Simulating a Gaussian stochastic gravitational wave background signal in pulsar timing arrays

TL;DR

This paper develops a comprehensive frequency- and Fourier-domain framework for simulating and analyzing Gaussian stochastic gravitational wave backgrounds in pulsar timing arrays, emphasizing the role of temporal correlations.

Contribution

It introduces transfer functions that relate the SGWB power spectrum to pulsar timing residual correlations, providing explicit forms and validation methods for improved modeling.

Findings

Transfer functions accurately describe pulsar residual correlations.

Validation confirms the theoretical model matches simulations.

Framework enhances future PTA gravitational wave data analysis.

Abstract

We revisit the theoretical modeling and simulation of a Gaussian stochastic gravitational wave background (SGWB) signal in a pulsar timing array (PTA). We show that the correlation between Fourier components of pulsar timing residuals can be expressed using transfer functions; that are indicative of characteristic temporal correlations in a SGWB signal observed in a finite time window. These transfer functions, when convolved with the SGWB power spectrum and spatial correlation (Hellings \& Downs curve), describe the variances and correlations of the pulsar timing residuals' Fourier coefficients. The convolutions are the exact frequency- and Fourier-domain representations of the time-domain covariance function. We derive explicit forms for the transfer functions for unpolarized and circularly polarized SGWB signals. We validate our results by comparing Gaussian theoretical expectation values with standard simulations based on point sources and our own covariance-matrix-based approach. The unified frequency- and Fourier-domain formalism provides a robust foundation for future PTA precision analyses and highlights the importance of temporal correlations in interpreting GW signals.