Binary system delays and timing noise in searches for gravitational waves from known pulsars
Abstract
The majority of fast millisecond pulsars are in binary systems, so that any periodic signal they emit is modulated by both Doppler and relativistic effects. Here we show how wellestablished binary models can be used to account for these effects in searches for gravitational waves from known pulsars within binary systems. A seperate issue affecting certain pulsar signals is that of timing noise and we show how this, with particular reference to the Crab pulsar, can be compensated for by using regularly updated timing ephemerides.
pacs:
97.60.Gb, 04.80.Nn, 07.05.KfI Introduction
Neutron stars are thought to be strong candidates for the emission of detectable continuous gravitational waves Schutz:1989 , including the 1627 pulsars currently discovered^{1}^{1}1As given by the Australia Telescope National Facility  ATNF  online pulsar catalogue ATNF ; Manchester:2005 as of Nov 2006, from which time all subsequent pulsar numbers will be taken.. The majority of these pulsars have been discovered through dedicated radio surveys. Surveys are ongoing, but estimates of the number of active pulsars in the galaxy can be made by inference from the current population, taking into account biasing from selection effects, and the supernova rate. Estimates give values of active pulsars within our galaxy (see Ref. Lorimer:2001 ).
Pulsars are found in a wide range of environments. Some are directly associated with the supernova remnants (SNRs) in which they were born. These are typically young pulsars whose birth velocity has not yet caused a large displacement from the remnant and the SNR has not dissipated into the interstellar medium (ISM). Some pulsars are found in binary systems as the companions of a wide range of astronomical bodies from planets, through main sequence stars, to white dwarfs and other neutron stars. The fastest, ‘millisecond’, pulsars (pulsars with rotation periods of ms) are usually found within binary systems, and often within globular clusters, their rapid rotation rate a consequence of being spunup by accretion of material from a stellar companion vandenHeuvel:1984 . Pulsars are also seen without any association, and in this paper we will classify any pulsar not in a binary system as isolated.
Pulsars are generally seen to spindown as they lose rotational energy through a variety of emission mechanisms, but the primary loss mechanism is thought to be magnetic dipole radiation. Other potential mechanisms include particle acceleration and gravitational radiation. Whatever the mechanisms at work, the rotational phase evolution of a pulsar can generally be well described by a short Taylor expansion,
(1)  
where is the initial phase, and its time derivatives are the pulsar frequency and spindown coefficients at an epoch , and is the time in a frame comoving with the pulsar. For the vast majority of pulsars the value of is very small and is unmeasurable or swamped by timing noise (see §IV). Note that gravitational waves emitted from a triaxial, nonprecessing, neutron star come from the quadrupolar component of the rotating body and so will have exactly twice the phase evolution described by Eq. 1.
The expected gravitational wave signal from a triaxial neutron star is given by JKS:1998
(2)  
where is that given in Eq. (1), and are the detector beam patterns for the plus and cross polarisations of the gravitational waves, is the wave polarisation angle, and is the angle between the rotation axis of the pulsar and the lineofsight. For a gravitational wave signal impinging on the Earth the signal arrival time at the detector, , will be modulated by Doppler, time delay and relativistic effects caused by the motions of the Earth and other bodies in the solar system, so we want a reference frame which will not be affected by these. In general, for isolated pulsars^{2}^{2}2assuming negligible proper motion and acceleration within a globular clusters., such a frame is the solar system barycentre (SSB). To convert from to the time at the SSB, , we must include a series of time corrections,
(3) 
where is the position of the detector with respect to the solar system barycentre (SSB), is the unit vector pointing to the pulsar, is the special relativistic Einstein delay, and is the general relativistic Shapiro delay (see Ref. TaylorWeisberg:1989 for definitions of these delay terms). For isolated pulsars we can therefore set in Eq. (1). For pulsars in binary systems there will be additional time delays as discussed in §III.
In this paper we describe the addition of such binary system time delays with respect to current searches for gravitational waves from known pulsars. In a seperate issue we also discuss how pulsar timing noise can cause deviations from the simple phase model in Eq. 1, and how to account for this in the analysis.
Ii Search method
Searches for gravitational waves from a selection of known pulsars have been performed using data from the LIGO Abbott:2003 and GEO 600 Willke:2002 gravitational wave detectors from the four science runs (S14), which have taken place since late 2002 Abbott:2004 ; Abbott:2005 ; Abbott:2006 . The search method used is outlined briefly here, but is described more fully in Refs. Dupuis:2004 ; DupuisWoan:2005 .
