Abstract
We present the result of searches for gravitational waves from 200 pulsars using data from the first observing run of the Advanced LIGO detectors. We find no significant evidence for a gravitational-wave signal from any of these pulsars, but we are able to set the most constraining upper limits yet on their gravitational-wave amplitudes and ellipticities. For eight of these pulsars, our upper limits give bounds that are improvements over the indirect spin-down limit values. For another 32, we are within a factor of 10 of the spin-down limit, and it is likely that some of these will be reachable in future runs of the advanced detector. Taken as a whole, these new results improve on previous limits by more than a factor of two.
1. Introduction
The recent observations of gravitational waves from the inspiral and merger of binary black holes herald the era of gravitational-wave astronomy (Abbott et al. 2016b, 2016c). Such cataclysmic, transient, and extragalatic events are not however the only potential sources of observable gravitational waves. Galactic neutron stars offer a more local, and continuous, quasi-monochromatic source of gravitational radiation. Although intrinsically far weaker than the transient sources that have been observed, their continuous nature allows their signals to be found buried deep in the noise by coherently integrating over the long observing runs of the gravitational-wave observatories.
The subset of known pulsars, identified through electromagnetic observations, provides an important possible source of continuous gravitational waves. They are often timed with exquisite precision, allowing their rotational phase evolution, sky location and, if required, binary orbital parameters to be determined very accurately. In turn, these timings allow us to carry out fully phase-coherent and computationally cheap gravitational-wave searches over the length of our observation runs. A selection of known pulsars have already been targeted using data from the initial LIGO, Virgo, and GEO 600 detectors (summarized in Aasi et al. 2014), setting upper limits on their signal amplitudes, though without any detections.
An important milestone is passed when this upper limit falls below the so-called spin-down limit on gravitational strain for the targeted pulsar. This spin-down limit is determined by equating the power radiated through gravitational-wave emission to the pulsar's observed spin-down luminosity (attributed to its loss in rotational kinetic energy), i.e., as would be the case if it were a gravitar (Palomba 2005; Knispel & Allen 2008), and determining the equivalent strain expected at Earth.154 It can be calculated (see, e.g., Aasi et al. 2014) using
where and are the pulsar's frequency and first frequency derivative, Izz is the principal moment of inertia (for which we generally assume a canonical value of 1038 kg m2), and d is the pulsar's distance. In previous searches, this limit has been surpassed (i.e., a smaller limit on the strain amplitude has been obtained) for two pulsars: PSR J0534+2200 (the Crab pulsar; Abbott et al. 2008) and PSR J0835−4510 (the Vela pulsar; Abadie et al. 2011).
In this paper, we provide results from a search for gravitational waves from 200 known pulsars using data from the first observing run (O1) of Advanced LIGO (aLIGO). For the LIGO Hanford Observatory (H1) and LIGO Livingston Observatory (L1), we used data starting on 2015 September 11 at 01:25:03 UTC and 18:29:03 UTC, respectively, and finishing on 2016 January 19 at 17:07:59 UTC at both sites. With duty factors of 60% and 51% for H1 and L1, this run provided 78 days and 66 days of data respectively for analysis. The estimated sensitivity of this search as a function of source frequency is shown in Figure 1.155 We see that, even with its comparatively short observation time, the O1 data provide a significant sensitivity improvement over the previous runs, particularly at lower frequencies.
Figure 1. Stars show 95% credible upper limits on gravitational-wave amplitude, , for 200 pulsars using data from the O1 run. give the spin-down limits for all pulsars (based on distance values taken from the ATNF pulsar catalog (Manchester et al. 2005), unless otherwise stated in Tables 1 and 4) and assuming the canonical moment of inertia. The upper limits shown within the shaded circles are those for which the spin-down limits (linked via the dashed vertical lines) are surpassed with our observations. The gray curve gives an estimate of the expected strain sensitivity for O1, combining representative amplitude spectral density measurements for both H1 and L1. This estimate is an angle-averaged value and for particular sources is representative only, while the broader range over all angles for such an estimate is shown, for example, in Figure 4 of Abbott et al. (2010). Previous initial detector run results (Aasi et al. 2014) for 195 pulsars are shown as red circles, with 122 of these sources corresponding to sources searched for in O1.
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Standard image High-resolution image1.1. The Signal
We model the source as a rigidly rotating triaxial star, generating a strain signal at the detector of (e.g., Jaranowski et al. 1998)
where h0 is the gravitational-wave strain amplitude, and and are the antenna responses of observatory D to the "+" and "×" polarizations. These are dependent on the source sky position (right ascension α and declination δ) and polarization angle ψ. ι is the inclination of the star's rotation axis to the line of sight, and represents the evolution of the sinusoidal signal phase with time.
This phase evolution is usefully represented as a Taylor expansion, so that
where is the initial gravitational-wave phase at time epoch T0, and is the time derivative of the gravitational-wave frequency defined at T0. is the time delay from the observatory to the solar system barycenter, and can also include binary system barycentering corrections to put the observatory and source in inertial frames. For the majority of pulsars, expansions to N = 1 or 2 are all that are required, but for some young sources, with significant timing noise, expansions to higher orders may be used. For the case of a source rotating around a principal axis of inertia and producing emission from the (spherical harmonic) mass quadrupole mode (e.g., a rigidly rotating star with a triaxial moment of inertia ellipsoid), the gravitational-wave frequencies and frequency derivatives are all twice their rotational values, e.g., .
2. Pulsar Selection
To reflect the improved sensitivity of LIGO during O1, we targeted pulsars with rotation frequencies, , greater than about 10 Hz, but also included seven promising sources with large spin-down luminosities156 with just below 10 Hz. The quadrupolar emission frequencies of these targets are therefore greater than ∼20 Hz and within the band of good sensitivity for the instruments. We did not impose an upper limit on target frequency.
We have obtained timings for 200 known pulsars in this band. Timing was performed using the 42 ft telescope and Lovell telescope at Jodrell Bank (UK), the 26 m telescope at Hartebeesthoek (South Africa), the Parkes radio telescope (Australia), the Nançay Decimetric Radio Telescope (France), the Arecibo Observatory (Puerto Rico) and the Fermi Large Area Telescope (LAT). Of these, 122 have been targeted in previous campaigns (Aasi et al. 2014), while 78 are new to this search.
For the vast majority of these, we have obtained timing solutions using pulse time-of-arrival (TOA) observations that spanned the O1 run. For those pulsars whose TOAs did not span O1, we still expect them to maintain very good coherence when extrapolated to the O1 time. The tempo 157 or tempo2 (Hobbs et al. 2006) pulsar timing codes were used to produce these solutions, which provide us with precise information on the parameters defining each pulsars phase evolution, including their sky location and any binary system orbital dynamics if applicable.158
2.1. High-value Targets
We identified 11 sources (Table 1) for which we could either improve upon, or closely approach, the spin-down limit based on Equation (1). These are all young pulsars at the lower end of our sensitive frequency band and include the Crab and Vela pulsars for which the spin-down limit had already been surpassed (Abbott et al. 2008; Abadie et al. 2011; Aasi et al. 2014).
Table 1. The High-value Targets for Which the Spin-down Limit can be Improved Upon or Closely Approached
PSR | f (Hz) | d (kpc) | |
---|---|---|---|
J0205+6449a | 30.4 | 3.2 | |
J0534+2200 (Crab) | 59.3 | 2.0 | |
J0835−4510 (Vela) | 22.4 | 0.3 | |
J1302−6350b | 41.9 | 2.3 | |
J1809−1917 | 24.2 | 3.7 | |
J1813−1246 | 41.6 | 2.5c | |
J1826−1256 | 18.1 | 1.2d | |
J1928+1746 | 29.1 | 8.1 | |
J1952+3252 (CTB 80) | 50.6 | 3.0 | |
J2043+2740 | 20.8 | 1.1 | |
J2229+6114 | 38.7 | 3.0 |
Notes. Unless otherwise stated, all distances are those from v1.54 of the ATNF Pulsar Catalog (Manchester et al. 2005).
aThis pulsar was observed to glitch during O1 on MJD 57345. bThis pulsar is in a binary system and as such was not able to be searched for with the -vector method. cThis distance is a lower limit on the distance from Marelli et al. (2014). It is slightly higher than the distance of 1.9 kpc used for calculations in Aasi et al. (2014). dThis distance is that taken from the lower distance range from Voisin et al. (2016; using values from Wang 2011).Download table as: ASCIITypeset image
3. Analyses
Following Aasi et al. (2014), we used three largely independent methods for carrying out the search for the 11 high-value targets: the time-domain-based Bayesian (Dupuis & Woan 2005) and /-statistic (Jaranowski & Królak 2010) methods, and the frequency-domain-based 5n-vector method (Astone et al. 2010, 2012). For the other 189 targets only the Bayesian method was applied.
We refer the reader to Aasi et al. (2014) and references therein for more detailed descriptions of these methods. Generally, the methods were not modified for O1, although there have been some significant improvements to the Bayesian method, which are described in Appendix
In addition, the results from the -vector method used an earlier data release, with a slightly different instrumental calibration (Abbott et al. 2016a), than that used for the two other methods. The calibrations applied differ, however, by less than 3% in amplitude and less than 3◦ in phase for all high-value sources.
