Understanding the gravitational-wave Hellings and Downs curve for pulsar timing arrays in terms of sound and electromagnetic waves

A pulsar timing model predicts the arrival times of the pulses given values for the pulsar's spin frequency, frequency derivative, location on the sky, proper motion with respect to the solar system barycenter, its orbital parameters if the pulsar is in a binary, etc.² The values of these parameters are typically determined by an iterative least-squares fitting procedure, which minimizes the root-mean-squared (rms) deviation of the resultant timing residuals. Systematic errors in the timing model parameters can usually be identified by this iterative procedure, but unmodelled processes in the timing model will lead to errors in the timing residuals that cannot easily be removed.

We are assuming here that the two pulsars—even for the case ζ = 0—are distinct (i.e., they do not occupy the same physical location in space). If we consider the same pulsar, as would be the case for an autocorrelation calculation, then the right-hand-side of Eq. (1) should have an extra term equal to $δ(ζ)/2$⁠.

This statement is a generalization (to fields) of the mathematical result that the Fourier transform of the probability distribution p(x) for a random variable x (the so-called characteristic function of the random variable) can be written as a power series with coefficients given by the expectation values $⟨xk⟩$ for $k=1,2,…$⁠.¹⁹ Thus, the expectation values $⟨xk⟩$ for $k=1,2,…$ completely determine the probability distribution p(x) and hence the statistical properties of the random variable x.

Recall that $∇2=(∂2∂x2+∂2∂y2+∂2∂z2)$ in Cartesian coordinates.

The field might actually be a tensor field, like the gravitational-wave field $hab(t,x)$⁠, and hence should have tensor indices in general. But for simplicity, we will ignore that complication here.

Hint: work in the coordinate system shown in Fig. 3 with the Earth at the origin and the two pulsars located along the z-axis and in the xz-plane, respectively. Evaluate Eq. (59) in this frame using Eq. (60) and the expressions for $θ̂, ϕ̂$ given in Eq. (36). Finally, use contour integration to do the integral over the azimuthal angle $ϕ$⁠. It is a long calculation, but worth the effort. If you have trouble completing the calculation, see Appendix for more details.