5.2 A pulsar timing array 

The idea of using timing data for a number of pulsars distributed on the sky to detect gravitational waves was first proposed by Hellings & Downs [74]. Such a ``timing array'' of pulsars 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, which would be common to all pulsars in the array. To quantify this, consider the fractional frequency shift observed for the pulsar in the array:

In this expression is a geometric factor dependent on the line-of-sight direction to the pulsar and the propagation and polarisation vectors of the gravity wave of dimensionless amplitude . The timing noise intrinsic to the pulsar is characterised by the function . The result of a cross-correlation between pulsars i and j is then

where the bracketed terms indicate cross-correlations. Since the wave function and the noise contributions from the two pulsars are independent quantities, the cross correlation tends to 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 [73]. This allows, in principle, the separation of the effects of terrestrial clock and solar system ephemeris errors from the GWB [61].

  

Figure 17: Hammer-Aitoff projections showing the known Galactic disk millisecond pulsar population in 1990 (left) and in 1997 (right). The impact of the new discoveries is clearly seen comparing the situation circa 1990, in which all of the known sources had been found at Arecibo, with the present situation in which the sources are much more uniformly distributed on the sky.

Applying the timing array concept to the present database of long-term timing observations of millisecond pulsars does not improve on the limits on the GWB discussed above. The sky distribution of these pulsars, seen in the left panel of Fig.  17, shows that their angular separation is rather low. To achieve optimum sensitivity it is desirable to have an array consisting of pulsar clocks distributed isotropically over the whole sky. The flood of recent discoveries of nearby binary and millisecond pulsars by the all-sky searches has resulted in essentially such a distribution, shown in the right panel of Fig.  17 . Continued timing of these pulsars in the coming years should greatly improve the sensitivity and will perhaps allow the detection of gravity waves, as opposed to upper limits, in the future.

Binary and Millisecond Pulsars D. R. Lorimer (dunc@mpifr-bonn.mpg.de) http://www.livingreviews.org/lrr-1998-10 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de