For one high-value target, PSR J1302−6350, the 5n-vector method was not used. This pulsar is in a binary system, which is not currently handled by this method. PSR J0205+6449 underwent a glitch on MJD 57345 (2015 November 19), causing the rotation frequency to increase by Hz. Because of the uncertain relation between the gravitational-wave and electromagnetic signal phases over a glitch, we analyzed both the pre-and-post-glitch periods independently and combined these incoherently to give the final result. To the best of our knowledge, none of our other sources glitched during the course of O1.
The results from the Bayesian method incorporate uncertainties into the pulsars' phase evolutions. If the fits to pulsar TOAs from electromagnetic observations provided uncertainties on any fitted parameters, then these parameters were also included in the search space (in addition to the four main unknown signal parameters, h0, , and ψ, defined with Equations (2) and (3)). Prior probabilities for these additional parameters were defined as Gaussian distributions, using their best-fit values and associated errors as means and standard deviations (see, e.g., Abbott et al. 2010). Upper limits are produced from the posterior probability distributions on h0 by marginalizing all other parameters over their prior ranges (see Appendix
Observations of pulsar-wind nebulae (PWNe) around several pulsars allow us to put prior constraints on their orientation angles ι and ψ, detailed in Appendix
4. Results
For all pulsars, we quote 95% credible/confidence upper limits on the gravitational-wave amplitude h0 set using coherently combined data from both H1 and L1.159 We use this value to also set limits on the mass quadrupole moment Q22 of the mode of the star (Owen 2005) via
In turn, this is related to the star's fiducial equatorial ellipticity ε through
To calculate ε, we use the canonical moment of inertia of kg m2 (see, e.g., Chapter 6 of Condon & Ransom 2016). We also quote the ratio of our observed h0 limits to the spin-down limits calculated using Equation (1). The distances used to calculate Q22 and ε are (unless otherwise stated in Tables 1 or 4) taken from v1.54 of the ATNF pulsar catalog (Manchester et al. 2005),160 and in most cases are calculated from the observed dispersion measure (noting that distance uncertainties of 20% or more are not uncommon; see, e.g., Figure 12 of Cordes & Lazio 2002). For the spin-down limit calculation, we generally use values of provided from the electromagnetic-pulse-arrival-time fits used in our search. If, however, an intrinsic period derivative, i.e., a period derivative corrected for proper motion effects (Shklovskii 1969) or globular cluster accelerations, is given in the ATNF catalog, then that value is used. If an intrinsic period derivative is not given for a globular cluster pulsar, then the spin-down limit is instead based on an assumed characteristic spin-down age of yr. The characteristic age (see, e.g., Chapter 6 of Condon & Ransom 2016) is defined as
where n is the braking index (), which has a value of n = 3 for purely magnetic dipole radiation, while we adopt the n = 5 case for purely gravitational radiation.
The calibration procedure for the aLIGO instruments and their amplitude uncertainties during the initial part of O1 are described in detail in Abbott et al. (2016a). After O1 was completed, the calibration was updated, and the maximum calibration uncertainties estimated over the whole run give a limit on the combined H1 and L1 amplitude uncertainties of . This is the conservative level of uncertainty on the h0 upper limits, and any quantities derived linearly from them, from the gravitational-wave observations alone.
The results for all targets, except the high-value targets discussed in Section 2.1, are shown in Table 4. For each pulsar, we produce two probability ratios, or odds (discussed in Appendix
Figure 2. Distributions of the probability ratios and for the observed pulsars.
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Standard image High-resolution imageThe h0 upper limits from this analysis (including those from the high-value targets) are shown in Figure 1. The figure also contains the upper limits obtained for the 195 pulsars targeted using data from the initial detector era (Aasi et al. 2014). We find that, on average, for pulsars that were both analyzed here and in previous runs, our new results have over two and a half times better sensitivity. The largest improvement is a factor of eight for PSR J0024−7204C at . For four pulsars, the new results are slightly less sensitive than the previous analyses, although in the worst case this is only by .162
Figure 3 shows corresponding limits on the fiducial ellipticity ε and mass quadrupole moment Q22. Figure 4 shows a histogram of the ratios between our upper limits and the spin-down limits.
Figure 3. Limits on fiducial ellipticities (ε) and mass quadrupole moments (Q22). show the values based on the spin-down limits for these pulsars. The pulsars for which the spin-down limit is surpassed are highlighted within larger shaded circles and linked to their spin-down limit values with dashed vertical lines. Also shown are diagonal lines of constant characteristic age, τ, for gravitars (with braking indices of n = 5) calculated via , where I38 is the principal moment of inertia in units of 1038 kg m2 (where we set ).
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Standard image High-resolution imageFigure 4. Ratio between our observed limits and the spin-down limits for all pulsars.
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Standard image High-resolution imageThe accelerations that pulsars experience in the cores of globular clusters can mask their true spin-down values. It is sometimes possible to determine these accelerations and correct for their effect on spin-down. As mentioned above, when such a correction is available, we have calculated the spin-down limits based on this corrected spin-down value. In cases where the correction is not available, we have instead assumed each pulsar has a characteristic age of yr and under the assumption of gravitational-radiation-dominated spin-down, calculated a naïve spin-down via Equation (6), which has then been used for the spin-down limit calculation. As proposed in Pitkin (2011), for these pulsars we could instead invert the process and use the h0 upper limit to set a limit on the spin-down of the pulsars (at least under the assumption that they are gravitars, with n = 5). Given that the maximum observed spin-up for a globular cluster pulsar is Hz s−1, we can say that the negative of this can be used as an approximation for the largest magnitude spin-down that could be masked by intracluster accelerations.163 Of the globular cluster pulsars for which the intrinsic spin-down is not known, we find that our upper limits on h0 give the smallest limit on the absolute spin-down value, due to gravitational waves for PSR J1623−2631 of Hz s−1. Although this value is probably too large to be masked by accelerations, it is of the same order as the spin-downs for two globular cluster millisecond pulsars, PSRs J1823-3021A (Freire et al. 2011) and J1824-2452A (Johnson et al. 2013), both with apparently large intrinsic spin-down values.
4.1. High-value Targets
Table 2 shows the results for the high-value targets (Section 2.1) for each of the three analysis methods discussed in Section 3. The results from the methods are broadly consistent. For pulsars that have restricted priors on orientations, the results using these are shown alongside the results from the full prior orientation range. We find that for eight of these pulsars, we achieve a sensitivity that surpasses the indirect spin-down limit.
Table 2. Limits on the Gravitational-wave Amplitude, and Other Derived Quantities, for the 11 High-value Pulsars
Analysis | ε | Q22 | cma | b/ | |||
---|---|---|---|---|---|---|---|
(10−25) | (10−4) | (1034 kg m2) | FAPc/p-valued | ||||
PSR J0205+6449 (Pre-glitch) | |||||||
Bayesian | 1.1 (1.3) | 3.6 (4.3) | 2.8 (3.3) | 0.25 (0.31) | 0.064 (0.093) | 90 (110) | −1.2 (−1.1) |
/-statistic | 1.8 (2.4) | 5.9 (7.9) | 4.5 (6.1) | 0.42 (0.55) | 0.17 (0.31) | 150 (200) | 0.27 (0.16) |
-vector | 0.75 (1.1) | 2.5 (3.7) | 1.9 (2.8) | 0.17 (0.25) | 0.030 (0.064) | 60 (90) | 0.95 |
(Post-glitch) | |||||||
Bayesian | 2.0 (2.6) | 6.4 (8.4) | 5.0 (6.5) | 0.45 (0.60) | 0.21 (0.36) | 160 (210) | −1.0 (−0.7) |
/-statistic | 1.7 (1.4) | 5.6 (4.6) | 4.3 (3.5) | 0.39 (0.32) | 0.15 (0.10) | 140 (120) | 0.49 (0.91) |
-vector | 1.1 (1.7) | 3.6 (5.4) | 2.8 (4.2) | 0.25 (0.38) | 0.065 (0.15) | 90 (140) | 0.32 |
(Incoherently Combined) | |||||||
Bayesian | 1.0 (1.3) | 3.4 (4.4) | 2.6 (3.4) | 0.24 (0.31) | 0.058 (0.097) | 90 (110) | −0.6 (−0.5) |
/-statistic | 1.2 (1.6) | 3.9 (5.2) | 3.0 (4.0) | 0.28 (0.37) | 0.077 (0.14) | 100 (130) | 0.36 (0.48) |
-vector | 0.73 (1.1) | 2.3 (3.5) | 1.8 (2.7) | 0.17 (0.25) | 0.028 (0.064) | 60 (90) | 0.95 |
