Discovery of a Gamma-Ray Black Widow Pulsar by GPU-accelerated Einstein@Home

The Fermi Large Area Telescope (LAT) source 4FGL J1653.6−0158 is a bright gamma-ray source, and the brightest remaining unassociated source (Saz Parkinson et al. 2016). It was first seen by the Energetic Gamma Ray Experiment Telescope (EGRET; Hartman et al. 1999), and was also listed in the LAT Bright Gamma-ray source list (Abdo et al. 2009) more than a decade ago. While pulsars were discovered in several other sources from this list (see, e.g., Ransom et al. 2011), the origin of 4FGL J1653.6−0158 remained unidentified. The detection of a variable X-ray and optical candidate counterpart with 75 minute period consistent with the gamma-ray position of 4FGL J1653.6−0158 provided strong evidence of it being a binary gamma-ray pulsar (Kong et al. 2014; Romani et al. 2014).

To identify the neutron star in 4FGL J1653.6−0158, we carried out a binary-pulsar search of the gamma-rays, using the powerful GPU-accelerated distributed volunteer computing system Einstein@Home. Such searches are very computationally demanding, and would take decades to centuries on a single computer while still taking weeks or months on Einstein@Home. Thus, the search methods are specifically designed to ensure efficiency (Nieder et al. 2020). One key element is the use of constraints derived from optical observations. The companion's pulsar-facing side is heated by the pulsar wind, leading to a periodically varying optical light curve. This permits the orbital period P_orb and other orbital parameters to be tightly constrained (for a feasible search the uncertainty ΔP_orb needs to be less than a few milliseconds). In addition, because the sky position of the optical source is typically known to high precision (submilliarcsecond level), a search over position parameters is not needed.

Here we present the discovery and analysis of gamma-ray pulsations from PSR J1653−0158 in 4FGL J1653.6−0158. The pulsar is spinning very rapidly, at a rotational frequency of 508 Hz. The inferred surface magnetic-field strength is one of the lowest of all known pulsars. The discovery also confirms the 75 minute orbital period. This very short orbital period raises interesting questions about the evolutionary path which created the system.

This Letter is organized as follows. In Section 2, we describe the gamma-ray search, detection, and analysis within LAT data. The optical analysis of the pulsar's companion, radio pulsation searches, and a continuous gravitational-wave follow-up search are presented in Section 3. We discuss the results and conclude in Section 4.

2.1. Data Preparation

We searched for gamma-ray pulsations in the arrival times of photons observed by the Fermi-LAT (Atwood et al. 2009) between 2008 August 3 and 2018 April 16 (MJDs 54,681 and 58,224). We included SOURCE-class photons according to the P8R2_SOURCE_V6 (Atwood et al. 2012) instrument response functions (IRFs),²⁸ with reconstructed incidence angles within a 5° region of interest (RoI) around the putative pulsar position, energies above 100 MeV, and zenith angles below 90°. Here, we used the presumptive companion's position as reported in the Gaia DR2 Catalog (hereafter Gaia catalog; Gaia Collaboration et al. 2018). The celestial parameters (J2000.0) are α = 16^h53^m3805381(5) and δ = −01°58'368930(5), with 1σ uncertainties on the last digits reported in parentheses.

Using the photon incidence angles and energies, we constructed a probability or weight for each photon, w_j ∈ [0, 1], where j labels the photon: w_j is the probability that the jth photon originated from the posited source, as opposed to a fore- or background source. These weights were computed by gtsrcprob, using the preliminary Fermi-LAT 8 yr source catalog²⁹ as a model for the flux within the RoI without performing a full spectral fit. Weighting the contribution of each photon to a detection statistic in this way greatly increases the search sensitivity (Kerr 2011), and the distribution of weights can be used to predict expected signal-to-noise ratios (Nieder et al. 2020).

The data set used here consisted of N = 354,009 photons, collected over a period of 3542 days. The properties of the detection statistics (semicoherent power S₁, coherent power P₁, and H statistic) depend upon the lowest moments of the weights, which are

$$\begin{eqnarray*}&&\displaystyle \sum _{j=1}^{N}{w}_{j}\approx 10266,\ \displaystyle \sum _{j=1}^{N}{w}_{j}^{2}\approx 2464,\ \mathrm{and}\ \displaystyle \sum _{j=1}^{N}{w}_{j}^{4}\approx 931.\end{eqnarray*}$$

These moments determine the ultimate sensitivity to a particular pulse profile and pulsed fraction, as given in Equation (11) in Nieder et al. (2020).

