SEARCHES FOR GRAVITATIONAL WAVES FROM KNOWN PULSARS WITH SCIENCE RUN 5 LIGO DATA

Within our Galaxy, some of the best targets for gravitational wave searches in the sensitive frequency band of current interferometric gravitational wave detectors (∼40–2000 Hz) are millisecond and young pulsars. There are currently just over 200 known pulsars with spin frequencies greater than 20 Hz, which therefore are within this band. In this paper, we describe the latest results from the ongoing search for gravitational waves from these known pulsars using data from the Laser Interferometric Gravitational-Wave Observatory (LIGO). As this search looks for objects with known positions and spin-evolutions, it can use long time spans of data in a fully coherent way to dig deeply into the detector noise. Here we use data from the entire two-year run of the three LIGO detectors, entitled Science Run 5 (S5), during which the detectors reached their design sensitivities (Abbott et al. 2009b). This run started on 2005 November 4 and ended on 2007 October 1. The detectors (the 4 km and 2 km detectors at LIGO Hanford Observatory, H1 and H2, and the 4 km detector at the LIGO Livingston Observatory, L1) had duty factors of 78% for H1, 79% for H2, and 66% for L1. The GEO 600 detector also participated in S5 (Grote & The LIGO Scientific Collaboration 2008), but at lower sensitivities that meant it was not able to enhance this search. The Virgo detector also had data overlapping with S5 during Virgo Science Run 1 (VSR1; Acernese et al. 2008). However, this was also generally at a lower sensitivity than the LIGO detectors and had an observation time of only about four months, meaning that no significant sensitivity improvements could be made by including these data. Due to its multi-stage seismic isolation system Virgo does have better sensitivity than the LIGO detectors below about 40 Hz, opening the possibility of searching for more young pulsars, including the Vela pulsar. These lower frequency searches will be explored more in the future.

This search assumes that the pulsars are triaxial stars emitting gravitational waves at precisely twice their observed spin frequencies, i.e., the emission mechanism is an ℓ = m = 2 quadrupole, and that gravitational waves are phase-locked with the electromagnetic signal. We use the so-called spin-down limit on strain tensor amplitude h^sd₀ as a sensitivity target for each pulsar in our analysis. This can be calculated, by assuming that the observed spin-down rate of a pulsar is entirely due to energy loss through gravitational radiation from an ℓ = m = 2 quadrupole, as

$$\begin{equation} h_0^{\rm sd} = 8.06\!\times \!10^{-19}I_{38}r^{-1}_{\rm kpc}(|\dot{\nu }|/\nu)^{1/2}, \end{equation} \tag{ 1 }$$

where I₃₈ is the pulsar's principal moment of inertia (I_zz) in units of 10³⁸ kg m², r_kpc is the pulsar distance in kpc, ν is the spin-frequency in Hz, and $\dot{\nu }$ is the spin-down rate in Hz s⁻¹. Due to uncertainties in I_zz and r, h^sd₀ is typically uncertain by about a factor of 2. Part of this is due to the uncertainty in I_zz which, though predicted to lie roughly in the range 1–3 × 10³⁸ kg m², has not been measured for any neutron star; and the best (though still uncertain) prospect is star A of the double pulsar system J0737 − 3039 with 20 years more observation (Kramer & Wex 2009). Distance estimates based on dispersion measure can also be wrong by a factor of 2–3, as confirmed by recent parallax observations of the double pulsar (Deller et al. 2009). For pulsars with measured braking indices, $n=\nu \ddot{\nu }/\dot{\nu }^2$, the assumption that spin-down is dominated by gravitational wave emission is known to be false (the braking index for quadrupolar gravitational wave emission should be 5, but all measured ns are less than 3) and a stricter indirect limit on gravitational wave emission can be set. A phenomenological investigation of some young pulsars (Palomba 2000) indicates that this limit is lower than h^sd₀ by a factor of 2.5 or more, depending on the pulsar. See Abbott et al. (2007, 2008) for more discussion of the uncertainties in indirect limits. Recycled millisecond pulsars have intrinsically small spin-downs, so for the majority of pulsars in our search these spin-down limits will be well below our current sensitivities, making detection unlikely. However, our search also covers four young pulsars with large spin-down luminosities, and for these we can potentially beat or reach their spin-down limits using current data.

The LIGO band covers the fastest (highest ν) known pulsars, and the quadrupole formula for strain tensor amplitude

$$\begin{equation} h_0 = 4.2\times 10^{-26} \nu _{100}^2 I_{38} \varepsilon _{-6} r_\mathrm{kpc}^{-1} \end{equation} \tag{ 2 }$$

indicates that these pulsars are the best gravitational wave emitters for a given equatorial ellipticity ε = (I_xx − I_yy)/I_zz (here ν₁₀₀ = ν/(100 Hz) and ε₋₆ = ε/10⁻⁶). The pulsars with high spin-downs are almost all less than ∼10⁴ years old. Usually this is interpreted as greater electromagnetic activity (including particle winds) in younger objects, but it could also mean that they are more active in gravitational wave emission. This is plausible on theoretical grounds too. Strong internal magnetic fields may cause significant ellipticities (Cutler 2002) which would then decay as the field decays or otherwise changes (Goldreich & Reisenegger 1992). The initial crust may be asymmetric if it forms on a timescale on which the neutron star is still perturbed by its violent formation and aftermath, including a possible lengthy perturbation due to the fluid r modes (Lindblom et al. 2000; Wu et al. 2001), and asymmetries may slowly relax due to mechanisms such as viscoelastic creep. Also the fluid r modes may remain unstable to gravitational wave emission for up to a few thousand years after the neutron star's birth, depending on its composition, viscosity, and initial spin frequency (Owen et al. 1998; Bondarescu et al. 2009). Such r modes are expected to have a gravitational wave frequency about 4/3 the spin frequency. However, we do not report on r-mode searches in this paper.

1.1. Previous Analyses

1.2. Electromagnetic Observations

The radio pulsar parameters used for our searches are based on ongoing radio pulsar monitoring programs, using data from the Jodrell Bank Observatory (JBO), the NRAO 100 m Green Bank Telescope (GBT), and the Parkes radio telescope of the Australia Telescope National Facility. We used radio data coincident with the S5 run as these would reliably represent the pulsars' actual phase evolution during our searches. We obtained data for 44 pulsars from JBO (including the Crab pulsar ephemeris; Lyne et al. 1993, 2009), 39 pulsars within the Terzan 5 and M28 globular clusters from GBT, and 47 from Parkes, including pulsars timed as part of the Parkes Pulsar Timing Array (Manchester 2008). For 15 of these pulsars there were observations from more than one site, making a total of 115 radio pulsars in the analysis (see Table 1 for list of the pulsars, including the observatory and time span of the observations). For the pulsars observed at JBO and Parkes, we have obtained parameters fit to data overlapping with the entire S5 run. For the majority of pulsars observed at GBT, the parameters have been fit to data overlapping approximately the first quarter of S5.

