Within the next several years pulsar timing arrays (PTAs) are positioned to detect the stochastic gravitational-wave background (GWB) likely produced by the collection of inspiralling supermassive black holes binaries, and potentially constrain some exotic physics. Searches for a GWB in real PTA data rely on Markov-Chain Monte Carlo (MCMC) analyses, which are computationally demanding and not easily accessible to non-experts. In order to develop a more intuitive understanding of what PTAs may (or may not) be able to detect, we built a simple yet realistic Fisher formalism for GWB searches with PTAs. Our formalism is able to accommodate realistic noise properties of PTAs, and allows to forecast their sensitivity not only to an isotropic GWB, but also, looking ahead, to GWB anisotropies. It moreover provides a useful tool to guide and optimize real data analysis. In this talk, I will describe the basic physics behind PTAs, then the Fisher formalism, and illustrate some applications to a real-life PTA. This talk is based on arXiv:2006.14570 and 2010.13958.
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