Following the pulsar discovery, we extended this data set to 2020 February 23 (MJD 58,902), using the latest P8R3_SOURCE_V2 IRFs (Bruel et al. 2018), a larger maximum zenith angle of 105°, and using the Fermi-LAT Fourth Source Catalog (hereafter 4FGL; Abdollahi et al. 2020) as the RoI model for the photon probability weight computations.

2.2. Search

2.3. Timing

The parameters used in the phase model to describe the pulsar's rotation are measured in a timing analysis. We use the timing methods as explained in Clark et al. (2017), which are an extension of the methods by Kerr et al. (2015). The basic principle is that the parameter space around the discovery parameters is explored using a Monte Carlo sampling algorithm with a template pulse profile.

To marginalize over the pulse-profile template, we vary the template parameters as described in Nieder et al. (2019). In the case of PSR J1653−0158, we used a template consisting of two symmetrical, wrapped Gaussian peaks. We used constraints on the peaks' FWHM, such that the peaks must be broader than 5% of a rotation, and narrower than half a rotation.

Our timing solution over 11 yr of LAT data is shown in Table 1. The folded gamma-ray data and the pulse profile are portrayed in Figure 1.

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Table 1.  Timing Solution for PSR J1653−0158

Parameter	Value
Range of observational data (MJD)	54682–58902
Reference epoch (MJD)	56100.0
Celestial Parameters from Gaia Catalog
R.A., α (J2000.0)	16h53m3805381(5)
Decl., δ (J2000.0)	−01°58'368930(5)
Positional epoch (MJD)	57205.875
Proper motion in R.A., ${\mu }_{\alpha }\cos \delta$ (mas yr−1)	−19.62 ± 1.86
Proper motion in decl., μδ (mas yr−1)	−3.74 ± 1.12
Parallaxa, ϖ (mas)	1.88 ± 1.01
Timing Parameters
Spin frequency, f (Hz)	508.21219457426(6)
Spin-frequency derivative, $\dot{f}$ (Hz s−1)	−6.204(8) × 10−16
Spin period, P (ms)	1.9676820247057(2)
Spin-period derivative, $\dot{P}$ (s s−1)	2.402(3) × 10−21
Proj. semimajor axis, x (s)	0.01071(1)
Orbital period, Porb (days)	0.0519447575(4)
Epoch of ascending node, Tasc (MJD)	56513.479171(8)
Derived Parameters for Distance d = 840 pc
Shklovskii spin-down, ${\dot{P}}_{\mathrm{Shk}}$ (s s−1)	1.6 × 10−21
Galactic acceleration spin-down, ${\dot{P}}_{\mathrm{Gal}}$ (s s−1)	−4.8 × 10−23
Spin-down power, $\dot{E}$ (erg s−1)	4.4 × 1033
Surface B-field, Bsurf (G)	4.1 × 107
Light-cylinder B-field, BLC (G)	5.0 × 104
Characteristic age, τc (Gyr)	37
Gamma-ray luminosityb, Lγ (erg s−1)	2.9 × 1033
Gamma-ray efficiency, nγ = ${L}_{\gamma }/\dot{E}$	0.66

Notes. The JPL DE405 solar system ephemeris has been used, and times refer to TDB.

^aCorresponds to a model-independent distance $d={533}_{-187}^{+625}$ pc, but for the derived parameters the consistent distance $d={840}_{-40}^{+40}$ pc derived from optical modeling is used (see Table 2). ^bTaken from 4FGL Source Catalog (Abdollahi et al. 2020).

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The observed spin-down $\dot{P}$ is one of the lowest of all known pulsars. To estimate the intrinsic $\dot{P}$ we account for the Shklovskii effect (Shklovskii 1970), and the Galactic acceleration (see, e.g., Damour & Taylor 1991). The results are summarized in Table 1. The observed contribution due to the difference in Galactic acceleration of the Sun and the pulsar is computed with R_Sun = 8.21 kpc, z_Sun = 14 pc, and the Galactic potential model PJM17_best.Tpot (McMillan 2017), as implemented in their code.³² For PSR J1653−0158, we used R_J1653 = 7.48 kpc, and z_J1653 = 367 pc, assuming d = 840 pc (see Table 2). The contributions parallel and perpendicular to the Galactic disk nearly cancel each other, so that the choice of the potential and its relevant parameters have a seemingly large effect on the actual small value of ${\dot{P}}_{\mathrm{Gal}}$, and can even change the sign. However, the overall kinematic contribution to the observed $\dot{P}$ is dominated by the Shklovskii term, and its uncertainty by the uncertainty in the distance estimate. The estimated intrinsic spin-down is ${\dot{P}}_{\mathrm{int}}=8.5\times {10}^{-22}$ s s⁻¹ for distance d = 840 pc.