Table 1. Information on the Non-glitching Pulsars in Our Search, Including the Start and End Times of the Radio Observations Used in Producing the Parameter Fits for Our Search, and Upper Limit Results

Pulsar	Start–End (MJD)	ν (Hz)	$\dot{\nu }$ (Hz s−1)	Distance (kpc)	Spin-down Limit	Joint h95%0	Ellipticity	h95%0/hsd0
J0024 − 7204Ccp	48383–54261	173.71	1.5 × 10−15	4.9	6.55 × 10−28	5.88 × 10−25	2.26 × 10−5	898
J0024 − 7204Dcp	48465–54261	186.65	1.2 × 10−16	4.9	6.55 × 10−28	4.45 × 10−26	1.48 × 10−6	68
J0024 − 7204Ebcp	48465–54261	282.78	−7.9 × 10−15	4.9	6.55 × 10−28	9.97 × 10−26	1.44 × 10−6	152
J0024 − 7204Fcp	48465–54261	381.16	−9.4 × 10−15	4.9	6.55 × 10−28	8.76 × 10−26	6.98 × 10−7	134
J0024 − 7204Gcp	48600–54261	247.50	2.6 × 10−15	4.9	6.55 × 10−28	1.00 × 10−25	1.90 × 10−6	153
J0024 − 7204Hbcp	48518–54261	311.49	1.8 × 10−16	4.9	6.55 × 10−28	6.44 × 10−26	7.69 × 10−7	98
J0024 − 7204Ibcp	50684–54261	286.94	3.8 × 10−15	4.9	6.55 × 10−28	5.19 × 10−26	7.30 × 10−7	79
J0024 − 7204Jbcp	48383–54261	476.05	2.2 × 10−15	4.9	6.55 × 10−28	1.04 × 10−25	5.34 × 10−7	159
J0024 − 7204Lcp	50687–54261	230.09	6.5 × 10−15	4.9	6.55 × 10−28	5.82 × 10−26	1.27 × 10−6	89
J0024 − 7204Mcp	48495–54261	271.99	2.8 × 10−15	4.9	6.55 × 10−28	6.14 × 10−26	9.61 × 10−7	94
J0024 − 7204Ncp	48516–54261	327.44	2.4 × 10−15	4.9	6.55 × 10−28	8.35 × 10−26	9.02 × 10−7	128
J0024 − 7204Qbcp	50690–54261	247.94	−2.1 × 10−15	4.9	6.55 × 10−28	5.74 × 10−26	1.08 × 10−6	88
J0024 − 7204Rbcp	50743–54261	287.32	−1.2 × 10−14	4.9	6.55 × 10−28	5.53 × 10−26	7.76 × 10−7	84
J0024 − 7204Sbcp	50687–54241	353.31	1.5 × 10−14	4.9	6.55 × 10−28	6.82 × 10−26	6.33 × 10−7	104
J0024 − 7204Tbcp	50684–54261	131.78	−5.1 × 10−15	4.9	6.55 × 10−28	3.34 × 10−26	2.23 × 10−6	51
J0024 − 7204Ubcp	48516–54261	230.26	−5.0 × 10−15	4.9	6.55 × 10−28	5.63 × 10−26	1.23 × 10−6	86
J0024 − 7204Ybcp	51504–54261	455.24	7.3 × 10−15	4.9	6.55 × 10−28	9.42 × 10−26	5.26 × 10−7	144
J0218+4232bj	49092–54520	430.46	−1.4 × 10−14a	5.8	7.91 × 10−28	1.47 × 10−25	1.10 × 10−6	186
J0407+1607bj	52719–54512	38.91	−1.2 × 10−16	4.1	3.48 × 10−28	6.18 × 10−26	3.93 × 10−5	178
J0437 − 4715bp	53683–54388	173.69	−4.7 × 10−16a	0.1	8.82 × 10−27	5.73 × 10−25	6.74 × 10−7	65
J0613 − 0200bjp	53406–54520	326.60	−9.8 × 10−16a	0.5	2.91 × 10−27	1.11 × 10−25	1.18 × 10−7	38
J0621+1002bj	52571–54516	34.66	−5.5 × 10−17a	1.9	5.40 × 10−28	1.53 × 10−25	5.65 × 10−5	284
J0711 − 6830p	53687–54388	182.12	−2.7 × 10−16a	1.0	9.52 × 10−28	5.00 × 10−26	3.71 × 10−7	53
J0737 − 3039Abj	53595–54515	44.05	−3.4 × 10−15a	1.1	6.17 × 10−27	7.87 × 10−26	1.10 × 10−5	13
J0751+1807bj	53405–54529	287.46	−6.3 × 10−16a	0.6	1.92 × 10−27	1.64 × 10−25	2.91 × 10−7	85
J1012+5307bj	53403–54523	190.27	−4.7 × 10−16a	0.5	2.43 × 10−27	6.94 × 10−26	2.36 × 10−7	29
J1022+1001bjp	53403–54521	60.78	−1.6 × 10−16	0.4	3.27 × 10−27	4.44 × 10−26	1.14 × 10−6	14
J1024 − 0719jp	53403–54501	193.72	−6.9 × 10−16	0.5	2.88 × 10−27	5.01 × 10−26	1.67 × 10−7	17
J1045 − 4509bp	53688–54386	133.79	−2.0 × 10−16a	3.2	3.00 × 10−28	4.37 × 10−26	1.87 × 10−6	145
J1455 − 3330bj	52688–54524	125.20	−2.5 × 10−16a	0.7	1.53 × 10−27	5.15 × 10−26	5.75 × 10−7	34
J1600 − 3053bp	53688–54386	277.94	−6.5 × 10−16a	2.7	4.62 × 10−28	5.57 × 10−26	4.55 × 10−7	121
J1603 − 7202bp	53688–54385	67.38	−5.9 × 10−17a	1.6	4.62 × 10−28	2.32 × 10−26	1.98 × 10−6	50
J1623 − 2631bcj	53403–54517	90.29	−5.5 × 10−15	2.2	1.46 × 10−27	5.81 × 10−26	3.71 × 10−6	40
J1640+2224bj	53410–54506	316.12	−1.6 × 10−16a	1.2	4.86 × 10−28	6.65 × 10−26	1.87 × 10−7	137
J1643 − 1224bjp	52570–54517	216.37	−6.8 × 10−16a	4.9	2.94 × 10−28	4.35 × 10−26	1.07 × 10−6	148
J1701 − 3006Abcp	53590–54391	190.78	4.8 × 10−15	6.9	4.65 × 10−28	5.82 × 10−26	2.61 × 10−6	125
J1701 − 3006Bbcp	53650–54391	278.25	2.7 × 10−14	6.9	4.65 × 10−28	7.63 × 10−26	1.61 × 10−6	164
J1701 − 3006Cbcp	53590–54396	131.36	1.1 × 10−15	6.9	4.65 × 10−28	3.52 × 10−26	3.32 × 10−6	76
J1713+0747bjp	53406–54509	218.81	−3.8 × 10−16a	1.1	9.54 × 10−28	4.44 × 10−26	2.45 × 10−7	47
J1730 − 2304jp	52571–54519	123.11	−3.1 × 10−16	0.5	2.49 × 10−27	5.93 × 10−26	4.72 × 10−7	24
J1732 − 5049bp	53725–54386	188.23	−4.9 × 10−16	1.8	7.18 × 10−28	5.25 × 10−26	6.34 × 10−7	73
J1744 − 1134jp	52604–54519	245.43	−4.1 × 10−16a	0.5	2.18 × 10−27	1.10 × 10−25	2.07 × 10−7	50
J1748 − 2446Abcjg	52320–54453	86.48	2.2 × 10−16	5.5	5.83 × 10−28	3.89 × 10−26	6.77 × 10−6	67
J1748 − 2446Ccjg	53403–54516	118.54	8.5 × 10−15	5.5	5.83 × 10−28	5.00 × 10−26	4.63 × 10−6	86
J1748 − 2446Dcg	50851–53820	212.13	−5.7 × 10−15	5.5	5.83 × 10−28	6.78 × 10−26	1.96 × 10−6	116
J1748 − 2446Ebcg	53193–53820	455.00	3.8 × 10−15	5.5	5.83 × 10−28	8.95 × 10−26	5.62 × 10−7	153
J1748 − 2446Fcg	53193–53820	180.50	−1.3 × 10−16	5.5	5.83 × 10−28	8.37 × 10−26	3.34 × 10−6	143
J1748 − 2446Gcg	51884–53820	46.14	−8.4 × 10−16	5.5	5.83 × 10−28	5.82 × 10−26	3.56 × 10−5	100
J1748 − 2446Hcg	51884–53820	203.01	3.4 × 10−15	5.5	5.83 × 10−28	7.81 × 10−26	2.46 × 10−6	134
J1748 − 2446Ibcg	50851–54195	104.49	7.3 × 10−16	5.5	5.83 × 10−28	3.54 × 10−26	4.21 × 10−6	61
J1748 − 2446Kcg	51884–53820	336.74	1.1 × 10−14	5.5	5.83 × 10−28	6.67 × 10−26	7.65 × 10−7	114
J1748 − 2446Lcg	51884–53820	445.49	3.4 × 10−15	5.5	5.83 × 10−28	1.39 × 10−25	9.09 × 10−7	238
J1748 − 2446Mbcg	51884–53820	280.15	−3.9 × 10−14	5.5	5.83 × 10−28	1.01 × 10−25	1.68 × 10−6	173
J1748 − 2446Nbcg	53193–54195	115.38	−7.4 × 10−15	5.5	5.83 × 10−28	5.85 × 10−26	5.71 × 10−6	100
J1748 − 2446Obcg	52500–53957	596.43	2.5 × 10−14	5.5	5.83 × 10−28	2.65 × 10−25	9.68 × 10−7	454
