Distinguishability of binary extreme-mass-ratio inspirals in low frequency band

The inspiral of compact stellar objects into massive black holes are one of the main astrophysical sources for the Laser Interferometer Space Antenna (LISA) and Taiji. These extreme-mass-ratio inspirals (EMRIs) have great potential for cosmology and fundamental physics. A binary extreme-mass-ratio inspiral (b-EMRI) describes the case where binary black holes (BBHs) are captured by a supermassive black hole. The b-EMRIs serve as multi-band gravitational wave sources and provide insights into the dynamics of nuclei and tests of general relativity. However, if the b-EMRIs can be distinguished from the normal EMRIs or not is still not clear. In this work, with a few of assumptions, and using the Teukolsky equation, we calculate the approximate gravitational waves of b-EMRIs and assess their detectability by space-based detectors. We also decouple the secondary object information from the Teukolsky equation, enabling us to calculate the energy fluxes and generate the waveforms more conveniently. Variations in the quadrupole of the binary result in small but non-negligible changes in energy fluxes and waveforms, making it possible to distinguish b-EMRI signals with data analysis. This opens up the potential of using b-EMRIs to test gravity theories and for further astrophysical studies.