Table 2.  Light-curve Fit Results for PSR J1653−0158

Parameters	Veiled	Veiled+HS
Inclination, i (deg)	${79.4}_{-6.8}^{+5.7}$	${72.3}_{-4.9}^{+5.0}$
Filling factor, fc	${0.97}_{-0.02}^{+0.02}$	${0.88}_{-0.03}^{+0.03}$
Heating luminosity, LP (1033 erg s−1)	${3.33}_{-0.34}^{+0.39}$	${3.15}_{-0.27}^{+0.26}$
Night-side temperature, TN (K)	${3250}_{-331}^{+243}$	${3295}_{-300}^{+227}$
V-band extinction, AV	${1.06}_{-0.10}^{+0.08}$	${1.06}_{-0.09}^{+0.07}$
Distance, d (pc)	${830}_{-50}^{+50}$	${840}_{-40}^{+40}$
Veiling flux norm, fA (μJy)	${101.7}_{-11.1}^{+11.4}$	${99.9}_{-11.4}^{+11.7}$
Veiling flux index, p	${0.50}_{-0.03}^{+0.05}$	${0.49}_{-0.03}^{+0.03}$
Spot azimuth, θc (deg)	⋯	${286.8}_{-6.9}^{+5.8}$
Spot co-latitude, ϕc (deg)	⋯	$-{50.5}_{-8.4}^{+9.2}$
Gaussian spot width, σc (deg)	⋯	${25.2}_{-4.9}^{+5.0}$
Spot temperature increase, Ac	⋯	${0.66}_{-0.21}^{+0.21}$
Neutron star mass, MNS (M⊙)	${1.99}_{-0.08}^{+0.18}$	${2.17}_{-0.15}^{+0.21}$
Companion mass, Mc (M⊙)	${0.013}_{-0.001}^{+0.001}$	${0.014}_{-0.001}^{+0.001}$
χ2/DoF	1.72	1.38

Note. Parameters from the best-fit light-curve/radial-velocity models, with and without a surface hot spot, including MCMC errors.

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3.1. Optical Light-curve Modeling and System Masses

By modeling the optical light curves and radial velocities we can constrain the binary mass and distance and the system viewing angle. Comparing the individual filters between nights suggest small δm ≈ 0.05 shifts in zero-points, consistent with the systematic estimates above. Correcting to match the individual filters, we then rebinned the light curve, placing the photometry on a regular grid with points spaced by δϕ = 0.004, using the Python package Lightkurve; after excision of a few obviously discrepant points, we retain 248 u', 239 g', 220 r', and 245 i' points for light-curve fitting (Figure 2). This fitting is done with a version of the Icarus code of Breton et al. (2013) modified to include the effect of hot spots on the companion surface, likely generated by precipitation of particles from the intrabinary shock (IBS) to companion magnetic poles (Sanchez & Romani 2017). All parameter values and errors are determined by Markov Chain Monte Carlo (MCMC) modeling.

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The very shallow modulation of these light curves might normally be interpreted as indicating a small inclination i. However given the large companion radial-velocity amplitude K = 666.9 ± 7.5 km s⁻¹, implying a mass function f(M) = 1.60 ± 0.05 M_⊙, measured by Romani et al. (2014), a small inclination would give an unphysical, large neutron star mass. As noted in that paper, the light curves and spectra show that a strong blue nonthermal veiling flux dominates at orbital minimum. With increasingly shallow modulation for the bluer colors, this is also evident in the present photometry. Thus, the minimal model for this pulsar must include a nonthermal veiling flux. Although this is likely associated with the IBS, we model it here as a simple power law with form f_ν = f_A (ν/10¹⁴ Hz)^−p. This flux is nearly constant through the orbit, although there are hints of phase structure, e.g., in r' and i' at ϕ_B = 0.72 (see Figure 2). Any model without such a power-law component is completely unacceptable. These fits prefer an A_V slightly higher than, but consistent with, the maximum in this direction (obtained by ∼300 pc; Green et al. 2019).³³