J1748 − 2446Pbcg	53193–54557	578.50	−8.7 × 10−14	5.5	5.83 × 10−28	1.56 × 10−25	6.08 × 10−7	267
J1748 − 2446Qbcg	53193–54139	355.62	4.6 × 10−15	5.5	5.83 × 10−28	8.80 × 10−26	9.05 × 10−7	151
J1748 − 2446Rcg	52500–53820	198.86	−1.9 × 10−14	5.5	5.83 × 10−28	8.23 × 10−26	2.71 × 10−6	141
J1748 − 2446Scg	53193–53820	163.49	−1.7 × 10−15	5.5	5.83 × 10−28	4.46 × 10−26	2.17 × 10−6	76
J1748 − 2446Tcg	51884–53819	141.15	−6.1 × 10−15	5.5	5.83 × 10−28	5.12 × 10−26	3.34 × 10−6	88
J1748 − 2446Vbcg	53193–53820	482.51	2.2 × 10−14	5.5	5.83 × 10−28	1.26 × 10−25	7.04 × 10−7	216
J1748 − 2446Wbcg	52500–53820	237.80	−7.1 × 10−15	5.5	5.83 × 10−28	9.57 × 10−26	2.20 × 10−6	164
J1748 − 2446Xbcg	51884–54139	333.44	−6.5 × 10−15	5.5	5.83 × 10−28	8.18 × 10−26	9.57 × 10−7	140
J1748 − 2446Ybcg	53193–53820	488.24	−4.0 × 10−14	5.5	5.83 × 10−28	2.10 × 10−25	1.15 × 10−6	360
J1748 − 2446Zbcg	53193–54139	406.08	1.4 × 10−14	5.5	5.83 × 10−28	8.43 × 10−26	6.65 × 10−7	145
J1748 − 2446aacg	51884–53819	172.77	1.3 × 10−14	5.5	5.83 × 10−28	2.28 × 10−25	9.92 × 10−6	391
J1748 − 2446abcg	51884–53819	195.32	−1.6 × 10−14	5.5	5.83 × 10−28	4.67 × 10−26	1.59 × 10−6	80
J1748 − 2446accg	52500–53819	196.58	−8.8 × 10−15	5.5	5.83 × 10−28	7.19 × 10−26	2.42 × 10−6	123
J1748 − 2446adbcg	53204–54557	716.36	1.7 × 10−14	5.5	5.83 × 10−28	1.77 × 10−25	4.48 × 10−7	303
J1748 − 2446aebcg	53193–53820	273.33	4.3 × 10−14	5.5	5.83 × 10−28	6.57 × 10−26	1.14 × 10−6	113
J1748 − 2446afcg	53193–53820	302.63	2.1 × 10−14	5.5	5.83 × 10−28	1.07 × 10−25	1.52 × 10−6	183
J1748 − 2446agcg	53193–53819	224.82	−6.3 × 10−16	5.5	5.83 × 10−28	9.49 × 10−26	2.44 × 10−6	163
J1748 − 2446ahcg	53193–53819	201.40	−2.3 × 10−14	5.5	5.83 × 10−28	5.49 × 10−26	1.76 × 10−6	94
J1756 − 2251bj	53403–54530	35.14	−1.3 × 10−15	2.9	1.65 × 10−27	9.70 × 10−26	5.42 × 10−5	59
J1801 − 1417j	53405–54505	275.85	−4.0 × 10−16	1.8	5.42 × 10−28	6.15 × 10−26	3.44 × 10−7	113
J1803 − 30cp	53654–54379	140.83	−1.0 × 10−15	7.8	4.11 × 10−28	5.51 × 10−26	5.12 × 10−6	134
J1804 − 0735bcj	52573–54518	43.29	−8.8 × 10−16	8.4	3.82 × 10−28	8.44 × 10−26	8.95 × 10−5	221
J1804 − 2717bj	52574–54453	107.03	−4.7 × 10−16	1.2	1.44 × 10−27	2.40 × 10−26	5.79 × 10−7	17
J1807 − 2459Abcp	53621–54462	326.86	4.8 × 10−16	2.7	1.19 × 10−27	1.53 × 10−25	9.13 × 10−7	129
J1810 − 2005bj	53406–54508	30.47	−1.4 × 10−16	4.0	4.28 × 10−28	2.22 × 10−25	2.28 × 10−4	519
J1823 − 3021Acj	53403–54530	183.82	−1.1 × 10−13	7.9	4.06 × 10−28	3.93 × 10−26	2.17 × 10−6	97
J1824 − 2452Acjpg	53403–54509	327.41	−1.7 × 10−13	4.9	3.79 × 10−27	7.80 × 10−26	8.43 × 10−7	21
J1824 − 2452Bcg	53629–54201	152.75	5.6 × 10−15	4.9	6.55 × 10−28	4.26 × 10−26	2.11 × 10−6	65
J1824 − 2452Cbcg	52335–54202	240.48	−9.8 × 10−15	4.9	6.55 × 10−28	6.48 × 10−26	1.30 × 10−6	99
J1824 − 2452Ecg	53629–54201	184.53	3.7 × 10−15	4.9	6.55 × 10−28	7.51 × 10−26	2.55 × 10−6	115
J1824 − 2452Fcg	52497–54114	407.97	−1.6 × 10−15	4.9	6.55 × 10−28	9.74 × 10−26	6.78 × 10−7	149
J1824 − 2452Gbcg	53629–54202	169.23	−5.2 × 10−15	4.9	6.55 × 10−28	7.23 × 10−26	2.93 × 10−6	110
J1824 − 2452Hbcg	53629–54202	216.01	−3.6 × 10−15	4.9	6.55 × 10−28	8.27 × 10−26	2.05 × 10−6	126
J1824 − 2452Jbcg	53629–54201	247.54	4.7 × 10−15	4.9	6.55 × 10−28	1.07 × 10−25	2.03 × 10−6	163
J1841+0130bj	53405–54513	33.59	−9.2 × 10−15	3.2	4.19 × 10−27	1.65 × 10−25	1.10 × 10−4	39
J1843 − 1113j	53353–54508	541.81	−2.8 × 10−15	2.0	9.33 × 10−28	1.64 × 10−25	2.61 × 10−7	176
J1857+0943bjp	53409–54517	186.49	−6.0 × 10−16a	0.9	1.59 × 10−27	7.27 × 10−26	4.50 × 10−7	46
J1905+0400j	53407–54512	264.24	−3.4 × 10−16	1.3	6.81 × 10−28	7.40 × 10−26	3.36 × 10−7	109
J1909 − 3744bp	53687–54388	339.32	−3.1 × 10−16a	1.1	6.78 × 10−28	8.09 × 10−26	1.89 × 10−7	119
J1910 − 5959Abcp	53666–54380	306.17	−2.8 × 10−16	4.5	7.13 × 10−28	7.71 × 10−26	8.75 × 10−7	108
J1910 − 5959Bcp	53609–54473	119.65	1.1 × 10−14	4.5	7.13 × 10−28	3.81 × 10−26	2.83 × 10−6	53
J1910 − 5959Ccp	53666–54390	189.49	−7.8 × 10−17	4.5	7.13 × 10−28	4.34 × 10−26	1.29 × 10−6	61
J1910 − 5959Dcp	53621–54460	110.68	−1.2 × 10−14	4.5	7.13 × 10−28	3.03 × 10−26	2.63 × 10−6	42
J1910 − 5959Ecp	53610–54441	218.73	2.1 × 10−14	4.5	7.13 × 10−28	4.77 × 10−26	1.06 × 10−6	67
J1911+1347j	53403–54530	216.17	−8.0 × 10−16	1.6	9.63 × 10−28	7.00 × 10−26	5.70 × 10−7	73
J1911 − 1114bj	53407–54512	275.81	−4.8 × 10−16a	1.6	6.66 × 10−28	5.62 × 10−26	2.78 × 10−7	84
J1913+1011j	53745–54911	27.85	−2.6 × 10−12	4.5	5.51 × 10−26	2.14 × 10−25	2.93 × 10−4	3.9
J1939+2134jp	53407–54519	641.93	−4.3 × 10−14a	3.5	1.86 × 10−27	1.79 × 10−25	3.65 × 10−7	96
J1955+2908bj	53403–54524	163.05	−7.6 × 10−16a	5.4	3.23 × 10−28	7.07 × 10−26	3.39 × 10−6	219
J2019+2425bj	53599–54505	254.16	−1.7 × 10−16a	0.9	7.14 × 10−28	9.23 × 10−26	3.07 × 10−7	129
J2033+17bj	53702–54522	168.10	−3.1 × 10−16	1.4	7.93 × 10−28	7.49 × 10−26	8.65 × 10−7	94
J2051 − 0827bj	53410–54520	221.80	−6.1 × 10−16a	1.3	1.04 × 10−27	7.57 × 10−26	4.65 × 10−7	73
J2124 − 3358jp	53410–54510	202.79	−5.1 × 10−16a	0.2	5.13 × 10−27	4.85 × 10−26	6.96 × 10−8	9.4
J2129 − 5721bp	53687–54388	268.36	−2.0 × 10−15a	2.5	8.71 × 10−28	6.12 × 10−26	5.13 × 10−7	70
J2145 − 0750bjp	53409–54510	62.30	−1.0 × 10−16a	0.5	2.05 × 10−27	3.83 × 10−26	1.17 × 10−6	19
J2229+2643bj	53403–54524	335.82	−1.6 × 10−16	1.4	3.95 × 10−28	9.89 × 10−26	2.96 × 10−7	250
J2317+1439bj	53406–54520	290.25	−1.3 × 10−16a	1.9	2.82 × 10−28	8.83 × 10−26	4.68 × 10−7	313
J2322+2057j	53404–54519	207.97	−1.8 × 10−16a	0.8	9.55 × 10−28	1.12 × 10−25	4.78 × 10−7	117