All current data from these detectors is sampled at 16 384 Hz, giving a range of 0–8192 Hz available for searches, although in practice this is limited to the most sensitive region of the detectors between –4000 Hz. The rotation frequencies of known pulsars are, of course, known very precisely from radio and/or Xray observations, so the vast majority of this frequency space is redundant and the speed of any search can be increased by removing it. Knowledge of the pulsar parameters allows us to perform a complex heterodyne on the data, using the precise pulsar phase evolution, and downsample it to Hz. Following this heterodyne a Bayesian parameter estimation for the unknown pulsar parameters , , and can be performed on the massively reduced data set.
The method relies on the accuracy of the signal model, and it is very important that the phase evolution in the heterodyne is sufficiently good. Any drift from the true pulsar signal phase could nullify the search if it becomes too severe. This paper will discuss how to ensure that this phase model is sufficiently accurate for pulsars in binary systems and those strongly affected by timing noise. Most recently, these methods have been used in Ref. Abbott:2006 to obtain limits on the gravitational wave emission from 78 pulsars using data from the LIGO and GEO 600 S3 and S4 runs.
Iii Pulsars in binary systems
Of the 1627 pulsars in the ATNF catalogue, 124 are in binary systems. The first of these to be discovered was the highly relativistic pulsar PSR J1915+1606 found by Hulse and Taylor in 1974 HulseTaylor:1975 . Of these 124 pulsars, 98 have spin frequencies greater than 25 Hz (out of a total 163 isolated and binary pulsars with frequencies greater than 25 Hz), and therefore gravitational wave frequencies Hz, putting them into the sensitive band of the LIGO detectors. Indeed, for the reason stated in §I, the majority of millisecond pulsars are in binary systems.
iii.1 Pulsar timing
A brief discussion on how pulsar timing information is obtained is relevant here. The majority of pulsars have been both discovered and monitored in radio. Pulsar surveys, discussed in more detail in Ref. PulsarAstronomy , use Fourier transform methods to look for periodic signals in the radio data, taking into account the effect of interstellar dispersion across the receiver band. Once the pulsar period has been determined, the radio time series data can be folded with this cadence to build up the signaltonoise ratio of a mean pulse. Once a stable pulse is obtained^{3}^{3}3Individual pulses can vary in shape, but the summation of many gives a generally stable pulse shape. the time of arrival (TOA) can be measured at the peak of the pulse. These pulse times can then be used to extract more precise information about the pulsar parameters, including its position and frequency parameters.
The most prevalent tool used to fit timing measurements is the Tempo software suite TEMPO , however others have been developed, including psrtime PSRtime at Jodrell Bank Observatory. Tempo requires precise solar system ephemerides, containing the positions and velocities of the major solar system bodies, to convert TOAs at a detector to the rest frame of the pulsar. It computes the pulsar phase at each TOA, , over the range of pulsar parameters (, , , , etc), and uses a goodnessoffit statistic to determine the best model via minimization. A starting point for the fit is obtained through a rough knowledge of the position and frequency from the initial discovery, but it can still be quite complex as there can be many other parameters that could be contributing. Below it is seen how a pulsar in a binary system requires a complex model with many more parameters than an isolated object.
iii.2 Binary pulsar timing
The majority of pulsars within the groundbased gravitational wave frequency band are in binary systems, so we must consider the effects of binary timing carefully. Techniques to create filters to match a binary signal (in the frequency domain) have been described before, for example in Ref. DhurandharVecchio:2001 , and these have been applied to the search for gravitational waves from the neutron star in Sco X1 FstatPaper:2006 , but there has been no equivalent treatment in the time domain, suitable for the search outlined above. Eq. (3) shows the timing corrections needed to take account of Doppler and relativistic delays of a signal and transform it to the SSB. Any constant Doppler delays from the pulsar’s actual motion relative to the SSB are unimportant, and the SSB frame can be considered as the rest frame of the pulsar. For a pulsar in a binary system however, its motion within the system will need to be taken into account with a transform from the binary system barycentre to the pulsar proper time.
