough theoretical upper limit for normal neutron stars (Johnson-McDaniel & Owen 2013), and reach as low as ≈ 6 × 10⁻⁸ for the closest source G266.2–1.2/Vela Jr., well below the theoretical limits. However, these limits are model dependent; the uncertainties on the star's geometry and composition, like the internal equation of state, the moment of inertia, and the magnetic field, play a significant role when deriving these limits. For example, Woan et al. (2018) show that an ellipticity of ∼ 10⁻⁹ can be sustained by neutron stars with a buried magnetic field of ∼10¹¹ G. The most stringent constraints on the r-mode amplitude obtained above ∼100 Hz arrive at the theoretical prediction level of α ∼ 10⁻³, expected for the nonlinear saturation mechanisms (Bondarescu et al. 2009), and reach as low as α ∼ 10⁻⁵ at higher frequencies.

We first describe the predefined veto procedures in this section.

A.1.1. Known-line Veto

A.1.2. Interferometer Veto

A.1.3. Doppler-shift (and Spin-down-shift) Veto

A.1.4. Sky-position Veto

A.1.5. Cumulative-significance Veto

A.1.6. Subband Veto

A.1.7. Coherence-time Veto

After the hierarchy of well-defined veto steps, we discuss the additional verification conducted in each pipeline.

A.2.1. BSD Follow-up

A total of 35 candidates identified by the BSD pipeline survive the vetoes described in Appendix A.1. This is consistent given the low CR threshold chosen in Section 4.1; indeed, we are exposed to false-alarm candidates, and most of them could arise from noise fluctuations. The ρ_CR,thr chosen corresponds to the probability of picking, on average, more than one noise candidate. Indeed, the CR threshold corresponding to the selection of only one false candidate over the total number of points in the parameter space would be ρ_CR,thr ∼ 5.7, while, for instance, in the search described in Piccinni et al. (2020) the CR threshold used is 6.5. In this section, we describe the extra steps taken to investigate and disqualify the surviving candidates (listed in Table 5). We repeat the Doppler-shift (and spin-down-shift) veto (Appendix A.1.3), the cumulative-significance veto (Appendix A.1.5), and the interferometer veto (Appendix A.1.2) using the full O3 (O3a and O3b) C01 data. We remind the reader that the CR computed by the FH is based on the number count associated with the pixel in the FH map where the candidate has been found, while the CR in the 5-vector is computed from the ${ \mathcal S }$ statistic. In this step, we use the CR from the ${ \mathcal S }$ statistic. None of the candidates in Table 5 survive the full O3 vetos, and as can be seen, the original CR associated with the candidate in the FH map is below the ρ_CR,thr ∼ 5.7, hence with a very low significance compatible with noise.

Table 5. Surviving Candidates from the BSD Pipeline after Vetos in Appendix A.1

Source	Original CR (FH)	f0 (Hz)	${\dot{f}}_{0}$ (Hz s−1)
G189.1+3.0	5.40	65.6458598	−3.6202 × 10−10
	4.80	75.2342077	−7.9666 × 10−10
	5.44	117.6621941	−7.3779 × 10−10
	4.89	321.8914027	−3.3898 × 10−9
	4.96	355.9654409	−1.1674 × 10−9
	4.90	498.1299761	−1.9622 × 10−9
G65.7+1.2	5.56	97.5785468	−8.5222 × 10−10
	4.92	120.7226480	−1.0230 × 10−9
G266.2–1.2	5.07	90.3241483	4.0128 × 10−11
	5.48	101.8200802	−6.1815 × 10−10
	5.18	139.6310303	−1.1902 × 10−10
	5.07	142.8760455	−2.7103 × 10−12
	4.87	217.9480484	2.1858 × 10−10
	5.15	274.5506302	1.1958 × 10−10
G93.3+6.9	5.61	208.8871413	−9.6487 × 10−10
	5.25	758.8185308	−4.4728 × 10−9
	4.83	807.0006637	−2.6116 × 10−9
	4.86	851.3276865	−3.9519 × 10−9
	5.49	858.9144800	−3.9696 × 10−9
G18.9–1.1	5.24	65.0196186	−4.7373 × 10−11
	5.18	91.4561088	−4.2390 × 10−10
	5.78	332.4543731	−3.6398 × 10−10
	4.88	488.0994499	−2.5631 × 10−9
	4.72	841.6011602	−2.8949 × 10−9
	4.95	844.8167574	−4.1613 × 10−9
G39.2–0.3	4.98	135.6997524	−7.5073 × 10−10
	4.77	333.8910938	−2.1219 × 10−9
	4.76	334.0656179	−1.7914 × 10−9
	4.76	700.7408355	−3.9014 × 10−9
	5.12	721.1474692	−2.9249 × 10−9
	5.35	802.0985150	−3.0294 × 10−9
	4.68	831.8034158	−6.6128 × 10−10
	5.31	902.3896162	−4.9226 × 10−9
G353.6–0.7	5.36	64.6267590	−9.1944 × 10−12
	4.99	754.3285519	−4.0769 × 10−10

Note. The candidates investigated were excluded using the full O3 data in the BSD search. The columns list the source name, the original mean CR from the FH map, the candidate frequency f₀, and the spin-down ${\dot{f}}_{0}$ at the time of the coincidences. The initial set of candidates has been selected using ρ_CR,thr = 4.7 for G65.7+1.2, G189.1+3.0, and G266.2–1.2; ρ_CR,thr = 4.6 for G18.9–1.1 and G93.3+6.9; and ρ_CR,thr = 4.5 for G353.6–0.7 and G39.2–0.3.

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