Notes. ^aThe pulsar's spin-down is corrected for proper motion effects. ^bThe pulsar is within a binary system. ^cThe pulsar is within a globular cluster. ^gThe pulsar was observed by the GBT. ^jThe pulsar was observed by the JBO. ^pThe pulsar was observed by the Parkes Observatory.

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Pulsars generally exhibit timing noise on long timescales. Over tens of years this can cause correlations in the pulse time of arrivals which can give systematic errors in the parameter fits produced, by the standard pulsar timing package TEMPO,⁹⁰ of order 2–10 times the regular errors that TEMPO assigns to each parameter (Verbiest et al. 2008), depending on the amplitude of the noise. For our pulsars, with relatively short observation periods of around two years, the long-term timing noise variations should be largely folded in to the parameter fitting, leaving approximately white uncorrelated residuals. Also millisecond pulsars, in general, have intrinsically low levels of timing noise, showing comparatively white residuals. This should mean that the errors produced by TEMPO are approximately the true 1σ errors on the fitted values.

The regular pulse timing observations of the Crab pulsar (Lyne et al. 1993, 2009) indicate that the 2006 August 23 glitch was the only glitch during the S5 run. One other radio pulsar, J1952+3252, was observed to glitch during the run (see Section 5.1.3). Independent ephemerides are available before and after each glitch.

We include one pulsar in our analysis that is not observed as a radio pulsar. This is PSR J0537 − 6910 in the Large Magellanic Cloud, for which only X-ray timings currently exist. Data for this source come from dedicated time on the Rossi X-ray Timing Explorer (RXTE; Middleditch et al. 2006) giving ephemerides covering the whole of S5. These ephemerides comprise seven inter-glitch segments, each of which produces phase-stable timing solutions. The segments are separated by times when the pulsar was observed to glitch. Due to the complexity of the pulsar behavior near glitches, which is not reflected in the simple model used to predict pulse times of arrival, sometimes up to ∼30 days around them are not covered by the ephemerides.

The details of the search method are discussed in Dupuis & Woan (2005) and Abbott et al. (2007), but we will briefly review them here. Data from the gravitational wave detectors are heterodyned using twice the known electromagnetic phase evolution of each pulsar, which removes this rapidly varying component of the signal, leaving only the daily varying amplitude modulation caused by each detector's antenna response. Once heterodyned the (now complex) data are low-pass filtered at 0.25 Hz, and then heavily down-sampled, by averaging, from the original sample rate of 16 384 Hz to 1/60 Hz. Using these down-sampled data (B_k, where k represents the kth sample), we perform parameter estimation over the signal model y_k(a) given the unknown signal parameters a. This is done by calculating the posterior probability distribution (Abbott et al. 2007)

$$\begin{eqnarray} p(\boldsymbol{a}|\lbrace B_k\rbrace) &\propto& \prod _j^M \bigg(\sum _{k}^{n} (\Re \lbrace B_k\rbrace -\Re \lbrace y_k(\boldsymbol{a}) \rbrace)^2 \nonumber \\ &&+ (\Im \lbrace B_k\rbrace -\Im \lbrace y_k(\boldsymbol{a})\rbrace)^2\bigg)^{-m_j} \times p(\boldsymbol{a}), \end{eqnarray} \tag{ 3 }$$

where the first term on the right-hand side is the likelihood (marginalized over the data variance, giving a Student's-t-like distribution), p(a) is the prior distribution for a, M is the number of data segments into which the B_ks have been cut (we assume stationarity of the data during each segment), m_j is the number of data points in the jth segment (with a maximum value of 30, i.e., we only assume stationarity for periods less than, or equal to, 30 minutes in length), and n = ∑^j_{j = 1}m_j. The assumption of Gaussianity and stationarity of the segments holds well for this analysis (see Section 4.5 of Dupuis (2004) for examples of χ² and Kolmogorov–Smirnov tests performed to assess these in previous analyses).

We have previously (Abbott et al. 2004, 2005, 2007) performed parameter estimation over the four unknown gravitational wave signal parameters of amplitude h₀, initial phase ϕ₀, cosine of the orientation angle cos ι, and polarization angle ψ, giving a = {h₀, ϕ₀, cos ι, ψ}. Priors on each parameter are set to be uniform over their allowed ranges, with the upper end of the range for h₀ set empirically from the noise level of the data. We choose a uniform prior on h₀ for consistency with our previous analyses (Dupuis & Woan 2005; Abbott et al. 2004, 2005, 2007) and to facilitate straightforward comparison of sensitivity. Extensive trials with software injections have shown this to be a very reasonable choice, returning a conservative (i.e., high) upper limit consistent with the data and any possible signal.

