4.7 Pulsar timing and gravitational wave detection

We have seen in the Section 4.4 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 [32398Jump 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 timescale 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 πœ–TOA, this “detector” would be sensitive to waves with dimensionless amplitudes h ≳ πœ–TOAβˆ•T and frequencies f ∼ 1βˆ•T [35Jump To The Next Citation Point43Jump To The Next Citation Point].

4.7.1 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 [282]. A significant component [301161] is the gravitational radiation from the inspiral prior to supermassive 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 σ 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 ± fβˆ•2 there is an additional contribution to the timing residuals σg [98]. When fT ≫ 1, the corresponding wave energy density is

243 π3f4σ2g ρg = -----------. (17 ) 208G
An upper limit to ρg can be obtained from a set of timing residuals by assuming the rms scatter is entirely due to this effect (σ = σg). These limits are commonly expressed as a fraction of the energy density required to close the Universe
3H2 ρc = ---0-≃ 2 × 10 −29h2 g cm −3, (18 ) 8πG
where the Hubble constant H0 = 100 h km s−1 Mpc.

This technique was first applied [318] 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 [102]. 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 (σ = 240 ms) to the GWB, the limit is ρgβˆ•ρc ≲ 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 [35Jump To The Next Citation Point43Jump To The Next Citation Point349] 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 [35Jump To The Next Citation Point] it is convenient to define

1-d-log-ρg Ωg = ρ d logf , (19 ) c
i.e. the energy density of the GWB per logarithmic frequency interval relative to ρc. With this definition, the power spectrum of the GWB [15143] is
-Gρg-- H20Ωg- 4 2 −5 2 𝒫(f ) = 3π3f4 = 8π4f5 = 1.34 × 10 Ωgh fyr−1 μs 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, Kaspi et al. [188Jump To The Next Citation Point] placed an upper limit of Ωgh2 < 1.1 × 10− 7. This limit is difficult to reconcile with most cosmic string models for galaxy formation [52379].

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 [35]. The most stringent results presently available are based on the B1855+09 limit Ω h2 < 2.7 × 10−4 g in the frequency range 10−11 < f < 4.4 × 10−9 Hz [198].

4.7.2 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 [217]. 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 [351].

A simulation of the expected modulations of the timing residuals for the putative binary system, with a total mass of 5.4 × 1010 M βŠ™, is shown along with the observed timing residuals in Figure 30View 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 [163Jump To The Next Citation Point].

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Figure 30: 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 [163].

4.7.3 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 [141]. 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

δ-νi νi = αiπ’œ(t) + 𝒩i(t), (21 )
where αi 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 π’œ. 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 αiαjβŸ¨π’œ ⟩ + αiβŸ¨π’œπ’©j ⟩ + αjβŸ¨π’œ 𝒩i⟩ + βŸ¨π’©i𝒩j ⟩, (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 2 αiαj βŸ¨π’œ ⟩ 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 [140] and allows the separation of the effects of clock and ephemeris errors from the GWB [114].

A recent analysis [162] applying the timing array concept to data for seven millisecond pulsars has reduced the energy density limit to Ωgh2 < 1.9 × 10− 8 for a background of supermassive black hole sources. The corresponding limits on the background of relic gravitational waves and cosmic strings are −8 2.0 × 10 and − 8 1.9 × 10 respectively. These limits can be used to constrain the merger rate of supermassive black hole binaries at high redshift, investigate inflationary parameters and place limits on the tension of currently proposed cosmic string scenarios.

The region of the gravitational wave energy density spectrum probed by the current pulsar timing array is shown in Figure 31View Image where it can be seen that the pulsar timing regime is complementary to the higher frequency bands of LISA and LIGO.

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Figure 31: 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 [200Jump To The Next Citation Point] from an original design by Richard Battye.

A number of long-term timing projects are now underway to make a large-scale pulsar timing array a reality. The Parkes pulsar timing array [373147Jump To The Next Citation Point] observes twenty millisecond pulsars twice a month. The European Pulsar Timing Array [364] uses the Lovell, Westerbork, Effelsberg and Nancay radio telescopes to regularly observe a similar number. Finally, at Arecibo and Green Bank, regular timing of a dozen or more millisecond pulsars is carried out by a North American consortium [372]. It is expected that a combined analysis of all these efforts could reach a limits of 2 − 10 Ωgh < 10 before 2010 [147Jump To The Next Citation Point]. Looking further ahead, the increase in sensitivity provided by the Square Kilometre Array [374200] should further improve the limits of the spectrum probed by pulsar timing. As Figure 31View Image shows, the SKA could provide up to two orders of magnitude improvement over current capabilities.

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