Comparing gravitational wave background predictions from cosmological simulations to pulsar timing observations
The recent detection of a gravitational wave background (GWB) by pulsar timing arrays (PTAs) may represent the first evidence of gravitational waves from merging supermassive black hole binaries, opening a new window on the low-frequency end of the gravitational wave spectrum. These inspiralling binaries are expected to dominate the signal, although most theoretical models seem to predict somewhat lower amplitudes than what is observed. We present the first comprehensive statistical framework to quantify the tension between PTA measurements and theoretical predictions, maximising the constraining power of current data and allowing straightforward application to future PTA datasets. We further investigate how different assumptions in the observational inference, particularly the use of a power-law model for the GWB spectrum, can bias tension estimates and potentially overstate discrepancies with theory. We apply our framework to compare predictions from the FABLE cosmological simulation with the NANOGrav 15-year dataset. For our fiducial black hole population, we find tension values of $1\sigma$-$2.5\sigma$, indicating no statistically significant disagreement with the observations. We further explore physically motivated modifications to the merging black hole population, guided by electromagnetic observations and theoretical uncertainties. In particular, scenarios with boosted black hole masses at high redshift and more equal-mass mergers substantially increase the predicted GWB amplitude, improving agreement with PTA data. Finally, we investigate the high-mass end of the black hole mass function and the impact of finite simulation volume. We find that the $(100 \, \mathrm{cMpc} \, h^{-1})^3$ FABLE box is sufficient to robustly predict the GWB signal at the most constraining frequency.