Using a uniformly spaced grid on this four-dimensional parameter space the posterior is calculated at each point. To obtain a posterior for each individual parameter we marginalize over the three others. Using the marginalized posterior on h₀ we can set an upper limit by calculating the value that, integrating up from zero, bounds the required cumulative probability (which we have taken as 95%). We also combine the data from multiple detectors to give a joint posterior. To do this, we simply take the product of the likelihoods for each detector and multiply this joint likelihood by the prior. This is possible due to the phase coherence between detectors. Again we can marginalize to produce posteriors for individual parameters.

Below, in Section 2.1, we discuss exploring and expanding this parameter space to more dimensions using a Markov Chain Monte Carlo (MCMC) technique.

2.1. MCMC Parameter Search

When high resolutions are needed it can be computationally time consuming to calculate the posterior over an entire grid as described above, and redundant areas of parameter space with very little probability are explored for a disproportionately large amount of time. A more efficient way to carry out such a search is with an MCMC technique, in which the parameter space is explored more efficiently and without spending much time in the areas with very low probability densities.

An MCMC integration explores the parameter space by stepping from one position in parameter space to another, comparing the posterior probability of the two points and using a simple algorithm to determine whether the step should be accepted. If accepted it moves to that new position and repeats; if it is rejected it stays at the current position and repeats. Each iteration of the chain, whether it stays in the same position or not, is recorded and the amount of time the chain spends in a particular part of parameter space is directly proportional to the posterior probability density there. The new points are drawn randomly from a specific proposal distribution, often given by a multivariate Gaussian with a mean set as the current position, and a predefined covariance. For an efficient MCMC the proposal distribution should reflect the underlying posterior it is sampling, but any proposal (that does not explicitly exclude the posterior), given enough time, will sample the posterior and deliver an accurate result. We use the Metropolis–Hastings (MH) algorithm to set the acceptance/rejection ratio. Given a current position a_i MH accepts the new position a_{i + 1} with probability

$$\begin{equation} \alpha (\boldsymbol{a}_{i+1}|\boldsymbol{a}_i) = {\rm min}\left(1, \frac{p(\boldsymbol{a}_{i+1}|d)}{p(\boldsymbol{a}_i|d)} \frac{q(\boldsymbol{a}_i|\boldsymbol{a}_{i+1})}{q(\boldsymbol{a}_{i+1} |\boldsymbol{a}_i)} \right), \end{equation} \tag{ 4 }$$

where p(a|d) is the posterior value at a given data d and q(a|b) is the proposal distribution defining how we choose position a given a current position b. In our case we have symmetric proposal distributions, so q(a_{i + 1}|a)/q(a_i|a_{i + 1}) = 1 and therefore only the ratio of the posteriors is needed.

A well-tuned MCMC will efficiently explore the parameter space and generate chains that, in histogram form, give the marginalized posterior distribution for each parameter. Defining a good set of proposal distributions for the parameters in a has been done experimentally assuming that they are uncorrelated and therefore have independent distributions. (There are in fact correlations between the h₀ and cos ι parameters and the ϕ₀ and ψ parameters, but in our studies these do not significantly alter the efficiency from assuming independent proposals.) The posterior distributions of these parameters will also generally not be Gaussian, especially in low signal-to-noise ratio (S/N) cases (which is the regime in which we expect to be), but a Gaussian proposal is easiest to implement and again does not appear to significantly affect the chain efficiency. We find that, for the angular parameters, Gaussian proposal distributions with standard deviations of an eighth the allowed parameter range (i.e., $\sigma _{\phi _0} = \pi /4$ rad, σ_cos ι = 1/4, and σ_ψ = π/16 rad) provide a good exploration of the parameter space (as determined from the ratio of accepted to rejected jumps in the chain) for low S/N signals. We have performed many simulations comparing the output of the MCMC and grid-based searches, both on simulated noise and simulated signals, and both codes give results consistent to within a few percent. In these tests, we find that the computational speed of the MCMC code is about three times faster than the grid-based code, although this can vary by tuning the codes.

An MCMC integration may take time to converge on the bulk of the probability distribution to be sampled, especially if the chains start a long way in parameter space from the majority of the posterior probability. Chains are therefore allowed a burn-in phase, during which the positions in the chain are not recorded. For low S/N signals, where the signal amplitude is close to zero and the posteriors are reasonably broad, this burn-in time can be short. To aid the convergence we use simulated annealing in which a temperature parameter is used to flatten the posterior during burn-in to help the chain explore the space more quickly. We do however use techniques to assess whether our chains have converged (see Brooks & Roberts 1998 for a good overview of convergence assessment tests for MCMCs). We use two such tests: the Geweke test is used on individual chains to compare the means of two independent sections of the chain; and the Gelman and Rubins test is used to compare the variances between and within two or more separate chains. These tests are never absolute indicators of convergence, so each chain also has to be examined manually. The acceptance/rejection ratio of each chain is also looked at as another indicator of convergence. For all our results, discussed in Sections 4 and 5, we have run these tests on the output chains, and conclude that all have converged.

2.2. Adding Phase Parameters

2.3. Setting up the MCMC

In our search we have information on the likely range over which to explore each parameter as given by the error from the fit (which we take as being the 1σ value of a Gaussian around the best-fit value), and the associated parameter covariance matrix. We use this information to set a prior on these parameters b given by a multivariate Gaussian

$$\begin{equation} p(\boldsymbol{b}) \propto \exp {\left\lbrace -\frac{1}{2}(\boldsymbol{b} - \boldsymbol{b}_0)^{\rm T} C^{-1} (\boldsymbol{b}-\boldsymbol{b}_0)\right\rbrace }, \end{equation} \tag{ 5 }$$

with a covariance matrix C. For the vast majority of pulsars, we expect that the uncertainties on the parameters are narrow enough that within their ranges they all give essentially the same phase model (see the discussion of phase mismatch in Section 3). In this case, the posterior on the parameters should be dominated by this prior, therefore a good proposal distribution to efficiently explore the space with is the same multivariate Gaussian.

An example of the posteriors produced when searching over additional parameters (in this case changes in declination and right ascension) can be seen in Figure 1. The figure shows the multivariate Gaussian used as priors on the two parameters and how the posterior is essentially identical to the prior (i.e., the data, which contains no signal, is adding no new information on those parameters, but the full prior space is being explored). We have assessed this technique on many simulations of noise and found that, as expected, the posteriors on the additional phase parameters match the priors. We have also tested this technique on many software simulated signals (using real pulsar phase parameters) by offsetting the injected signal parameters from the "best-fit" values by amounts consistent with the parameter covariance matrix. In these tests, we have been able to extract the parameters as expected.

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2.4. Hardware Injections

During all LIGO science runs, except the first, fake pulsar signals have been injected into the detectors by direct actuation of the end test masses. These have provided end-to-end validation of the analysis codes with coherence between the different detectors. During S5, as with S3 and S4 (Abbott et al. 2007), 10 signals with different source parameters were injected into the detectors. As a demonstration of the analysis method these have been extracted using the MCMC (over the four main parameters only) and all have been recovered with their known injection parameter values. The extracted parameters for the two strongest signals are shown in Figure 2. From these injections it can be seen that the parameters can be extracted to a high accuracy and consistently between detectors. The offsets in the extracted values from the injected values are within a few percent. These are well within the expected uncertainties of the detector calibration, which are approximately 10%, 10%, and 13% in amplitude, and 43 (0.08 rad), 34 (0.06 rad), and 23 (0.04 rad) in phase for H1, H2, and L1, respectively.

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For the majority of pulsars we have parameter correlation matrices that have been produced during the fit to the radio data, and (as discussed in Section 2.2) we can use these to search over the uncertainties on the phase parameters. For some pulsars no correlation matrix was produced with the radio observations, so for these we instead construct conservative correlation matrices. These assume no correlations between any parameters, except in the case of binary systems for which we assume a correlation of 1 between the angle and time of periastron. This gives a slightly conservative overestimate of the parameter errors, but is still a useful approximation for our purposes. From these correlation matrices and the given parameter standard deviations we produce a covariance matrix for each pulsar.