In Figure 2, one notices that the orbital maximum is slightly delayed from ϕ_B = 0.75, especially in the bluer colors. Such asymmetric heating is most easily modeled adding a polar hot spot with location (θ_c, ϕ_c) and local temperature increase A_c in a Gaussian pattern of width σ_c; when we include such a component, the fit improves greatly, with Δχ²/DoF = −0.34. The Akaike information criterion comparison of the two models indicates that the model with a hot spot is preferred at the 10⁻¹⁸ level, despite the extra degrees of freedom. We give the fit parameters for both models in Table 2. Note that with the fine structure near maximum, the model is not yet fully acceptable (χ²/DoF ∼ 1.4). More detailed models, including direct emission from the IBS or possibly the effects of companion global winds (Kandel & Romani 2020), may be needed to fully model the light curves. Such modeling would be greatly helped by light curves over an even broader spectral range, with IBS effects increasingly dominant in the UV, and low-temperature companion emission better constrained in the IR. With many cycles we could also assess the reality (and stability) of the apparent fine structure and test for hot-spot motion.

Our fit distance may be cross-checked with two other quantities. (1) With the 4FGL energy flux f_γ = 3.5 × 10⁻¹¹ erg cm⁻² s⁻¹ between 100 MeV and 100 GeV, our fit distance gives an isotropic gamma-ray luminosity L_γ = 3 × 10³³ erg s⁻¹, in good agreement with the L_γ ≈ ${({10}^{33}\mathrm{erg}{{\rm{s}}}^{-1}\dot{E})}^{1/2}$ heuristic luminosity law (Abdo et al. 2013), as a function of the spin-down power $\dot{E}$. This luminosity is consistent with the model for direct radiative heating of the companion. (2) Our fit distance is also consistent with the model-independent, but lower-accuracy, distance from the Gaia parallax. Thus, the 840 pc distance seems reliable, although systematic effects probably dominate over the rather small ∼50 pc statistical errors.

Armed with the fits, we can estimate the companion masses, correcting the observed radial-velocity amplitude (fit with a K-star template) for the temperature-dependent weighting of the absorption lines across the companion face as in Kandel & Romani (2020). The results indicate substantial mass accretion, as expected for these ultrashort-period systems. With the preferred Veiled+HS model the mass significantly exceeds 2.0 M_⊙, adding to the growing list of spider binaries in this mass range. Note that the inclination i uncertainty dominates the error in this mass determination. Broader range photometric studies, with better constraint on the heating pattern, can reduce the i uncertainty.

3.2. Radio Pulsation Searches

3.3. Continuous Gravitational Waves

PSR J1653−0158 is the second binary pulsar (Pletsch et al. 2012) and the fourth MSP (Clark et al. 2018) to be discovered through periodicity searches of gamma-rays. This pulsar is remarkable in many ways. It is only the second rotationally powered MSP from which no radio pulsations have been detected. It is among the fastest-rotating known pulsars with spin frequency f = 508 Hz. The 75 minute orbital period is shorter than for any other known rotation-powered pulsar, with the previous record being PSR J1311−3430 with a 93 minute orbit (Pletsch et al. 2012). The inferred surface magnetic field is possibly the weakest, depending on the Shklovskii correction.

The discovery was enabled by constraints on the sky position and orbital parameters from optical observations, together with efficient search techniques and the large computing power of the distributed volunteer computing system Einstein@Home. The detection proves that the optically variable candidate counterpart (Kong et al. 2014; Romani et al. 2014) is indeed the black-widow-type binary companion to PSR J1653−0158, and it conclusively resolves the nature of the brightest remaining unidentified gamma-ray source, first found more than 2 decades ago (Hartman et al. 1999).

The distance to PSR J1653−0158 and its proper motion are well constrained. Gaia measurements of the parallax, ϖ = 1.88 ± 1.01 mas, imply a distance $d={530}_{-200}^{+470}$ pc. A consistent, but tighter constraint is given by our optical modeling with $d={840}_{-40}^{+40}$ pc. The proper motion (see Table 1) is also measured with good precision (Gaia and our timing are in agreement).