PSR J0534+2200 (Crab) | |||||||
Bayesian | 0.67 (0.61) | 0.36 (0.33) | 0.28 (0.25) | 0.05 (0.04) | 0.0022 (0.0018) | 9 (8) | −0.7 (−0.7) |
/-statistic | 0.42 (0.24) | 0.23 (0.13) | 0.17 (0.10) | 0.03 (0.02) | 0.00087 (0.00028) | 6 (3) | 0.62 (0.31) |
-vector | 0.52 (0.50) | 0.28 (0.27) | 0.22 (0.21) | 0.04 (0.04) | 0.0013 (0.0012) | 7 (7) | 0.21 |
PSR J0835−4510 (Vela) | |||||||
Bayesian | 3.2 (2.8) | 1.7 (1.5) | 1.3 (1.2) | 0.10 (0.08) | 0.0090 (0.0070) | 40 (40) | −0.9 (−0.9) |
/-statistic | 3.8 (3.3) | 2.0 (1.7) | 1.6 (1.3) | 0.11 (0.10) | 0.012 (0.0094) | 50 (40) | 0.37 (0.58) |
-vector | 2.9 (2.9) | 1.5 (1.5) | 1.2 (1.2) | 0.09 (0.09) | 0.0073 (0.0073) | 40 (40) | 0.66 |
PSR J1302−6350 | |||||||
Bayesian | 0.77 | 0.96 | 0.74 | 1.0 | 1.0 | 20 | −1.0 |
-statistic | 0.60 | 0.74 | 0.58 | 0.78 | 0.61 | 20 | 0.44 |
PSR J1809−1917 | |||||||
Bayesian | 3.0 | 18 | 14 | 2.5 | ⋯ | 450 | −1.0 |
-statistic | 2.4 | 14 | 11 | 2.0 | ⋯ | 350 | 0.72 |
-vector | 2.5 | 15 | 12 | 2.1 | ⋯ | 380 | 0.62 |
PSR J1813−1246 | |||||||
Bayesian | 0.44 | 0.60 | 0.46 | 0.23 | 0.051 | 20 | −1.2 |
-statistic | 0.55 | 0.75 | 0.58 | 0.28 | 0.079 | 20 | 0.61 |
-vector | 0.55 | 0.75 | 0.58 | 0.28 | 0.079 | 20 | 0.69 |
PSR J1826−1256 | |||||||
Bayesian | 15 | 52 | 40 | 2.1 | ⋯ | 1300 | −0.9 |
-statistic | 17 | 59 | 45 | 2.4 | ⋯ | 1500 | 0.29 |
-vector | 18 | 62 | 48 | 2.6 | ⋯ | 1600 | 0.21 |
PSR J1928+1746 | |||||||
Bayesian | 1.4 | 12 | 9.5 | 3.1 | ⋯ | 300 | −1.0 |
-statistic | 1.5 | 14 | 11 | 3.4 | ⋯ | 350 | 0.42 |
-vector | 1.3 | 12 | 9.1 | 3.0 | ⋯ | 300 | 0.70 |
PSR J1952+3252 | |||||||
Bayesian | 0.47 (0.50) | 0.52 (0.56) | 0.40 (0.43) | 0.45 (0.49) | 0.20 (0.24) | 10 (10) | −1.1 (−1.1) |
-statistic | 0.48 | 0.53 | 0.41 | 0.46 | 0.22 | 10 | 0.44 |
-vector | 0.37 (0.39) | 0.41 (0.43) | 0.32 (0.33) | 0.36 (0.38) | 0.13 (0.14) | 10 (10) | 0.57 |
PSR J2043+2740 | |||||||
Bayesian | 5.3 | 13 | 10 | 0.57 | 0.32 | 330 | −0.8 |
-statistic | 5.6 | 14 | 11 | 0.61 | 0.37 | 350 | 0.41 |
-vector | 6.0 | 15 | 11 | 0.65 | 0.43 | 380 | 0.18 |
PSR J2229+6114 | |||||||
Bayesian | 0.50 (0.34) | 0.95 (0.64) | 0.73 (0.49) | 0.15 (0.10) | 0.023 (0.010) | 20 (20) | −1.3 (−1.4) |
/-statistic | 0.49 (0.45) | 0.93 (0.85) | 0.72 (0.66) | 0.15 (0.14) | 0.022 (0.018) | 20 (20) | 0.73 (0.35) |
-vector | 0.56 (0.43) | 1.1 (0.84) | 0.82 (0.63) | 0.17 (0.13) | 0.029 (0.017) | 30 (20) | 0.59 |
Notes. Limits with constrained orientations (see Appendix
Table 2 also contains an estimate of the maximum surface deformation of the mode, , for each of the pulsars. This is based on Figure 2 of Johnson-McDaniel (2013), where we adopt a scaling of cm maximized over equations of state and possible stellar masses. We also find that for five of these pulsars (PSRs J0534+2200, J1302−6350, J1813−1246, J1952+3252, and J2229+6114) the surface deformations are smaller than the rotational (l = 2, m = 0) surface deformation for all equations of state.164 For the Vela pulsar (PSR J0835−4510) and PSR J0205+6449, the surface deformations are smaller than the rotational deformations for roughly half of the equations of state used in Johnson-McDaniel (2013). There is no expected relation between the scales of these two deformations, but it is intriguing to compare them nonetheless.
5. Discussion
We have searched for gravitational-wave emission from the quadrupole mode of 200 known pulsars. There is no significant evidence for a signal from any of the sources. We have been able to set 95% credible upper limits on the gravitational-wave amplitudes from all these sources, and from these derived limits on each star's fiducial ellipticity and quadrupole moment.
In earlier analyses, the indirect spin-down limits on the gravitational-wave amplitude had been surpassed for two pulsars: PSR J0534+2200 (the Crab pulsar; Abbott et al. 2008) and PSR J0835−4510 (the Vela pulsar; Abadie et al. 2011). We improve upon the previous limits for these two pulsars by factors of ≳3. We find that for the Crab and Vela pulsars, less than and of the spin-down luminosity is being lost via gravitational radiation, respectively (assuming the distance is precisely known and using the fiducial moment of inertia of 1038 kg m2). The observed braking indices of these pulsars provide constraints on the contribution of gravitational-wave emission to the spin-down, under the assumption that the spin-down is due only to a combination of electromagnetic and gravitational-wave losses. These braking index constraints are more stringent, i.e., give smaller limits on the gravitational-wave emission, than the naïve spin-down limit given in Equation (1) (see Palomba 2000). Our results, however, surpass even these more stringent limits and are therefore compatible with the observed braking indices. We surpass the spin-down limits of six further pulsars. All these are young pulsars with large spin-down luminosities, and as such our limits translate to large ellipticities/quadrupole moments that are at the upper end of some maximally allowed values (see e.g., Owen 2005; Pitkin 2011; Johnson-McDaniel & Owen 2013). If we assume that internal toroidal magnetic fields are the source of any stellar mass quadrupole (Bonazzola & Gourgoulhon 1996), then we can use our limits on ellipticities as constraints on the magnitude of the internal field strength. For the Crab pulsar, PSR J1813−1246, PSR J1952+3252, and PSR J2229+6114, which have roughly comparable ellipticity limits, the internal magnetic field strength is limited to G (e.g., Cutler 2002; Haskell et al. 2008). For comparison, the Crab pulsar's inferred external polar magnetic field at its surface is G. Due to this being a rough order of magnitude estimate, this value is the same as that previously quoted for the Crab pulsar in Aasi et al. (2014), although the limit is now valid for several more pulsars.
For any neutron star equation of state, the lower bound on the mass quadrupole (due to the internal magnetic field, which may be very weak) is many orders of magnitude less than the upper bound. Therefore, it is always important to acknowledge that these upper limits on particular stars do not allow us to place constraints on neutron star equations of state.
Of all the pulsars, the smallest 95% credible limit on h0 that we find is for PSR J1918−0642. The smallest ellipticity and Q22 quadrupole moments are and kg m2, respectively, for J0636+5129, which is a relatively nearby pulsar at ∼0.20 kpc. Although neither of these pulsars surpasses their fiducial spin-down limits, it is interesting to note that there are 32 that we are able to constrain to within a factor of 10 of their spin-down limits (see Figure 4). For PSR J0437−4715 (which is nearby, at 0.16 kpc), we are in fact only 1.4 times above the spin-down limit. Therefore, an equivalent increase in detector sensitivity of that factor, or a times longer run, would allow us to surpass the spin-down limit. Alternatively, the spin-down limit would be surpassed if the true moment of inertia for PSR J0437−4715 were a factor of 1.9 times larger than I38, which is well within plausible values. As this is a millisecond pulsar, it would give ellipticity constraints of less than a few 10−8, or quadrupole moment constraints of kg m2, compared to the much larger constraints typically found for the young pulsars in Table 1. Using the conversion in Cutler (2002) the constraints on the internal toroidal fields for this pulsar would be G, which is similar to the external field strengths of young pulsars.
This search has imposed a model in which the gravitational-wave signal phase evolutions must be tightly locked to the pulsars' rotational evolutions determined through electromagnetic observations. There are mechanisms (discussed in, e.g., Abbott et al. 2008), however, that could lead to small deviations between the phase evolution and observed rotation. Additionally, there are many pulsars for which highly accurate timings do not exist or are not available from observations coincident with ours.165 There are several such sources for which the spin-down limit could be surpassed and these are being searched for in O1 data using narrow-band searches (see, e.g., Aasi et al. 2015), covering a small range in frequency and frequency derivative to account for uncertainties in the exact parameters (LIGO Scientific Collaboration & Virgo Collaboration 2017, in preparation). All-sky broadband searches for unknown rotating neutron stars are also underway.
In the near future, increasing sensitivities and considerably longer observing runs are planned for aLIGO and Advanced Virgo (Abbott et al. 2016d). This will give us several times greater sensitivity with which to search for gravitational-wave signals, and in any event will allow us to surpass the spin-down limits for 10 or more pulsars. Future searches will also address gravitational-wave emission at not just twice the rotation frequency, but also at the rotation frequency (e.g., Pitkin et al. 2015), further increasing the likelihood of a first detection of continuous gravitational waves.
