Pulsar timing and the upper limits on a gravitational wave background: A Bayesian approach

Stringent limits on $\Omega$, the energy density in a gravitational wave background per logarithmic frequency interval in units of the closure density, have recently been suggested by Thorsett and Dewey using observational data of PSR B1855+09. We show that their use of the Neyman-Pearson test of hypotheses cannot, in the general case, provide reliable upper limits on an unknown parameter. The alternative presented here is the calculation of the probability distribution and repartition function for $\Omega$ using a Bayesian formalism. A prior distribution must be specified and the choice of "Jeffreys' prior" is justified on the grounds that it best represents a total lack of prior knowledge about the parameter. The Bayesian approach yields an upper limit at 95% confidence of 9.3×${10}^{-8\mathrm{}}$ for $\Omega h^2$. This limit is less stringent by a factor of 10 than that placed by Thorsett and Dewey.