Forecasting constraints on the no-hair theorem from the stochastic gravitational wave background

Although the constraints on general relativity (GR) from each individual gravitational-wave (GW) event can be combined to form a cumulative estimate of the deviations from GR, the ever-increasing number of GW events used also leads to the ever-increasing computational cost during the parameter estimation. Therefore, in this paper, we will introduce the deviations from GR into GWs from all events in advance and then create a modified stochastic gravitational-wave background (SGWB) to perform tests of GR. More precisely, we use the $\mathtt{pSEOBNRv4HM\_PA}$ model to include the model-independent hairs and calculate the corresponding SGWB with a given merger rate. Then we turn to the Fisher information matrix to forecast the constraints on the no-hair theorem from SGWB at frequency $10[{\rm Hz}]\lesssim f\lesssim10^3[{\rm Hz}]$ detected by the third-generation ground-based GW detectors, such as the Cosmic Explorer. We find that the forecasting constraints on hairs at $68\%$ confidence range are $\delta\omega_{220}=0\pm0.1296$ and $\delta\tau_{220}=0\pm0.0678$ when the flat priors about the merger rate are added but $\delta\omega_{220}=0\pm0.0903$ and $\delta\tau_{220}=0\pm0.0608$ when the non-flat priors about the merger rate are added.