The authors gratefully acknowledge the support of the United States National Science Foundation (NSF) for the construction and operation of the LIGO Laboratory and Advanced LIGO as well as the Science and Technology Facilities Council (STFC) of the United Kingdom, the Max-Planck-Society (MPS), and the State of Niedersachsen/Germany for support of the construction of Advanced LIGO and construction and operation of the GEO600 detector. Additional support for Advanced LIGO was provided by the Australian Research Council. The authors gratefully acknowledge the Italian Istituto Nazionale di Fisica Nucleare (INFN), the French Centre National de la Recherche Scientifique (CNRS), and the Foundation for Fundamental Research on Matter supported by the Netherlands Organisation for Scientific Research, for the construction and operation of the Virgo detector and the creation and support of the EGO consortium. The authors also gratefully acknowledge research support from these agencies as well as by the Council of Scientific and Industrial Research of India, Department of Science and Technology, India, Science & Engineering Research Board (SERB), India, Ministry of Human Resource Development, India, the Spanish Ministerio de Economía y Competitividad, the Vicepresidència i Conselleria d'Innovació Recerca i Turisme and the Conselleria d'Educació i Universitat of the Govern de les Illes Balears, the National Science Centre of Poland, the European Commission, the Royal Society, the Scottish Funding Council, the Scottish Universities Physics Alliance, the Hungarian Scientific Research Fund (OTKA), the Lyon Institute of Origins (LIO), the National Research Foundation of Korea, Industry Canada and the Province of Ontario through the Ministry of Economic Development and Innovation, the Natural Science and Engineering Research Council Canada, Canadian Institute for Advanced Research, the Brazilian Ministry of Science, Technology, and Innovation, Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), Russian Foundation for Basic Research, the Leverhulme Trust, the Research Corporation, Ministry of Science and Technology (MOST), Taiwan, and the Kavli Foundation. The authors gratefully acknowledge the support of the NSF, STFC, MPS, INFN, CNRS, and the State of Niedersachsen/Germany for provision of computational resources.
Pulsar observations with the Lovell telescope and their analyses are supported through a consolidated grant from the STFC in the UK. The Nançay Radio Observatory is operated by the Paris Observatory, associated with the French CNRS. A. Ridolfi and P. C. C. Freire gratefully acknowledge financial support by the European Research Council for the ERC Starting grant BEACON under contract No. 279702. A. Ridolfi is member of the International Max Planck research school for Astronomy and Astrophysics at the Universities of Bonn and Cologne and acknowledges partial support through the Bonn-Cologne Graduate School of Physics and Astronomy.
LIGO Document No. LIGO-P1600159.
Appendix A: The Application of the Bayesian Method
The Bayesian Method used in our known pulsar searches involves a data processing stage and a parameter estimation stage. For a given source, the data processing stage takes calibrated strain data from H1 and L1 (sampled at a rate of 16,384 Hz), heterodynes it to remove a best fit for the source's phase evolution, and then low-pass filters and heavily downsamples the data to one sample per minute (Dupuis & Woan 2005). This leaves a complex time series with a bandwidth, in which a signal would have the form
where the remaining modulation will be due to the detector's diurnal antenna pattern and any slowly varying phase difference caused by potential differences between the best-fit phase evolution and the true signal phase evolution .166
These combined processed data sets for each detector are used to estimate the joint posterior probability distribution of the unknown source signal parameters, , using a Bayesian framework via
where is the likelihood of the data given the specific signal model () parameters, is the joint prior probability distribution of the parameters, and is the evidence (marginal likelihood) of observing our data, given a signal of the type we defined. In the cases where are negligible, this corresponds to just estimating four parameters, . In general, offsets between the best-fit phase parameters and true signal parameters can also be estimated, provided that they do not cause the signal to drift out of the bandwidth available. When using timing solutions calculated using tempo(2), uncertainties in the fitted parameters are produced, and when available these fitted parameters will be included in our estimation for the gravitational-wave signal.
In previous searches (e.g., Abbott et al. 2010; Aasi et al. 2014), a Markov Chain Monte Carlo (MCMC) method has been used to sample and estimate for the unknown parameters. However, the simple proposal distribution used for the MCMC was not well-tuned and was therefore inefficient, especially when searching over additional phase parameters. Furthermore, the MCMC did not naturally produce a value for the evidence . To allow the calculation of and, as a natural by-product, the joint parameter posterior probability distribution, we have adopted the nested sampling method (Skilling 2006). In particular our analysis code (Pitkin et al. 2012; M. Pitkin et al. 2017, in preparation) uses the nested sampling implementation of Veitch & Vecchio (2010) as provided in the LALIinference library (Veitch et al. 2015) within the LIGO Algorithm Library (LAL) suite.167 This implements more intelligent and efficient proposals than previously used. The code has been validated by extracting both software and hardware (Biwer et al. 2016) signal injections into gravitational-wave detector data.
A.1. The Likelihood
The likelihood, , is a Student's t-like probability distribution and is given in, e.g., Abbott et al. (2007). It assumes that the noise in the data may be non-stationary, but consists of stationary Gaussian segments, each with unknown variance. The analysis uses a Bayesian Blocks-type method (Scargle 1998) to divide the data into stationary segments, although those containing fewer than five points are discarded. Any segments longer than a day (1440 points given our 1/60 Hz sample rate) are split such that no segments are longer than 1440 points. This differs from previous analyses in which the data were automatically split into segments containing 30 points.
In cases where the search requires the recalculation of when evaluating the likelihood, this can be computationally expensive; the phase, including solar system and binary system barycentring time delays, is needed and the log-likelihood calculation requires summations over all data points. To make this considerably more efficient, we have adopted a Reduced Order Quadrature scheme (e.g., Antil et al. 2013; Canizares et al. 2013) to approximate the likelihood via interpolation of a reduced model basis.
A.2. The Priors
In Equation (8), a prior probability distribution for the parameters is required. For the parameters , and ψ, we generally have no prior knowledge of their values, and so use flat priors within their allowed ranges:
These ranges do not necessarily span the full physically allowable range of source values, but are a degenerate range that will contain all possible observable signal waveforms (Jones 2015; Pitkin et al. 2015). In some cases, there is information about the inclination and/or polarization angle of the source (see Appendix B). Where present this can be incorporated into the prior by using a Gaussian distribution based on this information. For the cases where the inclination is recovered from a PWN image, there is no information about the rotation direction of the source, so in fact a bimodal Gaussian prior on ι is required (Jones 2015; see Appendix B).168
For a prior on the gravitational-wave amplitude h0, the analysis in Aasi et al. (2014) used a flat distribution bounded at zero and some value that was large compared to the observed standard deviation of the data, or a distribution on h0 and based on previous searches (e.g., Abbott et al. 2010). In this analysis, inspired by that used in Middleton et al. (2016), we have adopted a different prior based on the Fermi–Dirac distribution:
where μ gives the point at which the distribution falls to half its maximum value, and σ defines the rate at which the distribution falls off. If we define a value at which the cumulative distribution function of Equation (10) is at 95%, and require that the probability density function falls from 97.5% to 2.5% of its maximum over a range that is , we are able to define μ and σ. In this analysis, there are two ways in which we define to calculate μ and σ: for pulsars where we already have a 95% h0 upper limit from previous searches, we use this value as for new pulsars, we have based on the the 95% upper limit that would have been expected if the pulsar had been searched for in the previous S6/VSR2,4 analysis.169 For small values of h0 this prior looks flat, while for large values it approximates an exponential distribution. Unlike the flat priors used previously, it is continuous for positive values and penalizes excessively large values.
If searching over the phase parameters defining in Equation (7), i.e., frequency, sky position, and binary system parameters, the prior distribution on the parameters is based on the uncertainties provided by the tempo(2) fits to TOAs. We take the uncertainties as the standard deviations for a multivariate Gaussian prior on these parameters. We conservatively have the parameters as uncorrelated, except in two specific cases for low eccentricity () binary systems. If there are uncertainties on the time and angle of periastron, or if there are uncertainties on the binary period and time derivative of the angle of periastron, then these pairs of parameters are set to be fully correlated.
A.3. The Evidence
The evidence allows a Bayesian model comparison to be performed, i.e., the comparison of the relative probabilities of different signal models given the data, which provides a way of assessing if an observed signal is real (see, e.g., the -statistic of Prix & Krishnan 2009, for the use of a Bayesian model comparison in this context). For example, we can calculate the ratio of the probability that the data contains a signal to the probability that the data is purely Gaussian noise:
where the first term on the right-hand side is called the Bayes factor and is the prior odds of the two models, which we set to unity. To calculate the likelihood can be evaluated with the signal set to zero.
Given more than one detector, we are also able to compare the probability that the data contain a coherent signal between detectors (as would be expected from an astrophysical source) versus independent (and therefore incoherent) signals in each detector or the data consisting of non-stationary (see Appendix A.1) Gaussian noise alone (e.g., Keitel et al. 2014). If we take the combined data to be , then we can form four incoherent-signal-or-noise hypotheses (where for compactness we have removed the implicit I dependence):
- an independent signal in both detectors,
- a signal in H1, but non-stationary Gaussian noise in L1,
- a signal in L1, but non-stationary Gaussian noise in H1,
- independent non-stationary Gaussian noise in both detectors, ,
where represents the hypothesis of our signal model/noise in the given detector. This gives a ratio
We choose the five hypothesis priors (, , , , and ) such that they have equal probabilities, and they therefore factorize out of the calculation.170 Such a probability ratio (i.e., the odds) obviously penalizes single detector detections, in which one detector may be considerably more sensitive than the other.
