Statistic threshold of distinguishing the environmental effects and modified theory of gravity with multiple massive black-hole binaries

In future space-borne \ac{GW} observations, matter arround the sources might influence the evolution and \ac{GW} signals from \ac{BBH} inspirals, which can be mistaken as deviations from \ac{GR}. Former research \cite{yuan2024} proposed a statistic $F$ that characterizes the dispersion of measured parameters to distinguish environmental effect(\ac{DF} from \ac{DM} spike) and theory of modified gravity effect(varying $G$). In this work we use the statistic to distinguish other couples of effects with \ac{GW} corrections at $-4$ PN order: \ac{DF} from \ac{DM} spike and the extra dimension theory, additionally try to determine the distinguishing threshold in more reasonable way to avoid arbitrariness, especially when the two effects to compare have more overlap in the $F$ distribution. Sources of different astronomical models are also considered, and two effects are still distinguishable but not much as in former work\cite{yuan2024}, so the threshold should be carefully selected. Following these procedures, we finally obtain the statistic thresholds of distinguishment between the three effects with \ac{GW} corrections at $-4$ PN order: \ac{DF} from \ac{DM} spike, the extra dimension theory, and varying $G$ theory. The method can be used to distinguish other effects among environmental effects and modified theories of gravity effects with the detections of \ac{GW} events.