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4.5 Pulsar timing and gravitational wave detection

We have seen in the previous Section4.4 how pulsar timing can be used to provide indirect evidence for the existence of gravitational waves from coalescing stellar-mass binaries. In this final section, we look at how pulsar timing might soon be used to detect gravitational radiation directly. The idea to use pulsars as natural gravitational wave detectors was first explored in the late 1970s [27785Jump To The Next Citation Point]. The basic concept is to treat the solar system barycentre and a distant pulsar as opposite ends of an imaginary arm in space. The pulsar acts as the reference clock at one end of the arm sending out regular signals which are monitored by an observer on the Earth over some time-scale T. The effect of a passing gravitational wave would be to perturb the local spacetime metric and cause a change in the observed rotational frequency of the pulsar. For regular pulsar timing observations with typical TOA uncertainties of eTOA, this “detector” would be sensitive to waves with dimensionless amplitudes h >~~ eTOA/T and frequencies f ~ 1/T [31Jump To The Next Citation Point39Jump To The Next Citation Point].

4.5.1 Probing the gravitational wave background

Many cosmological models predict that the Universe is presently filled with a low-frequency stochastic gravitational wave background (GWB) produced during the big bang era [243]. A significant component [260134] is the gravitational radiation from massive black hole mergers. In the ideal case, the change in the observed frequency caused by the GWB should be detectable in the set of timing residuals after the application of an appropriate model for the rotational, astrometric and, where necessary, binary parameters of the pulsar. As discussed in Section 4.2, all other effects being negligible, the rms scatter of these residuals s would be due to the measurement uncertainties and intrinsic timing noise from the neutron star.

For a GWB with a flat energy spectrum in the frequency band f ± f/2 there is an additional contribution to the timing residuals s g [85]. When fT » 1, the corresponding wave energy density is

243p3f 4s2g rg = -----------. (17) 208G
An upper limit to rg can be obtained from a set of timing residuals by assuming the rms scatter is entirely due to this effect (s = sg). These limits are commonly expressed as a fraction of the energy density required to close the Universe
3H2 rc = ---0- -~ 2 × 10 -29h2 g cm -3, (18) 8pG
where the Hubble constant H0 = 100 h km s-1 Mpc.

This technique was first applied [273] to a set of TOAs for PSR B1237+25 obtained from regular observations over a period of 11 years as part of the JPL pulsar timing programme [89]. This pulsar was chosen on the basis of its relatively low level of timing activity by comparison with the youngest pulsars, whose residuals are ultimately plagued by timing noise (see Section 4.3). By ascribing the rms scatter in the residuals (s = 240 ms) to the GWB, the limit rg/rc <~ 4× 10-3h- 2 for a centre frequency f = 7 nHz.

This limit, already well below the energy density required to close the Universe, was further reduced following the long-term timing measurements of millisecond pulsars at Arecibo (see Section 4.3). In the intervening period, more elaborate techniques had been devised [31Jump To The Next Citation Point39Jump To The Next Citation Point303] to look for the likely signature of a GWB in the frequency spectrum of the timing residuals and to address the possibility of “fitting out” the signal in the TOAs. Following [31Jump To The Next Citation Point] it is convenient to define

1-d-log-rg _O_g = r d logf , (19) c
i.e. the energy density of the GWB per logarithmic frequency interval relative to rc. With this definition, the power spectrum of the GWB [12539] is
-Grg-- H20_O_g- 4 2 -5 2 P(f ) = 3p3f4 = 8p4f5 = 1.34 × 10 _O_gh fyr-1 ms yr, (20)
where fyr- 1 is frequency in cycles per year. The long-term timing stability of B1937+21, discussed in Section 4.3, limits its use for periods >~~ 2 yr. Using the more stable residuals for PSR B1855+09, and a rigorous statistical analysis [319Jump To The Next Citation Point], the current 95% confidence upper limit is _O_ h2 < 10-8 g. This limit is difficult to reconcile with most cosmic string models for galaxy formation [45319].

For binary pulsars, the orbital period provides an additional clock for measuring the effects of gravitational waves. In this case, the range of frequencies is not limited by the time span of the observations, allowing the detection of waves with periods as large as the light travel time to the binary system [31]. The most stringent results presently available are based on the B1855+09 limit 2 -4 _O_gh < 2.7× 10 in the frequency range -11 -9 10 < f < 4.4 × 10 Hz [160].

