Projections of the uncertainty on the compact binary population background using

The LIGO-Virgo-KAGRA collaboration has announced the detection to date of almost 100 binary black holes that have been used in several studies to infer the features of the underlying binary black hole population. From these objects it is possible to predict the overall gravitational-wave (GW) fractional energy density contributed by black holes throughout the Universe, and thus estimate the gravitational-wave background (GWB) spectrum emitted in the current GW detector band. These predictions are fundamental in our forecasts for background detection and characterisation, with both present and future instruments. The uncertainties in the inferred population strongly impact the predicted energy spectrum, and in this paper we present a new flexible method to quickly calculate the energy spectrum for varying black hole population features, such as the mass spectrum and redshift distribution. We have implemented this method in an open-access package, <ns0:monospace>popstock</ns0:monospace>, and extensively tested its capabilities. Using <ns0:monospace>popstock</ns0:monospace>, we investigated how uncertainties in these distributions impact our detection capabilities, and present several caveats for background estimation. In particular, we find that the standard assumption that the background signal follows a two-thirds power law at low frequencies is both waveform and mass-model dependent, and that the power-law signal is likely shallower than previously modelled, given the current waveform and population knowledge.