PSR J1653−0158 has one of the lowest observed spin-period derivatives of all known pulsars ($\dot{P}=2.4\,\times {10}^{-21}\,{\rm{s}}\,{{\rm{s}}}^{-1}$). The intrinsic $\dot{P}=8.5\times {10}^{-22}\,{\rm{s}}\,{{\rm{s}}}^{-1}$ (accounting for Galactic acceleration and Shklovskii effects) is even smaller. In Figure 3, PSR J1653−0158 is shown in a P–$\dot{P}$ diagram, alongside the known radio and gamma-ray pulsar population outside of globular clusters.

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The intrinsic $\dot{P}$ can be used to estimate the pulsar's spin-down power $\dot{E}$, surface magnetic-field strength B_surf, magnetic-field strength at the light cylinder B_LC, and characteristic age τ_c. These are given in Table 1 for d = 840 pc. Constant lines of $\dot{E}$, B_surf, and τ_c are displayed in Figure 3 to show the distance-dependent ranges.

Spider pulsars in very-short-period orbits are difficult to discover with traditional radio searches. Even though we can now fold the radio data with the exact parameters, PSR J1653−0158 is still not visible. There are two simple explanations for the nondetection of radio pulsations. (1) Radio emission is blocked by material produced by the pulsar evaporating its companion. Eclipses for large fractions of the orbit would be expected, since they have been seen for many spider pulsars (see, e.g., Fruchter et al. 1988; Archibald et al. 2009; Polzin et al. 2020). This is further supported by the observed extremely compact orbit and the strong IBS. Radio imaging observations could be used to check whether there is any continuum radio flux at the sky position of PSR J1653−0158, but previous experience is not encouraging. The eclipses of a few other spider systems have been imaged at low frequencies, showing that, during the eclipse, the continuum flux from the pulsar disappears in tandem with the pulsed flux (Broderick et al. 2016; Polzin et al. 2018). (2) PSR J1653−0158 is intrinsically radio-quiet, in that its radio beam does not cross the line of sight, or it has a very low luminosity. There is one other radio-quiet MSP known (Clark et al. 2018).

The minimum average density of the companion 64 g cm⁻³ is very high, assuming a filled Roche lobe (Eggleton 1983). Using the filling factor from optical modeling, the average companion density 73 g cm⁻³ is even higher. The high density and the compact orbit suggest that the companion may be a helium white-dwarf remnant, and that the system may have evolved from an ultracompact X-ray binary (Sengar et al. 2017; Kaplan et al. 2018). In addition, simulations predict evolved ultracompact X-ray binaries to have orbital periods of around 70–80 minutes (van Haaften et al. 2012), consistent with the 75 minute orbital period from PSR J1653−0158. Future analysis of optical spectroscopic data may give additional insight into the evolution and composition of the companion.

The discovery of PSR J1653−0158 is the result of a multiwavelength campaign. The pulsar-like gamma-ray spectrum, and the nondetection of radio pulsations, motivated the search for a visible companion. This was subsequently discovered in optical and X-ray observations. Further optical observations provided constraints on the orbital parameters that were precise enough to enable a successful gamma-ray pulsation search.

We are deeply grateful to the thousands of volunteers who donated their computing time to Einstein@Home, and to those whose computers first detected PSR J1653−0158: Yi-Sheng Wu of Taoyuan, Taiwan; and Daniel Scott of Ankeny, Iowa, USA.

This work was supported by the Max-Planck-Gesellschaft (MPG), by the Deutsche Forschungsgemeinschaft (DFG) through an Emmy Noether Research grant, No. PL 710/1-1 (PI: Holger J. Pletsch) and by National Science Foundation grants 1104902 and 1816904. L.N. was supported by an STSM Grant from COST Action CA16214. C.J.C. and R.P.B. acknowledge support from the ERC under the European Union's Horizon 2020 research and innovation program (grant agreement No. 715051; Spiders). V.S.D. and ULTRACAM are supported by the STFC. R.W.R. and D.K. were supported in part by NASA grant 80NSSC17K0024. S.M.R. is a CIFAR Fellow and is supported by the NSF Physics Frontiers Center award 1430284 and the NASA Fermi GO Award NNX16AR55G. Fermi research at NRL is funded by NASA. J.W.T.H. is an NWO Vici fellow.

The ULTRACAM photometry was obtained as part of program WHT/2015A/35. The William Herschel Telescope is operated on the island of La Palma by the Isaac Newton Group of Telescopes in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias. Based on observations made with the Isaac Newton Telescope (program I17BN005) operated on the island of La Palma by the Isaac Newton Group of Telescopes in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias. This paper makes use of data obtained from the Isaac Newton Group of Telescopes Archive which is maintained as part of the CASU Astronomical Data Centre at the Institute of Astronomy, Cambridge.