The basic transformation and binary models below are summarised in Ref. TaylorWeisberg:1989 and used in the pulsar timing program Tempo TEMPO . The transformation from SSB time to pulsar proper time follows the form of Eq. (3) and is
(4) 
where is the Roemer time delay giving the propagation time across the binary orbit; is the Einstein delay and gives gravitational redshift and time dilation corrections; is the Shapiro delay which gives the gravitational propagation delay due to the signal propagation through the curved spacetime of the companion; and is the aberration delay caused by the pulsar’s rotation.
The majority of binary orbits can be wellmodelled as Keplerian. Keplerian orbits are defined by five parameters,  the time of periastron (closest approach in the binary orbit);  the longitude of periastron;  the orbital period;  the orbital eccentricity (where and and are the semimajor and semiminor axis of the orbital ellipse respectively); and is the projected semimajor axis, with being the orbital inclination. For some of the most extreme binary systems, with rapidlyspinning pulsars in tight orbits, the maximum (gravitational wave) orbital Doppler frequency shift can be up to Hz. For example, PSR J07373039A with light sec, days and Hz gives Hz.
We will now consider the three most commonly used models used to characterise the TOA of pulses from pulsars in binary systems.
iii.2.1 BlandfordTeukolsky model
The Blandford and Teukolsky model BlandfordTeukolsky:1976 (BT) makes no assumptions about the correct theory of gravity. Instead, it assumes a simple Keplerian orbit with slow precession, into which additional relativistic effects have been added. Other phenomena can be taken into account through time derivatives of the four main orbital elements, excluding . The BT model has been used to fit data for 53 of the binary pulsars with Hz, and is the most common model used. Exceptionally, one of these systems is modelled using the Tempo model BT2P which accommodates three orbits, the first of which can be relativistic, but the second and third are Keplerian. The system is a multiple system, described in Ref. Wolszczan:2000 , in which three, or possibly four, planets orbit the pulsar. Although these additional orbits complicate the above equations, the standard BT model is sufficient for our purposes.
iii.2.2 Low eccentricity model
The second most common model used in fitting radio observations of binaries is the low eccentricity model (called ELL1 in Tempo) developed in Ref. Lange:2001 . It is used as a fit for pulsars in very low eccentricity orbits () and is the appropriate model for 38 of our target pulsars with Hz. An almost circular orbit makes and nearly degenerate, so these parameters, along with , are replaced with the noncovariant parameters of the time of the ascending node of the orbit () and the first and second LaplaceLagrange parameters and . The time delays for this model are defined in Lange:2001 and Tempo.
iii.2.3 DamourDeruelle model
The third most common model is that of Damour and Deruelle (DD) DamourDeruelle:1986 . This model uses a method for solving the relativistic twobody problem to postNewtonian order and is valid under very general assumptions about the nature of gravity in strong field regimes. It is useful for highly relativistic systems, although in only mildly relativistic systems this model should be no different to the BT model. There are seven pulsars with Hz in the ATNF catalogue that use this model. This model is again summarised in TaylorWeisberg:1989 .
iii.3 Comparison with TEMPO
The above three models are all implemented in the pulsar timing software package Tempo. In our search for gravitational waves from binary systems we also require these additional time corrections to correctly calculate the phase of the pulsar for heterodyning. Code to calculate the binary time delays for each model has been adapted from the Tempo counterparts and is available under CVS in the LIGO Algorithm Library (LAL) repository LAL . Some consistency tests have been performed between the two codes, which are described below. It is acknowledged that Tempo has some uncertainties in its timing models due to simplifying assumptions, and these limit its accuracy to 100 ns. These effects are discussed in Refs. Hobbs:2006a ; Edwards:2006 , but as errors at this level have no significant impact on our search we will neglect them.
iii.3.1 Psr J1012+5307
We performed a validation check on the new LAL code by demodulating radio data from a known pulsar. With no known gravitational calibrators, this is a crucial step in the development of any gravitational wave detection code. A set of TOAs for PSR J1012+5307 obtained with the Effelsberg 100 m radio telescope in Bonn, Germany, was kindly supplied by Michael Kramer for this purpose. This pulsar has the second most circular orbit known and is therefore fitted by the ELL1 model. The data intermittently spanned just over 5 years from January 2000 to February 2005 and comprised TOAs (in Modified Julian Date format, where MJD = Julian Date2400000.5), together with the Tempoderived pulsar parameters (Table 1).
