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Abstract The dynamical formation channels of gravitational wave (GW) sources typically involve a stage when the compact object binary source interacts with the environment, which may excite its eccentricity, yielding efficient GW emission. For the wide eccentric compact object binaries, the GW emission happens mostly near the pericenter passage, creating a unique, burst-like signature in the waveform. This work examines the possibility of stellar-mass bursting sources in the mHz band for future LISA detections. Because of their long lifetime (∼10⁷yr) and promising detectability, the number of mHz bursting sources can be large in the local Universe. For example, based on our estimates, there will be ∼3–45 bursting binary black holes in the Milky Way, with ∼10²–10⁴bursts detected during the LISA mission. Moreover, we find that the number of bursting sources strongly depends on their formation history. If certain regions undergo active formation of compact object binaries in the recent few million years, there will be a significantly higher bursting source fraction. Thus, the detection of mHz GW bursts not only serves as a clue for distinguishing different formation channels, but also helps us understand the star formation history in different regions of the Milky Way. 

Many gravitational wave (GW) sources in the LISA band are expected to have non-negligible eccentricity. Furthermore, many of them can undergo acceleration because they reside in the presence of a tertiary. Here we develop analytical and numerical methods to quantify how the compact binary's eccentricity enhances the detection of its peculiar acceleration. We show that the general relativistic precession pattern can disentangle the binary's acceleration-induced frequency shift from the chirp-mass-induced frequency shift in GW template fitting, thus relaxing the signal-to-noise ratio requirement for distinguishing the acceleration by a factor of 10 ∼100 . Moreover, by adopting the GW templates of the accelerating eccentric compact binaries, we can enhance the acceleration measurement accuracy by a factor of ∼100 , compared to the zero-eccentricity case, and detect the source's acceleration even if it does not change during the observational time. For example, a stellar-mass binary black hole (BBH) with moderate eccentricity in the LISA band yields an error of the acceleration measurement ∼10-7 m .s−2 for SNR =20 and observational time of 4 yr. In this example, we can measure the BBHs' peculiar acceleration even when it is ∼1 pc away from a 4 ×106M⊙ supermassive black hole. Our results highlight the importance of eccentricity to the LISA-band sources and show the necessity of developing GW templates for accelerating eccentric compact binaries.