Characterizing gravitational wave stochastic background anisotropy with pulsar timing arrays

Detecting a stochastic gravitational wave background, particularly radiation from individually unresolvable supermassive black hole binary systems, is one of the primary targets for pulsar timing arrays. Increasingly more stringent upper limits are being set on these signals under the assumption that the background radiation is isotropic. However, some level of anisotropy may be present and the characterization of the gravitational wave energy density at different angular scales carries important information. We show that the standard analysis for isotropic backgrounds can be generalized in a conceptually straightforward way to the case of generic anisotropic background radiation by decomposing the angular distribution of the gravitational wave energy density on the sky into multipole moments. We introduce the concept of generalized overlap reduction functions which characterize the effect of the anisotropy multipoles on the correlation of the timing residuals from the pulsars timed by a pulsar timing array. In a search for a signal characterized by a generic anisotropy, the generalized overlap reduction functions play the role of the so-called Hellings and Downs curve used for isotropic radiation. We compute the generalized overlap reduction functions for a generic level of anisotropy and pulsar timing array configuration. We also provide an order of magnitude estimate of the level of anisotropy that can be expected in the background generated by supermassive black hole binary systems.

Figure 1

The real-value overlap reduction functions ${\Gamma }_{00}$ and ${\Gamma }_{21}$ for 18 EPTA pulsars in the cosmic rest frame. Note that for illustrative purposes, we have not included the autocorrelation term ($\zeta =0$).

Figure 2

The Earth-term only, complex-valued, generalized overlap reduction functions ${\Gamma }_l^m$ in the computational frame for $l=0$, 1, 2, 3 as a function of the angular separation of pulsar pairs. In the computational frame the functions are all real (see the Appendix for more details) and ${\Gamma }_l^{-m}=(-1{)}^m{\Gamma }_l^m$. For the case of the monopole ($l=0$), the overlap reduction function ${\Gamma }_0^0$ is the Hellings and Downs curve up to the multiplicative constant $4\sqrt{\pi }/3$.

Figure 3

Generalized overlap reduction functions (ORF) with the pulsar term. (a) The difference between the exact solution and the Earth-term only solution for $fL=10$ in the computational frame. These oscillations are already quite small for $\zeta =60°$ and rapidly converge to zero for larger values of $\zeta$. (b) The value of the complex component of the pulsar term for $fL=10$ in the computational frame. Recall that the Earth-term only solution is always real, but introducing the pulsar term gives rise to complex-valued overlap reduction functions, even in the computational frame. Notice that these oscillations induced by the pulsar term are at least an order of magnitude smaller than the real part but do not, however, converge as quickly as the real component. The ${\Gamma }_0^0$ function has no imaginary component.