PSR J1012+5307  

PMRA  2.38(3) mas/yr 
PMDEC  (5) mas/yr 
190.267837621884(9) Hz  
Hz/s  
Hz/  
Frequency epoch  MJD 50700 
Dispersion measure  9.0233(7) pc 
Observing Frequency  1408.6 MHz 
Binary model  ELL1 
0.581817(1) s  
0.6046727136(2) days  
MJD 50700.0816289(4)  
Some minor transformations are necessary to convert TOAs measured at the telescope to the GPS time stamps used in gravitational wave data analysis software. First the raw TOAs are corrected for the drifts between the hydrogen maser clock at Effelsberg and coordinated Universal Time of the National Institute of Science and Technology UTC(NIST) reference. This correction (supplied with the data) was typically a few microseconds. The difference between UTC(NIST) and UTC has been less than ns since July, 1994 BIPM and was neglected for this work. The conversion between the time scales therefore becomes
(5) 
where the 44244 corresponds to the MJD of the GPS time epoch ( January, 1980) and is the accumulated number of leap seconds included in the definition of UTC. For the timespan of these TOAs, .
The TOAs can now be corrected for interstellar dispersion time delay PulsarAstronomy
(6) 
where DM is the dispersion measure in pc and is the radio observation frequency in MHz (see Table 1 for values). This correction is subtracted from the TOAs to give observations at infinite frequency with no dispersion.
One of the major differences between our binary time domain code and the Tempo code is the time system used. All epochs in Tempo are defined as MJD Barycentric Dynamical Time (TDB  a timescale generally used for ephemerides referenced to the solar solar barycentre) whereas the general reference time for our gravitational wave data is GPS time. Epochs therefore have to be converted to GPS time on the TDB timescale. This TDB timescale is related to Terrestrial Time (TT  formerly Terrestrial Dynamical Time TDT), which represents a time consistent with relativity for an observer on the Earth’s surface, by a small factor, , no greater than a couple of milliseconds and given by
(7) 
where is the mean anomaly, or phase, of the Earth’s orbit at the given Modified Julian Date Times . TT is offset from International Atomic Time (TAI), so that TT = TAI + 32.184 seconds^{4}^{4}4There are many definitions of time used in astronomy and very careful attention of which one is being used and how to convert between them is essential when high precision timings are being made. A good guide for these is Times .. The conversion thus goes , where the 51.184 s comprises the 32.184 s difference between TT and TAI and 19 second difference between TAI and GPS.
The LAL code to calculate the SSB time delay uses the pulsar’s position, the telescope position and a solar system ephemeris JPLEphemeris and was used for each pulsar TOA to correct to the time at the SSB. This code was written by Curt Cutler and has been independently tested against Tempo Abbott:2004 ; Dupuis:2004 showing no more than s difference between the two. For a pulsar with a gravitational wave frequency at 1 kHz a 4 s timing error would give a phase offset between any signal and our model template of 0.025 rads. This would reduce our sensitivity to a signal by of order , which is negligible.
Once corrected to the SSB the TOAs are further corrected to the pulsar proper time by calculating the time delays in the binary system using the binary system parameters (see Table 1). The binary and solar system time delays for a selection of TOAs covering part of the binary orbit are shown in Fig. 1.
Once these corrections to the TOAs are applied we can compare the LAL barycentring codes with Tempo by inserting the Tempoderived pulsar parameters into the LAL barycentring routines and examining the predicted phase at the corrected TOAs. TOAs converted incorrectly by the barycentring codes would show up as a phase drift. The phase at each TOA was calculated using the supplied frequency and frequency derivatives in Eq. (1), with and the frequency epoch as . Fig. 2 demonstrates the serious effect of neglecting these binary time delays.
In contrast, Fig. 3 shows how the TOAs barycentred using our code stay well in phase over the observation time when the binary delays are included, with a residual phase slope of rads/yr.
A yearly periodicity is also present possibly showing up the slight difference in the LAL solar system barycentring code and Tempo, although these effects are at a very low level. Several points clearly show large phase residuals and correspond to times when the level of noise on the TOA measurements was high.
The parameters for PSR J1012+5307 were generated using the ELL1 model, so the above test only checked the ELL1 code. We can also check the two other models by converting to and the LaplaceLagrange parameters and to and . This pulsar has a low eccentricity, so can be set equal to and and set to zero for practical purposes. Doing this we can again produce the phase plots for the BT and DD models (Fig. 4).