4.5.2 Constraints on massive black hole binaries

In addition to probing the GWB, pulsar timing is beginning to place interesting constraints on the existence of massive black hole binaries. Arecibo data for PSRs B1937+21 and J1713+0747 already make the existence of an equal-mass black hole binary in Sagittarius A* unlikely [176]. More recently, timing data from B1855+09 have been used to virtually rule out the existence of a proposed supermassive black hole as the explanation for the periodic motion seen at the centre of the radio galaxy 3C66B [304].

A simulation of the expected modulations of the timing residuals for the putative binary system, with a total mass of 10 5.4 × 10 Mo ., is shown along with the observed timing residuals in Figure 29View Image. Although the exact signature depends on the orientation and eccentricity of the binary system, Monte Carlo simulations show that the existence of such a massive black hole binary is ruled out with at least 95% confidence [135Jump To The Next Citation Point].

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Figure 29: Top panel: Observed timing residuals for PSR B1855+09. Bottom panel: Simulated timing residuals induced from a putative black hole binary in 3C66B. Figure provided by Rick Jenet [135].

4.5.3 A millisecond pulsar timing array

A natural extension of the single-arm detector concept discussed above is the idea of using timing data for a number of pulsars distributed over the whole sky to detect gravitational waves [118]. Such a “timing array” would have the advantage over a single arm in that, through a cross-correlation analysis of the residuals for pairs of pulsars distributed over the sky, it should be possible to separate the timing noise of each pulsar from the signature of the GWB common to all pulsars in the array. To illustrate this, consider the fractional frequency shift of the ith pulsar in an array

dni = a A(t) + N (t), (21) ni i i
where ai is a geometric factor dependent on the line-of-sight direction to the pulsar and the propagation and polarisation vectors of the gravitational wave of dimensionless amplitude A. The timing noise intrinsic to the pulsar is characterised by the function Ni. The result of a cross-correlation between pulsars i and j is then
2 aiaj<A > + ai<ANj > + aj<ANi > + <NiNj >, (22)
where the bracketed terms indicate cross-correlations. Since the wave function and the noise contributions from the two pulsars are independent, the cross correlation tends to aiaj <A2> as the number of residuals becomes large. Summing the cross-correlation functions over a large number of pulsar pairs provides additional information on this term as a function of the angle on the sky [117] and allows the separation of the effects of clock and ephemeris errors from the GWB [97].
View Image

Figure 30: Summary of the gravitational wave spectrum showing the location in phase space of the pulsar timing array (PTA) and its extension with the Square Kilometre Array (SKA). Figure updated by Michael Kramer [162Jump To The Next Citation Point] from an original design by Richard Battye.
A preliminary analysis applying the timing array concept to Arecibo data for three millisecond pulsars (B1937+21, B1855+09 and J1737+0747) now spanning 17 years has reduced the energy density limit to 2 -9 _O_gh < 2 × 10 [175]3. This limit, and the region of the gravitational wave energy density spectrum probed by the current pulsar timing array is shown in Figure 30View Image where it can be seen that the pulsar timing regime is complementary to the higher frequency bands of LISA and LIGO. A number of long-term timing projects are now underway to make a large-scale pulsar timing array a reality. At Arecibo and Green Bank, regular timing of a dozen or more millisecond pulsars are now carried out using on-line data acquisition systems by groups at Berkeley [325], Princeton [258] and UBC [323]. A similar system is also being commissioned at Jodrell Bank [142]. The Berkeley group have installed identical sets of data taking equipment at Effelsberg and Nançay which has enabled these telescopes to make regular high-quality millisecond pulsar timing observations [17468]. In the southern hemisphere, a high-precision timing program at Parkes [333] has been in existence since the discovery bright nearby millisecond pusar J0437-4715 [138Jump To The Next Citation Point]. Since February 2004, weekly Parkes observations of ~20 millisecond pulsars are already achieving a limit of _O_ h2 < 10-4 g, with the ultimate goal being to reach a limits of _O_ h2 < 10-10 g before 2010 [121Jump To The Next Citation Point].

Looking further ahead, the increase in sensitivity provided by the Square Kilometre Array [132Jump To The Next Citation Point162] should further improve the limits of the spectrum probed by pulsar timing. As Figure 30View Image shows, the SKA could provide up to two orders of magnitude improvement over current capabilities.

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