Using these covariance matrices we can also assess the potential phase mismatch that might occur between the true signal and the best-fit signal used in the heterodyne due to errors in the pulsar phase parameters (as discussed in Section 2.2.) We take the mismatch (i.e., the loss in signal power caused by the heterodyne phase being offset, over time, from the true signal phase) to be

$$\begin{equation} M = 1 - \left(\frac{1}{T}\int _0^T \cos {\lbrace \phi (\boldsymbol{b}+\delta \boldsymbol{b},t) - \phi (\boldsymbol{b}, t)\rbrace } dt\right)^2, \end{equation} \tag{ 6 }$$

where ϕ is the phase given a vector of phase parameters b at time t and T is the total observation time. To get offsets in the phase parameters, δb, we draw random values from the multivariate Gaussian defined by the covariance matrix. For each pulsar, we drew 100,000 random points and calculated the mean and maximum mismatch over the period of the S5 run. The mean mismatch is a good indicator of whether the given best-fit parameter values are adequate for our search, or whether the potential offset is large and a search including the phase parameters is entirely necessary. The maximum mismatch represents a potential worst case in which the true signal parameters are at a point in parameter space approximately 4.5σ from the best-fit values (the maximum mismatch will obviously increase if one increases the number of randomly drawn values). Of our 113 non-glitching pulsars there are only 16 pulsars with mean mismatches greater than 1%, three of which are greater than 10%: J0218+4232 at ∼43%, J0024 − 7204H at ∼15%, and J1913+1011 at ∼37%. Given the exclusion criterion for the S3 and S4 analyses (see Section 2.2), these three pulsars would have been removed from this analysis. There are 16 pulsars with maximum mismatches greater than 10% with three of these being at almost 100%—i.e., if the signal parameters truly were offset from their best-fit values by this much, and these parameters were not searched over, then the signal would be completely missed. This suggests that for the majority of pulsars the search over the four main parameters of h₀, ϕ₀, cos ι, and ψ is all that is necessary. But for a few, and in particular the three with large mean mismatches (J0218+4232, J0024 − 7204H, and J1913+1011), the search over the extra parameters is needed to ensure not losing signal power.

We have used the MCMC search over all phase parameters, where they have given errors, for all but three pulsars (see below). For the majority of pulsars it is unnecessary (though harmless) to include these extra parameters in the search as their priors are so narrow, but it does provide an extra demonstration of the flexibility of the method. To double check, we also produced results for each pulsar using the four dimensional grid of the earlier analyses and found that, for pulsars with negligible mismatch, the results were consistent to within a few percent.

For our full analysis we produced three independent MCMC chains with burn-in periods of 100,000 iterations, followed by 100,000 iterations to sample the posterior. The three chains were used to assess convergence using the tests discussed above in Section 2.1. All chains were seen to converge and were therefore combined to give a total chain length of 300,000 iterations from which the posteriors were generated.

For the three pulsars that glitched during S5 (the Crab, J1952+3252, and J0539 − 6910), the MCMC was not used to search over the position and frequency parameters as above, as the uncertainties on these parameters gave negligible potential mismatch. Our analysis of these pulsars did however include extra parameters in the MCMC to take into account potential uncertainties in the model caused by the glitches. We analyzed the data coherently over the full run as well as in stretches separated by the glitches that are treated separately. We also included a model that allowed for a fixed but unknown phase jump Δϕ at the time of each glitch, keeping other physical parameters fixed across the glitch. Results for all of these cases are given in Section 5.1.

The maximum detector calibration uncertainties are approximately 10%, 10%, and 13% in amplitude, and 43 (0.08 rad), 34 (0.06 rad), and 23 (0.04 rad) in phase for H1, H2, and L1, respectively.

No evidence of a gravitational wave signal was seen for any of the pulsars. In light of this, we present joint 95% upper limits on h₀ for each pulsar (see Table 1 for the results for the non-glitching pulsars). We also interpret these as limits on the pulsar ellipticity, given by

$$\begin{equation} \varepsilon = 0.237\,h_{-24}r_{\rm kpc}\nu ^{-2}I_{38}, \end{equation} \tag{ 7 }$$

where h₋₂₄ is the h₀ upper limit in units of 1 × 10⁻²⁴, I₃₈ = 1, and r_kpc is the pulsar distance in kpc. For the majority of pulsars, this distance is taken as the estimate value from Australia Telescope National Facility Pulsar Catalogue⁹¹ (Manchester et al. 2005), but for others more up-to-date distances are known. For pulsars in Terzan 5 (those with the name J1748 − 2446), a distance of 5.5 kpc is used (Ortolani et al. 2007). For J1939+2134 the best estimate distance is a highly uncertain value of ∼8.3 ± 5 kpc based on parallax measurements (Kaspi et al. 1994), but it is thought to be a large overestimate, so instead we use a value of 3.55 kpc derived from the Cordes & Lazio (2002) NE2001 galactic electron density model. The observed spin-down rate for globular cluster pulsars is contaminated by the accelerations within the cluster, which can lead to some seeming to spin-up as can be seen in column four of Table 1. For all pulsars in globular clusters, except J1824 − 2452A, we instead calculate a conservative spin-down limit by assuming all pulsars have a characteristic age $\tau = \nu /2\dot{\nu }$ of 10⁹ years. Note that using the characteristic age gives a spin-down limit that is independent of frequency and only depends on τ and r. Globular cluster pulsar J1824 − 2452A has a large spin-down that is well above what could be masked by cluster accelerations. Therefore, for this pulsar we use its true spin-down for our limit calculation. For some nearby pulsars there is a small, but measurable, Shklovskii effect which will contaminate the observed spin-down. For these if a value of the intrinsic spin-down is known then this is used when calculating the spin-down limit.

The results are plotted in histogram form in Figure 3. Also shown for comparison are the results of the previous search using combined data from the S3 and S4 science runs. The median upper limit on h₀ for this search, at 7.2 × 10⁻²⁶, is about an order of magnitude better than in the previous analysis (6.3 × 10⁻²⁵). A large part of this increased sensitivity (about a factor of 4–5) is due the longer observation time. The median ellipticity is ε = 1.1 × 10⁻⁶, an improvement from 9.1 × 10⁻⁶, and the median ratio to the spin-down limit is 108, improved from 870. If one excludes the conservatively estimated spin-down limits for the globular cluster pulsars, the median ratio is 73. In Figure 4, the upper limits on h₀ are also plotted overlaid onto an estimate of the search sensitivity.⁹² The expected uncertainties in these results due to the calibrations are given in Section 2.4.

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The smallest upper limit on h₀ for any pulsar is 2.3 × 10⁻²⁶ for J1603 − 7202, which has a gravitational wave frequency of 135 Hz and is in the most sensitive part of the detectors' bands. The lowest ellipticity upper limit is 7.0 × 10⁻⁸ for J2124 − 3358, which has a gravitational wave frequency of 406 Hz and a best estimate distance of 0.2 kpc. Of the millisecond recycled pulsars this is also the closest to its spin-down limit, at a value of 9.4 times greater than this limit. Of all pulsars which did not glitch during S5, the young pulsar J1913+1011 is the closest to its spin-down limit, at only 3.9 times greater than it.

5.1. Glitching Pulsars

For the three pulsars that glitched, we have chosen to take into account three different models related to the coherence of the gravitational wave signal and the electromagnetic signal over the glitch: (1) there is coherence between them over the glitch (i.e., the glitch causes no discontinuity between the electromagnetic and gravitational wave phases); (2) there is decoherence (in terms of a phase jump) between them at the time of the glitch, but the phase discontinuity is included as an extra search parameter (i.e., a Δϕ parameter is added at the time of the glitch); and (3) the data stretch before, between, and after the glitches are treated separately and analyzed independently.