The phase is again well described for these two models, with the slope and periodicity still present. This suggests that the slope and periodicities are not caused by the binary timing correction code (as each model is independent), but may be a results of slight errors in the other timing corrections, the solar system barycentring code, the ephemerides, or the parameters used.
The effects of inaccuracies in these binary parameters can be shown by offsetting one from its true value. In Fig. 5 the true value of the parameter has been offset by 5, 10 and 20 seconds and the phase at each TOA recalculated.
Even for a 5 s mismatch we start to depart from the true phase by up to 0.5 radians. This would not be disastrous for the analysis but would degrade the search. For a 20 s offset we start to see phase errors of up to 2 radians. At this level, all sensitivity is lost. In real analyses as in Ref. Abbott:2006 the effect of any errors on the parameters, as to whether they could cause our heterodyne phase to depart significantly from the signal phase, is thoroughly checked.
iii.3.2 Direct check against TEMPO
In common with the solar system barycentring code Dupuis:2004 ; Abbott:2005 , the binary timing code was tested directly against Tempo. Tempo can be run in predictive mode, to use a set of pulsar parameters to predict the pulsar phase over a period of time. This predicted phase can be then be compared with that calculated using our binary timing code. This was done for each model with a set of 100 randomly generated binary pulsar systems over a period of 100 days. The detector location was set to be at the SSB, so the solar system time delay errors would not be included. Histograms of the time residuals between the codes are shown in Fig. 6 for each model.
These show that the time difference between the two codes is generally less than s, which is sufficiently small to ensure any signal and template remain in phase over the frequency range considered.
Iv The problem of timing noise
A rather different issue that if unaddressed could potentially cause problems for our analysis is that of timing noise.
Pulsars are generally very stable over periods of several days, but there are phenomena which can cause deviations in this timing stability. With the very high accuracy of pulsar timing any random timing irregularities will start to become evident. These phenomena show up as glitches and timing noise. Timing noise has been known about since the early days of pulsar observations and represents a random walk in phase, frequency or frequency derivative of the pulsar about the regular spindown model given in Eq. (1) CordesHelfand:1980 . If our search method cannot take account of these phase deviations they could reduce our sensitivity or even invalidate the search. It is therefore important to see firstly if timing noise produces large enough phase deviations to cause a problem for our method and if so how it can be countered.
The strength of this effect has been quantitatively defined in Ref. CordesHelfand:1980 as the activity parameter , as referenced to that of the Crab pulsar, and in Ref. Arzoumanian:1994 as the stability parameter . There is however no real consensus on how to quantitatively define a measure of the level of timing noise, with the magnitude and sign of maybe providing the measure which includes the least other assumptions. A thorough study of timing noise, comparing and contrasting the various measures used, is given in Ref. Hobbs:2006 (also see Refs. Hobbs:2004 ; Hobbs:2005 ). There is a definite correlation between these parameters ( and ) and the pulsar’s spindown rate, therefore possibly the pulsar’s age. Young pulsars, like the Crab pulsar, generally show the most timing noise activity.
The Crab pulsar is the youngest known pulsar targeted by current gravitational wave detectors and it is important to considered how timing noise may be countered for this pulsar. A method was first proposed in Ref. PitkinWoan:2004 and has been used in the analysis of Abbott et al. Abbott:2005 , but has not previously been described in detail. A brief look into the effects of timing noise for other pulsars will be discussed later.
iv.1 Timing noise in the Crab pulsar
The pulsar (J0534+2200) in the Crab nebula (M1) has undergone intense study since its discovery in 1968. Its parameters are given in Table 2.
PSR J0534+2200  

Right ascension  
Declination  
proper motion in  mas/yr 
proper motion in  7 mas/yr 
Position epoch  MJD 40675 
29.7670971390 Hz  
Hz/s  
Hz/  
Frequency epoch  MJD 53993 
Distance  2.0 kpc 
It has an observed age of 951 years (the formation of the Crab nebula is associated with a supernova observed in AD 1054) and a spindown age of years. Analyses of longterm timing observation of the Crab pulsar are given in Refs. Lyne:1993 ; Wong:2001 . These analyses show some of the timing features which make the Crab pulsar such an interesting object: the timing noise and glitches.
