The birefringence mode-locked fiber laser-based dual-comb sources have strong potential in the fast spectroscopy, ranging and imaging, due to its large repetition rate difference, configuration compactness, and common-mode noise suppression. However, the intrinsic effects of the mutual dynamics between the dual-polarization solitons (DPSs) output from the dual-comb source on the coherence and common-mode noise still have attracted few attentions. In this work, the polarization mode competition dynamics and the statistical correlation of the intensity jitter between the DPSs are studied, and the short-time common-mode noise of the dual-comb source is revealed. The noise floor of relative timing jitter is as low as $0.01\mathrm{fs}/{\mathrm{Hz}}^{1/2}$, and the integrated jitter from 200 kHz to 96 MHz reaches only 71 fs. The mutual dynamics shows that asynchronous sampling between DPSs can obtain effective time-domain interferometry signals. Eventually, the acoustical vibration signal carried by the sub-fs level relative timing jitter can be measured by calculating the phase of the interference pattern on the shot-to-shot spectrum with an update frame of up to the kHz level. This work provides a comprehensive understanding of the DPSs mutual dynamics and opens the avenue for its application in the time-resolved spectroscopy-based precise measurements.

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• Acoustic measurements
• Frequency combs & self-phase locking
• Noise
• Photonics
• Fiber lasers
• Optical sources & detectors
• Solitons
• Optical spectroscopy