Appendix B: Orientation Angle Priors
For several pulsars in our search, there are observations of their PWN. Under the assumption that a pulsar's orientation is aligned with its surrounding nebula, we can use the fits to the pulsar orientation given in Ng & Romani (2004, 2008) as restricted priors on ψ and ι. For the Bayesian and 5n-vector methods, the prior probability distributions on ψ and ι are Gaussian distributions based on the PWN fits, while the -statistic uses a δ-function prior at the best-fit value. Table 3 shows the means and standard deviations used for the parameter priors. In general, these are taken from Table 2 of Ng & Romani (2008), where Ψ is equivalent to our ψ, and ζ is equivalent to our ι.171 Statistical and systematic uncertainties are added in quadrature (for non-symmetric uncertainties, the larger value is used). For the Crab pulsar and PSR J0205+6449, Ng & Romani (2008) give fits to the inner and outer PWN torii, so in these cases our mean value is the average of the inner and outer fits, and the quadrature-combined systematic and statistical errors for each are combined via .
Table 3. Means and Standard Deviations for Restricted Priors on ψ and ι Based on Table 2 of Ng & Romani (2008)
PSR | ψ (rad) | (rad) | (rad) |
---|---|---|---|
J0205+6449 | 1.5760 ± 0.0078 | 1.5896 ± 0.0219 | 1.5519 ± 0.0219 |
J0534+2200 (Crab) | 2.1844 ± 0.0016 | 1.0850 ± 0.0149 | 2.0566 ± 0.0149 |
J0835−4510 (Vela) | 2.2799 ± 0.0015 | 1.1048 ± 0.0105 | 2.0368 ± 0.0105 |
J1709−4429 (B1706−44) | 2.8554 ± 0.0305 | 0.9303 ± 0.0578 | 2.2113 ± 0.0578 |
J1952+3252 | −0.2007 ± 0.1501 | ⋯ | ⋯ |
J2229+6114 | 1.7977 ± 0.0454 | 0.8029 ± 0.1100 | 2.3387 ± 0.1100 |
Note. For PSR J1952+3252, the values for ψ are not from PWN fitting but are from the mean of a value derived from proper motion measurements and observations of Hα "lobes" bracketing the bow shock (Ng & Romani 2004).
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Table 4. Limits on the Gravitational-wave Amplitude, and Other Derived Quantities, for Known Pulsars
PSR | f (Hz) | d(kpc) | ε | Q22 | Spin-down Ratio | ||||
---|---|---|---|---|---|---|---|---|---|
kg m2) | |||||||||
J0023+0923 | 655.69 | 1.0 | 0.016 | 0.36 | 0.79 | 0.61 | 23 | −0.5 | −1.0 |
J0024−7204AA | 1083.79 | 4.0 | 0.0057b | 0.90 | 2.9 | 2.2 | 160 | 0.0 | −0.5 |
J0024−7204AB | 539.86 | 4.0 | 0.0057b | 0.33 | 4.4 | 3.4 | 58 | −0.4 | −0.9 |
J0024−7204C | 347.42 | 4.0 | 0.0057b | 0.22 | 6.8 | 5.2 | 38 | −1.1 | −1.2 |
J0024−7204D | 373.30 | 4.0 | 0.0057b | 0.18 | 5.0 | 3.9 | 32 | −0.5 | −0.8 |
J0024−7204E | 565.56 | 4.0 | 0.0042a | 0.23 | 2.7 | 2.1 | 54 | −0.9 | −1.0 |
J0024−7204F | 762.32 | 4.0 | 0.0057b | 0.35 | 2.3 | 1.8 | 62 | −0.5 | −0.8 |
J0024−7204G | 495.00 | 4.0 | 0.0057b | 0.24 | 3.7 | 2.9 | 43 | −0.7 | −0.9 |
J0024−7204H | 622.99 | 4.0 | 0.0044a | 0.40 | 3.9 | 3.0 | 90 | −0.1 | −0.6 |
J0024−7204I | 573.89 | 4.0 | 0.0057b | 0.34 | 3.9 | 3.0 | 60 | −0.4 | −0.7 |
J0024−7204J | 952.09 | 4.0 | 0.0057b | 0.52 | 2.2 | 1.7 | 91 | −0.3 | −0.7 |
J0024−7204L | 460.18 | 4.0 | 0.0057b | 0.22 | 3.9 | 3.0 | 38 | −0.5 | −0.8 |
J0024−7204M | 543.97 | 4.0 | 0.0057b | 0.38 | 4.9 | 3.8 | 67 | −0.3 | −0.7 |
J0024−7204N | 654.89 | 4.0 | 0.0057b | 0.28 | 2.4 | 1.9 | 49 | −0.5 | −0.8 |
J0024−7204O | 756.62 | 4.0 | 0.0057b | 0.42 | 2.8 | 2.1 | 74 | −0.3 | −0.8 |
J0024−7204Q | 495.89 | 4.0 | 0.0040a | 0.21 | 3.2 | 2.5 | 53 | −0.5 | −1.0 |
J0024−7204R | 574.64 | 4.0 | 0.0057b | 0.29 | 3.3 | 2.5 | 51 | −0.3 | −0.8 |
J0024−7204S | 706.61 | 4.0 | 0.0045a | 0.34 | 2.6 | 2.0 | 77 | −0.4 | −0.8 |
J0024−7204T | 263.56 | 4.0 | 0.011a | 0.22 | 12 | 9.1 | 19 | −0.4 | −0.8 |
J0024−7204U | 460.53 | 4.0 | 0.0042a | 0.37 | 6.6 | 5.1 | 88 | 0.1 | −0.4 |
J0024−7204W | 850.22 | 4.0 | 0.0057b | 0.38 | 2.0 | 1.5 | 66 | −0.6 | −1.2 |
J0024−7204X | 419.15 | 4.0 | 0.0051a | 0.47 | 10 | 7.8 | 93 | −0.2 | −0.5 |
J0024−7204Y | 910.47 | 4.0 | 0.0029a | 0.42 | 1.9 | 1.5 | 150 | −0.4 | −0.7 |
J0024−7204Z | 439.13 | 4.0 | 0.0057b | 0.23 | 4.5 | 3.5 | 40 | −0.5 | −0.8 |
J0030+0451 | 411.06 | 0.3 | 0.039 | 0.27 | 0.46 | 0.36 | 7.1 | −0.6 | −0.8 |
J0034−0534 | 1065.43 | 1.0 | 0.013 | 0.49 | 0.40 | 0.31 | 36 | −0.8 | −1.0 |
J0102+4839 | 674.74 | 4.0 | 0.0039 | 0.26 | 2.2 | 1.7 | 67 | −0.7 | −0.9 |
J0218+4232 | 860.92 | 3.1 | 0.015 | 0.42 | 1.7 | 1.3 | 28 | −0.5 | −0.8 |
J0340+4130 | 606.18 | 2.7 | 0.0044 | 0.31 | 2.1 | 1.7 | 70 | −0.5 | −0.9 |
J0348+0432 | 51.12 | 2.1 | 0.0095 | 0.48 | 360 | 280 | 50 | −0.8 | −0.9 |
J0407+1607 | 77.82 | 4.1 | 0.0037 | 0.22 | 140 | 110 | 61 | −0.6 | −1.0 |
J0437−4715 | 347.38 | 0.2 | 0.16 | 0.22 | 0.28 | 0.21 | 1.4 | −1.0 | −1.1 |
J0453+1559 | 43.69 | 1.8 | 0.0089 | 0.66 | 600 | 460 | 74 | −0.8 | −1.0 |
J0605+37 | 733.15 | 1.2 | 0.0091 | 0.33 | 0.67 | 0.52 | 36 | −0.6 | −0.8 |
J0609+2130 | 35.91 | 1.8 | 0.0091 | 1.0 | 1300 | 1000 | 110 | −0.9 | −1.2 |
J0610−2100 | 517.96 | 5.6 | 0.0026 | 0.29 | 5.8 | 4.5 | 110 | −0.7 | −0.9 |
J0613−0200 | 653.20 | 1.1 | 0.013 | 0.37 | 0.90 | 0.70 | 28 | −0.3 | −0.7 |
J0614−3329 | 635.19 | 1.0 | 0.019 | 0.46 | 1.1 | 0.85 | 25 | −0.2 | −0.7 |
J0621+1002 | 69.31 | 1.9 | 0.0055 | 0.30 | 110 | 85 | 54 | −0.8 | −1.0 |
J0636+5129 | 697.12 | 0.2 | 0.044 | 0.32 | 0.13 | 0.097 | 7.3 | −0.6 | −0.8 |
J0645+5158 | 225.90 | 0.8 | 0.0078 | 0.23 | 3.3 | 2.5 | 30 | −0.2 | −0.7 |
J0711−6830 | 364.23 | 1.0 | 0.013 | 0.27 | 2.0 | 1.5 | 21 | −0.3 | −0.7 |
J0721−2038 | 128.68 | 3.9 | 0.0035 | 0.40 | 89 | 68 | 120 | −0.2 | −0.7 |
J0737−3039A | 88.11 | 1.1 | 0.065 | 0.26 | 34 | 27 | 4.0 | −0.6 | −0.9 |
J0742+66 | 693.06 | 0.9 | 0.019 | 0.40 | 0.68 | 0.53 | 21 | −0.6 | −0.8 |
J0751+1807 | 574.92 | 0.4 | 0.030 | 0.46 | 0.53 | 0.41 | 15 | 0.3 | −0.6 |
J0900−3144 | 180.02 | 0.8 | 0.021 | 0.89 | 21 | 17 | 43 | −0.6 | −1.0 |
J0908−4913 | 18.73 | 1.0 | 2.6 | 14 | 37,000 | 29,000 | 5.3 | −0.7 | −0.9 |