Since 1982 there has been a regular monitoring program of the Crab pulsar at Jodrell Bank Observatory, and timing ephemerides from this are publicly available online CrabEphemeris . The ephemeris gives the pulsar frequency and frequency derivative and associated errors, and the associated epoch. The epochs, generally given on the of each month, represent the time of the peak of the first pulse after midnight on that day. They therefore represent zero of modulus phase of the electromagnetic pulse. Notes are given in the event of a timing irregularity or glitch being observed. The Crab’s timing noise can be clearly seen in the ephemeris once a bestfit quadratic timing model (Eq. (1)) has been subtracted (see Fig. 7). The section of data used was chosen to be free of glitches as these can be much larger than any timing noise frequency deviations.
Fig. 7 compares well with that given in Ref. Lyne:1993 , although some difference can be expected due to the different lengths of data and epochs used in the fitting. It can be seen that on scales of several months there is quite a large variation in the timing residual. It is shown in Ref. Lyne:1993 that on smaller time scales the variation is far smoother. It is important to consider whether timing noise at this level will cause a loss in sensitivity in any search for gravitational wave emission from the Crab pulsar. Jones Jones:2004 constructs a decoherence timescale, , defined as the time over which the timing noise will cause the phase to deviate by 1 radian from the second order Taylor expansion of phase. This makes use of the “activity parameter” and is calculated to be yr for the Crab pulsar. Though useful as an indicator, this statistic does not take account of permanent changes to spindown caused by glitches or other secular variations in the pulsar.
Although it is reasonable that a third order fit to the entire data would improve the model, there is no need. The Crab pulsar ephemeris provides timing every month, which is sufficient to track the phase excursions in the timing noise. All that is necessary is that these be interpolated between the ephemeris times. By using the phase, frequency and frequency derivative for each entry in the ephemeris as boundary conditions to a set of simultaneous equations the full phase evolution between each month can be calculated, giving a fifth order polynomial,
(8) 
Indeed, for much of the time even this method is unnecessarily complicated and a simple linear interpolation between months would be sufficient.
These corrections can be included in the method detailed in §II as an extra heterodyne step PitkinWoan:2004 . The initial heterodyne (described in §II), uses a third order fit to the the phase with values of and taken from the ephemeris at the closest time before the timestamp on the data to be analysed, and taken from the ATNF pulsar catalogue value given in Table 3. Then, assuming that any gravitational wave signal would show the same timing noise (see Refs. Jones:2004 and Abbott:2006 for discussions of this), we apply a second heterodyne to the data using the phase difference between Eqs. (1) and (IV.1)
(9) 
where the factor of two in the phase is due to the gravitational wave frequency being twice the spin frequency. This step can be performed on the data after downsampling as the rate of change of this phase difference will be very low.
The effect of this extra heterodyne can be seen in search for a signal from the Crab in the S2 data. This science run of the LIGO interferometers lasted approximately two months and overlapped three entries in the Crab pulsar ephemeris. The S2 run started on Feb 2003, so values of the frequency and spindown used in the initial heterodyning were chosen to be those given in the first ephemeris entry prior to the run (15th Jan 2003). The second derivative was set to be that taken from the ATNF catalogue. The values are shown in Table 3.
PSR J0534+2200  

29.8102713888 Hz  
Hz/s  
Hz/  
Frequency epoch  GPS 726624013 
Once the data were produced the ephemeris values were used to calculate the phase given in Eq. (IV.1). The difference between the initial heterodyne phase and the order phase is shown in Fig. 8.
This phase difference is used in the extra heterodyne to remove the variation. It can be seen in Fig. 8 how a linear fit between ephemeris values would be acceptable for these times, with only small deviations in phase from the fifth order fit. The black crosses in Fig. 8 provide the first step in checking the code used for the extra heterodyne stage. The red points represent the phase difference used in our extra heterodyne step (Eq. (9)) to heterodyne each S2 data point as calculated using our code, whereas the black crosses just show the phase difference between the initial heterodyne and the individual Crab pulsar ephemeris data points. The fact that these overlap provides a check that the heterodyne code is producing the correct phase difference.
If we simulate a signal from the Crab pulsar over the period of S2, with parameters , , and , we can see how including a timing noise heterodyne step affects the parameter estimation. Fig. 9 shows the extracted probability distribution functions (pdfs) of and for the signal with and without the timing noise removed. There is very little difference between the amplitudes for the two cases because the slope of the phase difference are not too steep over the period of S2.