5.1.1. Crab Pulsar

For this search, we include the three different models described above relating to the observed glitch in the pulsar on 2006 August 3. Potentially all the four main gravitational wave signal parameters could be changed during the glitch if it is large enough to cause major disruption to the star, but this is not the case for the observed glitch. It had a fractional frequency change of order Δν/ν ∼ 5 × 10⁻⁹, which is unlikely to be energetic enough to cause changes in the gravitational wave amplitude near our current levels of sensitivity. To model the signal phase, we use the regularly updated Crab pulsar Monthly Ephemeris (Lyne et al. 1993, 2009), which is needed to take into account the phase variations caused by timing noise.

As for the previous Crab pulsar search (Abbott et al. 2008), we have information on the orientation of the pulsar from the orientation of the pulsar wind nebula (PWN; Ng & Romani 2008). We use this information to set Gaussian priors on the ψ and ι parameters of ψ = 125155 ± 1355 (the ψ dependence wraps around at ±45°, so the actual value used is 35155) and ι = 6217 ± 220.

There is reason to believe that the PWN orientation reflects that of the central pulsar, but in case this is not an accurate description we present results using both a uniform prior over all parameters, and using the restricted prior ranges. The MCMC only searches over the four main parameters and does not include errors on the pulsar position, frequency or frequency derivatives as these are negligible. The results are summarized in Table 3, where for the ellipticity and spin-down limit calculations a distance of 2 kpc was used. The orientation angle suggested by the restricted priors is slightly favorable in terms of the observable gravitational wave emission it would produce, and this allows us to set a better upper limit.

In Figure 5, we plot the result from model (1), with restricted priors, as an exclusion region on the moment of inertia–ellipticity plane. It can been seen that for all the allowed regions in moment of inertia we beat the spin-down limit. If one assumes the Crab pulsar to have a moment of inertia at the upper end of the allowed range our result beats the spin-down limit by over an order of magnitude.

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5.1.2. PSR J0537 − 6910

For pulsar J0537 − 6910, which has so far been timed only in X-rays, we rely on data from the RXTE satellite. This pulsar is young, has a high spin-down rate, and is a prolific glitcher, and therefore we require observations overlapping with our data to produce a coherent template. Middleditch et al. (2006) have published observations covering from the beginning of S5 up to 2006 August 21, during which time the pulsar was seen to glitch three times. Further observations have been made which span the rest of the S5 run and show another three glitches during this time. The epochs and parameters for the seven ephemeris periods overlapping with our data run are given in Table 2. For the first epoch there was no data for L1, so the joint result only uses H1 and H2 data. For the analyses using all the data (models 1 and 2) we have 474 days of H1 data, 475 days of H2 data, and 397 days of L1 data. Due to the glitches we perform parameter estimation for the same three models given above.

Table 2. Ephemeris Information for PSR J0537 − 6910

S. No.	Start–End (MJD)	ν (Hz)	$\dot{\nu }$ (Hz s−1)	$\ddot{\nu }$ (Hz s−2)	Epoch (MJD)
1.	53551–53687	62.000663106	−1.994517 × 10−10	11.2 × 10−21	53557.044381976239
2.	53711–53859	61.996292178	−1.993782 × 10−10	8.1 × 10−21	53812.224185839386
3.	53862–53950	61.995120229	−1.994544 × 10−10	9.6 × 10−21	53881.096123033291
4.	53953–53996	61.993888026	−1.995140 × 10−10	9.6 × 10−21	53952.687378384607
5.	54003–54088	61.992869785	−1.994516 × 10−10	2.2 × 10−21	54013.061576594146
6.	54116–54273	61.990885506	−1.994577 × 10−10	8.1 × 10−21	54129.540333159754
7.	54277–54441	61.988307647	−1.994958 × 10−10	7.7 × 10−21	54280.918705402011

Note. The first four values are taken from Middleditch et al. (2006).

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As with the Crab pulsar, there is also information on the orientation of J0537 − 6910 from model fits to its PWN (Ng & Romani 2008). These are used to set Gaussian priors on ψ and ι of ψ = 1310 ± 22 (equivalently 410) and ι = 928 ± 09. We again quote results using uniform priors over all parameters and with these restricted priors. The distance used in the ellipticity and spin-down limits for J0537 − 6910 is 49.4 kpc. The results are summarized in Table 3. Using the restricted priors we obtain a worse upper limit on h₀ than for uniform priors. This is because the nebula suggests that the star has its spin axis perpendicular to the line of sight, and therefore the gravitational radiation is linearly polarized and the numerical strain amplitude is lower than average for a given strain tensor amplitude h₀.

Table 3. Results of the Analysis for the Crab Pulsar, J0537 − 6910, and J1952+3252

Epoch	h95%0		Ellipticity		h95%0/hsd0
	Uniform	Restricteda	Uniform	Restricteda	Uniform	Restricteda
Crab pulsar
Model (1)b	2.6 × 10−25	2.0 × 10−25	1.4 × 10−4	1.1 × 10−4	0.18	0.14
Model (2)c	2.4 × 10−25	1.9 × 10−25	1.3 × 10−4	9.9 × 10−5	0.17	0.13
1.	4.9 × 10−25	3.9 × 10−25	2.6 × 10−4	2.1 × 10−4	0.34	0.27
2.	2.4 × 10−25	1.9 × 10−25	1.3 × 10−4	1.0 × 10−4	0.15	0.13
J0537 − 6910
Model (1)b	2.9 × 10−26	3.9 × 10−26	8.9 × 10−5	1.2 × 10−4	1.0	1.3
Model (2)c	4.1 × 10−26	4.6 × 10−26	1.2 × 10−4	1.4 × 10−4	1.4	1.5
1.	9.1 × 10−25	1.2 × 10−24	2.8 × 10−3	3.7 × 10−3	31.2	41.2
2.	1.2 × 10−25	1.5 × 10−25	3.6 × 10−4	4.6 × 10−4	4.1	5.2
3.	1.4 × 10−25	1.3 × 10−25	4.2 × 10−4	3.8 × 10−4	4.7	4.3
4.	8.3 × 10−26	1.1 × 10−25	2.5 × 10−4	3.3 × 10−4	2.8	3.7
5.	1.4 × 10−25	1.3 × 10−25	4.1 × 10−4	4.0 × 10−4	4.7	4.6
6.	4.4 × 10−26	5.8 × 10−26	1.3 × 10−4	1.8 × 10−4	1.5	2.0
7.	4.9 × 10−26	6.1 × 10−26	1.5 × 10−4	1.9 × 10−4	1.7	2.1
J1952+3252
Model (1)b	2.5 × 10−25	⋅⋅⋅	2.3 × 10−4	⋅⋅⋅	2.0	⋅⋅⋅
Model (2)c	3.6 × 10−25	⋅⋅⋅	3.3 × 10−4	⋅⋅⋅	2.9	⋅⋅⋅
1.	4.7 × 10−25	⋅⋅⋅	4.3 × 10−4	⋅⋅⋅	3.8	⋅⋅⋅
2.	4.4 × 10−25	⋅⋅⋅	4.0 × 10−4	⋅⋅⋅	3.5	⋅⋅⋅

Notes. ^aUses observationally restricted priors on the orientation angle and polarization angle. ^bUses the full data set for a coherent analysis. ^cUses the full data set, but searches over an extra phase parameter at each glitch.

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In Figure 5, we again plot the result from model (1), with restricted priors, on the moment of inertia–ellipticity plane. It can be seen that the spin-down limit is beaten if we assume a moment of inertia to be 2 × 10³⁸ kg m² or greater.

5.1.3. PSR J1952+3252

PSR J1952+3252 is another young pulsar with a high spin-down rate (although a couple of orders of magnitude less than for the Crab pulsar and J0537 − 6910)—it has spin parameters of ν = 25.30 Hz and $\dot{\nu }=-3.73\times 10^{-12}$ Hz s⁻¹. Jodrell Bank observations of this pulsar were made over the whole of S5, but it was observed to glitch at some point between 2007 January 1 and January 12. For both the pre- and post-glitch epochs, we have coherent timing solutions and again perform analyses as above. There are no constraints on the orientation of this pulsar, so we do not use any restricted priors. The results for this pulsar are given in Table 3. We reach about a factor of 2 above the spin-down limit using a distance of 2.5 kpc. The result from model (1) is also plotted on the moment of inertia–ellipticity plane in Figure 5. It can be seen that I₃₈ just over 4, which is above the expected maximum allowable value, would be needed to beat the spin-down limit.