J0931−1902 | 431.22 | 3.6 | 0.0020 | 0.21 | 4.0 | 3.1 | 110 | −0.6 | −0.9 |
J0940−5428 | 22.84 | 4.3 | 1.2 | 3.9 | 30,000 | 23,000 | 3.4 | −0.7 | −1.0 |
J1012+5307 | 380.54 | 0.7 | 0.021 | 0.36 | 1.6 | 1.3 | 17 | −0.0 | −0.8 |
J1016−5819 | 22.77 | 4.6 | 0.16 | 3.2 | 27,000 | 21,000 | 20 | −0.7 | −1.2 |
J1016−5857 | 18.62 | 9.3 | 1.1 | 14 | 360,000 | 280,000 | 13 | −0.9 | −1.1 |
J1017−7156 | 855.24 | 0.7 | 0.011 | 0.36 | 0.33 | 0.25 | 34 | −0.7 | −0.9 |
J1022+1001 | 121.56 | 0.7 | 0.017 | 0.18 | 8.5 | 6.6 | 11 | −0.7 | −1.0 |
J1024−0719 | 387.43 | 1.1 | 0.014 | 0.33 | 2.3 | 1.8 | 24 | −0.1 | −0.5 |
J1028−5819 | 21.88 | 2.8 | 1.2 | 9.7 | 53,000 | 41,000 | 7.9 | 0.6 | −0.6 |
J1038+0032 | 69.32 | 2.4 | 0.0052 | 0.29 | 130 | 100 | 56 | −0.9 | −1.0 |
J1045−4509 | 267.59 | 0.3 | 0.036 | 0.19 | 0.87 | 0.67 | 5.3 | −0.4 | −0.8 |
J1055−6028 | 20.07 | 30.0 | 0.16 | 11 | 800,000 | 620,000 | 72 | −1.1 | −1.3 |
J1105−6107 | 31.65 | 7.1 | 0.60 | 1.8 | 12,000 | 9200 | 3.0 | −0.8 | −1.0 |
J1112−6103 | 30.78 | 30.0 | 0.19 | 1.5 | 44,000 | 34,000 | 7.8 | −1.7 | −1.7 |
J1122+78 | 476.01 | 0.6 | 0.016 | 0.23 | 0.61 | 0.47 | 14 | −0.6 | −0.9 |
J1125−6014 | 760.35 | 1.9 | 0.0045 | 0.53 | 1.7 | 1.3 | 120 | −0.2 | −0.6 |
J1142+0119 | 394.07 | 2.0 | 0.0068 | 0.29 | 3.7 | 2.8 | 43 | −0.3 | −0.6 |
J1231−1411 | 542.91 | 0.5 | 0.044 | 0.30 | 0.43 | 0.33 | 6.7 | −0.6 | −0.8 |
J1300+1240 | 321.62 | 0.6 | 0.058 | 0.22 | 1.2 | 0.94 | 3.9 | −0.6 | −0.9 |
J1302−3258 | 530.38 | 1.9 | 0.0057 | 0.22 | 1.4 | 1.0 | 38 | −0.7 | −1.0 |
J1312+0051 | 473.03 | 1.1 | 0.014 | 0.26 | 1.3 | 0.99 | 18 | −0.5 | −1.1 |
J1327−0755 | 746.85 | 2.2 | 0.0031 | 0.38 | 1.4 | 1.1 | 120 | −0.5 | −0.8 |
J1410−6132 | 39.96 | 30.0 | 0.21 | 0.77 | 14,000 | 11,000 | 3.6 | −1.0 | −1.1 |
J1418−6058 | 18.08 | 1.6 | 6.2 | 14 | 63,000 | 49,000 | 2.2 | 0.2 | −0.5 |
J1446−4701 | 911.29 | 2.0 | 0.0082 | 0.45 | 1.0 | 0.80 | 54 | −0.4 | −1.1 |
J1453+1902 | 345.29 | 0.9 | 0.012 | 0.32 | 2.4 | 1.8 | 26 | −0.7 | −0.8 |
J1455−3330 | 250.40 | 0.7 | 0.019 | 0.21 | 2.3 | 1.8 | 11 | −0.5 | −0.8 |
J1509−5850 | 22.49 | 3.9 | 0.67 | 2.5 | 18,000 | 14,000 | 3.7 | −1.0 | −1.1 |
J1518+4904 | 48.86 | 0.7 | 0.0094 | 0.37 | 100 | 79 | 39 | −1.1 | −1.2 |
J1524−5625 | 25.57 | 3.8 | 1.5 | 4.0 | 22,000 | 17,000 | 2.7 | −0.5 | −0.8 |
J1531−5610 | 23.75 | 3.1 | 1.1 | 3.3 | 17,000 | 13,000 | 3.1 | −0.6 | −1.1 |
J1537+1155 | 52.76 | 1.1 | 0.061 | 0.46 | 160 | 130 | 7.4 | −0.3 | −0.7 |
J1545−4550 | 559.40 | 2.0 | 0.015 | 0.27 | 1.6 | 1.3 | 18 | −0.6 | −0.9 |
J1551−0658 | 281.94 | 1.5 | 0.0094 | 0.27 | 4.7 | 3.6 | 28 | −0.3 | −1.0 |
J1600−3053 | 555.88 | 1.8 | 0.0073 | 0.27 | 1.5 | 1.2 | 38 | −0.6 | −1.0 |
J1603−7202 | 134.75 | 0.5 | 0.016 | 0.18 | 5.0 | 3.8 | 11 | −0.3 | −0.9 |
J1614−2230 | 634.76 | 0.7 | 0.020 | 0.51 | 0.84 | 0.65 | 25 | 0.4 | −0.1 |
J1618−3921 | 166.84 | 4.8 | 0.0036 | 0.19 | 30 | 23 | 52 | −0.4 | −0.8 |
J1623−2631 | 180.57 | 1.8 | 0.013b | 0.27 | 14 | 11 | 21 | −0.3 | −0.7 |
J1630+37 | 602.75 | 0.8 | 0.017 | 0.31 | 0.68 | 0.53 | 18 | −0.5 | −0.8 |
J1640+2224 | 632.25 | 1.4 | 0.0053 | 0.54 | 1.9 | 1.4 | 100 | 0.3 | −0.3 |
J1643−1224 | 432.75 | 0.7 | 0.022 | 0.27 | 1.0 | 0.77 | 12 | −0.3 | −0.7 |
J1653−2054 | 484.36 | 2.6 | 0.0050 | 0.26 | 2.7 | 2.1 | 51 | −0.5 | −0.9 |
J1708−3506 | 443.94 | 3.5 | 0.0037 | 0.25 | 4.2 | 3.2 | 68 | −0.5 | −0.9 |
J1709+2313 | 431.85 | 1.8 | 0.0039 | 0.30 | 2.8 | 2.1 | 76 | −0.6 | −0.9 |
J1709−4429 | 19.51 | 2.6 | 3.0 | 6.1 | 40,000 | 31,000 | 2.1 | −0.7 | −1.0 |
J1709−4429c | 19.51 | 2.6 | 3.0 | 4.6 | 30,000 | 23,000 | 1.5 | −0.8 | −1.0 |
J1710+49 | 621.07 | 0.4 | 0.049 | 0.28 | 0.27 | 0.21 | 5.8 | −0.6 | −1.0 |
J1713+0747 | 437.62 | 1.2 | 0.0093 | 0.36 | 2.1 | 1.6 | 38 | 0.1 | −0.4 |
J1718−3825 | 26.78 | 4.2 | 0.80 | 1.7 | 9800 | 7500 | 2.2 | −0.8 | −1.1 |
J1719−1438 | 345.41 | 1.6 | 0.0058 | 0.24 | 3.1 | 2.4 | 42 | −1.0 | −1.1 |
J1721−2457 | 571.98 | 1.6 | 0.0065 | 0.32 | 1.4 | 1.1 | 49 | −0.5 | −0.8 |
J1727−2946 | 73.85 | 1.6 | 0.015 | 0.29 | 82 | 63 | 19 | −0.8 | −1.1 |
J1729−2117 | 30.17 | 1.4 | 0.0093 | 1.0 | 1500 | 1100 | 110 | −1.0 | −1.1 |
J1730−2304 | 246.22 | 0.6 | 0.020 | 0.17 | 1.6 | 1.3 | 8.2 | −0.7 | −1.1 |
J1731−1847 | 853.04 | 4.0 | 0.0066 | 0.56 | 2.9 | 2.3 | 84 | −0.5 | −0.7 |
J1732−5049 | 376.47 | 1.8 | 0.0073 | 0.18 | 2.1 | 1.6 | 24 | −0.7 | −0.9 |
J1738+0333 | 341.87 | 1.5 | 0.011 | 0.23 | 2.8 | 2.1 | 21 | −0.9 | −1.1 |
J1741+1351 | 533.74 | 1.1 | 0.021 | 0.50 | 1.8 | 1.4 | 24 | 0.5 | −0.2 |
J1744−1134 | 490.85 | 0.4 | 0.030 | 0.40 | 0.63 | 0.49 | 13 | −0.0 | −0.5 |
J1745+1017 | 754.11 | 1.4 | 0.0063 | 0.45 | 1.0 | 0.79 | 71 | 0.1 | −0.4 |
J1745−0952 | 103.22 | 2.4 | 0.0074 | 0.22 | 46 | 35 | 29 | −0.7 | −0.9 |
J1748−2446A | 172.96 | 5.5 | 0.0041b | 0.21 | 36 | 28 | 50 | −0.4 | −0.9 |
J1748−3009 | 206.53 | 6.0 | 0.0026 | 0.21 | 27 | 21 | 79 | −0.4 | −0.8 |
J1750−2536 | 57.55 | 3.5 | 0.0035 | 0.41 | 410 | 320 | 120 | −0.8 | −1.0 |
J1751−2857 | 510.87 | 1.4 | 0.0095 | 0.52 | 2.7 | 2.1 | 55 | −0.1 | −0.6 |
J1753−1914 | 31.77 | 2.8 | 0.016 | 1.3 | 3400 | 2600 | 80 | −0.5 | −0.7 |
J1753−2240 | 21.02 | 3.5 | 0.0071 | 5.5 | 40,000 | 31,000 | 770 | −0.8 | −1.0 |
J1756−2251 | 70.27 | 0.7 | 0.066 | 0.36 | 50 | 39 | 5.4 | −0.5 | −1.0 |
J1757−27 | 113.08 | 5.4 | 0.0012 | 0.34 | 140 | 100 | 290 | −0.2 | −0.5 |
J1801−1417 | 551.71 | 1.8 | 0.0054 | 0.38 | 2.1 | 1.6 | 69 | −0.4 | −0.7 |
J1801−3210 | 268.33 | 5.1 | 0.000046 | 0.25 | 17 | 13 | 5500 | −0.3 | −0.8 |
J1802−2124 | 158.13 | 3.3 | 0.0058 | 0.17 | 21 | 16 | 29 | −0.7 | −0.9 |
J1804−0735 | 86.58 | 7.8 | 0.0029b | 0.28 | 280 | 210 | 97 | −0.7 | −0.9 |
J1804−2717 | 214.06 | 1.2 | 0.014 | 0.23 | 5.5 | 4.3 | 16 | 0.2 | −0.5 |
J1810+1744 | 1202.82 | 2.5 | 0.0053 | 0.49 | 0.80 | 0.61 | 92 | −0.6 | −0.9 |
J1811−2405 | 751.71 | 1.7 | 0.011 | 0.30 | 0.86 | 0.66 | 28 | −0.7 | −0.9 |
