The effect of small inter-pulsar distance variations in stochastic gravitational wave background searches with Pulsar Timing Arrays

One of the primary objectives for Pulsar Timing Arrays (PTAs) is to detect a stochastic background generated by the incoherent superposition of gravitational waves (GWs), in particular from the cosmic population of supermassive black hole binaries. Current stochastic background searches assume that pulsars in a PTA are separated from each other and the Earth by many GW wavelengths. As more millisecond pulsars are discovered and added to PTAs, some may be separated by only a few radiation wavelengths or less, resulting in correlated GW phase changes between close pulsars in the array. Here we investigate how PTA overlap reduction functions (ORFs), up to quadrupole order, are affected by these additional correlated phase changes, and how they are in turn affected by relaxing the assumption that all pulsars are equidistant from the solar system barycenter. We find that in the low frequency GW background limit of $f\sim10^{-9}$~Hz, and for pulsars at varying distances from the Earth, that these additional correlations only affect the ORFs by a few percent for pulsar pairs at large angular separations, as expected. However when nearby (order 100 pc) pulsars are separated by less than a few degrees, the correlated phase changes can introduce variations of a few tens of percent in the magnitude of the isotropic ORF, and much larger fractional differences in the anisotropic ORFs-- up to 188 in the $m=0$, $l=2$ ORF for equidistant pulsars separated by 3 degrees. In fact, the magnitude of most of the anisotropic ORFs is largest at small, but non-zero, pulsar separations. Finally, we write down a small angle approximation for the correlated phase changes which can easily be implemented in search pipelines, and for completeness, examine the behavior of the ORFs for pulsars which lie at a radiation wavelength from the Earth.Comment: 16 pages, 8 figures, submitted to PR

Characterising gravitational wave stochastic background anisotropy with Pulsar Timing Arrays

Detecting a stochastic gravitational wave background, particularly radiation from individually unresolvable super-massive 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 characterisation of the power at different angular scales carries important information. We show that the standard analysis for isotropic backgrounds can be generalised in a conceptually straightforward way to the case of generic anisotropic background radiation by decomposing the angular distribution of the gravitational wave power on the sky into multipole moments. We introduce the concept of generalised overlap reduction functions which characterise 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 characterised by a generic anisotropy, the generalised overlap reduction functions play the role of the so-called Hellings and Downs curve used for isotropic radiation. We compute the generalised 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 super-massive black hole binary systems.Comment: 12 pages plus 5 page Appendix. Accepted to PR

Analysis of the first IPTA Mock Data Challenge by the EPTA timing data analysis working group

This is a summary of the methods we used to analyse the first IPTA Mock Data Challenge (MDC), and the obtained results. We have used a Bayesian analysis in the time domain, accelerated using the recently developed ABC-method which consists of a form of lossy linear data compression. The TOAs were first processed with Tempo2, where the design matrix was extracted for use in a subsequent Bayesian analysis. We used different noise models to analyse the datasets: no red noise, red noise the same for all pulsars, and individual red noise per pulsar. We sampled from the likelihood with four different samplers: "emcee", "t-walk", "Metropolis-Hastings", and "pyMultiNest". All but emcee agreed on the final result, with emcee failing due to artefacts of the high-dimensionality of the problem. An interesting issue we ran into was that the prior of all the 36 (red) noise amplitudes strongly affects the results. A flat prior in the noise amplitude biases the inferred GWB amplitude, whereas a flat prior in log-amplitude seems to work well. This issue is only apparent when using a noise model with individually modelled red noise for all pulsars. Our results for the blind challenges are in good agreement with the injected values. For the GWB amplitudes we found h_c = 1.03 +/- 0.11 [10^{-14}], h_c = 5.70 +/- 0.35 [10^{-14}], and h_c = 6.91 +/- 1.72 [10^{-15}], and for the GWB spectral index we found gamma = 4.28 +/- 0.20, gamma = 4.35 +/- 0.09, and gamma = 3.75 +/- 0.40. We note that for closed challenge 3 there was quite some covariance between the signal and the red noise: if we constrain the GWB spectral index to the usual choice of gamma = 13/3, we obtain the estimates: h_c = 10.0 +/- 0.64 [10^{-15}], h_c = 56.3 +/- 2.42 [10^{-15}], and h_c = 4.83 +/- 0.50 [10^{-15}], with one-sided 2 sigma upper-limits of: h_c <= 10.98 [10^{-15}], h_c <= 60.29 [10^{-15}], and h_c <= 5.65 [10^{-15}].Comment: 10 pages, 5 figure