An upper limit to can be obtained from a set of timing residuals by assuming the rms scatter is entirely due to this effect (). These limits are commonly expressed as a fraction of the energy density required to close the Universe:
where the Hubble constant .
Romani & Taylor  applied this technique to a set of TOAs for PSR B1237+12 obtained from regular observations over a period of 11 years as part of the JPL pulsar timing programme . 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 (§ 4.3). By ascribing the rms scatter in the residuals ( ms) to the GWB, Romani & Taylor placed a limit of for a centre frequency .
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 by Taylor and collaborators (§ 4.3). In the intervening period, more elaborate techniques had been devised [29, 36, 228] 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  it is convenient to define , the energy density of the GWB per logarithmic frequency interval relative to . With this definition, the power spectrum of the GWB, , can be written [99, 36] as
where is frequency in cycles per year. The timing residuals for B1937+21 shown in Fig. 25 are clearly non-white and, as we saw in § 4.3, limit its timing stability for periods yr. The residuals for PSR B1855+09 clearly show no systematic trends and are in fact consistent with the measurement uncertainties alone. Based on these data, and using a rigorous statistical analysis, Thorsett & Dewey  place a 95% confidence upper limit of for . This limit is difficult to reconcile with most cosmic string models for galaxy formation [40, 249].
For those pulsars in binary systems, an additional clock for measuring the effects of gravitational waves is the orbital period. 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 . The most stringent results presently available are based on the B1855+09 limit in the frequency range . Kopeikin  has recently presented this limit and discusses the methods in detail.
|Binary and Millisecond Pulsars at the New Millennium
Duncan R. Lorimer
© Max-Planck-Gesellschaft. ISSN 1433-8351
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