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Although the constraints on general relativity (GR) from each individualgravitational-wave (GW) event can be combined to form a cumulative estimate ofthe deviations from GR, the ever-increasing number of GW events used also leadsto the ever-increasing computational cost during the parameter estimation.Therefore, in this paper, we will introduce the deviations from GR into GWsfrom all events in advance and then create a modified stochasticgravitational-wave background (SGWB) to perform tests of GR. More precisely, weuse the $\mathtt{pSEOBNRv4HM\_PA}$ model to include the model-independent hairsand calculate the corresponding SGWB with a given merger rate. Then we turn tothe Fisher information matrix to forecast the constraints on the no-hairtheorem from SGWB at frequency $10[{\rm Hz}]\lesssim f\lesssim10^4[{\rm Hz}]$detected by the third-generation ground-based GW detectors, such as the CosmicExplorer. We find that the forecasting constraints on hairs are$\delta\omega_{220}=0\pm0.02$ and $\delta\tau_{220}=0\pm0.04$ at $68\%$confidence range for the parameter space with only two parameters.

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