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Direct detection of gravitational waves by pulsar timing arrays will becomefeasible over the next few years. In the low frequency regime ($10^{-7}$ Hz --$10^{-9}$ Hz), we expect that a superposition of gravitational waves from manysources will manifest itself as an isotropic stochastic gravitational wavebackground. Currently, a number of techniques exist to detect such a signal;however, many detection methods are computationally challenging. Here weintroduce an approximation to the full likelihood function for a pulsar timingarray that results in computational savings proportional to the square of thenumber of pulsars in the array. Through a series of simulations we show thatthe approximate likelihood function reproduces results obtained from the fulllikelihood function. We further show, both analytically and throughsimulations, that, on average, this approximate likelihood function givesunbiased parameter estimates for astrophysically realistic stochasticbackground amplitudes.

An Efficient Approximation to the Likelihood for Gravitational Wave Stochastic Background Detection Using Pulsar Timing Data