However, the extracted value of the phase is strongly affected, mainly due to the the phase offset between the start of S2 and the epoch of the initial heterodyne parameters seen in Fig. 8.
We can simulate a Crab pulsar signal and analyse it with and without the timing noise heterodyne step over longer periods than just S2 to show its importance. The same process as above has been carried out over the period of the S3 run, using the same initial heterodyne parameters and pulsar injection parameters. The extracted pdfs are shown in Fig. 10 and demonstrate that without the extra heterodyne the signal is completely lost.
The fact that this signal is not seen without the extra heterodyne may appear at odds with the approximate 2.6 year decoherence time stated above, as S3 was only about 8 months after S2. However, the initial heterodyne values used were still those from the Crab ephemeris closest to the start of S2 and not those from a more general fit to the data over an extended period, as was used to calculate , so it is not the timing noise causing the decoherence in this case, but badly chosen initial heterodyne parameters. This does however demonstrate the importance of having welldefined heterodyne parameter for all pulsars. If we perform a fit to the Crab pulsar ephemeris over the whole of 2003 (see Table 4 for fit values), when the S2 and S3 runs took place, we can again check the impact of the timing noise in S3 (see Fig. 11).
PSR J0534+2200  

29.81027139567395 Hz  
Hz/s  
Hz/  
Frequency epoch  GPS 726624013.0597030 
We see that, with an extended fit for the Crab parameters, timing noise makes little difference over S3, although a slight phase offset is present.
iv.2 Timing noise in other pulsars
For the majority of pulsars timing noise is most prominent in the second derivative of frequency, but for millisecond pulsars this value is often so small as to be unmeasurable. For these pulsars, the value of the parameter can be used to estimate the cumulative phase contribution of timing noise via the empirical relationship between and given in Ref. Arzoumanian:1994 . Abbott et al. Abbott:2006 used this as a test of the coherence of the pulsars phases over the periods of the LIGO and GEO 600 S3 and S4 data runs. The generally small values of for the majority of other pulsars suggest that there is not significant timing noise.
There is however, one known interesting target pulsar within our frequency range that has similar properties to the Crab pulsar. This pulsar, PSR J05376910, is young and has the second largest spindown after the Crab pulsar, making it a very interesting candidate for our search. It is distant, lying in the Large Magellanic Cloud, and has so far only been observed in Xrays. It is also a prolific glitcher, showing high levels of timing noise Marshall:2004 . As dedicated time on the Rossi Xray Timing Explorer satellite is required to time this pulsar, it has not been observed as regularly as the Crab pulsar and a comprehensive ephemeris does not exist. A regularly updated ephemeris would however allow us to track the interglitch phase and perform an analysis similar to that of the Crab pulsar.
V Conclusions
We have shown how to take account of time delays due to the motion of pulsars within binary systems in searches for gravitational waves . This is particularly important as a large number of the pulsars within the sensitivity band of current interferometric gravitational wave detectors are in binaries. However, as discussed in Ref. Abbott:2006 , we have demonstrated the importance of knowing the binary parameters and their covariances.
At present our search code contains the three main binary models described above. These are sufficient for the majority of pulsars, although Tempo contains many more models which could be incorporated in the future if needed.
For the cases where large levels of timing noise are seen it is important that uptodate parameters are obtained allowing us to track the rotational variations. Fortunately, the Crab pulsar is constantly monitored in radio and therefore has a very welldetermined phase evolution. Other pulsars in our band are not so well monitored and this is especially pertinent if we want to target young pulsars (generally the best candidates for gravitational waves due to there large spindown rates), because these will generally also be the most affected by timing noise. In particular, the young Xray pulsar PSR J05376910 Marshall:2004 has high levels of glitchinduced timing noise, and an ability to regularly monitor this object would allow a more detailed study of this prime gravitational wave candidate.
Acknowledgements.
The authors are very grateful to Michael Kramer for supplying pulsar timing information and useful discussions on its use. We would also like to acknowledge members of the LIGO Scientific Collaboration for useful advise during the writing of this manuscript. This work is to a large extent an extension of the work of Réjean Dupuis and to him we extend many thanks. This work has been supported by the UK Particle Physics and Astronomy Research Council.References
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