We acknowledge support of the Department of Atomic Energy, Government of India, under project No. 12-R&D-TFR-5.02-0700 for the GMRT observations. The GMRT is run by the National Centre for Radio Astrophysics of the Tata Institute of Fundamental Research, India. The Nançay Radio Observatory is operated by the Paris Observatory, associated with the French Centre National de la Recherche Scientifique (CNRS). We acknowledge financial support from the "Programme National Hautes Energies" (PNHE) of CNRS/INSU, France. This Letter is based (in part) on data obtained with the International LOFAR Telescope (ILT) under project code LC7_018. LOFAR (van Haarlem et al. 2013) is the Low Frequency Array designed and constructed by ASTRON. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. The Green Bank Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. FAST is a Chinese national mega-science facility, built and operated by NAOC. Partly based on observations with the 100 m telescope of the MPIfR (Max-Planck-Institut für Radioastronomie) at Effelsberg.

The Fermi-LAT Collaboration acknowledges generous ongoing support from a number of agencies and institutes that have supported both the development and the operation of the LAT as well as scientific data analysis. These include the National Aeronautics and Space Administration and the Department of Energy in the United States, the Commissariat à l'Energie Atomique and the Centre National de la Recherche Scientifique/Institut National de Physique Nucléaire et de Physique des Particules in France, the Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), High Energy Accelerator Research Organization (KEK) and Japan Aerospace Exploration Agency (JAXA) in Japan, and the K. A. Wallenberg Foundation, the Swedish Research Council, and the Swedish National Space Board in Sweden.

Additional support for science analysis during the operations phase is gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy and the Centre National d'Études Spatiales in France. This work performed in part under DOE Contract DE-AC02-76SF00515.

The authors thank the LIGO Scientific Collaboration for access to the data and gratefully acknowledge the support of the United States National Science Foundation (NSF) for the construction and operation of the LIGO Laboratory and Advanced LIGO as well as the Science and Technology Facilities Council (STFC) of the United Kingdom, and the Max-Planck-Society (MPS) for support of the construction of Advanced LIGO. Additional support for Advanced LIGO was provided by the Australian Research Council. This research has made use of data, software, and/or web tools obtained from the LIGO Open Science Center (https://losc.ligo.org), a service of LIGO Laboratory, the LIGO Scientific Collaboration and the Virgo Collaboration, to which the authors have also contributed. LIGO is funded by the U.S. National Science Foundation. Virgo is funded by the French Centre National de Recherche Scientifique (CNRS), the Italian Istituto Nazionale della Fisica Nucleare (INFN), and the Dutch Nikhef, with contributions by Polish and Hungarian institutes.

Software: Fermi Science Tools, MultiNest (Feroz et al. 2019), ULTRACAM software pipelines, Icarus (Breton et al. 2012), psrqpy (Manchester et al. 2005; Pitkin 2018), Astropy (Astropy Collaboration et al. 2013, 2018), matplotlib (Hunter 2007), NumPy (Oliphant 2006; van der Walt et al. 2011), GalPot (McMillan 2017), Lightkurve (Lightkurve Collaboration et al. 2018), PRESTO (Ransom et al. 2002), LALSuite (LIGO Scientific Collaboration 2018).

Acknowledging the possibility of mismatches between the pulsar rotation frequency and the gravitational-wave frequency, we perform an ${ \mathcal F }$-statistic search in a ∼2 Hz band around twice the rotation frequency, a factor of 10⁻³ of the gravitational-wave frequency, similarly to what was done in Abbott et al. (2019a) and also extend the spin-down search to the range $2\dot{f}\in (-1.260,-1.2216)\times {10}^{-15}\,\mathrm{Hz}\,{{\rm{s}}}^{-1}$. Overall, we use 2.4 × 10⁹ templates resulting in an average mismatch of 1%. We examine the results in 10 mHz-wide bands. The most significant $2{ \mathcal F }$ values from each band are consistent with the noise-only expectation, apart from six outliers that can be ascribed to a disturbance in L1 around ≈1016.32 Hz. We set upper limits in each band. The values are plotted in Figure 4 and are provided as data behind the figure. The mean value is 1.3 × 10⁻²⁵ and it is higher than the targeted search upper limit, consistently with the larger volume of searched wave shapes.

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