In this paper, we have searched for continuous gravitational waves from an unprecedented number of pulsars with unprecedented sensitivity, using coincident electromagnetic observations of many millisecond and young pulsars. Our direct upper limits have beaten the indirect limits (including the spin-down limit) for the Crab pulsar and are at the canonical spin-down limit for J0537 − 6910. For several more pulsars our upper limits are within roughly 1 order of magnitude of the spin-down limits, and therefore we expect a comparable search of Advanced LIGO and Virgo data to beat the latter.

For the Crab pulsar we improve upon the previous upper limit, which already beats the spin-down limit and other indirect limits; and we also provide results for different scenarios in which the signals may not be fully coherent over the run. By assuming that the observationally constrained priors on the angular parameters are correct we can limit the power output of the pulsar (see Abbott et al. 2008) to be less than ∼2% of the canonical spin-down luminosity.

Pulsar J0537 − 6910 is an interesting case due to its youth and glitchiness. Assuming that the gravitational wave signal phase is coherent with the X-ray pulses over glitches, our upper limit is within a few percent of the canonical spin-down limit for this pulsar. If we assume that the phase is incoherent over the glitches, then using the full data set our upper limit is ∼1.2 times the canonical spin-down limit. Even if we assume that none of the four main pulsar parameters remains coherent during a glitch (i.e., there is a major rearrangement in the star's structure) then the results from the final two epochs individually are still only approximately two times the spin-down limit. All of these direct upper limits are within the uncertainties of the spin-down limit, and (unlike the Crab) there is no firmly observed braking index and thus no substantially stricter indirect limit.

Pulsar J1952+3252 gets close to its spin-down limit, but even taking into account moment of inertia and distance uncertainties, probably would still just miss beating it. It does however make an exciting target for future runs.

Our observational upper limits on the ellipticities of the Crab and J0537 − 6910 are in the vicinity of 10⁻⁴. Elastic deformations of this magnitude have been predicted as sustainable (Owen 2005; Haskell et al. 2007; Lin 2007; Knippel & Sedrakian 2009) not for normal neutron stars but only for exotic forms of crystalline quark matter (Xu 2003; Mannarelli et al. 2007). Such ellipticities could also be sustained by internal toroidal magnetic fields of order 10¹⁶ G depending on the field configuration, equation of state, and superconductivity of the star (Cutler 2002; Akgun & Wasserman 2007; Haskell et al. 2008; Colaiuda et al. 2008). Therefore, our observations have achieved the sensitivity to detect some of the more extreme possibilities, and have constrained the internal magnetic fields of the Crab and J0537 − 6910 to be less than of order 10¹⁶ G. However, it is important to be clear that the upper limits reported here do not constrain the properties of crystalline quark matter, because gravitational wave observations constrain the true ellipticity rather than the maximum ellipticity. This distinction was made clear in Owen (2005), but has not been properly enforced in more recent work (Lin 2007; Knippel & Sedrakian 2009). Also, while it is the case that, as anticipated by Haskell et al. (2007), the upper bounds on ellipticity reported here for some pulsars, notably J2124 − 3358, have pushed into a regime where elastic strains might support such deformations, detection was not expected, as such large ellipticities would conflict with the spin-down limit.

Recent work by Horowitz & Kadau (2009) suggests that the breaking strain of neutron star crusts may be an order of magnitude higher (10⁻¹) than the highest values (10⁻²) previously used in estimates of maximum sustainable ellipticities. Their simulations are strictly applicable only to the outer crust (i.e., no neutron drip), but since the reason for the high breaking strain is very generic—the extreme pressure simply crushes away defects that contribute to early fracture—it may apply to the inner crust (the major contributor to ellipticity) as well. In that case, normal neutron stars could sustain ellipticities close to 10⁻⁵, which is beyond the reach of the present search (for those stars where our results are at or near the spin-down limit) but would be accessible with data from advanced interferometers. If the high breaking strain holds for mixed phases in hybrid stars (which is not clear due to the increasing importance of the strong interaction at high densities), it would bring their maximum ellipticities estimated by Owen (2005) up an order of magnitude to about the 10⁻⁴ values achieved here.

In 2009 July, the LIGO 4 km detectors (featuring some upgraded systems and titled Enhanced LIGO) and the upgraded Virgo detector began their sixth and second science runs, respectively (S6 and VSR2). These upgrades aim to provide significantly better strain sensitivities than S5/VSR1. For LIGO these improvements will primarily be at frequencies greater than 40 Hz, but Virgo will improve at lower frequencies too and should outperform LIGO below about 40 Hz. LIGO and Virgo will closely approach and could potentially beat the spin-down limits not only for the Crab and J0537 − 6910 but for six more known pulsars: J1952+3252, J1913+1011, J0737 − 3039A, J0437 − 4715,⁹³ and the recently discovered J1747 − 2809 (Camilo et al. 2009) and J1813 − 1749 (Gotthelf & Halpern 2009). Below 40 Hz, Virgo could beat the spin-down limits for three more pulsars: the Vela pulsar, J0205+6449, and J1833 − 1034. Most of these 11 pulsars are young and may be prone to large levels of timing noise and glitches. Therefore, to reach the best sensitivities it is essential to have radio and X-ray observations of these sources made in coincidence with the S6/VSR2 run. In particular, the extension of the RXTE satellite is crucial to our ability to perform targeted searches of J0537 − 6910 during this time of enhanced detector performance.

On a timescale of a few years, the Advanced LIGO and Virgo interferometers are expected to achieve strain sensitivities more than 10 times better for the pulsars in the current band, and will extend that band downward to approximately 10 Hz. Searches at such sensitivities will beat the spin-down limits from dozens of known pulsars and also enter the range of ellipticities predicted for normal neutron stars, improving the prospects for direct gravitational wave observations from these objects.

The authors gratefully acknowledge the support of the United States National Science Foundation for the construction and operation of the LIGO Laboratory, the Science and Technology Facilities Council of the United Kingdom, the Max-Planck-Society, and the State of Niedersachsen/Germany for support of the construction and operation of the GEO600 detector, and the Italian Istituto Nazionale di Fisica Nucleare and the French Centre National de la Recherche Scientifique for the construction and operation of the Virgo detector. The authors also gratefully acknowledge the support of the research by these agencies and by the Australian Research Council, the Council of Scientific and Industrial Research of India, the Istituto Nazionale di Fisica Nucleare of Italy, the Spanish Ministerio de Educación y Ciencia, the Conselleria d'Economia Hisenda i Innovació of the Govern de les Illes Balears, the Foundation for Fundamental Research on Matter supported by the Netherlands Organisation for Scientific Research, the Royal Society, the Scottish Funding Council, the Polish Ministry of Science and Higher Education, the FOCUS Programme of Foundation for Polish Science, the Scottish Universities Physics Alliance, the National Aeronautics and Space Administration, the Carnegie Trust, the Leverhulme Trust, the David and Lucile Packard Foundation, the Research Corporation, and the Alfred P. Sloan Foundation. LIGO Document No. LIGO-P080112-v5. Pulsar research at UBC is supported by a Natural Sciences and Engineering Research Council of Canada Discovery Grant. The Parkes radio telescope is part of the Australia Telescope which is funded by the Commonwealth Government for operation as a National Facility managed by CSIRO. The National Radio Astronomy Observatory is a facility of the United States National Science Foundation operated under cooperative agreement by Associated Universities, Inc. We thank Maura McLaughlin for useful discussions.