J1813−2621 | 451.47 | 3.4 | 0.0040 | 0.24 | 3.7 | 2.9 | 60 | −0.5 | −0.8 |
J1823−3021A | 367.65 | 8.6 | 0.023 | 0.27 | 23 | 17 | 11 | −0.3 | −0.9 |
J1824−2452A | 654.81 | 5.1 | 0.036 | 0.45 | 5.5 | 4.2 | 12 | −0.0 | −0.5 |
J1825−0319 | 439.22 | 3.3 | 0.0031 | 0.26 | 4.1 | 3.2 | 83 | −0.5 | −0.8 |
J1826−1334 | 19.71 | 4.1 | 1.7 | 12 | 120,000 | 93,000 | 7.2 | −0.3 | −0.7 |
J1828−1101 | 27.76 | 7.3 | 0.50 | 3.3 | 30,000 | 23,000 | 6.7 | −0.6 | −0.9 |
J1832−0836 | 735.53 | 1.4 | 0.010 | 0.34 | 0.83 | 0.64 | 34 | −0.6 | −0.9 |
J1833−0827 | 23.45 | 4.5 | 0.62 | 17 | 130,000 | 100,000 | 28 | 12.4 | −5.5 |
J1837−0604 | 20.77 | 6.2 | 0.89 | 7.9 | 110,000 | 83,000 | 8.8 | −0.4 | −0.7 |
J1840−0643 | 56.21 | 6.7 | 0.0020 | 0.33 | 660 | 510 | 160 | −1.0 | −1.1 |
J1843−1113 | 1083.62 | 2.0 | 0.0094 | 0.46 | 0.73 | 0.56 | 49 | −0.7 | −1.0 |
J1845−0743 | 19.10 | 5.8 | 0.082 | 6.5 | 98,000 | 76,000 | 79 | −0.8 | −1.0 |
J1853+1303 | 488.78 | 1.6 | 0.0074 | 0.32 | 2.0 | 1.6 | 43 | −0.6 | −0.9 |
J1853−0004 | 19.72 | 6.6 | 0.29 | 18 | 280,000 | 220,000 | 62 | −0.2 | −0.6 |
J1856+0245 | 24.72 | 10.3 | 0.69 | 2.6 | 41,000 | 32,000 | 3.7 | −0.8 | −1.0 |
J1857+0943 | 372.99 | 0.7 | 0.021 | 0.51 | 2.4 | 1.9 | 24 | 0.3 | −0.6 |
J1903+0327 | 930.27 | 6.5 | 0.0037 | 0.41 | 2.9 | 2.2 | 110 | −0.8 | −1.0 |
J1903−7051 | 555.88 | 1.1 | 0.029 | 0.42 | 1.4 | 1.1 | 14 | −0.3 | −0.8 |
J1909−3744 | 678.63 | 1.1 | 0.015 | 0.28 | 0.65 | 0.50 | 18 | −0.6 | −0.9 |
J1910+1256 | 401.32 | 1.9 | 0.0058 | 0.30 | 3.4 | 2.7 | 52 | −0.6 | −0.9 |
J1910−5959A | 612.33 | 4.5 | 0.0051b | 0.26 | 2.9 | 2.2 | 50 | 0.0 | −0.5 |
J1910−5959C | 378.98 | 4.5 | 0.0051b | 0.21 | 6.3 | 4.9 | 42 | −0.4 | −1.2 |
J1910−5959D | 221.35 | 4.5 | 0.0051b | 0.16 | 14 | 11 | 32 | −0.3 | −0.8 |
J1911+1347 | 432.34 | 1.6 | 0.0096 | 0.29 | 2.4 | 1.8 | 30 | −0.4 | −0.7 |
J1911−1114 | 551.61 | 1.6 | 0.010 | 0.27 | 1.3 | 1.0 | 27 | −0.6 | −1.2 |
J1915+1606 | 33.88 | 7.1 | 0.014 | 0.86 | 5000 | 3900 | 63 | −0.8 | −1.0 |
J1918−0642 | 261.58 | 0.9 | 0.016 | 0.16 | 2.1 | 1.6 | 10 | −0.7 | −0.9 |
J1923+2515 | 527.96 | 1.0 | 0.013 | 0.25 | 0.83 | 0.64 | 19 | −0.7 | −1.1 |
J1925+1721 | 26.43 | 9.6 | 0.31 | 3.0 | 38,000 | 30,000 | 9.5 | −0.6 | −0.7 |
J1932+17 | 47.81 | 2.7 | 0.023 | 1.3 | 1400 | 1100 | 54 | 1.6 | 0.9 |
J1935+2025 | 24.96 | 8.6 | 0.81 | 3.3 | 44,000 | 34,000 | 4.1 | −0.6 | −1.0 |
J1939+2134 | 1283.86 | 1.5 | 0.044 | 0.48 | 0.42 | 0.32 | 11 | −0.7 | −1.0 |
J1943+2210 | 393.38 | 8.3 | 0.0013 | 0.26 | 13 | 10 | 200 | −0.4 | −0.8 |
J1944+0907 | 385.71 | 1.3 | 0.012 | 0.30 | 2.4 | 1.9 | 26 | −0.4 | −0.7 |
J1946+3417 | 630.89 | 6.4 | 0.0013 | 0.32 | 4.9 | 3.8 | 260 | −0.5 | −0.8 |
J1949+3106 | 152.23 | 7.8 | 0.0028 | 0.20 | 64 | 49 | 72 | −0.5 | −0.8 |
J1950+2414 | 464.60 | 7.3 | 0.0023 | 0.36 | 11 | 8.9 | 150 | −0.2 | −0.7 |
J1955+2527 | 410.44 | 9.1 | 0.0012 | 0.35 | 18 | 14 | 280 | −0.0 | −0.6 |
J1955+2908 | 326.10 | 5.4 | 0.0033 | 0.19 | 9.2 | 7.1 | 58 | −0.7 | −0.9 |
J1959+2048 | 1244.24 | 1.5 | 0.017 | 0.74 | 0.69 | 0.54 | 44 | −0.4 | −0.8 |
J2007+2722 | 81.64 | 6.8 | 0.0074 | 0.39 | 380 | 290 | 53 | −0.4 | −0.9 |
J2010−1323 | 382.90 | 1.3 | 0.0060 | 0.28 | 2.3 | 1.8 | 46 | −0.5 | −0.8 |
J2017+0603 | 690.56 | 1.3 | 0.010 | 0.46 | 1.2 | 0.92 | 45 | −0.4 | −1.1 |
J2019+2425 | 508.32 | 0.9 | 0.012 | 0.55 | 1.8 | 1.4 | 46 | −0.1 | −0.7 |
J2033+1734 | 336.19 | 1.4 | 0.0081 | 0.27 | 3.1 | 2.4 | 34 | −0.6 | −1.0 |
J2043+1711 | 840.38 | 1.2 | 0.0096 | 0.32 | 0.53 | 0.41 | 33 | −0.6 | −1.0 |
J2047+1053 | 466.64 | 2.2 | 0.0080 | 0.21 | 2.0 | 1.6 | 26 | −0.7 | −0.9 |
J2051−0827 | 443.59 | 1.3 | 0.011 | 0.21 | 1.3 | 1.0 | 20 | −0.7 | −0.9 |
J2124−3358 | 405.59 | 0.4 | 0.041 | 0.29 | 0.68 | 0.52 | 7.1 | −0.3 | −1.0 |
J2129+1210A | 18.07 | 12.9 | 0.0018b | 14 | 530,000 | 410,000 | 8000 | −0.6 | −0.8 |
J2129+1210B | 35.63 | 12.9 | 0.0018b | 0.90 | 8600 | 6700 | 510 | −1.1 | −1.2 |
J2129+1210C | 65.51 | 10.0 | 0.010a | 0.34 | 970 | 750 | 33 | −0.8 | −1.0 |
J2129+1210D | 416.42 | 12.9 | 0.0018b | 0.23 | 16 | 12 | 130 | −0.6 | −0.9 |
J2129+1210E | 429.97 | 12.9 | 0.0018b | 0.34 | 22 | 17 | 190 | −0.1 | −0.5 |
J2129−5721 | 536.72 | 3.2 | 0.0060 | 0.28 | 2.9 | 2.3 | 47 | 0.1 | −0.4 |
J2145−0750 | 124.59 | 0.5 | 0.021 | 0.19 | 6.2 | 4.8 | 9.2 | −0.4 | −0.8 |
J2214+3000 | 641.18 | 1.0 | 0.018 | 0.56 | 1.3 | 1.0 | 32 | −0.1 | −0.6 |
J2222−0137 | 60.94 | 0.3 | 0.040 | 0.30 | 21 | 16 | 7.5 | −1.1 | −1.2 |
J2229+2643 | 671.63 | 1.4 | 0.0040 | 0.39 | 1.2 | 0.90 | 96 | −0.4 | −0.9 |
J2234+06 | 559.19 | 1.1 | 0.013 | 0.32 | 1.1 | 0.85 | 25 | −0.5 | −0.8 |
J2235+1506 | 33.46 | 1.1 | 0.011 | 0.80 | 780 | 600 | 71 | −1.0 | −1.1 |
J2241−5236 | 914.62 | 0.7 | 0.021 | 0.45 | 0.35 | 0.27 | 21 | −0.5 | −0.9 |
J2302+4442 | 385.18 | 0.8 | 0.018 | 0.19 | 0.89 | 0.69 | 11 | −0.1 | −0.4 |
J2317+1439 | 580.51 | 1.9 | 0.0036 | 0.40 | 2.1 | 1.6 | 110 | −0.2 | −0.6 |
J2322+2057 | 415.94 | 0.8 | 0.015 | 0.25 | 1.1 | 0.84 | 17 | −0.6 | −0.8 |
Notes. This does not include the high-value targets already listed in Table 2. For PSR J0023+0923 and PSR J0340+4130, intrinsic period derivatives are available in the ATNF pulsar catalog (v. 1.54) (Manchester et al. 2005); however, they are incorrect and therefore the spin-down limits have been calculated using the observed spin-down. For eight pulsars in the globular cluster 47 Tuc (PSRs J0024−7204E, H, Q, S, T, U, X, and Y). we have obtained (P. C. C. Freire 2016, private communication) intrinsic period derivatives to calculate the spin-down limits, with that for X being the upper limit from Ridolfi et al. (2016) given that it gives a characteristic age older than 109 years. For PSR J1823−3021A (in globular cluster NGC 6624) and PSR J1824−2452A (in globular cluster M28), we follow Freire et al. (2011) and Johnson et al. (2013) and calculate the spin-down limit assuming that the contributions to the observed are negligibly affected by cluster accelerations. The intrinsic spin-down for PSR J2129+1210C (in globular cluster M15) is taken from McNamara et al. (2004), which shows that the observed spin-down is negligibly affected by accelerations (it is in the outskirts of the cluster as is shown in Anderson 1993). The following pulsars use distance estimates that are not taken from the values given in the ATNF pulsar catalog: PSR J1017−7156 (updated parallax distance provided by R. M. Shannon 2016, private communication), PSR J1418−6058 (distance to more distant association in Yadigaroglu & Romani 1997), PSR J1813−1246 (lower limit on distance from Marelli et al. 2014), PSR J1823−3021A (distance for NGC 6624 in Valenti et al. 2007), PSR J1824−2452A (distance for M28 in Rees & Cudworth 1991), PSR J1826−1256 (lower distance range from Wang 2011; Voisin et al. 2016), PSRs J1910−5959A, C, and D (distances of 4.45 kpc calculated from the distance modulus to NGC 6752 in Table 4 of Gratton et al. 2003), PSR J2129+1210C (McNamara et al. 2004), and PSR J2234+06 (P. C. C. Freire 2016, private communication).
aThe pulsar's spin-down is corrected for proper motion effects. bThe pulsar's spin-down is calculated using a characteristic spin-down age of 109 years and a braking index, n, of 5 (i.e., braking due to gravitational radiation). cUses a restricted prior on orientation parameters (see Appendix B).Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.
When these restricted priors were used in the previous analyses of Aasi et al. (2014), there has been an implicit (and at the time unrealized) assumption about the rotation of the star. As noted in Jones (2015), constraining ι and ψ to particular values implicitly forces a rotation direction on the signal, while the PWN observations (or indeed the electromagnetic timing observations) give us no knowledge of the actual rotation direction. To incorporate this unknown rotation direction in the search, while maintaining the convenient minimal range in ψ of radians, there must be a bimodal distribution on ι with the additional mode at radians. The mean and standard deviations of Gaussian prior distributions used for ψ and the two modes for ι are given in Table 3.
Footnotes
- 154
This is known to be a naïve limit. For several young pulsars where the braking index (see Section 4) is measured (Lyne et al. 2015; Archibald et al. 2016), we know that it is not consistent with pure gravitational-wave emission, and other energy-loss mechanisms can be dominant. Effects of this on spin-down limit calculations are discussed in Palomba (2000). Figures 9 and 10 of Abdo et al. (2013) also show that for pulsars observed as Fermi gamma-ray sources, a not insignificant proportion of their spin-down luminosity is emitted through gamma-rays.
- 155
The sensitivity is taken as , where is the harmonic mean of the observation-time-weighted one-sided power spectral densities, , for H1 and L1 (see https://dcc.ligo.org/LIGO-G1600150/public and https://dcc.ligo.org/LIGO-G1600151/public, respectively). The factor of 10.8 gives the 95% credible upper limit on gravitational-wave strain amplitude averaged over orientation angles assuming Gaussian noise (Dupuis & Woan 2005).
- 156
PSRs J0908−4913, J1418−6058, J1709−4429, J1826−1334, J1845−0743, J1853−0004, and J2129+1210A.
- 157
- 158
Of the 200 pulsars, 119 are in binary systems.
- 159
For the Bayesian results, these are credible limits bounded from zero, while for the frequentist results these are confidence limits.
- 160
- 161
For each source, a different prior volume was used, so directly comparing odds values between sources should be treated with caution.
- 162
This is for PSR J1833−0827 at 23.4 Hz, for which there appears to be a large amount of incoherent interference between the detectors.
- 163
In Owen (2006) and Pitkin (2011), it is stated that Hz s−1 is roughly the largest magnitude spin-down that could be masked by globular cluster accelerations. This is mainly based on the maximum observed spin-up for a globular cluster pulsar (PSR J2129+1210D) being Hz s−1 as given in v1.54 of the ATNF pulsar catalog (Manchester et al. 2005). However, this value appears to be wrong, with the original observations for PSR J2129+1210D (Anderson 1993) giving a value of just under Hz s−1. This is still the maximum observed spin-up for any globular cluster pulsar.
- 164
For this we have assumed a 1.4 star and used approximate scalings calculated from Table 1 of Johnson-McDaniel (2013), taking into account that the rotational deformation scales with .
- 165
One desirable source that we no longer have accurate timings for is PSR J0537−6910, an X-ray pulsar in the Large Magellanic Cloud, for which we relied on the now-defunct RXTE satellite.
- 166
- 167
- 168
A bimodal prior was not used in Aasi et al. (2014), but subsequently its inclusion was found to have minimal effect on the upper limits produced.
- 169
For two pulsars, PSR J0024−7204X and PSR J0721−2038, the priors set using an estimated 95% upper limit from the S6/VSR2,4 analysis were found to be too narrow and unduly narrowed the posterior. So, to maintain a more conservative upper limit dominated by the likelihood, as has been the case in previous searches, the priors were widened by a factor of three.
- 170
The hypothesis contains all the other hypotheses as its subsets, but within it the other hypotheses will all be downweighted by their tiny prior volumes in comparison to the full volume. Therefore, to provide more weight to the alternative noise hypotheses, we explicitly include them with equal weight.
- 171
ψ can be rotated by integer numbers of radians and still give signals within our search parameter space, although for each rotation any signal would have equivalently rotated by π radians.